Research ArticleWarm Dark Matter Sterile Neutrinos in ElectronCapture and Beta Decay Spectra

O Moreno1 E Moya de Guerra1 and M Ramoacuten Medrano2

1Departamento de Fısica Atomica Molecular y Nuclear Facultad de Ciencias Fısicas Universidad Complutense de Madrid28040 Madrid Spain2Departamento de Fısica Teorica I Facultad de Ciencias Fısicas Universidad Complutense de Madrid 28040 Madrid Spain

Correspondence should be addressed to O Moreno osmorenomitedu

Received 1 July 2016 Revised 21 September 2016 Accepted 3 October 2016

Academic Editor Theocharis Kosmas

Copyright copy 2016 O Moreno et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We briefly review the motivation to search for sterile neutrinos in the keV mass scale as dark matter candidates and the prospectsto find them in beta decay or electron capture spectra with a global perspective We describe the fundamentals of the neutrinoflavor-mass eigenstate mismatch that opens the possibility of detecting sterile neutrinos in such ordinary nuclear processes Resultsare shown and discussed for the effect of heavy neutrino emission in electron capture in Holmium 163 and in two isotopes of Lead202 and 205 as well as in the beta decay of TritiumWe study the deexcitation spectrum in the considered cases of electron captureand the charged lepton spectrum in the case of Tritium beta decay For each of these cases we define ratios of integrated transitionrates over different regions of the spectrum under study and give new results that may guide and facilitate the analysis of possiblefuture measurements paying particular attention to forbidden transitions in Lead isotopes

1 Introduction

There is an inconsistency between the amount of matterinferred from gravitational effects and the one we see onobservable scales This fact leads to considering the existenceof dark matter (DM) Evidence arises from astrophysical andcosmological probes [1ndash4] such as the kinematics of viriallybound systems rotation curves of spiral galaxies strongand weak lensing cosmic microwave background (CMB)information onmatter density and geometry of the Universemass-to-light ratios in dwarf spheroidal galaxies (dSph) andlarge surveys to measure Universe structures Because of thelimits inferred from Big Bang nucleosynthesis an importantfraction of DM should be nonbaryonic Although the natureof DM is still unknown a popular hypothesis considers thatDM consists of elementary particles [3 4] DM is the maincomponent of galaxies being in average at least six timesmore abundant than baryonic matter (more than 81 ofthe matter in the Universe) but its nature is still unknownDM self-interactions have been unobserved so far and DM

particle candidates are bound by gravitational interactionsVery distinct predictions for small scale structures in theUniverse (below 100 kpc) are obtained by DM particles withdifferentmass scales [5] On the largemass scale sideWIMPs(weakly interacting massive particles) are popular candidatesto DM that have masses of the order of GeV or even TeVThis type of particles fall in the category of cold dark matter(CDM) which predicts for small scales too many galaxysatellites in the Milky Way and cusped profiles for themass density of galaxies contradicting present observationalevidences On the contrary particles with mass in the keVscale namely warm dark matter (WDM) [6ndash15] are able toreproduce the number of observed satellite galaxies as wellas the cored profiles found in DM-dominated objects such asdwarf spheroidal galaxies

Concerning the satellite problem cosmic structureswould form from the gravitational collapse of overdenseregions in the DM primordial field Free relativistic particlesdo not cluster and structures at scales smaller than the par-ticle free-streaming length 119897fs are erased The free-streaming

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 6318102 12 pageshttpdxdoiorg10115520166318102

2 Advances in High Energy Physics

length is the distance travelled freely by a relativistic particleafter decoupling from the primordial plasma due to Universeexpansion (approximately the distance travelled before thetransition to nonrelativistic velocities) WDM particles (keVscale) give 119897fs sim 100 kpc while CDM (GeV-TeV scale)which are heavier and slower than WDM would give a 119897fs amillion times smaller and lead to the existence of a host ofsmall scale structures [16] On the galaxy density profiles (120588)CDM gives a steep cusp at the center (120588 sim 119903minus1) [17] On thecontrary WDM gives a finite constant density core at thecenter (120588 sim 1205880) in agreement with observations [18ndash24]WDM quantum effects [25] could be important inside thegalaxy core (below 100 pc) showing a fermionic nature forthe DM particle through the manifestation of quantumnonvanishing pressure versus gravity

Astrophysical observations fromDM-dominated objectsas well as theoretical analysis lead to aDM fermionic thermalparticle with a mass around 2 keV [26] Chandra and XMM-Newton detections in the X-ray spectra of theM31 galaxy andthe Perseus cluster (both DM-dominated) seem to be con-sistent with sterile neutrinos of a few keV In particular anunidentified 355 keV line has been observed which seemscompatible with the decay or annihilation of sterile relicneutrinos [27 28] and that does not correspond to any knownatomic emissionAlthough the interpretation of this line is stilsubject to debate it could be considered an indication of thedecay of a 71 keV nonthermal sterile neutrino [29 30]

As it is well known the Standard Model (SM) of elemen-tary particles does not describe DM particles nor does itprovide a mass for active neutrinos (]119890 ]120583 ]120591) In this modelneutrino eigenstates are left-handed and they transform asdoublets under the weak SU(2) gauge group In extended SMmodels [31 32] additional SU(3)times SU(2)timesU(1) singlet right-handed neutrinos are introduced Neutrino mass eigenstateswill be linear combinations of the left and right-handed statesandmassmatrix eigenvalues will split into lighter and heavierstates (seesaw mechanism) The lighter mass eigenstateswould be the main components of the active flavor eigen-states of the neutrinos whereas the heavier ones would bedominant in sterile neutrino flavors [33] Some extended SMmodels add three extra sterile neutrinos one with a mass ofthe order of keV (see [34] and references therein) Neutrinosin the keVmass scale open the possibility to detect warmdarkmatter in nuclear electron capture and beta decay Heaviersterile neutrinos cannot be produced in weak nuclear decaysand there are of coursemany other candidate particles forDMunconnected in principle to weak nuclear processes such asthe lightest supersymmetric particles [35 36]

In this paper we assume that a keV neutrino a soundcandidate for DM [37] is produced in ordinary weak nuclearprocesses such as beta decay and electron capture vianeutrino mixing Experiments searching for active neutrinomasses also look for sterile neutrinos in the keV range [38]MARE [39ndash41] (that used Rhenium 187 beta decay and is notactive anymore) KATRIN [42ndash46] PTOLEMY [47 48] andProject8 [49] (using Tritium beta decay) and ECHo [50ndash52]and HOLMES [53] (using Holmium 163 electron capture)The mass of the neutrino (or antineutrino) emitted in a weak

nuclear decay has an effect on the energy spectrum of theprocess as was first shown by Enrico Fermi in the early1930s [54 55] For the active neutrinos this effect showsup at the endpoint of the spectrum whereas for the sterileneutrinos considered here it may be expected to appear at afew keVbelow the endpoint Clearly in order to leave a finger-print the sterile neutrino mass must be within the119876 windowthat is lower than the energy 119876 available in the decay Theenergy spectrum to be analyzed corresponds to the emittedcharged lepton in the case of beta decay namely the spon-taneous conversion of a neutron into a proton or vice versawith emission of the charged lepton and an antineutrino orneutrino In the case of electron capture only a neutrinois emitted and no charged lepton comes out so that thespectrum to be measured corresponds to the deexcitation ofthe daughter atom

A summary of current experimental studies of neutrinoproperties in the frontiers of intensities and sensitivitiesis given in [62] Experiments are under way to determinedirectly the active neutrino mass from neutrinoless doublebeta decay [63] as well as from single beta decay [42ndash48] andelectron capture [50ndash53] The last two types of experimentsalso provide a way to search for WDM sterile neutrinos inthe measured spectra In electron capture experiments thespectrum collected in a calorimeter is directly linked to theexcitation energies of the daughter atoms or molecules In acalorimeter the active source is embedded in the detectorwhich collects the energy of all the particles emitted inthe deexcitation processes that take place in the sourceexcept that of the neutrinos In beta decay the fact that theatoms or molecules may remain excited poses a challenge onthe interpretation of the electron spectrum Electron cap-ture experiments take advantage of the larger statistics inthe spectrum regions around the capture resonances Alimitation of electron capture measurements is that in acalorimeter where the full energy range of the spectrum ismeasured pile-up can be a problem that should be preventedby limiting the activity in the experimental setup We referalways in this paper to Earth-based experiments where boththe emission and the detection of particles take place inthe laboratory Other possible scenarios like the search forsterile neutrinos in stellar matter (stellar beta decay andelectron capture rates) are beyond the scope of this paperThephase space for these rates increases manifold in stellarenvironment For a reference on stellar weak rates see [6465]

2 Heavy Mass and SterileFlavor in Neutrino States

Already observed neutrino oscillations (see eg [66ndash69])among light active neutrinos are due to the fact that the neu-trino flavor eigenstates and the mass eigenstates are not thesame Eachflavor eigenstate associatedwith a charged leptoncan be written as a combination of mass eigenstates andvice versa [70] For instance the neutrino flavor eigenstateemitted after electron capture called electron neutrino ]119890 canbe written as a combination of the three light SM mass

Advances in High Energy Physics 3

eigenstates and hypothetically of one or more extra heaviermass eigenstates as [71ndash77]

1003816100381610038161003816]119890⟩ = 1198801198901 1003816100381610038161003816]1⟩ + 1198801198902 1003816100381610038161003816]2⟩ + 1198801198903 1003816100381610038161003816]3⟩ +sumℎ

119880119890ℎ 1003816100381610038161003816]ℎ⟩ (1)

where the subscript ℎ = 4 5 stands for extra (heavier)mass eigenstates and the quantities 119880 belong to the unitaryneutrino mixing matrix The masses of the three light SMmass eigenstates are so close to each other that so far nomea-surement has been able to discern which of them has beenemitted in a given process As a result the three are emittedas a coherent superposition which is at the origin of theneutrino oscillation phenomenologyThe phase of each masseigenstate changes at a different rate while travelling givingrise to a different superposition at each location that translatesinto a varying (oscillatory) probability of detection of a givenflavor eigenstate

To simplify things we will consider here the linear combi-nation of light mass eigenstates as a single effective neutrinomass eigenstate called ldquolightrdquo with mass

119898119897 = 119898]119890 = (1198802119890111989821 + 1198802119890211989822 + 1198802119890311989823)12 (2)

and just one extra mass eigenstate clearly heavier than theothers called ldquoheavyrdquo (withmass119898ℎ whichwe shall considerin the keV range)

1003816100381610038161003816]119890⟩ = cos 120577 1003816100381610038161003816]119897⟩ + sin 120577 1003816100381610038161003816]ℎ⟩ (3)

where the mixing amplitudes have been written in terms of amixing angle 120577 between the light and the heavy neutrinomasseigenstate The other possible combination of these two masseigenstates would be the sterile flavor eigenstate

1003816100381610038161003816]119904⟩ = minus sin 120577 1003816100381610038161003816]119897⟩ + cos 120577 1003816100381610038161003816]ℎ⟩ (4)

such that ⟨]119890 | ]119904⟩ = 0 For the sterile neutrinos that canbe relevant as WDM cosmological constraints based on theobserved average darkmatter density suggest that the value ofthe mixing angle could approximately be 120577 = 0006∘ [71ndash74]corresponding to a flavor-mass amplitude119880119890ℎ = sin 120577 asymp 10minus4Other recent cosmological constraints give values of sin2(2120577)between 2 sdot 10minus11 and 2 sdot 10minus10 [27 28]

Sterile neutrino states with masses close to the ones ofthe active neutrinos can also have an impact on the patternsmeasured in oscillation experiments [78] Heavier sterileneutrinos as in particular in the keV scale do not modify theoscillation patterns since they are not emitted coherently withthe active neutrinos due to the large mass splitting

The differential energy spectrum of a weak processwhere an electronic neutrino (or antineutrino) is emittedcan therefore be decomposed in a term for light neutrinoemission and another term for heavy neutrino emission asfollows [75ndash77]

119889120582119889119864 = 119889120582119897

119889119864 cos2120577 + 119889120582ℎ

119889119864 sin2120577 (5)

This energy spectrum can be the one of the electron emittedin beta decay or the one of the daughter atom deexcitations in

electron captureThe heavymass eigenstate if it exists wouldbe emitted independently of the other masses (noncoher-ently) as long as the energy resolution of the detector is betterthan the mass difference Δ120598 lt (119898ℎ minus 119898119890) Its contribution tothemeasured spectrum 119889120582ℎ119889119864 would show up in the rangewhere the collected energy is small enough for the heavyneutrino mass to have been produced namely when 0 le 119864 le(119876minus119898ℎ) where119876 is the difference between the masses of theinitial and the final atoms when the reaction takes place invacuumAt the edge of that region at119864 = 119876minus119898ℎ a kink in thespectrum would be found with the size of the heavy neutrinocontribution namely proportional to sin2120577 sim 1205772 (the latterapproximation valid for small mixing angles as is the case inrealistic scenarios)

3 Theory of Electron Capture

Let us consider the capture of an atomic electron by thenucleus119883 (119885119873) to turn into the nucleus119884 (119885minus1119873+1)Thereactions involving the corresponding atoms (represented bythe symbol of the nucleus within brackets) begin withelectron capture

(119883) 997888rarr (119884)119867 + ]119894 (6)

followed by deexcitation of the daughter atom In (6) ]119894 is aneutrino mass eigenstate (]119897 or ]ℎ) and the superscript 119867accounts for the excited state of the atom (119884) correspondingto an electron hole in the shell 119867 due to the electroncapture from this shell in the parent atom The deexcitationof the daughter atom after electron capture can happen eitherthrough emission of a photon (X-ray emission)

(119884)119867 997888rarr (119884) + 120574 (7)

or through emission of electrons (Auger or Coster-Kronigprocess) and photons

(119884)119867 997888rarr (119884+)1198671015840 11986710158401015840 + 119890minus 997888rarr (119884+) + 120574 + 119890minus (8)

where1198671015840 and11986710158401015840 represent holes in the electron shells of theion (119884+)

The energies carried by the emitted photons and electronsare deposited in the calorimeter that surrounds the sourceand that measures the full energy spectrum of these particlesprovided that the corresponding deexcitation lifetimes aremuch smaller than the detector time response [79] The pro-cess in (7) and (8) yields the same calorimeter spectrum thatis the peaks appear at the same energy values correspondingto the excitation energies 119864119867 of the excited daughter atom(119884)119867 The excitation energy is the difference between thebinding energies of the captured electron shell and theadditional electron in the outermost shell The former refersto the daughter atom whereas the latter refers to the parent119864119867 asymp |119861(119884)119867 | minus |119861(119883)out | The reason is that for the shells abovethe vacancy the effective charge is closer to the one in theparent atom (there is one proton less in the nucleus but alsoone electron less in an inner shell) and therefore its bindingenergies should be used [80]

4 Advances in High Energy Physics

An alternative capture process to the one in (6) involvesthe instantaneous formation of a second hole 1198671015840 due to themismatch between the spectator electron wave functions inthe parent and in the daughter atom Whereas the hole 119867 isleft by the captured electron and therefore fulfils the energyand angular momentum conditions for such process theextra hole 1198671015840 has a different origin namely the abovementioned incomplete overlap of electron wave functionswhichmay occur inmany electron shellsThe electron leavingthe extra hole may have been ldquoshaken uprdquo (excited) to anunoccupied orbital in the daughter atom

(119883) 997888rarr (119884)1198671198671015840 + ]119894 (9)

or it may have been ldquoshaken offrdquo to the continuum (ejected)

(119883) 997888rarr (119884+)1198671198671015840 + 119890minus + ]119894 (10)

In a shake-up process the deexcitation of the final atomcontributes to the calorimeter energywith a peak at an energyapproximately equal to the sum of the binding energies of the119867 and 1198671015840 electrons 1198641198671198671015840 asymp |119861(119884)119867 | + |119861(119883)1198671015840 | minus |119861(119883)out | Thussatellite peaks show up located after the single-hole peak at119864119867 In the case of a shake-off the calorimeter energy is thecombination of 1198641198671198671015840 and the electron kinetic energy thelatter showing a continuous distribution as corresponds to thethree-body decay of (10)

It is important to notice that some electron emissionprocesses of the type

(119883) 997888rarr (119884+)1198671015840 11986710158401015840 + 119890minus + ]119894 (11)

where none of the holes in the daughter atom correspondto the captured electron 119867 are quantum mechanicallyindistinguishable [81 82] from the three-step process in (6)and (8) and therefore they contribute to the same one-hole peaks at 119864119867 The contributions to the spectrum of thetwo-hole excitations that are actually distinct from the one-hole excitations ((9) and (10)) have been a subject of recentstudies devoted to search for the electron neutrino mass andare therefore focused on the endpoint of the spectrum (seeeg [80ndash83]) For heavy neutrino searches a similar analysisshould be performed at other regions of the energy spectrumnot just at the endpoint In this paper we restrict thecalculations to the one-hole states

The density of final states of the emitted neutrino inelectron capture is proportional to

120588 = 120588] (119864]) prop 119901]119864] = (1198642] minus 1198982])12 119864] (12)

where 119864] 119901] and 119898] are the neutrino total energy momen-tum and mass respectively On the other hand the proba-bility of capture of a bound electron follows a Breit-Wignerdistribution of width Γ119909 and peaks at the energy 119864119909

119875 (119864) = Γ1199092120587(119864 minus 119864119909)2 + Γ21199094 (13)

Using Fermirsquos Golden Rule the differential reaction ratewith respect to the neutrino energy yields

119889120582119889119864] = 119870EC (1198642] minus 1198982])12 119864]sum119867

119882(])119867sdot Γ1198672120587[119864] minus (119876 minus 119864119867)]2 + Γ21198674

(14)

where 119870EC contains among other factors the weak interac-tion coupling constant and the nuclear matrix element Thefactor119882(])119867 is the squared leptonic matrix element for a givenhole state119867

We write 119882(])119867 = 119862119867119878(])119867 to have a general expressionfor allowed and forbidden transitions The factor 119862119867 is thesquared amplitude of the bound-state electron radial wavefunction at the nuclear interior containing also the squaredoverlap between the initial and the final atom orbital wavefunctions and the effect of electron exchange (as defined inAppendix F-2 of [56]) The cases discussed in the nextsections involve allowed Gamow-Teller transitions (163Ho (72minus)rarr 163Dy (52minus)) and first forbiddenGamow-Teller transi-tions (202Pb (0+) rarr 202Tl (2minus) and 205Pb (52minus) rarr 205Tl (12+)) In the first case the factor 119878(])119867 = 1 for all 119867 states Inthe second case the factor 119878(])119867 = 1 for electrons captured withorbital angularmomentum 119897 = 1 (and neutrinos emitted with119897 = 0) and 119878(])119867 = 1199012] for electrons captured with orbitalangular momentum 119897 = 0 (and neutrinos emitted with 119897 =1) The reason for this is that due to conservation of totalangularmomentum first forbidden transitions have 119871 = 1 forthe leptonic pair Hence there is a linear momentum depen-dence in the leptonic matrix element corresponding to thelepton that carries the angular momentum in each specificcaptureThe corresponding 1199012119890 factor for the 119897 = 1 electrons isembedded in the 119862119867 factor [56]

Equation (14) includes every possible orbital electroncapture the probability of each peaking at 119864] = 119876 minus 119864119867 asdictated by energy conservation where 119864119867 is the excitationenergy of the final atom due to the electron hole119867 resultingfrom capture The energy collected by a calorimeter from theatomic deexcitations is 119864119888 = 119876 minus 119864] namely all the availableenergy except for the one carried away by the neutrino Thedifferential rate can be written in terms of the calorimeterenergy as

119889120582119889119864119888 = 119870EC [(119876 minus 119864119888)2 minus 1198982]]12 (119876 minus 119864119888)sum119867

119862119867119878(])119867sdot Γ1198672120587(119864119888 minus 119864119867)2 + Γ21198674

(15)

In the next section we shall also consider integrated ratesover particular energy ranges which can be calculatednumerically from (15) A convenient compact analyticalexpression of the integrated rate can be obtained from (15)by replacing the Breit-Wigner distributions by Dirac deltas

Advances in High Energy Physics 5

With this approximation the integral of the differential ratecan be written as

120582 = 119870ECsum119867

119862119867119878(])119867 [(119876 minus 119864119867)2 minus 1198982]]12 (119876 minus 119864119867) (16)

In this theoretical model we have neglected two-holepeaks deexcitations through virtual intermediate states andinterferences between deexcitation channels The theoreticalcalorimeter spectrum is thus a single-hole approximationthat assumes full collection of de-excitation energy by thecalorimeter and no pile-up

4 Sterile Neutrino Effect in the Spectrum

Due to the smallness of the mixing angle it is useful toconsider ratios between rates that emphasize the contributionof the heavy neutrino In [84ndash86] for the case of beta decay aratio was considered between the contributions of the heavyand the light neutrino to the differential rates as we shall seein Section 6 For the case of electron capture the differentialdecay rates contain many peaks and it is more convenient toconsider integrated decay rates over specific energy rangesas discussed in [87] In either case (beta decay and electroncapture) this amounts to consider that the detector collectsevents within energy ranges 119864119894 plusmn Δ and 119864119895 plusmn Δ in two regionsof the spectrum such that the mass of the hypothetical heavyneutrino lies in between namely 119864119894 + Δ lt 119898ℎ lt 119864119895 minus Δ with119864119895 lt 119876minusΔ A ratio 119877 between the number of collected eventsin both regions is then performed which corresponds to thetheoretical expression

119877 = Λ 119894Λ 119895 =120581119897119894 + tan2120577120581ℎ119894120581119897119895 (17)

where Λ 119894 = cos2120577120581119897119894 + sin2120577120581ℎ119894 (in the region where both 119897and ℎ mass eigenstates contribute) and Λ 119895 = cos2120577120581119897119894 (in theregion where ℎ is not energetically allowed) The integrals 120581are defined as

120581]119894119895 = int119864119894119895+Δ119864119894119895minusΔ

119889120582119889119864119888 (119898]) 119889119864119888 (18)

In electron capture the energies 119894 and 119895 can be selected asthe energies of two peaks and the integration intervals canbe chosen as the width of the capture peaks If the peaks areapproximated by delta functions the integrals (using119864119867 = 119864119894and Δ rarr 0) can be written as

120581]EC119894119903 = 119870EC119862119894119903119878(])119894119903 (119876119903 minus 119864119894)2 [1 minus ( 119898]119876119903 minus 119864119894)2]12 (19)

where 119876119903 is the atomic mass difference in a given isotope 119903119864119894 is the energy position of a given peak 119894 in the calorimeterspectrum 119898] is 119898119897 asymp 0 the mass of a light neutrino or119898ℎ the mass of a heavy neutrino and 119878(])119894119903 contains theneutrino momentum dependence for the peak 119894 comingfrom the leptonic matrix element squared As explained inthe previous section for allowed transitions 119878(])119894119903 = 1 For

first forbidden transitions some of the capture peaks have119878(])119894119903 = 1199012]119894119903 for the 119894 peaks corresponding to 11990412 shells (aswell as for those corresponding to 11990112 shells that contributethrough their admixtures with the 119897 = 0 orbital due torelativistic corrections) The other peaks have 119878(])119894119903 = 1where 119894 corresponds to 11990132 shells (and 11988932 shells throughtheir admixtures with the 119897 = 1 orbital due to relativisticcorrections)

For electron capture it is convenient to define a ratiosimilar to the one in (17) but where the numerator andthe denominator are themselves ratios of numbers of eventswithin the same peak but for different isotopes of the sameelement 119903 and 119904 so that some atomic corrections cancel out[87]

1198771015840 = Λ 119894119903Λ 119894119904Λ 119895119903Λ 119895119904 = (119877119897119894119895119903119904)2(120574+1)(1 + 1205962120574+1119894119903 tan2120577

1 + 1205962120574+1119894119904 tan2120577) (20)

where one should notice that the factors 119862119894 and 119862119895 cancel outin addition to the factor119870EC which cancels out in both ratios(119877 and 1198771015840) The factors in (20) have very simple analyticalforms when one uses (19)

119877119897119894119895119903119904 = (119876119903 minus 119864119894) (119876119904 minus 119864119895)(119876119903 minus 119864119895) (119876119904 minus 119864119894)

120596119894119903(119904) = [1 minus ( 119898ℎ119876119903(119904) minus 119864119894)2]12

(21)

where 120574 depends on the angular momenta of each lepton in agivenΔ119869120587 nuclear transition For instance for allowed decays120574 = 0 whereas for first forbidden decays 120574 = 1 for 11990412 and11990112 peaks or 120574 = 0 for 11990132 and 11988932 peaks In the expressionsabove we have assumed that both peaks 119894 and 119895 are of thesame type namely 120574119894 = 120574119895 = 120574

By measuring the ratio in (20) the mixing angle can thenbe obtained as

120577 = arctan[[

(119877119897119894119895119903119904)2(120574+1) minus 1198771015840exp1198771015840exp1205962120574+1119894119904 minus (119877119897119894119895119903119904)2(120574+1) 1205962120574+1119894119903

]]12

(22)

where the theoretical values of 119877119897119894119895119903119904 120596119894119903 and 120596119894119904 requirean accurate experimental knowledge of the position of theselected peaks and more importantly of the atomic massdifferences 119876119903 and 119876119904 In addition one has to consider fixedinitially unknown values of the mass of the heavy neutrino119898ℎ that is searched for or otherwise be content with thegeneration of exclusion plots in case of no effect observation

In summary we consider here two types of ratios that canbe useful in the search for a signal of keV sterile neutrinos inelectron capture (1) the ratio between the intensity of twopeaks in the electron capture spectrum of a given nucleusas for the case of 163Ho discussed in Section 5 (2) The casein which two isotopes of a given element undergo electroncapture where one can use the ratio defined in (20) as isthe case of Lead isotopes discussed in Section 5 For the

6 Advances in High Energy Physics

10minus5

Neutrino phase spaceAtomic deexcitationTotal

M1M2

N1

N2O1

O2

0 05 1 15 2 25 3Ec (keV)

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

(a)

m = 25 keVm = 2 keVm = 15 keV

m = 1 keVm = 05 keVm = 0

0 05 1 15 2 25 3Ec (keV)

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

10minus5

(b)

0 05 1 15 2 25 3

No mixingMax mixing

Ec (keV)

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

(c)

Figure 1 Calorimeter spectrum after electron capture in 163Ho (a) Full spectrum (solid curve) and contributions emitted neutrino phasespace (dotted curve) and daughter atom deexcitation (dashed curve) (b) Full spectrum for different neutrino masses 0 05 1 15 2 and25 keV (c) full spectrum for light (119898119897 asymp 0 keV) - heavy (119898ℎ = 2 keV) neutrino mixing using a maximal mixing angle 120577 = 45∘ for illustrationwith (solid curve) and without (dotted curve) the heavy neutrino contribution For comparison the spectrumwithout heavy neutrinomixing(dashed curve) is also shownWe use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of the figure Realistic valuesof 120577 lt 001∘ result in a reduction of the difference between the dashed and the solid lines in (c) by more than seven orders of magnitude

beta decay case in Section 6 we consider the ratio betweenthe differential rates but the ratios between integrated ratesare also useful and may be more realistic for comparison toexperiments

5 Results for Electron Capture Spectra

Let us first consider the capture of an atomic electron by thenucleus Holmium 163 (163Ho 119885 = 67 119873 = 96) to turn intoDysprosium 163 (163Dy 119885 = 66119873 = 97) with ground statespins and parities 119869120587 = 72minus and 119869120587 = 52minus respectively Itis an allowed unique Gamow-Teller transition (Δ119869 = 1 andno parity change the leptons carry no orbital angularmomentum Δ119871 = 0 and couple to spin 119878 = 1) with atomicmass difference 119876 = 2833 plusmn 0030 plusmn 0015 keV [88] which isbeing currently measured at the Electron Capture Holmium(ECHo) experiment [50ndash52] Only electrons with principalquantum number 119899 larger than 2 have a binding energylower than the 119876-value and can therefore be captured Inaddition being an allowed decay only electrons from 11990412 and

11990112 orbitals can be captured the latter through admixtureswith the 119897 = 0 orbital due to relativistic corrections Thuscapture from the orbitals M1 M2 N1 N2 O1 O2 and P1 (inspectroscopic notation) is possible [80 89ndash91]

In Figure 1 we show the differential electron capture ratein 163Ho as a function of the calorimeter energy 119864119888 = 119876minus119864]In Table 1 we give theoretical values of the strengths andwidths of each capture peak and experimental values ofthe electron binding energy again for each capture peakThe strengths 119862119867 are obtained in [56 57] from relativisticHartree-Fock calculations of the electron wave functions atthe nuclear site accounting for exchange and overlap contri-butions In the upper plot the emitted neutrino phase spacecontribution and the daughter atom deexcitation contribu-tion are plotted separately together with the product of bothwhich is the full differential electron capture rate In themiddle plot we show the full differential rate for differentmasses of the emitted neutrino 119898] asymp 0 (realistic) and 119898] asymp05 1 15 2 and 25 keV (unrealistic) As can be seen in theplot each curve ends at 119876minus119898] Finally the lower plot showsthe spectrum resulting from the mixing of a light neutrino

Advances in High Energy Physics 7

Table 1 Atomic parameters for electron capture in Holmium from shells M1 to O2The values of 119862119867 [56 57] correspond to the atomic shellsin Holmium while the orbital electron widths Γ119867 [58] and binding energies 119861119867 (ltQ) [59ndash61] correspond to the atomic shells in Dysprosium

M1 M2 N1 N2 O1 O2119862119867 005377 0002605 001373 00005891 0001708 00001Γ119867 [keV] 0013 0006 0006 0005 0005 0002119861119867 [keV] 2047 1842 0416 0332 0063 0026

Table 2 Same as Table 1 (data from the same references) but for electron capture in Lead going toThalliumThe upper table summarizes thevalues of the relevant parameters for 11990412 and 11990112 atomic shells while the lower table shows the values for 11990132 and 11988932 atomic shells In thelatter case 119862119867 includes an extra factor 1199012119890

L1 L2 M1 M2 N1 N2 O1 O2119862119867 08781 006647 02125 001759 005849 0004431 001016 00008052Γ119867 [keV] 0011 0006 0015 0010 0009 0007 mdash mdash119864119867 [keV] 15346 14697 3703 3415 0845 0720 0136 0099L3 M3 M4 N3 N4 O3 O4119862119867 [keV2] 18978193 5366014 45383 1363307 12607 237280 1500Γ119867 [keV] 0006 0009 0002 0006 0004 0001 0001119861119867 [keV] 12657 2956 2484 0608 0406 0072 0015

mass eigenstate 119898119897 asymp 0 with a heavy mass eigenstate 119898ℎ asymp2 keV (solid curve) as in (5) Although the spectrum withmixing is realistic in the sense that it is the result expected if aheavy mass eigenstate exists it is unrealistic in the degree ofmixing shown in the figure which has been maximized herefor the sake of visibility of the ldquokinkrdquo in the scale of the figure50 light neutrino and 50 heavy neutrino correspondingto amixing angle 120577 = 45∘The ldquokinkrdquo can be seen in this curveat 119864119888 = 119876 minus 119898ℎ = 0833 keV Above this calorimeter energythe heavy mass eigenstate cannot be produced due to energyconservation For comparison the spectrum for119898] asymp 0 withno mixing with heavy states is shown in the dashed curve

Another example of electron capture under study here isthat of the Lead isotopes 202 (202Pb 119885 = 82 119873 = 120)and 205 (205Pb 119885 = 82 119873 = 123) going to the Thalliumisotopes 202 (202Tl 119885 = 81 119873 = 121) and 205 (205Tl119885 = 81 119873 = 124) respectively where the process is in bothcases a first forbidden Gamow-Teller unique transition with0+ rarr 2minus for 202Pb and 52minus rarr 12+ for 205Pb (Δ119869 = 2 andparity change the leptons carry Δ119871 = 1 and couple to 119878 = 1)The atomic mass differences obtained from the data in [92]are 119876 = 46 plusmn 14 keV and 119876 = 506 plusmn 18 keV respectivelyAccording to the explanations given in Section 3 capturesfrom orbitals L1 L2 M1 M2 N1 N2 O1 and O2 containan extra neutrino momentum dependence 119878(]) = 1199012] thatmodifies the calorimeter spectrum and corresponds to 120574 =1 On the other hand captures from orbitals L3 M3 M4N3 N4 O3 and O4 which correspond to 120574 = 0 containan extra electron momentum dependence 1199012119890 which beingfixed in bound electrons is included in the values of 119862119867They are given in Table 2 together with the widths and theexperimental electron binding energies in the daughter atomThallium

Figure 2 shows the capture differential rate in 205Pb (darkcurves) and in 202Pb (light curves) Position width and

strength of the capture peaks are assumed to be the same inboth isotopes but the spectra are different because of the dif-ferent 119876-values Solid curves are for light neutrino emission119898] asymp 0 and dashed lines are for heavy neutrino emissionwith119898] = 40 keV (results are given separately for each massbeforemixing) Due to the large119876-values electron capture inLead isotopes allows us to explore this mass value althoughit may be somewhat larger than the expected value fromcosmological reasons The analysis of the ratio in (20) canbe computed for example using the capture peaks L3 at119864119888 = 12657 keV and M3 at 119864119888 = 2956 keV both beingof the 120574 = 0 type For 119898ℎ = 40 keV one would obtain119877l1198723 1198713 202 205 = 1037 1205961198723202 = 0369 1205961198723205 = 0543These results should be introduced in (22) together with theexperimental ratio 1198771015840exp to obtain the value of the mixingangle 120577

It is important to remark that the theoretical quantities 119877119897and 120596 computed as described above contain several approxi-mations They correspond to Dirac-delta peaks ignoring theactual shapes and widths and thus they just refer to onesingle capture peak neglecting the possible effect of the tailsof nearby one-hole peaks Moreover the influence of two-hole peaks close to the main ones has also been neglectedOther effects that have not been taken into account butwhose influence is expected to be very small are multihole(more than two) peaks virtual intermediate states (influenceof transitions through atomic shells not energetically acces-sible) or interference between atomic transitions resultingfrom addition of amplitudes instead of intensities [81 82]Theaccurate experimental determination of the ratio 1198771015840exp entailsits own difficulties among them the possible existence ofmetastable atomic states whose delayed de-excitations failto contribute to the collected spectrum and the variety ofchemical environments resulting in a complex mixture of 119876-values

8 Advances in High Energy Physics

L1

L2

L3

M1M2

M3

M4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

6 162 8 10 12 144 18 200Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(a)

N1

N2

N3

O1

O2

O3

N4O4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

05 0603 0401 02 07 08 09 1000Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(b)

Figure 2 Calorimeter spectrum after electron capture in 202Pb (light curves) and in 205Pb (black curves) with emission of a light neutrino119898119897 = 0 (solid curves) and a heavy neutrino119898ℎ = 40 keV (dashed curves) (a) Full spectrum showing L and M capture peak labels (b) Lowenergy region (119864119888 = 0-1 keV) showing N and O capture peak labels

6 Results for Beta Decay Spectra

In addition to electron capture we also show here the effectof heavy neutrino emission in the electron spectrum of betadecays for one of the cases of most interest Tritium It hasbeen studied in depth in previous works [84ndash86 93] togetherwith another process that has also drawn considerable theo-retical end experimental attention the beta decay of Rhenium187 [37 39ndash41 84ndash86] The beta decay of Tritium (3H 119885 = 1119873 = 2) going to Helium 3 (3He 119885 = 2119873 = 1) is

3H 997888rarr 3He + 119890minus + ]119890 (23)

and has a 119876-value of 1859 keV [92] Both the initial and thefinal nuclear ground states have spin-parity 12+ and thetransition is allowed with the electron and the antineutrinoemitted in 119904-wave The Karlsruhe Tritium Neutrino Experi-ment (KATRIN) [42ndash46] is currently studying this decay inorder to determine the active neutrino mass and could alsostudy the production of a heavy neutrino of amass lower than18 keV

The differential decay rate with respect to the electronenergy is given by

119889120582119889119864119890 = 119870120573 (1198642119890 minus 1198982119890)12 [(119876 + 119898119890 minus 119864119890)2 minus 1198982]]12

sdot 119864119890 (119876 + 119898119890 minus 119864119890) (24)

where 119870120573 contains among others the weak interactioncoupling the nuclearmatrix element and the Fermi function

As an illustration of the heavy neutrino effect in the Tri-tium beta decay we plot in Figure 3 the differential decay ratefrom (24) and (5) using a maximal mixing with an unrealisticvalue of the mixing angle 120577 = 45∘ to show the effect more

distinctly in the scale of the figure We have used heavy masscomponents with 119898ℎ = 2 keV [26] (Figure 3(a)) and with119898ℎ = 7 keV [29 30] (Figure 3(b)) For comparison thespectra without the heavy neutrino contribution (120582ℎ = 0) andwithout mixing (120577 = 0∘) are also shown in this plot The kinkin the spectrum at 119864119890 minus 119898119890 = 119876 minus 119898ℎ can be observed if theexperimental relative error is lower than the size of the stepthe latter being very small in realistic situations

The effect of a heavy neutrino emission can be analyzedthrough the ratio between the heavy and the light neutrinocontributions to the spectrum

R equiv 119889120582ℎ119889119864119890119889120582119897119889119864119890 tan2120577 (25)

The full differential decay rate is also related to the ratio Rthrough

119889120582119889119864119890 =119889120582119897119889119864119890 [1 +R] cos2120577 (26)

In Figure 4 we plot R the ratio of the heavy neutrinocontribution over the light neutrino contribution to the decayrate as a function of the momentum of the emitted electronfor a heavy neutrino mass 119898ℎ = 2 keV and different mixingangles 120577 = 001∘ 0005∘ and 0001∘ The size of this ratio(lt10minus7) gives an idea of the difficulty of finding the kink in thespectrum due to the production of the heavymass eigenstateAs can be seen in the figure the ratio is different from zeroand almost constant in the range 0 le 119901119890 lt (119901119890)max wherethe maximum electron momentum is given by (119901119890)max =[(119876 minus 119898ℎ)(119876 minus 119898ℎ + 2119898119890)]12 For the heavy mass used inthe figure (119901119890)max ≃ 1313 keV The ratio R decreases as themixing angle decreases being approximately proportional to

Advances in High Energy Physics 9

Mixing but no heavy neutrino contribution

104

105

106

107

108d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 2 keV

(a)

104

105

106

107

108

d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

Mixing but no heavy neutrino contribution

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 7 keV

(b)

Figure 3 Electron spectrum of Tritium beta decay for a light-heavy neutrino mixing angle 120577 = 45∘ shown just for illustration and a heavyneutrino mass (solid curve)119898ℎ = 2 keV (a) and119898ℎ = 7 keV (b) The spectrum without heavy neutrino mass contribution (dotted curve) andwithout mixing (dashed curve) are also shown We use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of thefigure Realistic values of 120577 lt 001∘ result in a reduction of the difference between the dotted and the solid lines by more than seven orders ofmagnitude

10minus11

10minus10

10minus9

10minus8

10minus7

10minus12

120577 = 001∘

120577 = 0005∘

120577 = 0001∘

15090 100 110 120 130 1408070pe (keV)

ℛ

Figure 4 Ratio R defined in (25) for the Tritium beta decay as afunction of the electronmomentum for a heavy neutrinomass119898ℎ =2 keV and different values of the mixing angle 120577 = 001∘ (solid line)0005∘ (dashed line) and 0001∘ (dotted line)

1205772 It also decreases for larger heavy neutrino masses 119898ℎ Adifferent ratio between the spectra with mixing and without

mixing (Rlowast) has also been considered in [84ndash86 93] relatedwithR in (25) as

Rlowast = minussin2120577 +R cos2120577 = 119889120582119889119864119890119889120582119897119889119864119890 minus 1 (27)

7 Conclusions

Signatures of hypothetical keV sterile neutrinos which couldbe warm dark matter candidates (WDM) may be found inthe spectra of ordinary weak nuclear decays The electronspectrum in beta decay is cleaner for this purpose becauseit is a smooth curve while the deexcitation spectrum afterelectron capture shows many peaks On the other hand inbeta decay the possible excitation of the atoms or moleculesis not disentangled while in electron capture the calorimetercollects all the energy independently of the excitation ofthe atom and of the deexcitation path and one has higherstatistics around the capture peaks

Figures of electron spectrum for beta decay in 3H aswell as of calorimeter spectrum for electron capture in 163Hoand 202Pb and 205Pb are given considering various valuesof the heavy neutrino mass (2 keV 7 keV and 40 keV) thatmay be experimentally probed In both weak processes thesmall value of the light-heavy neutrino mixing angle requiresextremely high experimental precision and the use of sourceswith large stability to reduce systematic errors This is why it

10 Advances in High Energy Physics

is useful to consider relevant ratios between transition rates asfirst introduced in [84ndash87] to analyze the data We considertwo cases the case of a single isotope where one may use theratio of accumulated number of events in different regionsof the spectrum which allows us to remove uncertaintiesrelated to the nuclear matrix element and to the valuesof overall constants And in the case of two isotopes weconsider ratios of the above mentioned ratios that allow usto reduce uncertainties from atomic parameters We give aswell analytical expressions that can be used to obtain a goodtheoretical approximation to the experimental ratios ((19) to(21))

Both the experimental measurement and the theoreticalmodel must be accurate enough to detect differences in theexpected versus themeasured ratios of the order of themixingangle squared 1205772 ≲ 10minus8 This is also the size of the kinkexpected in the electron spectrumof beta decay located at thelimit of the regionwhere the production of a heavy neutrino isenergetically allowed In order to identify this kink among thestatistical fluctuations of themeasured spectrum the numberof collected events must be larger than the inverse of thesquared ratioR in (25)

Our results particularly on the electron capture spectraof Lead isotopes that include for the first time all thepossible peaks may help in designing and analyzing futureexperiments to search for sterile neutrinos in the keV massrangeThe use of the different types of ratios between numberof events discussed here for both electron capture and betadecay processes may also help in planning the experimentsby establishing the threshold of statistical and systematicuncertainties required for the detection of sterile neutrinosof given mass and mixing or to properly extract exclusionplots

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

E Moya de Guerra and O Moreno acknowledge MINECOfor partial financial support (FIS2014-51971-P) and IEM-CSIC for hospitality O Moreno acknowledges supportfrom a Marie Curie International Outgoing Fellowshipwithin the European Union Seventh Framework Pro-gramme under Grant Agreement PIOF-GA-2011-298364(ELECTROWEAK) M R M thanks N G Sanchez forcorrespondence and conversations

References

[1] F Zwicky ldquoDieRotverschiebung von extragalaktischenNebelnrdquoHelvetica Physica Acta vol 6 pp 110ndash127 1933

[2] M Roos ldquoAstrophysical and cosmological probes of darkmatterrdquo Journal of Modern Physics vol 3 pp 1152ndash1171 2012

[3] S Dodelson Modern Cosmology Academic Press New YorkNY USA 2003

[4] S Weinberg Cosmology Oxford University Press Oxford UK2008

[5] J Lesgourgues and S Pastor ldquoNeutrino mass from cosmologyrdquoAdvances in High Energy Physics vol 2012 Article ID 60851534 pages 2012

[6] D Boyanovsky H J de Vega and N G Sanchez ldquoConstraintson dark matter particles from theory galaxy observations andN-body simulationsrdquo Physical Review D vol 77 no 4 ArticleID 043518 2008

[7] D Boyanovsky H J de Vega and N G Sanchez ldquoDarkmatter transfer function free streaming particle statistics andmemory of gravitational clusteringrdquo Physical Review D vol 78no 6 Article ID 063546 2008

[8] C Destri H J de Vega and N G Sanchez ldquoWarm dark matterprimordial spectra and the onset of structure formation atredshift zrdquo Physical Review D vol 88 no 8 Article ID 0835122013

[9] H J De Vega P Salucci and N G Sanchez ldquoThe mass ofthe dark matter particle theory and galaxy observationsrdquo NewAstronomy vol 17 no 7 pp 653ndash666 2012

[10] H J de Vega P Salucci and N G Sanchez ldquoObservationalrotation curves and density profiles versus the Thomas-Fermigalaxy structure theoryrdquoMonthlyNotices of the Royal Astronom-ical Society vol 442 no 3 pp 2717ndash2727 2014

[11] H J de Vega and N G Sanchez ldquoModel-independent analysisof dark matter points to a particle mass at the keV scalerdquoMonthly Notices of the Royal Astronomical Society vol 404 no2 pp 885ndash894 2010

[12] H J de Vega and N G Sanchez ldquoConstant surface gravity anddensity profile of dark matterrdquo International Journal of ModernPhysics A vol 26 no 6 pp 1057ndash1072 2011

[13] H J de Vega and N G Sanchez ldquoCosmological evolutionof warm dark matter fluctuations I Efficient computationalframework with Volterra integral equationsrdquo Physical ReviewDvol 85 no 4 Article ID 043516 20 pages 2012

[14] H J de Vega and N G Sanchez ldquoThe dark matter distributionfunction and halo thermalization from the Eddington equationin galaxiesrdquo International Journal of Modern Physics A vol 31no 13 Article ID 1650073 2016

[15] NMenci S Paduroiu P Salucci N Sanchez and C RWatsonldquolectures at the 20th Paris Cosmology Colloquium Chalonge-Hector de Vegardquo 2016 httpchalongeobspmfr

[16] E W Kolb and M S Turner The Early Universe AddisonWesley Boston Mass USA 1990

[17] J F Navarro C S Frenk and S D M White ldquoA universal den-sity profile from hierarchical clusteringrdquo Astrophysical JournalLetters vol 490 no 2 pp 493ndash508 1997

[18] W J G de Blok ldquoThe core-cusp problemrdquo Advances in Astron-omy vol 2010 Article ID 789293 14 pages 2010

[19] G Gilmore M I Wilkinson R F GWyse et al ldquoThe observedproperties of dark matter on small spatial scalesrdquo AstrophysicalJournal vol 663 no 2 pp 948ndash959 2007

[20] P Salucci and Ch Frigerio Martins ldquoThe mass distribution inSpiral galaxiesrdquo EAS Publications Series vol 36 pp 133ndash1402009

[21] J van Eymeren C Trachternach B S Koribalski and R-JDettmar ldquoNon-circular motions and the cusp-core discrepancyin dwarf galaxiesrdquo Astronomy amp Astrophysics vol 505 no 1 pp1ndash20 2009

[22] M Walker and J Penarrubia ldquoA method for measuring (slopesof) the mass profiles of dwarf spheroidal galaxiesrdquo The Astro-physical Journal vol 742 no 1 p 20 2011

Advances in High Energy Physics 11

[23] R F G Wyse and G Gilmore ldquoObserved properties of darkmatter on small spatial scalesrdquo in Proceedings of the Proceedingsof the International Astronomical Union Symposium (IUA rsquo07)vol 3 no 244 pp 44ndash52 Cardiff UK June 2007

[24] A Burkert ldquoThe structure of dark matter halos in dwarfgalaxiesrdquo Astrophysical Journal vol 447 no 1 pp L25ndashL281995

[25] C Destri H J de Vega and N G Sanchez ldquoQuantum WDMfermions and gravitation determine the observed galaxy struc-turesrdquo Astroparticle Physics vol 46 pp 14ndash22 2013

[26] N Menci N G Sanchez M Castellano and A GrazianldquoConstraining the warm dark matter particle mass throughultra-deep UV luminosity functions at Z = 2rdquoTheAstrophysicalJournal vol 818 no 1 article 90 2016

[27] E Bulbul M Markevitch A Foster R K Smith M Loewen-stein and SW Randall ldquoDetection of an unidentified emissionline in the stackedX-ray spectrumof galaxy clustersrdquoAstrophys-ical Journal vol 789 article 13 2014

[28] A Boyarsky O Ruchayskiy D Iakubovskyi and J FranseldquoUnidentified line inX-ray spectra of the andromeda galaxy andperseus galaxy clusterrdquo Physical Review Letters vol 113 no 25Article ID 251301 2014

[29] K N Abazajian ldquoResonantly produced 7 keV sterile neutrinodark matter models and the properties of Milky Way satellitesrdquoPhysical Review Letters vol 112 no 16 Article ID 161303 5pages 2014

[30] N Menci A Grazian M Castellano and N G SanchezldquoShock connectivity in the 2010 August and 2012 July solarenergetic particle events inferred from observations and ENLILmodelingrdquo Astrophysical Journal Letters vol 825 no 1 article 12016

[31] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[32] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 481 no 1-2 pp 1ndash28 2009

[33] B Pontecorvo ldquoNeutrino experiments and the problem ofconservation of leptonic chargerdquo Soviet Physics-JETP vol 26no 5 pp 984ndash988 1968

[34] A Merle ldquokeV neutrino model buildingrdquo International Journalof Modern Physics D vol 22 no 10 Article ID 1330020 2013

[35] J DVergados andT S Kosmas ldquoSearching for cold darkmatterA case of coexistence of supersymmetry and nuclear physicsrdquoPhysics of Atomic Nuclei vol 61 p 1066 1998

[36] E Holmlund M Kortelainen T S Kosmas J Suhonen and JToivanen ldquoMicroscopic calculation of the LSP detection ratesfor the 71Ga 73Ge and 127I dark-matter detectorsrdquoPhysics LettersB vol 584 no 1-2 pp 31ndash39 2004

[37] R Adhikari M Agostini N Anh Ky et al ldquoA white paper onkeV sterile neutrino dark matterrdquo httpsarxivorgabs160204816

[38] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[39] MARE collaboration httpmaredfmuninsubriaitfrontendexecphp

[40] A Nucciotti ldquoNeutrino mass calorimetric searches in theMARE experimentrdquo httpsarxivorgabs10122290

[41] A Nucciotti ldquoOn behalf of the MARE collaboration lecture attheWorkshop ChalongeMeudonrdquo 2011 httpchalongeobspmfr

[42] KATRIN Collaboration httpwwwkatrinkitedu

[43] C Weinheimer ldquoDirect determination of neutrino mass fromtritium beta spectrumrdquo httpsarxivorgabs09121619

[44] C Weinheimer lecture at the 20th Paris Cosmology Chalonge-Hector de Vega Colloquium 2016 httpchalongeobspmfr

[45] G Drexlin lecture at the 19th Paris Cosmology Colloquium2015

[46] A Huber lecture at the Chalonge-de Vega Meudon Workshop2016 httpchalongeobspmfr

[47] C Tully lecture at the 19th Paris Cosmology Colloquium 2015[48] S Betts W R Blanchard R H Carnevale et al ldquoDevelop-

ment of a relic neutrino detection experiment at PTOLEMYprinceton tritiumobservatory for light early-universemassive-neutrino yieldrdquo httpsarxivorgabs13074738

[49] D M Asner R F Bradley L de Viveiros et al ldquoSingle-electrondetection and spectroscopy via relativistic cyclotron radiationrdquoPhysical Review Letters vol 114 no 16 Article ID 162501 2015

[50] P C-O Ranitzsch J-P Porst S Kempf et al ldquoDevelopmentof metallic magnetic calorimeters for high precision measure-ments of calorimetric 187Re and 163Ho spectrardquo Journal of LowTemperature Physics vol 167 no 5-6 pp 1004ndash1014 2012

[51] C Hassel ldquoLecture at the International School of AstrophysicsDaniel ChalongemdashHector de Vegardquo 2015

[52] L Gastaldo K Blaum A Doerr et al ldquoThe electron capture163Ho experiment ECHordquo Journal of Low Temperature Physicsvol 176 no 5 pp 876ndash884 2014

[53] B Alpert M Balata D Bennett et al ldquoThe electron capturedecay of 163Ho tomeasure the electron neutrino mass with sub-eV sensitivityrdquo The European Physical Journal C vol 75 article112 2015

[54] E Fermi ldquoRadioattivita prodotta da bombardamento di neu-tronirdquo Il Nuovo Cimento vol 11 no 7 pp 429ndash441 1934

[55] E Fermi ldquoVersuch einer theorie der 120573-strahlen Irdquo Zeitschriftfur Physik vol 88 no 3 pp 161ndash177 1934

[56] R B Firestone and V S Shirley Eds Table of Isotopes JohnWiley amp Sons New York NY USA 8th edition 1999

[57] W Bambynek H Behrens M H Chen et al ldquoOrbital electroncapture by the nucleusrdquo Reviews of Modern Physics vol 49 no1 pp 77ndash221 1977

[58] J L Campbell and T Papp ldquoWidths of the atomic KndashN7 levelsrdquoAtomic Data and Nuclear Data Tables vol 77 no 1 pp 1ndash562001

[59] J A Bearden and A F Burr ldquoReevaluation of X-ray atomicenergy levelsrdquo Reviews of Modern Physics vol 39 no 1 pp 125ndash142 1967

[60] M Cardona and L Ley Eds Photoemission in Solids I GeneralPrinciples Springer Berlin Germany 1978

[61] J C Fuggle and N Martensson ldquoCore-level binding energies inmetalsrdquo Journal of Electron Spectroscopy and Related Phenom-ena vol 21 no 3 pp 275ndash281 1980

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[64] J-U Nabi and H V Klapdor-Kleingrothaus ldquoWeak interactionrates of sd-shell nuclei in stellar environments calculated in theproton-neutron quasiparticle random-phase approximationrdquoAtomic Data andNuclear Data Tables vol 71 no 2 pp 149ndash3451999

12 Advances in High Energy Physics

[65] J-U Nabi and H V Klapdor-Kleingrothaus ldquoMicroscopic cal-culations of stellar weak interaction rates and energy losses forfp- and fpg-shell nucleirdquo Atomic Data and Nuclear Data Tablesvol 88 no 2 pp 237ndash476 2004

[66] Y Fukuda T Hayakawa E Ichihara et al ldquoEvidence foroscillation of atmospheric neutrinosrdquo Physical Review Lettersvol 81 no 8 pp 1562ndash1567 1998

[67] Q R Ahmad and SNOCollaboration ldquoMeasurement of the rateof V119890+119889 rarr 119901+119901+119890minus interactions produced by 8B solar neutrinosat the Sudbury Neutrino Observatoryrdquo Physical Review Lettersvol 87 no 7 Article ID 071301 6 pages 2001

[68] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 2002

[69] G J Feldman J Hartnell and T Kobayashi ldquoLong-baselineneutrino oscillation experimentsrdquo Advances in High EnergyPhysics vol 2013 Article ID 475749 30 pages 2013

[70] S T Petcov ldquoThenature ofmassive neutrinosrdquoAdvances inHighEnergy Physics vol 2013 Article ID 852987 20 pages 2013

[71] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[72] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 48 no 1-2 pp 1ndash28 2009

[73] F Munyaneza and P L Biermann ldquoDegenerate sterile neutrinodark matter in the cores of galaxiesrdquo Astronomy amp Astrophysicsvol 458 no 2 pp L9ndashL12 2006

[74] D Boyanovsky and C M Ho ldquoSterile neutrino production viaactive-sterile oscillations the quantum Zeno effectrdquo Journal ofHigh Energy Physics vol 2007 no 7 p 30 2007

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[76] R E Shrock ldquoGeneral theory of weak processes involving neu-trinos I Leptonic pseudoscalar-meson decays with associatedtests for and bounds on neutrino masses and lepton mixingrdquoPhysical Review D vol 24 no 5 pp 1232ndash1274 1981

[77] R E Shrock ldquoImplications of neutrino masses and mixing forweak processesrdquo in Proceedings of the Weak Interactions asProbes of Unification vol 72 of AIP Conference Proceedings p368 Blacksburg Va USA 1981

[78] J M Conrad C M Ignarra G Karagiorgi M H Shaevitzand J Spitz ldquoSterile neutrino fits to short-baseline neutrinooscillation measurementsrdquo Advances in High Energy Physicsvol 2013 Article ID 163897 2013

[79] A Nucciotti ldquoThe use of low temperature detectors for directmeasurements of themass of the electron neutrinordquoAdvances inHigh Energy Physics vol 2016 Article ID 9153024 41 pages2016

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[81] A De Rujula ldquoTwo old ways to measure the electron-neutrinomassrdquo httpsarxivorgabs13054857

[82] A De Rujula ldquoAn upper bound on the exceptional character-istics for Lusztigrsquos character formulardquo httpsarxivorgabs08111674v2

[83] A Faessler and F Simkovic ldquoCan electron capture tell us themass of the neutrinordquo Physica Scripta vol 91 no 4 Article ID043007 2016

[84] H J de Vega O Moreno E Moya de Guerra M RamonMedrano and N G Sanchez ldquoRole of sterile neutrino warmdark matter in rhenium and tritium beta decaysrdquo NuclearPhysics B vol 866 no 2 pp 177ndash195 2013

[85] E M de Guerra O Moreno P Sarriguren and M RamonMedrano ldquoTopics on nuclear structure with electroweakprobesrdquo Journal of Physics Conference Series vol 366 ArticleID 012011 2012

[86] J M Boillos O Moreno and E Moya de ldquoActive and sterileneutrino mass effects on beta decay spectrardquo in Proceedingsof the La Rabida International Scientific Meeting on NuclearPhysics Basic Concepts in Nuclear Physics Theory Experimentsand Applications vol 1541 ofAIP Conference Proceedings p 167La Rabida Spain September 2012

[87] P E Filianin K Blaum S A Eliseev et al ldquoOn the keVsterile neutrino search in electron capturerdquo Journal of PhysicsG Nuclear and Particle Physics vol 41 no 9 Article ID 0950042014

[88] S Eliseev K BlaumM Block et al ldquoDirect measurement of themass difference of 163Ho and 163Dy solves theQ-value puzzle forthe neutrino mass determinationrdquo Physical Review Letters vol115 no 6 Article ID 062501 2015

[89] A De Rujula ldquoA new way tomeasure neutrinomassesrdquoNuclearPhysics Section B vol 188 no 3 pp 414ndash458 1981

[90] M Lusignoli and M Vignati ldquoRelic antineutrino capture on163Hodecaying nucleirdquoPhysics Letters B vol 697 no 1 pp 11ndash142011

[91] A Faessler L Gastaldo and F Simkovic ldquoElectron capture in163Ho overlap plus exchange corrections and neutrino massrdquoJournal of Physics G Nuclear and Particle Physics vol 42 no 1Article ID 015108 2015

[92] G Audi F G Kondev M Wang et al ldquoThe Nubase2012evaluation of nuclear propertiesrdquo Chinese Physics C vol 36 no12 pp 1157ndash1286 2012

[93] S Mertens ldquoSensitivity of next-generation tritium beta-decayexperiments for keV-scale sterile neutrinosrdquo JCAP vol 1502 no2 p 20 2015

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2 Advances in High Energy Physics

length is the distance travelled freely by a relativistic particleafter decoupling from the primordial plasma due to Universeexpansion (approximately the distance travelled before thetransition to nonrelativistic velocities) WDM particles (keVscale) give 119897fs sim 100 kpc while CDM (GeV-TeV scale)which are heavier and slower than WDM would give a 119897fs amillion times smaller and lead to the existence of a host ofsmall scale structures [16] On the galaxy density profiles (120588)CDM gives a steep cusp at the center (120588 sim 119903minus1) [17] On thecontrary WDM gives a finite constant density core at thecenter (120588 sim 1205880) in agreement with observations [18ndash24]WDM quantum effects [25] could be important inside thegalaxy core (below 100 pc) showing a fermionic nature forthe DM particle through the manifestation of quantumnonvanishing pressure versus gravity

Astrophysical observations fromDM-dominated objectsas well as theoretical analysis lead to aDM fermionic thermalparticle with a mass around 2 keV [26] Chandra and XMM-Newton detections in the X-ray spectra of theM31 galaxy andthe Perseus cluster (both DM-dominated) seem to be con-sistent with sterile neutrinos of a few keV In particular anunidentified 355 keV line has been observed which seemscompatible with the decay or annihilation of sterile relicneutrinos [27 28] and that does not correspond to any knownatomic emissionAlthough the interpretation of this line is stilsubject to debate it could be considered an indication of thedecay of a 71 keV nonthermal sterile neutrino [29 30]

As it is well known the Standard Model (SM) of elemen-tary particles does not describe DM particles nor does itprovide a mass for active neutrinos (]119890 ]120583 ]120591) In this modelneutrino eigenstates are left-handed and they transform asdoublets under the weak SU(2) gauge group In extended SMmodels [31 32] additional SU(3)times SU(2)timesU(1) singlet right-handed neutrinos are introduced Neutrino mass eigenstateswill be linear combinations of the left and right-handed statesandmassmatrix eigenvalues will split into lighter and heavierstates (seesaw mechanism) The lighter mass eigenstateswould be the main components of the active flavor eigen-states of the neutrinos whereas the heavier ones would bedominant in sterile neutrino flavors [33] Some extended SMmodels add three extra sterile neutrinos one with a mass ofthe order of keV (see [34] and references therein) Neutrinosin the keVmass scale open the possibility to detect warmdarkmatter in nuclear electron capture and beta decay Heaviersterile neutrinos cannot be produced in weak nuclear decaysand there are of coursemany other candidate particles forDMunconnected in principle to weak nuclear processes such asthe lightest supersymmetric particles [35 36]

In this paper we assume that a keV neutrino a soundcandidate for DM [37] is produced in ordinary weak nuclearprocesses such as beta decay and electron capture vianeutrino mixing Experiments searching for active neutrinomasses also look for sterile neutrinos in the keV range [38]MARE [39ndash41] (that used Rhenium 187 beta decay and is notactive anymore) KATRIN [42ndash46] PTOLEMY [47 48] andProject8 [49] (using Tritium beta decay) and ECHo [50ndash52]and HOLMES [53] (using Holmium 163 electron capture)The mass of the neutrino (or antineutrino) emitted in a weak

nuclear decay has an effect on the energy spectrum of theprocess as was first shown by Enrico Fermi in the early1930s [54 55] For the active neutrinos this effect showsup at the endpoint of the spectrum whereas for the sterileneutrinos considered here it may be expected to appear at afew keVbelow the endpoint Clearly in order to leave a finger-print the sterile neutrino mass must be within the119876 windowthat is lower than the energy 119876 available in the decay Theenergy spectrum to be analyzed corresponds to the emittedcharged lepton in the case of beta decay namely the spon-taneous conversion of a neutron into a proton or vice versawith emission of the charged lepton and an antineutrino orneutrino In the case of electron capture only a neutrinois emitted and no charged lepton comes out so that thespectrum to be measured corresponds to the deexcitation ofthe daughter atom

A summary of current experimental studies of neutrinoproperties in the frontiers of intensities and sensitivitiesis given in [62] Experiments are under way to determinedirectly the active neutrino mass from neutrinoless doublebeta decay [63] as well as from single beta decay [42ndash48] andelectron capture [50ndash53] The last two types of experimentsalso provide a way to search for WDM sterile neutrinos inthe measured spectra In electron capture experiments thespectrum collected in a calorimeter is directly linked to theexcitation energies of the daughter atoms or molecules In acalorimeter the active source is embedded in the detectorwhich collects the energy of all the particles emitted inthe deexcitation processes that take place in the sourceexcept that of the neutrinos In beta decay the fact that theatoms or molecules may remain excited poses a challenge onthe interpretation of the electron spectrum Electron cap-ture experiments take advantage of the larger statistics inthe spectrum regions around the capture resonances Alimitation of electron capture measurements is that in acalorimeter where the full energy range of the spectrum ismeasured pile-up can be a problem that should be preventedby limiting the activity in the experimental setup We referalways in this paper to Earth-based experiments where boththe emission and the detection of particles take place inthe laboratory Other possible scenarios like the search forsterile neutrinos in stellar matter (stellar beta decay andelectron capture rates) are beyond the scope of this paperThephase space for these rates increases manifold in stellarenvironment For a reference on stellar weak rates see [6465]

2 Heavy Mass and SterileFlavor in Neutrino States

Already observed neutrino oscillations (see eg [66ndash69])among light active neutrinos are due to the fact that the neu-trino flavor eigenstates and the mass eigenstates are not thesame Eachflavor eigenstate associatedwith a charged leptoncan be written as a combination of mass eigenstates andvice versa [70] For instance the neutrino flavor eigenstateemitted after electron capture called electron neutrino ]119890 canbe written as a combination of the three light SM mass

Advances in High Energy Physics 3

eigenstates and hypothetically of one or more extra heaviermass eigenstates as [71ndash77]

1003816100381610038161003816]119890⟩ = 1198801198901 1003816100381610038161003816]1⟩ + 1198801198902 1003816100381610038161003816]2⟩ + 1198801198903 1003816100381610038161003816]3⟩ +sumℎ

119880119890ℎ 1003816100381610038161003816]ℎ⟩ (1)

where the subscript ℎ = 4 5 stands for extra (heavier)mass eigenstates and the quantities 119880 belong to the unitaryneutrino mixing matrix The masses of the three light SMmass eigenstates are so close to each other that so far nomea-surement has been able to discern which of them has beenemitted in a given process As a result the three are emittedas a coherent superposition which is at the origin of theneutrino oscillation phenomenologyThe phase of each masseigenstate changes at a different rate while travelling givingrise to a different superposition at each location that translatesinto a varying (oscillatory) probability of detection of a givenflavor eigenstate

To simplify things we will consider here the linear combi-nation of light mass eigenstates as a single effective neutrinomass eigenstate called ldquolightrdquo with mass

119898119897 = 119898]119890 = (1198802119890111989821 + 1198802119890211989822 + 1198802119890311989823)12 (2)

and just one extra mass eigenstate clearly heavier than theothers called ldquoheavyrdquo (withmass119898ℎ whichwe shall considerin the keV range)

1003816100381610038161003816]119890⟩ = cos 120577 1003816100381610038161003816]119897⟩ + sin 120577 1003816100381610038161003816]ℎ⟩ (3)

where the mixing amplitudes have been written in terms of amixing angle 120577 between the light and the heavy neutrinomasseigenstate The other possible combination of these two masseigenstates would be the sterile flavor eigenstate

1003816100381610038161003816]119904⟩ = minus sin 120577 1003816100381610038161003816]119897⟩ + cos 120577 1003816100381610038161003816]ℎ⟩ (4)

such that ⟨]119890 | ]119904⟩ = 0 For the sterile neutrinos that canbe relevant as WDM cosmological constraints based on theobserved average darkmatter density suggest that the value ofthe mixing angle could approximately be 120577 = 0006∘ [71ndash74]corresponding to a flavor-mass amplitude119880119890ℎ = sin 120577 asymp 10minus4Other recent cosmological constraints give values of sin2(2120577)between 2 sdot 10minus11 and 2 sdot 10minus10 [27 28]

Sterile neutrino states with masses close to the ones ofthe active neutrinos can also have an impact on the patternsmeasured in oscillation experiments [78] Heavier sterileneutrinos as in particular in the keV scale do not modify theoscillation patterns since they are not emitted coherently withthe active neutrinos due to the large mass splitting

The differential energy spectrum of a weak processwhere an electronic neutrino (or antineutrino) is emittedcan therefore be decomposed in a term for light neutrinoemission and another term for heavy neutrino emission asfollows [75ndash77]

119889120582119889119864 = 119889120582119897

119889119864 cos2120577 + 119889120582ℎ

119889119864 sin2120577 (5)

This energy spectrum can be the one of the electron emittedin beta decay or the one of the daughter atom deexcitations in

electron captureThe heavymass eigenstate if it exists wouldbe emitted independently of the other masses (noncoher-ently) as long as the energy resolution of the detector is betterthan the mass difference Δ120598 lt (119898ℎ minus 119898119890) Its contribution tothemeasured spectrum 119889120582ℎ119889119864 would show up in the rangewhere the collected energy is small enough for the heavyneutrino mass to have been produced namely when 0 le 119864 le(119876minus119898ℎ) where119876 is the difference between the masses of theinitial and the final atoms when the reaction takes place invacuumAt the edge of that region at119864 = 119876minus119898ℎ a kink in thespectrum would be found with the size of the heavy neutrinocontribution namely proportional to sin2120577 sim 1205772 (the latterapproximation valid for small mixing angles as is the case inrealistic scenarios)

3 Theory of Electron Capture

Let us consider the capture of an atomic electron by thenucleus119883 (119885119873) to turn into the nucleus119884 (119885minus1119873+1)Thereactions involving the corresponding atoms (represented bythe symbol of the nucleus within brackets) begin withelectron capture

(119883) 997888rarr (119884)119867 + ]119894 (6)

followed by deexcitation of the daughter atom In (6) ]119894 is aneutrino mass eigenstate (]119897 or ]ℎ) and the superscript 119867accounts for the excited state of the atom (119884) correspondingto an electron hole in the shell 119867 due to the electroncapture from this shell in the parent atom The deexcitationof the daughter atom after electron capture can happen eitherthrough emission of a photon (X-ray emission)

(119884)119867 997888rarr (119884) + 120574 (7)

or through emission of electrons (Auger or Coster-Kronigprocess) and photons

(119884)119867 997888rarr (119884+)1198671015840 11986710158401015840 + 119890minus 997888rarr (119884+) + 120574 + 119890minus (8)

where1198671015840 and11986710158401015840 represent holes in the electron shells of theion (119884+)

The energies carried by the emitted photons and electronsare deposited in the calorimeter that surrounds the sourceand that measures the full energy spectrum of these particlesprovided that the corresponding deexcitation lifetimes aremuch smaller than the detector time response [79] The pro-cess in (7) and (8) yields the same calorimeter spectrum thatis the peaks appear at the same energy values correspondingto the excitation energies 119864119867 of the excited daughter atom(119884)119867 The excitation energy is the difference between thebinding energies of the captured electron shell and theadditional electron in the outermost shell The former refersto the daughter atom whereas the latter refers to the parent119864119867 asymp |119861(119884)119867 | minus |119861(119883)out | The reason is that for the shells abovethe vacancy the effective charge is closer to the one in theparent atom (there is one proton less in the nucleus but alsoone electron less in an inner shell) and therefore its bindingenergies should be used [80]

4 Advances in High Energy Physics

An alternative capture process to the one in (6) involvesthe instantaneous formation of a second hole 1198671015840 due to themismatch between the spectator electron wave functions inthe parent and in the daughter atom Whereas the hole 119867 isleft by the captured electron and therefore fulfils the energyand angular momentum conditions for such process theextra hole 1198671015840 has a different origin namely the abovementioned incomplete overlap of electron wave functionswhichmay occur inmany electron shellsThe electron leavingthe extra hole may have been ldquoshaken uprdquo (excited) to anunoccupied orbital in the daughter atom

(119883) 997888rarr (119884)1198671198671015840 + ]119894 (9)

or it may have been ldquoshaken offrdquo to the continuum (ejected)

(119883) 997888rarr (119884+)1198671198671015840 + 119890minus + ]119894 (10)

In a shake-up process the deexcitation of the final atomcontributes to the calorimeter energywith a peak at an energyapproximately equal to the sum of the binding energies of the119867 and 1198671015840 electrons 1198641198671198671015840 asymp |119861(119884)119867 | + |119861(119883)1198671015840 | minus |119861(119883)out | Thussatellite peaks show up located after the single-hole peak at119864119867 In the case of a shake-off the calorimeter energy is thecombination of 1198641198671198671015840 and the electron kinetic energy thelatter showing a continuous distribution as corresponds to thethree-body decay of (10)

It is important to notice that some electron emissionprocesses of the type

(119883) 997888rarr (119884+)1198671015840 11986710158401015840 + 119890minus + ]119894 (11)

where none of the holes in the daughter atom correspondto the captured electron 119867 are quantum mechanicallyindistinguishable [81 82] from the three-step process in (6)and (8) and therefore they contribute to the same one-hole peaks at 119864119867 The contributions to the spectrum of thetwo-hole excitations that are actually distinct from the one-hole excitations ((9) and (10)) have been a subject of recentstudies devoted to search for the electron neutrino mass andare therefore focused on the endpoint of the spectrum (seeeg [80ndash83]) For heavy neutrino searches a similar analysisshould be performed at other regions of the energy spectrumnot just at the endpoint In this paper we restrict thecalculations to the one-hole states

The density of final states of the emitted neutrino inelectron capture is proportional to

120588 = 120588] (119864]) prop 119901]119864] = (1198642] minus 1198982])12 119864] (12)

where 119864] 119901] and 119898] are the neutrino total energy momen-tum and mass respectively On the other hand the proba-bility of capture of a bound electron follows a Breit-Wignerdistribution of width Γ119909 and peaks at the energy 119864119909

119875 (119864) = Γ1199092120587(119864 minus 119864119909)2 + Γ21199094 (13)

Using Fermirsquos Golden Rule the differential reaction ratewith respect to the neutrino energy yields

119889120582119889119864] = 119870EC (1198642] minus 1198982])12 119864]sum119867

119882(])119867sdot Γ1198672120587[119864] minus (119876 minus 119864119867)]2 + Γ21198674

(14)

where 119870EC contains among other factors the weak interac-tion coupling constant and the nuclear matrix element Thefactor119882(])119867 is the squared leptonic matrix element for a givenhole state119867

We write 119882(])119867 = 119862119867119878(])119867 to have a general expressionfor allowed and forbidden transitions The factor 119862119867 is thesquared amplitude of the bound-state electron radial wavefunction at the nuclear interior containing also the squaredoverlap between the initial and the final atom orbital wavefunctions and the effect of electron exchange (as defined inAppendix F-2 of [56]) The cases discussed in the nextsections involve allowed Gamow-Teller transitions (163Ho (72minus)rarr 163Dy (52minus)) and first forbiddenGamow-Teller transi-tions (202Pb (0+) rarr 202Tl (2minus) and 205Pb (52minus) rarr 205Tl (12+)) In the first case the factor 119878(])119867 = 1 for all 119867 states Inthe second case the factor 119878(])119867 = 1 for electrons captured withorbital angularmomentum 119897 = 1 (and neutrinos emitted with119897 = 0) and 119878(])119867 = 1199012] for electrons captured with orbitalangular momentum 119897 = 0 (and neutrinos emitted with 119897 =1) The reason for this is that due to conservation of totalangularmomentum first forbidden transitions have 119871 = 1 forthe leptonic pair Hence there is a linear momentum depen-dence in the leptonic matrix element corresponding to thelepton that carries the angular momentum in each specificcaptureThe corresponding 1199012119890 factor for the 119897 = 1 electrons isembedded in the 119862119867 factor [56]

Equation (14) includes every possible orbital electroncapture the probability of each peaking at 119864] = 119876 minus 119864119867 asdictated by energy conservation where 119864119867 is the excitationenergy of the final atom due to the electron hole119867 resultingfrom capture The energy collected by a calorimeter from theatomic deexcitations is 119864119888 = 119876 minus 119864] namely all the availableenergy except for the one carried away by the neutrino Thedifferential rate can be written in terms of the calorimeterenergy as

119889120582119889119864119888 = 119870EC [(119876 minus 119864119888)2 minus 1198982]]12 (119876 minus 119864119888)sum119867

119862119867119878(])119867sdot Γ1198672120587(119864119888 minus 119864119867)2 + Γ21198674

(15)

In the next section we shall also consider integrated ratesover particular energy ranges which can be calculatednumerically from (15) A convenient compact analyticalexpression of the integrated rate can be obtained from (15)by replacing the Breit-Wigner distributions by Dirac deltas

Advances in High Energy Physics 5

With this approximation the integral of the differential ratecan be written as

120582 = 119870ECsum119867

119862119867119878(])119867 [(119876 minus 119864119867)2 minus 1198982]]12 (119876 minus 119864119867) (16)

In this theoretical model we have neglected two-holepeaks deexcitations through virtual intermediate states andinterferences between deexcitation channels The theoreticalcalorimeter spectrum is thus a single-hole approximationthat assumes full collection of de-excitation energy by thecalorimeter and no pile-up

4 Sterile Neutrino Effect in the Spectrum

Due to the smallness of the mixing angle it is useful toconsider ratios between rates that emphasize the contributionof the heavy neutrino In [84ndash86] for the case of beta decay aratio was considered between the contributions of the heavyand the light neutrino to the differential rates as we shall seein Section 6 For the case of electron capture the differentialdecay rates contain many peaks and it is more convenient toconsider integrated decay rates over specific energy rangesas discussed in [87] In either case (beta decay and electroncapture) this amounts to consider that the detector collectsevents within energy ranges 119864119894 plusmn Δ and 119864119895 plusmn Δ in two regionsof the spectrum such that the mass of the hypothetical heavyneutrino lies in between namely 119864119894 + Δ lt 119898ℎ lt 119864119895 minus Δ with119864119895 lt 119876minusΔ A ratio 119877 between the number of collected eventsin both regions is then performed which corresponds to thetheoretical expression

119877 = Λ 119894Λ 119895 =120581119897119894 + tan2120577120581ℎ119894120581119897119895 (17)

where Λ 119894 = cos2120577120581119897119894 + sin2120577120581ℎ119894 (in the region where both 119897and ℎ mass eigenstates contribute) and Λ 119895 = cos2120577120581119897119894 (in theregion where ℎ is not energetically allowed) The integrals 120581are defined as

120581]119894119895 = int119864119894119895+Δ119864119894119895minusΔ

119889120582119889119864119888 (119898]) 119889119864119888 (18)

In electron capture the energies 119894 and 119895 can be selected asthe energies of two peaks and the integration intervals canbe chosen as the width of the capture peaks If the peaks areapproximated by delta functions the integrals (using119864119867 = 119864119894and Δ rarr 0) can be written as

120581]EC119894119903 = 119870EC119862119894119903119878(])119894119903 (119876119903 minus 119864119894)2 [1 minus ( 119898]119876119903 minus 119864119894)2]12 (19)

where 119876119903 is the atomic mass difference in a given isotope 119903119864119894 is the energy position of a given peak 119894 in the calorimeterspectrum 119898] is 119898119897 asymp 0 the mass of a light neutrino or119898ℎ the mass of a heavy neutrino and 119878(])119894119903 contains theneutrino momentum dependence for the peak 119894 comingfrom the leptonic matrix element squared As explained inthe previous section for allowed transitions 119878(])119894119903 = 1 For

first forbidden transitions some of the capture peaks have119878(])119894119903 = 1199012]119894119903 for the 119894 peaks corresponding to 11990412 shells (aswell as for those corresponding to 11990112 shells that contributethrough their admixtures with the 119897 = 0 orbital due torelativistic corrections) The other peaks have 119878(])119894119903 = 1where 119894 corresponds to 11990132 shells (and 11988932 shells throughtheir admixtures with the 119897 = 1 orbital due to relativisticcorrections)

For electron capture it is convenient to define a ratiosimilar to the one in (17) but where the numerator andthe denominator are themselves ratios of numbers of eventswithin the same peak but for different isotopes of the sameelement 119903 and 119904 so that some atomic corrections cancel out[87]

1198771015840 = Λ 119894119903Λ 119894119904Λ 119895119903Λ 119895119904 = (119877119897119894119895119903119904)2(120574+1)(1 + 1205962120574+1119894119903 tan2120577

1 + 1205962120574+1119894119904 tan2120577) (20)

where one should notice that the factors 119862119894 and 119862119895 cancel outin addition to the factor119870EC which cancels out in both ratios(119877 and 1198771015840) The factors in (20) have very simple analyticalforms when one uses (19)

119877119897119894119895119903119904 = (119876119903 minus 119864119894) (119876119904 minus 119864119895)(119876119903 minus 119864119895) (119876119904 minus 119864119894)

120596119894119903(119904) = [1 minus ( 119898ℎ119876119903(119904) minus 119864119894)2]12

(21)

where 120574 depends on the angular momenta of each lepton in agivenΔ119869120587 nuclear transition For instance for allowed decays120574 = 0 whereas for first forbidden decays 120574 = 1 for 11990412 and11990112 peaks or 120574 = 0 for 11990132 and 11988932 peaks In the expressionsabove we have assumed that both peaks 119894 and 119895 are of thesame type namely 120574119894 = 120574119895 = 120574

By measuring the ratio in (20) the mixing angle can thenbe obtained as

120577 = arctan[[

(119877119897119894119895119903119904)2(120574+1) minus 1198771015840exp1198771015840exp1205962120574+1119894119904 minus (119877119897119894119895119903119904)2(120574+1) 1205962120574+1119894119903

]]12

(22)

where the theoretical values of 119877119897119894119895119903119904 120596119894119903 and 120596119894119904 requirean accurate experimental knowledge of the position of theselected peaks and more importantly of the atomic massdifferences 119876119903 and 119876119904 In addition one has to consider fixedinitially unknown values of the mass of the heavy neutrino119898ℎ that is searched for or otherwise be content with thegeneration of exclusion plots in case of no effect observation

In summary we consider here two types of ratios that canbe useful in the search for a signal of keV sterile neutrinos inelectron capture (1) the ratio between the intensity of twopeaks in the electron capture spectrum of a given nucleusas for the case of 163Ho discussed in Section 5 (2) The casein which two isotopes of a given element undergo electroncapture where one can use the ratio defined in (20) as isthe case of Lead isotopes discussed in Section 5 For the

6 Advances in High Energy Physics

10minus5

Neutrino phase spaceAtomic deexcitationTotal

M1M2

N1

N2O1

O2

0 05 1 15 2 25 3Ec (keV)

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

(a)

m = 25 keVm = 2 keVm = 15 keV

m = 1 keVm = 05 keVm = 0

0 05 1 15 2 25 3Ec (keV)

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

10minus5

(b)

0 05 1 15 2 25 3

No mixingMax mixing

Ec (keV)

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

(c)

Figure 1 Calorimeter spectrum after electron capture in 163Ho (a) Full spectrum (solid curve) and contributions emitted neutrino phasespace (dotted curve) and daughter atom deexcitation (dashed curve) (b) Full spectrum for different neutrino masses 0 05 1 15 2 and25 keV (c) full spectrum for light (119898119897 asymp 0 keV) - heavy (119898ℎ = 2 keV) neutrino mixing using a maximal mixing angle 120577 = 45∘ for illustrationwith (solid curve) and without (dotted curve) the heavy neutrino contribution For comparison the spectrumwithout heavy neutrinomixing(dashed curve) is also shownWe use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of the figure Realistic valuesof 120577 lt 001∘ result in a reduction of the difference between the dashed and the solid lines in (c) by more than seven orders of magnitude

beta decay case in Section 6 we consider the ratio betweenthe differential rates but the ratios between integrated ratesare also useful and may be more realistic for comparison toexperiments

5 Results for Electron Capture Spectra

Let us first consider the capture of an atomic electron by thenucleus Holmium 163 (163Ho 119885 = 67 119873 = 96) to turn intoDysprosium 163 (163Dy 119885 = 66119873 = 97) with ground statespins and parities 119869120587 = 72minus and 119869120587 = 52minus respectively Itis an allowed unique Gamow-Teller transition (Δ119869 = 1 andno parity change the leptons carry no orbital angularmomentum Δ119871 = 0 and couple to spin 119878 = 1) with atomicmass difference 119876 = 2833 plusmn 0030 plusmn 0015 keV [88] which isbeing currently measured at the Electron Capture Holmium(ECHo) experiment [50ndash52] Only electrons with principalquantum number 119899 larger than 2 have a binding energylower than the 119876-value and can therefore be captured Inaddition being an allowed decay only electrons from 11990412 and

11990112 orbitals can be captured the latter through admixtureswith the 119897 = 0 orbital due to relativistic corrections Thuscapture from the orbitals M1 M2 N1 N2 O1 O2 and P1 (inspectroscopic notation) is possible [80 89ndash91]

In Figure 1 we show the differential electron capture ratein 163Ho as a function of the calorimeter energy 119864119888 = 119876minus119864]In Table 1 we give theoretical values of the strengths andwidths of each capture peak and experimental values ofthe electron binding energy again for each capture peakThe strengths 119862119867 are obtained in [56 57] from relativisticHartree-Fock calculations of the electron wave functions atthe nuclear site accounting for exchange and overlap contri-butions In the upper plot the emitted neutrino phase spacecontribution and the daughter atom deexcitation contribu-tion are plotted separately together with the product of bothwhich is the full differential electron capture rate In themiddle plot we show the full differential rate for differentmasses of the emitted neutrino 119898] asymp 0 (realistic) and 119898] asymp05 1 15 2 and 25 keV (unrealistic) As can be seen in theplot each curve ends at 119876minus119898] Finally the lower plot showsthe spectrum resulting from the mixing of a light neutrino

Advances in High Energy Physics 7

Table 1 Atomic parameters for electron capture in Holmium from shells M1 to O2The values of 119862119867 [56 57] correspond to the atomic shellsin Holmium while the orbital electron widths Γ119867 [58] and binding energies 119861119867 (ltQ) [59ndash61] correspond to the atomic shells in Dysprosium

M1 M2 N1 N2 O1 O2119862119867 005377 0002605 001373 00005891 0001708 00001Γ119867 [keV] 0013 0006 0006 0005 0005 0002119861119867 [keV] 2047 1842 0416 0332 0063 0026

Table 2 Same as Table 1 (data from the same references) but for electron capture in Lead going toThalliumThe upper table summarizes thevalues of the relevant parameters for 11990412 and 11990112 atomic shells while the lower table shows the values for 11990132 and 11988932 atomic shells In thelatter case 119862119867 includes an extra factor 1199012119890

L1 L2 M1 M2 N1 N2 O1 O2119862119867 08781 006647 02125 001759 005849 0004431 001016 00008052Γ119867 [keV] 0011 0006 0015 0010 0009 0007 mdash mdash119864119867 [keV] 15346 14697 3703 3415 0845 0720 0136 0099L3 M3 M4 N3 N4 O3 O4119862119867 [keV2] 18978193 5366014 45383 1363307 12607 237280 1500Γ119867 [keV] 0006 0009 0002 0006 0004 0001 0001119861119867 [keV] 12657 2956 2484 0608 0406 0072 0015

mass eigenstate 119898119897 asymp 0 with a heavy mass eigenstate 119898ℎ asymp2 keV (solid curve) as in (5) Although the spectrum withmixing is realistic in the sense that it is the result expected if aheavy mass eigenstate exists it is unrealistic in the degree ofmixing shown in the figure which has been maximized herefor the sake of visibility of the ldquokinkrdquo in the scale of the figure50 light neutrino and 50 heavy neutrino correspondingto amixing angle 120577 = 45∘The ldquokinkrdquo can be seen in this curveat 119864119888 = 119876 minus 119898ℎ = 0833 keV Above this calorimeter energythe heavy mass eigenstate cannot be produced due to energyconservation For comparison the spectrum for119898] asymp 0 withno mixing with heavy states is shown in the dashed curve

Another example of electron capture under study here isthat of the Lead isotopes 202 (202Pb 119885 = 82 119873 = 120)and 205 (205Pb 119885 = 82 119873 = 123) going to the Thalliumisotopes 202 (202Tl 119885 = 81 119873 = 121) and 205 (205Tl119885 = 81 119873 = 124) respectively where the process is in bothcases a first forbidden Gamow-Teller unique transition with0+ rarr 2minus for 202Pb and 52minus rarr 12+ for 205Pb (Δ119869 = 2 andparity change the leptons carry Δ119871 = 1 and couple to 119878 = 1)The atomic mass differences obtained from the data in [92]are 119876 = 46 plusmn 14 keV and 119876 = 506 plusmn 18 keV respectivelyAccording to the explanations given in Section 3 capturesfrom orbitals L1 L2 M1 M2 N1 N2 O1 and O2 containan extra neutrino momentum dependence 119878(]) = 1199012] thatmodifies the calorimeter spectrum and corresponds to 120574 =1 On the other hand captures from orbitals L3 M3 M4N3 N4 O3 and O4 which correspond to 120574 = 0 containan extra electron momentum dependence 1199012119890 which beingfixed in bound electrons is included in the values of 119862119867They are given in Table 2 together with the widths and theexperimental electron binding energies in the daughter atomThallium

Figure 2 shows the capture differential rate in 205Pb (darkcurves) and in 202Pb (light curves) Position width and

strength of the capture peaks are assumed to be the same inboth isotopes but the spectra are different because of the dif-ferent 119876-values Solid curves are for light neutrino emission119898] asymp 0 and dashed lines are for heavy neutrino emissionwith119898] = 40 keV (results are given separately for each massbeforemixing) Due to the large119876-values electron capture inLead isotopes allows us to explore this mass value althoughit may be somewhat larger than the expected value fromcosmological reasons The analysis of the ratio in (20) canbe computed for example using the capture peaks L3 at119864119888 = 12657 keV and M3 at 119864119888 = 2956 keV both beingof the 120574 = 0 type For 119898ℎ = 40 keV one would obtain119877l1198723 1198713 202 205 = 1037 1205961198723202 = 0369 1205961198723205 = 0543These results should be introduced in (22) together with theexperimental ratio 1198771015840exp to obtain the value of the mixingangle 120577

It is important to remark that the theoretical quantities 119877119897and 120596 computed as described above contain several approxi-mations They correspond to Dirac-delta peaks ignoring theactual shapes and widths and thus they just refer to onesingle capture peak neglecting the possible effect of the tailsof nearby one-hole peaks Moreover the influence of two-hole peaks close to the main ones has also been neglectedOther effects that have not been taken into account butwhose influence is expected to be very small are multihole(more than two) peaks virtual intermediate states (influenceof transitions through atomic shells not energetically acces-sible) or interference between atomic transitions resultingfrom addition of amplitudes instead of intensities [81 82]Theaccurate experimental determination of the ratio 1198771015840exp entailsits own difficulties among them the possible existence ofmetastable atomic states whose delayed de-excitations failto contribute to the collected spectrum and the variety ofchemical environments resulting in a complex mixture of 119876-values

8 Advances in High Energy Physics

L1

L2

L3

M1M2

M3

M4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

6 162 8 10 12 144 18 200Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(a)

N1

N2

N3

O1

O2

O3

N4O4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

05 0603 0401 02 07 08 09 1000Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(b)

Figure 2 Calorimeter spectrum after electron capture in 202Pb (light curves) and in 205Pb (black curves) with emission of a light neutrino119898119897 = 0 (solid curves) and a heavy neutrino119898ℎ = 40 keV (dashed curves) (a) Full spectrum showing L and M capture peak labels (b) Lowenergy region (119864119888 = 0-1 keV) showing N and O capture peak labels

6 Results for Beta Decay Spectra

In addition to electron capture we also show here the effectof heavy neutrino emission in the electron spectrum of betadecays for one of the cases of most interest Tritium It hasbeen studied in depth in previous works [84ndash86 93] togetherwith another process that has also drawn considerable theo-retical end experimental attention the beta decay of Rhenium187 [37 39ndash41 84ndash86] The beta decay of Tritium (3H 119885 = 1119873 = 2) going to Helium 3 (3He 119885 = 2119873 = 1) is

3H 997888rarr 3He + 119890minus + ]119890 (23)

and has a 119876-value of 1859 keV [92] Both the initial and thefinal nuclear ground states have spin-parity 12+ and thetransition is allowed with the electron and the antineutrinoemitted in 119904-wave The Karlsruhe Tritium Neutrino Experi-ment (KATRIN) [42ndash46] is currently studying this decay inorder to determine the active neutrino mass and could alsostudy the production of a heavy neutrino of amass lower than18 keV

The differential decay rate with respect to the electronenergy is given by

119889120582119889119864119890 = 119870120573 (1198642119890 minus 1198982119890)12 [(119876 + 119898119890 minus 119864119890)2 minus 1198982]]12

sdot 119864119890 (119876 + 119898119890 minus 119864119890) (24)

where 119870120573 contains among others the weak interactioncoupling the nuclearmatrix element and the Fermi function

As an illustration of the heavy neutrino effect in the Tri-tium beta decay we plot in Figure 3 the differential decay ratefrom (24) and (5) using a maximal mixing with an unrealisticvalue of the mixing angle 120577 = 45∘ to show the effect more

distinctly in the scale of the figure We have used heavy masscomponents with 119898ℎ = 2 keV [26] (Figure 3(a)) and with119898ℎ = 7 keV [29 30] (Figure 3(b)) For comparison thespectra without the heavy neutrino contribution (120582ℎ = 0) andwithout mixing (120577 = 0∘) are also shown in this plot The kinkin the spectrum at 119864119890 minus 119898119890 = 119876 minus 119898ℎ can be observed if theexperimental relative error is lower than the size of the stepthe latter being very small in realistic situations

The effect of a heavy neutrino emission can be analyzedthrough the ratio between the heavy and the light neutrinocontributions to the spectrum

R equiv 119889120582ℎ119889119864119890119889120582119897119889119864119890 tan2120577 (25)

The full differential decay rate is also related to the ratio Rthrough

119889120582119889119864119890 =119889120582119897119889119864119890 [1 +R] cos2120577 (26)

In Figure 4 we plot R the ratio of the heavy neutrinocontribution over the light neutrino contribution to the decayrate as a function of the momentum of the emitted electronfor a heavy neutrino mass 119898ℎ = 2 keV and different mixingangles 120577 = 001∘ 0005∘ and 0001∘ The size of this ratio(lt10minus7) gives an idea of the difficulty of finding the kink in thespectrum due to the production of the heavymass eigenstateAs can be seen in the figure the ratio is different from zeroand almost constant in the range 0 le 119901119890 lt (119901119890)max wherethe maximum electron momentum is given by (119901119890)max =[(119876 minus 119898ℎ)(119876 minus 119898ℎ + 2119898119890)]12 For the heavy mass used inthe figure (119901119890)max ≃ 1313 keV The ratio R decreases as themixing angle decreases being approximately proportional to

Advances in High Energy Physics 9

Mixing but no heavy neutrino contribution

104

105

106

107

108d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 2 keV

(a)

104

105

106

107

108

d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

Mixing but no heavy neutrino contribution

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 7 keV

(b)

Figure 3 Electron spectrum of Tritium beta decay for a light-heavy neutrino mixing angle 120577 = 45∘ shown just for illustration and a heavyneutrino mass (solid curve)119898ℎ = 2 keV (a) and119898ℎ = 7 keV (b) The spectrum without heavy neutrino mass contribution (dotted curve) andwithout mixing (dashed curve) are also shown We use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of thefigure Realistic values of 120577 lt 001∘ result in a reduction of the difference between the dotted and the solid lines by more than seven orders ofmagnitude

10minus11

10minus10

10minus9

10minus8

10minus7

10minus12

120577 = 001∘

120577 = 0005∘

120577 = 0001∘

15090 100 110 120 130 1408070pe (keV)

ℛ

Figure 4 Ratio R defined in (25) for the Tritium beta decay as afunction of the electronmomentum for a heavy neutrinomass119898ℎ =2 keV and different values of the mixing angle 120577 = 001∘ (solid line)0005∘ (dashed line) and 0001∘ (dotted line)

1205772 It also decreases for larger heavy neutrino masses 119898ℎ Adifferent ratio between the spectra with mixing and without

mixing (Rlowast) has also been considered in [84ndash86 93] relatedwithR in (25) as

Rlowast = minussin2120577 +R cos2120577 = 119889120582119889119864119890119889120582119897119889119864119890 minus 1 (27)

7 Conclusions

Signatures of hypothetical keV sterile neutrinos which couldbe warm dark matter candidates (WDM) may be found inthe spectra of ordinary weak nuclear decays The electronspectrum in beta decay is cleaner for this purpose becauseit is a smooth curve while the deexcitation spectrum afterelectron capture shows many peaks On the other hand inbeta decay the possible excitation of the atoms or moleculesis not disentangled while in electron capture the calorimetercollects all the energy independently of the excitation ofthe atom and of the deexcitation path and one has higherstatistics around the capture peaks

Figures of electron spectrum for beta decay in 3H aswell as of calorimeter spectrum for electron capture in 163Hoand 202Pb and 205Pb are given considering various valuesof the heavy neutrino mass (2 keV 7 keV and 40 keV) thatmay be experimentally probed In both weak processes thesmall value of the light-heavy neutrino mixing angle requiresextremely high experimental precision and the use of sourceswith large stability to reduce systematic errors This is why it

10 Advances in High Energy Physics

is useful to consider relevant ratios between transition rates asfirst introduced in [84ndash87] to analyze the data We considertwo cases the case of a single isotope where one may use theratio of accumulated number of events in different regionsof the spectrum which allows us to remove uncertaintiesrelated to the nuclear matrix element and to the valuesof overall constants And in the case of two isotopes weconsider ratios of the above mentioned ratios that allow usto reduce uncertainties from atomic parameters We give aswell analytical expressions that can be used to obtain a goodtheoretical approximation to the experimental ratios ((19) to(21))

Both the experimental measurement and the theoreticalmodel must be accurate enough to detect differences in theexpected versus themeasured ratios of the order of themixingangle squared 1205772 ≲ 10minus8 This is also the size of the kinkexpected in the electron spectrumof beta decay located at thelimit of the regionwhere the production of a heavy neutrino isenergetically allowed In order to identify this kink among thestatistical fluctuations of themeasured spectrum the numberof collected events must be larger than the inverse of thesquared ratioR in (25)

Our results particularly on the electron capture spectraof Lead isotopes that include for the first time all thepossible peaks may help in designing and analyzing futureexperiments to search for sterile neutrinos in the keV massrangeThe use of the different types of ratios between numberof events discussed here for both electron capture and betadecay processes may also help in planning the experimentsby establishing the threshold of statistical and systematicuncertainties required for the detection of sterile neutrinosof given mass and mixing or to properly extract exclusionplots

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

E Moya de Guerra and O Moreno acknowledge MINECOfor partial financial support (FIS2014-51971-P) and IEM-CSIC for hospitality O Moreno acknowledges supportfrom a Marie Curie International Outgoing Fellowshipwithin the European Union Seventh Framework Pro-gramme under Grant Agreement PIOF-GA-2011-298364(ELECTROWEAK) M R M thanks N G Sanchez forcorrespondence and conversations

References

[1] F Zwicky ldquoDieRotverschiebung von extragalaktischenNebelnrdquoHelvetica Physica Acta vol 6 pp 110ndash127 1933

[2] M Roos ldquoAstrophysical and cosmological probes of darkmatterrdquo Journal of Modern Physics vol 3 pp 1152ndash1171 2012

[3] S Dodelson Modern Cosmology Academic Press New YorkNY USA 2003

[4] S Weinberg Cosmology Oxford University Press Oxford UK2008

[5] J Lesgourgues and S Pastor ldquoNeutrino mass from cosmologyrdquoAdvances in High Energy Physics vol 2012 Article ID 60851534 pages 2012

[6] D Boyanovsky H J de Vega and N G Sanchez ldquoConstraintson dark matter particles from theory galaxy observations andN-body simulationsrdquo Physical Review D vol 77 no 4 ArticleID 043518 2008

[7] D Boyanovsky H J de Vega and N G Sanchez ldquoDarkmatter transfer function free streaming particle statistics andmemory of gravitational clusteringrdquo Physical Review D vol 78no 6 Article ID 063546 2008

[8] C Destri H J de Vega and N G Sanchez ldquoWarm dark matterprimordial spectra and the onset of structure formation atredshift zrdquo Physical Review D vol 88 no 8 Article ID 0835122013

[9] H J De Vega P Salucci and N G Sanchez ldquoThe mass ofthe dark matter particle theory and galaxy observationsrdquo NewAstronomy vol 17 no 7 pp 653ndash666 2012

[10] H J de Vega P Salucci and N G Sanchez ldquoObservationalrotation curves and density profiles versus the Thomas-Fermigalaxy structure theoryrdquoMonthlyNotices of the Royal Astronom-ical Society vol 442 no 3 pp 2717ndash2727 2014

[11] H J de Vega and N G Sanchez ldquoModel-independent analysisof dark matter points to a particle mass at the keV scalerdquoMonthly Notices of the Royal Astronomical Society vol 404 no2 pp 885ndash894 2010

[12] H J de Vega and N G Sanchez ldquoConstant surface gravity anddensity profile of dark matterrdquo International Journal of ModernPhysics A vol 26 no 6 pp 1057ndash1072 2011

[13] H J de Vega and N G Sanchez ldquoCosmological evolutionof warm dark matter fluctuations I Efficient computationalframework with Volterra integral equationsrdquo Physical ReviewDvol 85 no 4 Article ID 043516 20 pages 2012

[14] H J de Vega and N G Sanchez ldquoThe dark matter distributionfunction and halo thermalization from the Eddington equationin galaxiesrdquo International Journal of Modern Physics A vol 31no 13 Article ID 1650073 2016

[15] NMenci S Paduroiu P Salucci N Sanchez and C RWatsonldquolectures at the 20th Paris Cosmology Colloquium Chalonge-Hector de Vegardquo 2016 httpchalongeobspmfr

[16] E W Kolb and M S Turner The Early Universe AddisonWesley Boston Mass USA 1990

[17] J F Navarro C S Frenk and S D M White ldquoA universal den-sity profile from hierarchical clusteringrdquo Astrophysical JournalLetters vol 490 no 2 pp 493ndash508 1997

[18] W J G de Blok ldquoThe core-cusp problemrdquo Advances in Astron-omy vol 2010 Article ID 789293 14 pages 2010

[19] G Gilmore M I Wilkinson R F GWyse et al ldquoThe observedproperties of dark matter on small spatial scalesrdquo AstrophysicalJournal vol 663 no 2 pp 948ndash959 2007

[20] P Salucci and Ch Frigerio Martins ldquoThe mass distribution inSpiral galaxiesrdquo EAS Publications Series vol 36 pp 133ndash1402009

[21] J van Eymeren C Trachternach B S Koribalski and R-JDettmar ldquoNon-circular motions and the cusp-core discrepancyin dwarf galaxiesrdquo Astronomy amp Astrophysics vol 505 no 1 pp1ndash20 2009

[22] M Walker and J Penarrubia ldquoA method for measuring (slopesof) the mass profiles of dwarf spheroidal galaxiesrdquo The Astro-physical Journal vol 742 no 1 p 20 2011

Advances in High Energy Physics 11

[23] R F G Wyse and G Gilmore ldquoObserved properties of darkmatter on small spatial scalesrdquo in Proceedings of the Proceedingsof the International Astronomical Union Symposium (IUA rsquo07)vol 3 no 244 pp 44ndash52 Cardiff UK June 2007

[24] A Burkert ldquoThe structure of dark matter halos in dwarfgalaxiesrdquo Astrophysical Journal vol 447 no 1 pp L25ndashL281995

[25] C Destri H J de Vega and N G Sanchez ldquoQuantum WDMfermions and gravitation determine the observed galaxy struc-turesrdquo Astroparticle Physics vol 46 pp 14ndash22 2013

[26] N Menci N G Sanchez M Castellano and A GrazianldquoConstraining the warm dark matter particle mass throughultra-deep UV luminosity functions at Z = 2rdquoTheAstrophysicalJournal vol 818 no 1 article 90 2016

[27] E Bulbul M Markevitch A Foster R K Smith M Loewen-stein and SW Randall ldquoDetection of an unidentified emissionline in the stackedX-ray spectrumof galaxy clustersrdquoAstrophys-ical Journal vol 789 article 13 2014

[28] A Boyarsky O Ruchayskiy D Iakubovskyi and J FranseldquoUnidentified line inX-ray spectra of the andromeda galaxy andperseus galaxy clusterrdquo Physical Review Letters vol 113 no 25Article ID 251301 2014

[29] K N Abazajian ldquoResonantly produced 7 keV sterile neutrinodark matter models and the properties of Milky Way satellitesrdquoPhysical Review Letters vol 112 no 16 Article ID 161303 5pages 2014

[30] N Menci A Grazian M Castellano and N G SanchezldquoShock connectivity in the 2010 August and 2012 July solarenergetic particle events inferred from observations and ENLILmodelingrdquo Astrophysical Journal Letters vol 825 no 1 article 12016

[31] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[32] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 481 no 1-2 pp 1ndash28 2009

[33] B Pontecorvo ldquoNeutrino experiments and the problem ofconservation of leptonic chargerdquo Soviet Physics-JETP vol 26no 5 pp 984ndash988 1968

[34] A Merle ldquokeV neutrino model buildingrdquo International Journalof Modern Physics D vol 22 no 10 Article ID 1330020 2013

[35] J DVergados andT S Kosmas ldquoSearching for cold darkmatterA case of coexistence of supersymmetry and nuclear physicsrdquoPhysics of Atomic Nuclei vol 61 p 1066 1998

[36] E Holmlund M Kortelainen T S Kosmas J Suhonen and JToivanen ldquoMicroscopic calculation of the LSP detection ratesfor the 71Ga 73Ge and 127I dark-matter detectorsrdquoPhysics LettersB vol 584 no 1-2 pp 31ndash39 2004

[37] R Adhikari M Agostini N Anh Ky et al ldquoA white paper onkeV sterile neutrino dark matterrdquo httpsarxivorgabs160204816

[38] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[39] MARE collaboration httpmaredfmuninsubriaitfrontendexecphp

[40] A Nucciotti ldquoNeutrino mass calorimetric searches in theMARE experimentrdquo httpsarxivorgabs10122290

[41] A Nucciotti ldquoOn behalf of the MARE collaboration lecture attheWorkshop ChalongeMeudonrdquo 2011 httpchalongeobspmfr

[42] KATRIN Collaboration httpwwwkatrinkitedu

[43] C Weinheimer ldquoDirect determination of neutrino mass fromtritium beta spectrumrdquo httpsarxivorgabs09121619

[44] C Weinheimer lecture at the 20th Paris Cosmology Chalonge-Hector de Vega Colloquium 2016 httpchalongeobspmfr

[45] G Drexlin lecture at the 19th Paris Cosmology Colloquium2015

[46] A Huber lecture at the Chalonge-de Vega Meudon Workshop2016 httpchalongeobspmfr

[47] C Tully lecture at the 19th Paris Cosmology Colloquium 2015[48] S Betts W R Blanchard R H Carnevale et al ldquoDevelop-

ment of a relic neutrino detection experiment at PTOLEMYprinceton tritiumobservatory for light early-universemassive-neutrino yieldrdquo httpsarxivorgabs13074738

[49] D M Asner R F Bradley L de Viveiros et al ldquoSingle-electrondetection and spectroscopy via relativistic cyclotron radiationrdquoPhysical Review Letters vol 114 no 16 Article ID 162501 2015

[50] P C-O Ranitzsch J-P Porst S Kempf et al ldquoDevelopmentof metallic magnetic calorimeters for high precision measure-ments of calorimetric 187Re and 163Ho spectrardquo Journal of LowTemperature Physics vol 167 no 5-6 pp 1004ndash1014 2012

[51] C Hassel ldquoLecture at the International School of AstrophysicsDaniel ChalongemdashHector de Vegardquo 2015

[52] L Gastaldo K Blaum A Doerr et al ldquoThe electron capture163Ho experiment ECHordquo Journal of Low Temperature Physicsvol 176 no 5 pp 876ndash884 2014

[53] B Alpert M Balata D Bennett et al ldquoThe electron capturedecay of 163Ho tomeasure the electron neutrino mass with sub-eV sensitivityrdquo The European Physical Journal C vol 75 article112 2015

[54] E Fermi ldquoRadioattivita prodotta da bombardamento di neu-tronirdquo Il Nuovo Cimento vol 11 no 7 pp 429ndash441 1934

[55] E Fermi ldquoVersuch einer theorie der 120573-strahlen Irdquo Zeitschriftfur Physik vol 88 no 3 pp 161ndash177 1934

[56] R B Firestone and V S Shirley Eds Table of Isotopes JohnWiley amp Sons New York NY USA 8th edition 1999

[57] W Bambynek H Behrens M H Chen et al ldquoOrbital electroncapture by the nucleusrdquo Reviews of Modern Physics vol 49 no1 pp 77ndash221 1977

[58] J L Campbell and T Papp ldquoWidths of the atomic KndashN7 levelsrdquoAtomic Data and Nuclear Data Tables vol 77 no 1 pp 1ndash562001

[59] J A Bearden and A F Burr ldquoReevaluation of X-ray atomicenergy levelsrdquo Reviews of Modern Physics vol 39 no 1 pp 125ndash142 1967

[60] M Cardona and L Ley Eds Photoemission in Solids I GeneralPrinciples Springer Berlin Germany 1978

[61] J C Fuggle and N Martensson ldquoCore-level binding energies inmetalsrdquo Journal of Electron Spectroscopy and Related Phenom-ena vol 21 no 3 pp 275ndash281 1980

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[64] J-U Nabi and H V Klapdor-Kleingrothaus ldquoWeak interactionrates of sd-shell nuclei in stellar environments calculated in theproton-neutron quasiparticle random-phase approximationrdquoAtomic Data andNuclear Data Tables vol 71 no 2 pp 149ndash3451999

12 Advances in High Energy Physics

[65] J-U Nabi and H V Klapdor-Kleingrothaus ldquoMicroscopic cal-culations of stellar weak interaction rates and energy losses forfp- and fpg-shell nucleirdquo Atomic Data and Nuclear Data Tablesvol 88 no 2 pp 237ndash476 2004

[66] Y Fukuda T Hayakawa E Ichihara et al ldquoEvidence foroscillation of atmospheric neutrinosrdquo Physical Review Lettersvol 81 no 8 pp 1562ndash1567 1998

[67] Q R Ahmad and SNOCollaboration ldquoMeasurement of the rateof V119890+119889 rarr 119901+119901+119890minus interactions produced by 8B solar neutrinosat the Sudbury Neutrino Observatoryrdquo Physical Review Lettersvol 87 no 7 Article ID 071301 6 pages 2001

[68] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 2002

[69] G J Feldman J Hartnell and T Kobayashi ldquoLong-baselineneutrino oscillation experimentsrdquo Advances in High EnergyPhysics vol 2013 Article ID 475749 30 pages 2013

[70] S T Petcov ldquoThenature ofmassive neutrinosrdquoAdvances inHighEnergy Physics vol 2013 Article ID 852987 20 pages 2013

[71] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[72] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 48 no 1-2 pp 1ndash28 2009

[73] F Munyaneza and P L Biermann ldquoDegenerate sterile neutrinodark matter in the cores of galaxiesrdquo Astronomy amp Astrophysicsvol 458 no 2 pp L9ndashL12 2006

[74] D Boyanovsky and C M Ho ldquoSterile neutrino production viaactive-sterile oscillations the quantum Zeno effectrdquo Journal ofHigh Energy Physics vol 2007 no 7 p 30 2007

[75] R Shrock ldquoNew tests for and bounds on neutrino masses andlepton mixingrdquo Physics Letters B vol 96 no 1-2 pp 159ndash1641980

[76] R E Shrock ldquoGeneral theory of weak processes involving neu-trinos I Leptonic pseudoscalar-meson decays with associatedtests for and bounds on neutrino masses and lepton mixingrdquoPhysical Review D vol 24 no 5 pp 1232ndash1274 1981

[77] R E Shrock ldquoImplications of neutrino masses and mixing forweak processesrdquo in Proceedings of the Weak Interactions asProbes of Unification vol 72 of AIP Conference Proceedings p368 Blacksburg Va USA 1981

[78] J M Conrad C M Ignarra G Karagiorgi M H Shaevitzand J Spitz ldquoSterile neutrino fits to short-baseline neutrinooscillation measurementsrdquo Advances in High Energy Physicsvol 2013 Article ID 163897 2013

[79] A Nucciotti ldquoThe use of low temperature detectors for directmeasurements of themass of the electron neutrinordquoAdvances inHigh Energy Physics vol 2016 Article ID 9153024 41 pages2016

[80] R G H Robertson ldquoExamination of the calorimetric spectrumto determine the neutrino mass in low-energy electron capturedecayrdquo Physical Review C vol 91 no 3 Article ID 035504 2015

[81] A De Rujula ldquoTwo old ways to measure the electron-neutrinomassrdquo httpsarxivorgabs13054857

[82] A De Rujula ldquoAn upper bound on the exceptional character-istics for Lusztigrsquos character formulardquo httpsarxivorgabs08111674v2

[83] A Faessler and F Simkovic ldquoCan electron capture tell us themass of the neutrinordquo Physica Scripta vol 91 no 4 Article ID043007 2016

[84] H J de Vega O Moreno E Moya de Guerra M RamonMedrano and N G Sanchez ldquoRole of sterile neutrino warmdark matter in rhenium and tritium beta decaysrdquo NuclearPhysics B vol 866 no 2 pp 177ndash195 2013

[85] E M de Guerra O Moreno P Sarriguren and M RamonMedrano ldquoTopics on nuclear structure with electroweakprobesrdquo Journal of Physics Conference Series vol 366 ArticleID 012011 2012

[86] J M Boillos O Moreno and E Moya de ldquoActive and sterileneutrino mass effects on beta decay spectrardquo in Proceedingsof the La Rabida International Scientific Meeting on NuclearPhysics Basic Concepts in Nuclear Physics Theory Experimentsand Applications vol 1541 ofAIP Conference Proceedings p 167La Rabida Spain September 2012

[87] P E Filianin K Blaum S A Eliseev et al ldquoOn the keVsterile neutrino search in electron capturerdquo Journal of PhysicsG Nuclear and Particle Physics vol 41 no 9 Article ID 0950042014

[88] S Eliseev K BlaumM Block et al ldquoDirect measurement of themass difference of 163Ho and 163Dy solves theQ-value puzzle forthe neutrino mass determinationrdquo Physical Review Letters vol115 no 6 Article ID 062501 2015

[89] A De Rujula ldquoA new way tomeasure neutrinomassesrdquoNuclearPhysics Section B vol 188 no 3 pp 414ndash458 1981

[90] M Lusignoli and M Vignati ldquoRelic antineutrino capture on163Hodecaying nucleirdquoPhysics Letters B vol 697 no 1 pp 11ndash142011

[91] A Faessler L Gastaldo and F Simkovic ldquoElectron capture in163Ho overlap plus exchange corrections and neutrino massrdquoJournal of Physics G Nuclear and Particle Physics vol 42 no 1Article ID 015108 2015

[92] G Audi F G Kondev M Wang et al ldquoThe Nubase2012evaluation of nuclear propertiesrdquo Chinese Physics C vol 36 no12 pp 1157ndash1286 2012

[93] S Mertens ldquoSensitivity of next-generation tritium beta-decayexperiments for keV-scale sterile neutrinosrdquo JCAP vol 1502 no2 p 20 2015

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Advances in High Energy Physics 3

eigenstates and hypothetically of one or more extra heaviermass eigenstates as [71ndash77]

1003816100381610038161003816]119890⟩ = 1198801198901 1003816100381610038161003816]1⟩ + 1198801198902 1003816100381610038161003816]2⟩ + 1198801198903 1003816100381610038161003816]3⟩ +sumℎ

119880119890ℎ 1003816100381610038161003816]ℎ⟩ (1)

where the subscript ℎ = 4 5 stands for extra (heavier)mass eigenstates and the quantities 119880 belong to the unitaryneutrino mixing matrix The masses of the three light SMmass eigenstates are so close to each other that so far nomea-surement has been able to discern which of them has beenemitted in a given process As a result the three are emittedas a coherent superposition which is at the origin of theneutrino oscillation phenomenologyThe phase of each masseigenstate changes at a different rate while travelling givingrise to a different superposition at each location that translatesinto a varying (oscillatory) probability of detection of a givenflavor eigenstate

To simplify things we will consider here the linear combi-nation of light mass eigenstates as a single effective neutrinomass eigenstate called ldquolightrdquo with mass

119898119897 = 119898]119890 = (1198802119890111989821 + 1198802119890211989822 + 1198802119890311989823)12 (2)

and just one extra mass eigenstate clearly heavier than theothers called ldquoheavyrdquo (withmass119898ℎ whichwe shall considerin the keV range)

1003816100381610038161003816]119890⟩ = cos 120577 1003816100381610038161003816]119897⟩ + sin 120577 1003816100381610038161003816]ℎ⟩ (3)

where the mixing amplitudes have been written in terms of amixing angle 120577 between the light and the heavy neutrinomasseigenstate The other possible combination of these two masseigenstates would be the sterile flavor eigenstate

1003816100381610038161003816]119904⟩ = minus sin 120577 1003816100381610038161003816]119897⟩ + cos 120577 1003816100381610038161003816]ℎ⟩ (4)

such that ⟨]119890 | ]119904⟩ = 0 For the sterile neutrinos that canbe relevant as WDM cosmological constraints based on theobserved average darkmatter density suggest that the value ofthe mixing angle could approximately be 120577 = 0006∘ [71ndash74]corresponding to a flavor-mass amplitude119880119890ℎ = sin 120577 asymp 10minus4Other recent cosmological constraints give values of sin2(2120577)between 2 sdot 10minus11 and 2 sdot 10minus10 [27 28]

Sterile neutrino states with masses close to the ones ofthe active neutrinos can also have an impact on the patternsmeasured in oscillation experiments [78] Heavier sterileneutrinos as in particular in the keV scale do not modify theoscillation patterns since they are not emitted coherently withthe active neutrinos due to the large mass splitting

The differential energy spectrum of a weak processwhere an electronic neutrino (or antineutrino) is emittedcan therefore be decomposed in a term for light neutrinoemission and another term for heavy neutrino emission asfollows [75ndash77]

119889120582119889119864 = 119889120582119897

119889119864 cos2120577 + 119889120582ℎ

119889119864 sin2120577 (5)

This energy spectrum can be the one of the electron emittedin beta decay or the one of the daughter atom deexcitations in

electron captureThe heavymass eigenstate if it exists wouldbe emitted independently of the other masses (noncoher-ently) as long as the energy resolution of the detector is betterthan the mass difference Δ120598 lt (119898ℎ minus 119898119890) Its contribution tothemeasured spectrum 119889120582ℎ119889119864 would show up in the rangewhere the collected energy is small enough for the heavyneutrino mass to have been produced namely when 0 le 119864 le(119876minus119898ℎ) where119876 is the difference between the masses of theinitial and the final atoms when the reaction takes place invacuumAt the edge of that region at119864 = 119876minus119898ℎ a kink in thespectrum would be found with the size of the heavy neutrinocontribution namely proportional to sin2120577 sim 1205772 (the latterapproximation valid for small mixing angles as is the case inrealistic scenarios)

3 Theory of Electron Capture

Let us consider the capture of an atomic electron by thenucleus119883 (119885119873) to turn into the nucleus119884 (119885minus1119873+1)Thereactions involving the corresponding atoms (represented bythe symbol of the nucleus within brackets) begin withelectron capture

(119883) 997888rarr (119884)119867 + ]119894 (6)

followed by deexcitation of the daughter atom In (6) ]119894 is aneutrino mass eigenstate (]119897 or ]ℎ) and the superscript 119867accounts for the excited state of the atom (119884) correspondingto an electron hole in the shell 119867 due to the electroncapture from this shell in the parent atom The deexcitationof the daughter atom after electron capture can happen eitherthrough emission of a photon (X-ray emission)

(119884)119867 997888rarr (119884) + 120574 (7)

or through emission of electrons (Auger or Coster-Kronigprocess) and photons

(119884)119867 997888rarr (119884+)1198671015840 11986710158401015840 + 119890minus 997888rarr (119884+) + 120574 + 119890minus (8)

where1198671015840 and11986710158401015840 represent holes in the electron shells of theion (119884+)

The energies carried by the emitted photons and electronsare deposited in the calorimeter that surrounds the sourceand that measures the full energy spectrum of these particlesprovided that the corresponding deexcitation lifetimes aremuch smaller than the detector time response [79] The pro-cess in (7) and (8) yields the same calorimeter spectrum thatis the peaks appear at the same energy values correspondingto the excitation energies 119864119867 of the excited daughter atom(119884)119867 The excitation energy is the difference between thebinding energies of the captured electron shell and theadditional electron in the outermost shell The former refersto the daughter atom whereas the latter refers to the parent119864119867 asymp |119861(119884)119867 | minus |119861(119883)out | The reason is that for the shells abovethe vacancy the effective charge is closer to the one in theparent atom (there is one proton less in the nucleus but alsoone electron less in an inner shell) and therefore its bindingenergies should be used [80]

4 Advances in High Energy Physics

An alternative capture process to the one in (6) involvesthe instantaneous formation of a second hole 1198671015840 due to themismatch between the spectator electron wave functions inthe parent and in the daughter atom Whereas the hole 119867 isleft by the captured electron and therefore fulfils the energyand angular momentum conditions for such process theextra hole 1198671015840 has a different origin namely the abovementioned incomplete overlap of electron wave functionswhichmay occur inmany electron shellsThe electron leavingthe extra hole may have been ldquoshaken uprdquo (excited) to anunoccupied orbital in the daughter atom

(119883) 997888rarr (119884)1198671198671015840 + ]119894 (9)

or it may have been ldquoshaken offrdquo to the continuum (ejected)

(119883) 997888rarr (119884+)1198671198671015840 + 119890minus + ]119894 (10)

In a shake-up process the deexcitation of the final atomcontributes to the calorimeter energywith a peak at an energyapproximately equal to the sum of the binding energies of the119867 and 1198671015840 electrons 1198641198671198671015840 asymp |119861(119884)119867 | + |119861(119883)1198671015840 | minus |119861(119883)out | Thussatellite peaks show up located after the single-hole peak at119864119867 In the case of a shake-off the calorimeter energy is thecombination of 1198641198671198671015840 and the electron kinetic energy thelatter showing a continuous distribution as corresponds to thethree-body decay of (10)

It is important to notice that some electron emissionprocesses of the type

(119883) 997888rarr (119884+)1198671015840 11986710158401015840 + 119890minus + ]119894 (11)

where none of the holes in the daughter atom correspondto the captured electron 119867 are quantum mechanicallyindistinguishable [81 82] from the three-step process in (6)and (8) and therefore they contribute to the same one-hole peaks at 119864119867 The contributions to the spectrum of thetwo-hole excitations that are actually distinct from the one-hole excitations ((9) and (10)) have been a subject of recentstudies devoted to search for the electron neutrino mass andare therefore focused on the endpoint of the spectrum (seeeg [80ndash83]) For heavy neutrino searches a similar analysisshould be performed at other regions of the energy spectrumnot just at the endpoint In this paper we restrict thecalculations to the one-hole states

The density of final states of the emitted neutrino inelectron capture is proportional to

120588 = 120588] (119864]) prop 119901]119864] = (1198642] minus 1198982])12 119864] (12)

where 119864] 119901] and 119898] are the neutrino total energy momen-tum and mass respectively On the other hand the proba-bility of capture of a bound electron follows a Breit-Wignerdistribution of width Γ119909 and peaks at the energy 119864119909

119875 (119864) = Γ1199092120587(119864 minus 119864119909)2 + Γ21199094 (13)

Using Fermirsquos Golden Rule the differential reaction ratewith respect to the neutrino energy yields

119889120582119889119864] = 119870EC (1198642] minus 1198982])12 119864]sum119867

119882(])119867sdot Γ1198672120587[119864] minus (119876 minus 119864119867)]2 + Γ21198674

(14)

where 119870EC contains among other factors the weak interac-tion coupling constant and the nuclear matrix element Thefactor119882(])119867 is the squared leptonic matrix element for a givenhole state119867

We write 119882(])119867 = 119862119867119878(])119867 to have a general expressionfor allowed and forbidden transitions The factor 119862119867 is thesquared amplitude of the bound-state electron radial wavefunction at the nuclear interior containing also the squaredoverlap between the initial and the final atom orbital wavefunctions and the effect of electron exchange (as defined inAppendix F-2 of [56]) The cases discussed in the nextsections involve allowed Gamow-Teller transitions (163Ho (72minus)rarr 163Dy (52minus)) and first forbiddenGamow-Teller transi-tions (202Pb (0+) rarr 202Tl (2minus) and 205Pb (52minus) rarr 205Tl (12+)) In the first case the factor 119878(])119867 = 1 for all 119867 states Inthe second case the factor 119878(])119867 = 1 for electrons captured withorbital angularmomentum 119897 = 1 (and neutrinos emitted with119897 = 0) and 119878(])119867 = 1199012] for electrons captured with orbitalangular momentum 119897 = 0 (and neutrinos emitted with 119897 =1) The reason for this is that due to conservation of totalangularmomentum first forbidden transitions have 119871 = 1 forthe leptonic pair Hence there is a linear momentum depen-dence in the leptonic matrix element corresponding to thelepton that carries the angular momentum in each specificcaptureThe corresponding 1199012119890 factor for the 119897 = 1 electrons isembedded in the 119862119867 factor [56]

Equation (14) includes every possible orbital electroncapture the probability of each peaking at 119864] = 119876 minus 119864119867 asdictated by energy conservation where 119864119867 is the excitationenergy of the final atom due to the electron hole119867 resultingfrom capture The energy collected by a calorimeter from theatomic deexcitations is 119864119888 = 119876 minus 119864] namely all the availableenergy except for the one carried away by the neutrino Thedifferential rate can be written in terms of the calorimeterenergy as

119889120582119889119864119888 = 119870EC [(119876 minus 119864119888)2 minus 1198982]]12 (119876 minus 119864119888)sum119867

119862119867119878(])119867sdot Γ1198672120587(119864119888 minus 119864119867)2 + Γ21198674

(15)

In the next section we shall also consider integrated ratesover particular energy ranges which can be calculatednumerically from (15) A convenient compact analyticalexpression of the integrated rate can be obtained from (15)by replacing the Breit-Wigner distributions by Dirac deltas

Advances in High Energy Physics 5

With this approximation the integral of the differential ratecan be written as

120582 = 119870ECsum119867

119862119867119878(])119867 [(119876 minus 119864119867)2 minus 1198982]]12 (119876 minus 119864119867) (16)

In this theoretical model we have neglected two-holepeaks deexcitations through virtual intermediate states andinterferences between deexcitation channels The theoreticalcalorimeter spectrum is thus a single-hole approximationthat assumes full collection of de-excitation energy by thecalorimeter and no pile-up

4 Sterile Neutrino Effect in the Spectrum

Due to the smallness of the mixing angle it is useful toconsider ratios between rates that emphasize the contributionof the heavy neutrino In [84ndash86] for the case of beta decay aratio was considered between the contributions of the heavyand the light neutrino to the differential rates as we shall seein Section 6 For the case of electron capture the differentialdecay rates contain many peaks and it is more convenient toconsider integrated decay rates over specific energy rangesas discussed in [87] In either case (beta decay and electroncapture) this amounts to consider that the detector collectsevents within energy ranges 119864119894 plusmn Δ and 119864119895 plusmn Δ in two regionsof the spectrum such that the mass of the hypothetical heavyneutrino lies in between namely 119864119894 + Δ lt 119898ℎ lt 119864119895 minus Δ with119864119895 lt 119876minusΔ A ratio 119877 between the number of collected eventsin both regions is then performed which corresponds to thetheoretical expression

119877 = Λ 119894Λ 119895 =120581119897119894 + tan2120577120581ℎ119894120581119897119895 (17)

where Λ 119894 = cos2120577120581119897119894 + sin2120577120581ℎ119894 (in the region where both 119897and ℎ mass eigenstates contribute) and Λ 119895 = cos2120577120581119897119894 (in theregion where ℎ is not energetically allowed) The integrals 120581are defined as

120581]119894119895 = int119864119894119895+Δ119864119894119895minusΔ

119889120582119889119864119888 (119898]) 119889119864119888 (18)

In electron capture the energies 119894 and 119895 can be selected asthe energies of two peaks and the integration intervals canbe chosen as the width of the capture peaks If the peaks areapproximated by delta functions the integrals (using119864119867 = 119864119894and Δ rarr 0) can be written as

120581]EC119894119903 = 119870EC119862119894119903119878(])119894119903 (119876119903 minus 119864119894)2 [1 minus ( 119898]119876119903 minus 119864119894)2]12 (19)

where 119876119903 is the atomic mass difference in a given isotope 119903119864119894 is the energy position of a given peak 119894 in the calorimeterspectrum 119898] is 119898119897 asymp 0 the mass of a light neutrino or119898ℎ the mass of a heavy neutrino and 119878(])119894119903 contains theneutrino momentum dependence for the peak 119894 comingfrom the leptonic matrix element squared As explained inthe previous section for allowed transitions 119878(])119894119903 = 1 For

first forbidden transitions some of the capture peaks have119878(])119894119903 = 1199012]119894119903 for the 119894 peaks corresponding to 11990412 shells (aswell as for those corresponding to 11990112 shells that contributethrough their admixtures with the 119897 = 0 orbital due torelativistic corrections) The other peaks have 119878(])119894119903 = 1where 119894 corresponds to 11990132 shells (and 11988932 shells throughtheir admixtures with the 119897 = 1 orbital due to relativisticcorrections)

For electron capture it is convenient to define a ratiosimilar to the one in (17) but where the numerator andthe denominator are themselves ratios of numbers of eventswithin the same peak but for different isotopes of the sameelement 119903 and 119904 so that some atomic corrections cancel out[87]

1198771015840 = Λ 119894119903Λ 119894119904Λ 119895119903Λ 119895119904 = (119877119897119894119895119903119904)2(120574+1)(1 + 1205962120574+1119894119903 tan2120577

1 + 1205962120574+1119894119904 tan2120577) (20)

where one should notice that the factors 119862119894 and 119862119895 cancel outin addition to the factor119870EC which cancels out in both ratios(119877 and 1198771015840) The factors in (20) have very simple analyticalforms when one uses (19)

119877119897119894119895119903119904 = (119876119903 minus 119864119894) (119876119904 minus 119864119895)(119876119903 minus 119864119895) (119876119904 minus 119864119894)

120596119894119903(119904) = [1 minus ( 119898ℎ119876119903(119904) minus 119864119894)2]12

(21)

where 120574 depends on the angular momenta of each lepton in agivenΔ119869120587 nuclear transition For instance for allowed decays120574 = 0 whereas for first forbidden decays 120574 = 1 for 11990412 and11990112 peaks or 120574 = 0 for 11990132 and 11988932 peaks In the expressionsabove we have assumed that both peaks 119894 and 119895 are of thesame type namely 120574119894 = 120574119895 = 120574

By measuring the ratio in (20) the mixing angle can thenbe obtained as

120577 = arctan[[

(119877119897119894119895119903119904)2(120574+1) minus 1198771015840exp1198771015840exp1205962120574+1119894119904 minus (119877119897119894119895119903119904)2(120574+1) 1205962120574+1119894119903

]]12

(22)

where the theoretical values of 119877119897119894119895119903119904 120596119894119903 and 120596119894119904 requirean accurate experimental knowledge of the position of theselected peaks and more importantly of the atomic massdifferences 119876119903 and 119876119904 In addition one has to consider fixedinitially unknown values of the mass of the heavy neutrino119898ℎ that is searched for or otherwise be content with thegeneration of exclusion plots in case of no effect observation

In summary we consider here two types of ratios that canbe useful in the search for a signal of keV sterile neutrinos inelectron capture (1) the ratio between the intensity of twopeaks in the electron capture spectrum of a given nucleusas for the case of 163Ho discussed in Section 5 (2) The casein which two isotopes of a given element undergo electroncapture where one can use the ratio defined in (20) as isthe case of Lead isotopes discussed in Section 5 For the

6 Advances in High Energy Physics

10minus5

Neutrino phase spaceAtomic deexcitationTotal

M1M2

N1

N2O1

O2

0 05 1 15 2 25 3Ec (keV)

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

(a)

m = 25 keVm = 2 keVm = 15 keV

m = 1 keVm = 05 keVm = 0

0 05 1 15 2 25 3Ec (keV)

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

10minus5

(b)

0 05 1 15 2 25 3

No mixingMax mixing

Ec (keV)

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

(c)

Figure 1 Calorimeter spectrum after electron capture in 163Ho (a) Full spectrum (solid curve) and contributions emitted neutrino phasespace (dotted curve) and daughter atom deexcitation (dashed curve) (b) Full spectrum for different neutrino masses 0 05 1 15 2 and25 keV (c) full spectrum for light (119898119897 asymp 0 keV) - heavy (119898ℎ = 2 keV) neutrino mixing using a maximal mixing angle 120577 = 45∘ for illustrationwith (solid curve) and without (dotted curve) the heavy neutrino contribution For comparison the spectrumwithout heavy neutrinomixing(dashed curve) is also shownWe use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of the figure Realistic valuesof 120577 lt 001∘ result in a reduction of the difference between the dashed and the solid lines in (c) by more than seven orders of magnitude

beta decay case in Section 6 we consider the ratio betweenthe differential rates but the ratios between integrated ratesare also useful and may be more realistic for comparison toexperiments

5 Results for Electron Capture Spectra

Let us first consider the capture of an atomic electron by thenucleus Holmium 163 (163Ho 119885 = 67 119873 = 96) to turn intoDysprosium 163 (163Dy 119885 = 66119873 = 97) with ground statespins and parities 119869120587 = 72minus and 119869120587 = 52minus respectively Itis an allowed unique Gamow-Teller transition (Δ119869 = 1 andno parity change the leptons carry no orbital angularmomentum Δ119871 = 0 and couple to spin 119878 = 1) with atomicmass difference 119876 = 2833 plusmn 0030 plusmn 0015 keV [88] which isbeing currently measured at the Electron Capture Holmium(ECHo) experiment [50ndash52] Only electrons with principalquantum number 119899 larger than 2 have a binding energylower than the 119876-value and can therefore be captured Inaddition being an allowed decay only electrons from 11990412 and

11990112 orbitals can be captured the latter through admixtureswith the 119897 = 0 orbital due to relativistic corrections Thuscapture from the orbitals M1 M2 N1 N2 O1 O2 and P1 (inspectroscopic notation) is possible [80 89ndash91]

In Figure 1 we show the differential electron capture ratein 163Ho as a function of the calorimeter energy 119864119888 = 119876minus119864]In Table 1 we give theoretical values of the strengths andwidths of each capture peak and experimental values ofthe electron binding energy again for each capture peakThe strengths 119862119867 are obtained in [56 57] from relativisticHartree-Fock calculations of the electron wave functions atthe nuclear site accounting for exchange and overlap contri-butions In the upper plot the emitted neutrino phase spacecontribution and the daughter atom deexcitation contribu-tion are plotted separately together with the product of bothwhich is the full differential electron capture rate In themiddle plot we show the full differential rate for differentmasses of the emitted neutrino 119898] asymp 0 (realistic) and 119898] asymp05 1 15 2 and 25 keV (unrealistic) As can be seen in theplot each curve ends at 119876minus119898] Finally the lower plot showsthe spectrum resulting from the mixing of a light neutrino

Advances in High Energy Physics 7

Table 1 Atomic parameters for electron capture in Holmium from shells M1 to O2The values of 119862119867 [56 57] correspond to the atomic shellsin Holmium while the orbital electron widths Γ119867 [58] and binding energies 119861119867 (ltQ) [59ndash61] correspond to the atomic shells in Dysprosium

M1 M2 N1 N2 O1 O2119862119867 005377 0002605 001373 00005891 0001708 00001Γ119867 [keV] 0013 0006 0006 0005 0005 0002119861119867 [keV] 2047 1842 0416 0332 0063 0026

Table 2 Same as Table 1 (data from the same references) but for electron capture in Lead going toThalliumThe upper table summarizes thevalues of the relevant parameters for 11990412 and 11990112 atomic shells while the lower table shows the values for 11990132 and 11988932 atomic shells In thelatter case 119862119867 includes an extra factor 1199012119890

L1 L2 M1 M2 N1 N2 O1 O2119862119867 08781 006647 02125 001759 005849 0004431 001016 00008052Γ119867 [keV] 0011 0006 0015 0010 0009 0007 mdash mdash119864119867 [keV] 15346 14697 3703 3415 0845 0720 0136 0099L3 M3 M4 N3 N4 O3 O4119862119867 [keV2] 18978193 5366014 45383 1363307 12607 237280 1500Γ119867 [keV] 0006 0009 0002 0006 0004 0001 0001119861119867 [keV] 12657 2956 2484 0608 0406 0072 0015

mass eigenstate 119898119897 asymp 0 with a heavy mass eigenstate 119898ℎ asymp2 keV (solid curve) as in (5) Although the spectrum withmixing is realistic in the sense that it is the result expected if aheavy mass eigenstate exists it is unrealistic in the degree ofmixing shown in the figure which has been maximized herefor the sake of visibility of the ldquokinkrdquo in the scale of the figure50 light neutrino and 50 heavy neutrino correspondingto amixing angle 120577 = 45∘The ldquokinkrdquo can be seen in this curveat 119864119888 = 119876 minus 119898ℎ = 0833 keV Above this calorimeter energythe heavy mass eigenstate cannot be produced due to energyconservation For comparison the spectrum for119898] asymp 0 withno mixing with heavy states is shown in the dashed curve

Another example of electron capture under study here isthat of the Lead isotopes 202 (202Pb 119885 = 82 119873 = 120)and 205 (205Pb 119885 = 82 119873 = 123) going to the Thalliumisotopes 202 (202Tl 119885 = 81 119873 = 121) and 205 (205Tl119885 = 81 119873 = 124) respectively where the process is in bothcases a first forbidden Gamow-Teller unique transition with0+ rarr 2minus for 202Pb and 52minus rarr 12+ for 205Pb (Δ119869 = 2 andparity change the leptons carry Δ119871 = 1 and couple to 119878 = 1)The atomic mass differences obtained from the data in [92]are 119876 = 46 plusmn 14 keV and 119876 = 506 plusmn 18 keV respectivelyAccording to the explanations given in Section 3 capturesfrom orbitals L1 L2 M1 M2 N1 N2 O1 and O2 containan extra neutrino momentum dependence 119878(]) = 1199012] thatmodifies the calorimeter spectrum and corresponds to 120574 =1 On the other hand captures from orbitals L3 M3 M4N3 N4 O3 and O4 which correspond to 120574 = 0 containan extra electron momentum dependence 1199012119890 which beingfixed in bound electrons is included in the values of 119862119867They are given in Table 2 together with the widths and theexperimental electron binding energies in the daughter atomThallium

Figure 2 shows the capture differential rate in 205Pb (darkcurves) and in 202Pb (light curves) Position width and

strength of the capture peaks are assumed to be the same inboth isotopes but the spectra are different because of the dif-ferent 119876-values Solid curves are for light neutrino emission119898] asymp 0 and dashed lines are for heavy neutrino emissionwith119898] = 40 keV (results are given separately for each massbeforemixing) Due to the large119876-values electron capture inLead isotopes allows us to explore this mass value althoughit may be somewhat larger than the expected value fromcosmological reasons The analysis of the ratio in (20) canbe computed for example using the capture peaks L3 at119864119888 = 12657 keV and M3 at 119864119888 = 2956 keV both beingof the 120574 = 0 type For 119898ℎ = 40 keV one would obtain119877l1198723 1198713 202 205 = 1037 1205961198723202 = 0369 1205961198723205 = 0543These results should be introduced in (22) together with theexperimental ratio 1198771015840exp to obtain the value of the mixingangle 120577

It is important to remark that the theoretical quantities 119877119897and 120596 computed as described above contain several approxi-mations They correspond to Dirac-delta peaks ignoring theactual shapes and widths and thus they just refer to onesingle capture peak neglecting the possible effect of the tailsof nearby one-hole peaks Moreover the influence of two-hole peaks close to the main ones has also been neglectedOther effects that have not been taken into account butwhose influence is expected to be very small are multihole(more than two) peaks virtual intermediate states (influenceof transitions through atomic shells not energetically acces-sible) or interference between atomic transitions resultingfrom addition of amplitudes instead of intensities [81 82]Theaccurate experimental determination of the ratio 1198771015840exp entailsits own difficulties among them the possible existence ofmetastable atomic states whose delayed de-excitations failto contribute to the collected spectrum and the variety ofchemical environments resulting in a complex mixture of 119876-values

8 Advances in High Energy Physics

L1

L2

L3

M1M2

M3

M4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

6 162 8 10 12 144 18 200Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(a)

N1

N2

N3

O1

O2

O3

N4O4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

05 0603 0401 02 07 08 09 1000Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(b)

Figure 2 Calorimeter spectrum after electron capture in 202Pb (light curves) and in 205Pb (black curves) with emission of a light neutrino119898119897 = 0 (solid curves) and a heavy neutrino119898ℎ = 40 keV (dashed curves) (a) Full spectrum showing L and M capture peak labels (b) Lowenergy region (119864119888 = 0-1 keV) showing N and O capture peak labels

6 Results for Beta Decay Spectra

In addition to electron capture we also show here the effectof heavy neutrino emission in the electron spectrum of betadecays for one of the cases of most interest Tritium It hasbeen studied in depth in previous works [84ndash86 93] togetherwith another process that has also drawn considerable theo-retical end experimental attention the beta decay of Rhenium187 [37 39ndash41 84ndash86] The beta decay of Tritium (3H 119885 = 1119873 = 2) going to Helium 3 (3He 119885 = 2119873 = 1) is

3H 997888rarr 3He + 119890minus + ]119890 (23)

and has a 119876-value of 1859 keV [92] Both the initial and thefinal nuclear ground states have spin-parity 12+ and thetransition is allowed with the electron and the antineutrinoemitted in 119904-wave The Karlsruhe Tritium Neutrino Experi-ment (KATRIN) [42ndash46] is currently studying this decay inorder to determine the active neutrino mass and could alsostudy the production of a heavy neutrino of amass lower than18 keV

The differential decay rate with respect to the electronenergy is given by

119889120582119889119864119890 = 119870120573 (1198642119890 minus 1198982119890)12 [(119876 + 119898119890 minus 119864119890)2 minus 1198982]]12

sdot 119864119890 (119876 + 119898119890 minus 119864119890) (24)

where 119870120573 contains among others the weak interactioncoupling the nuclearmatrix element and the Fermi function

As an illustration of the heavy neutrino effect in the Tri-tium beta decay we plot in Figure 3 the differential decay ratefrom (24) and (5) using a maximal mixing with an unrealisticvalue of the mixing angle 120577 = 45∘ to show the effect more

distinctly in the scale of the figure We have used heavy masscomponents with 119898ℎ = 2 keV [26] (Figure 3(a)) and with119898ℎ = 7 keV [29 30] (Figure 3(b)) For comparison thespectra without the heavy neutrino contribution (120582ℎ = 0) andwithout mixing (120577 = 0∘) are also shown in this plot The kinkin the spectrum at 119864119890 minus 119898119890 = 119876 minus 119898ℎ can be observed if theexperimental relative error is lower than the size of the stepthe latter being very small in realistic situations

The effect of a heavy neutrino emission can be analyzedthrough the ratio between the heavy and the light neutrinocontributions to the spectrum

R equiv 119889120582ℎ119889119864119890119889120582119897119889119864119890 tan2120577 (25)

The full differential decay rate is also related to the ratio Rthrough

119889120582119889119864119890 =119889120582119897119889119864119890 [1 +R] cos2120577 (26)

In Figure 4 we plot R the ratio of the heavy neutrinocontribution over the light neutrino contribution to the decayrate as a function of the momentum of the emitted electronfor a heavy neutrino mass 119898ℎ = 2 keV and different mixingangles 120577 = 001∘ 0005∘ and 0001∘ The size of this ratio(lt10minus7) gives an idea of the difficulty of finding the kink in thespectrum due to the production of the heavymass eigenstateAs can be seen in the figure the ratio is different from zeroand almost constant in the range 0 le 119901119890 lt (119901119890)max wherethe maximum electron momentum is given by (119901119890)max =[(119876 minus 119898ℎ)(119876 minus 119898ℎ + 2119898119890)]12 For the heavy mass used inthe figure (119901119890)max ≃ 1313 keV The ratio R decreases as themixing angle decreases being approximately proportional to

Advances in High Energy Physics 9

Mixing but no heavy neutrino contribution

104

105

106

107

108d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 2 keV

(a)

104

105

106

107

108

d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

Mixing but no heavy neutrino contribution

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 7 keV

(b)

Figure 3 Electron spectrum of Tritium beta decay for a light-heavy neutrino mixing angle 120577 = 45∘ shown just for illustration and a heavyneutrino mass (solid curve)119898ℎ = 2 keV (a) and119898ℎ = 7 keV (b) The spectrum without heavy neutrino mass contribution (dotted curve) andwithout mixing (dashed curve) are also shown We use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of thefigure Realistic values of 120577 lt 001∘ result in a reduction of the difference between the dotted and the solid lines by more than seven orders ofmagnitude

10minus11

10minus10

10minus9

10minus8

10minus7

10minus12

120577 = 001∘

120577 = 0005∘

120577 = 0001∘

15090 100 110 120 130 1408070pe (keV)

ℛ

Figure 4 Ratio R defined in (25) for the Tritium beta decay as afunction of the electronmomentum for a heavy neutrinomass119898ℎ =2 keV and different values of the mixing angle 120577 = 001∘ (solid line)0005∘ (dashed line) and 0001∘ (dotted line)

1205772 It also decreases for larger heavy neutrino masses 119898ℎ Adifferent ratio between the spectra with mixing and without

mixing (Rlowast) has also been considered in [84ndash86 93] relatedwithR in (25) as

Rlowast = minussin2120577 +R cos2120577 = 119889120582119889119864119890119889120582119897119889119864119890 minus 1 (27)

7 Conclusions

Signatures of hypothetical keV sterile neutrinos which couldbe warm dark matter candidates (WDM) may be found inthe spectra of ordinary weak nuclear decays The electronspectrum in beta decay is cleaner for this purpose becauseit is a smooth curve while the deexcitation spectrum afterelectron capture shows many peaks On the other hand inbeta decay the possible excitation of the atoms or moleculesis not disentangled while in electron capture the calorimetercollects all the energy independently of the excitation ofthe atom and of the deexcitation path and one has higherstatistics around the capture peaks

Figures of electron spectrum for beta decay in 3H aswell as of calorimeter spectrum for electron capture in 163Hoand 202Pb and 205Pb are given considering various valuesof the heavy neutrino mass (2 keV 7 keV and 40 keV) thatmay be experimentally probed In both weak processes thesmall value of the light-heavy neutrino mixing angle requiresextremely high experimental precision and the use of sourceswith large stability to reduce systematic errors This is why it

10 Advances in High Energy Physics

is useful to consider relevant ratios between transition rates asfirst introduced in [84ndash87] to analyze the data We considertwo cases the case of a single isotope where one may use theratio of accumulated number of events in different regionsof the spectrum which allows us to remove uncertaintiesrelated to the nuclear matrix element and to the valuesof overall constants And in the case of two isotopes weconsider ratios of the above mentioned ratios that allow usto reduce uncertainties from atomic parameters We give aswell analytical expressions that can be used to obtain a goodtheoretical approximation to the experimental ratios ((19) to(21))

Both the experimental measurement and the theoreticalmodel must be accurate enough to detect differences in theexpected versus themeasured ratios of the order of themixingangle squared 1205772 ≲ 10minus8 This is also the size of the kinkexpected in the electron spectrumof beta decay located at thelimit of the regionwhere the production of a heavy neutrino isenergetically allowed In order to identify this kink among thestatistical fluctuations of themeasured spectrum the numberof collected events must be larger than the inverse of thesquared ratioR in (25)

Our results particularly on the electron capture spectraof Lead isotopes that include for the first time all thepossible peaks may help in designing and analyzing futureexperiments to search for sterile neutrinos in the keV massrangeThe use of the different types of ratios between numberof events discussed here for both electron capture and betadecay processes may also help in planning the experimentsby establishing the threshold of statistical and systematicuncertainties required for the detection of sterile neutrinosof given mass and mixing or to properly extract exclusionplots

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

E Moya de Guerra and O Moreno acknowledge MINECOfor partial financial support (FIS2014-51971-P) and IEM-CSIC for hospitality O Moreno acknowledges supportfrom a Marie Curie International Outgoing Fellowshipwithin the European Union Seventh Framework Pro-gramme under Grant Agreement PIOF-GA-2011-298364(ELECTROWEAK) M R M thanks N G Sanchez forcorrespondence and conversations

References

[1] F Zwicky ldquoDieRotverschiebung von extragalaktischenNebelnrdquoHelvetica Physica Acta vol 6 pp 110ndash127 1933

[2] M Roos ldquoAstrophysical and cosmological probes of darkmatterrdquo Journal of Modern Physics vol 3 pp 1152ndash1171 2012

[3] S Dodelson Modern Cosmology Academic Press New YorkNY USA 2003

[4] S Weinberg Cosmology Oxford University Press Oxford UK2008

[5] J Lesgourgues and S Pastor ldquoNeutrino mass from cosmologyrdquoAdvances in High Energy Physics vol 2012 Article ID 60851534 pages 2012

[6] D Boyanovsky H J de Vega and N G Sanchez ldquoConstraintson dark matter particles from theory galaxy observations andN-body simulationsrdquo Physical Review D vol 77 no 4 ArticleID 043518 2008

[7] D Boyanovsky H J de Vega and N G Sanchez ldquoDarkmatter transfer function free streaming particle statistics andmemory of gravitational clusteringrdquo Physical Review D vol 78no 6 Article ID 063546 2008

[8] C Destri H J de Vega and N G Sanchez ldquoWarm dark matterprimordial spectra and the onset of structure formation atredshift zrdquo Physical Review D vol 88 no 8 Article ID 0835122013

[9] H J De Vega P Salucci and N G Sanchez ldquoThe mass ofthe dark matter particle theory and galaxy observationsrdquo NewAstronomy vol 17 no 7 pp 653ndash666 2012

[10] H J de Vega P Salucci and N G Sanchez ldquoObservationalrotation curves and density profiles versus the Thomas-Fermigalaxy structure theoryrdquoMonthlyNotices of the Royal Astronom-ical Society vol 442 no 3 pp 2717ndash2727 2014

[11] H J de Vega and N G Sanchez ldquoModel-independent analysisof dark matter points to a particle mass at the keV scalerdquoMonthly Notices of the Royal Astronomical Society vol 404 no2 pp 885ndash894 2010

[12] H J de Vega and N G Sanchez ldquoConstant surface gravity anddensity profile of dark matterrdquo International Journal of ModernPhysics A vol 26 no 6 pp 1057ndash1072 2011

[13] H J de Vega and N G Sanchez ldquoCosmological evolutionof warm dark matter fluctuations I Efficient computationalframework with Volterra integral equationsrdquo Physical ReviewDvol 85 no 4 Article ID 043516 20 pages 2012

[14] H J de Vega and N G Sanchez ldquoThe dark matter distributionfunction and halo thermalization from the Eddington equationin galaxiesrdquo International Journal of Modern Physics A vol 31no 13 Article ID 1650073 2016

[15] NMenci S Paduroiu P Salucci N Sanchez and C RWatsonldquolectures at the 20th Paris Cosmology Colloquium Chalonge-Hector de Vegardquo 2016 httpchalongeobspmfr

[16] E W Kolb and M S Turner The Early Universe AddisonWesley Boston Mass USA 1990

[17] J F Navarro C S Frenk and S D M White ldquoA universal den-sity profile from hierarchical clusteringrdquo Astrophysical JournalLetters vol 490 no 2 pp 493ndash508 1997

[18] W J G de Blok ldquoThe core-cusp problemrdquo Advances in Astron-omy vol 2010 Article ID 789293 14 pages 2010

[19] G Gilmore M I Wilkinson R F GWyse et al ldquoThe observedproperties of dark matter on small spatial scalesrdquo AstrophysicalJournal vol 663 no 2 pp 948ndash959 2007

[20] P Salucci and Ch Frigerio Martins ldquoThe mass distribution inSpiral galaxiesrdquo EAS Publications Series vol 36 pp 133ndash1402009

[21] J van Eymeren C Trachternach B S Koribalski and R-JDettmar ldquoNon-circular motions and the cusp-core discrepancyin dwarf galaxiesrdquo Astronomy amp Astrophysics vol 505 no 1 pp1ndash20 2009

[22] M Walker and J Penarrubia ldquoA method for measuring (slopesof) the mass profiles of dwarf spheroidal galaxiesrdquo The Astro-physical Journal vol 742 no 1 p 20 2011

Advances in High Energy Physics 11

[23] R F G Wyse and G Gilmore ldquoObserved properties of darkmatter on small spatial scalesrdquo in Proceedings of the Proceedingsof the International Astronomical Union Symposium (IUA rsquo07)vol 3 no 244 pp 44ndash52 Cardiff UK June 2007

[24] A Burkert ldquoThe structure of dark matter halos in dwarfgalaxiesrdquo Astrophysical Journal vol 447 no 1 pp L25ndashL281995

[25] C Destri H J de Vega and N G Sanchez ldquoQuantum WDMfermions and gravitation determine the observed galaxy struc-turesrdquo Astroparticle Physics vol 46 pp 14ndash22 2013

[26] N Menci N G Sanchez M Castellano and A GrazianldquoConstraining the warm dark matter particle mass throughultra-deep UV luminosity functions at Z = 2rdquoTheAstrophysicalJournal vol 818 no 1 article 90 2016

[27] E Bulbul M Markevitch A Foster R K Smith M Loewen-stein and SW Randall ldquoDetection of an unidentified emissionline in the stackedX-ray spectrumof galaxy clustersrdquoAstrophys-ical Journal vol 789 article 13 2014

[28] A Boyarsky O Ruchayskiy D Iakubovskyi and J FranseldquoUnidentified line inX-ray spectra of the andromeda galaxy andperseus galaxy clusterrdquo Physical Review Letters vol 113 no 25Article ID 251301 2014

[29] K N Abazajian ldquoResonantly produced 7 keV sterile neutrinodark matter models and the properties of Milky Way satellitesrdquoPhysical Review Letters vol 112 no 16 Article ID 161303 5pages 2014

[30] N Menci A Grazian M Castellano and N G SanchezldquoShock connectivity in the 2010 August and 2012 July solarenergetic particle events inferred from observations and ENLILmodelingrdquo Astrophysical Journal Letters vol 825 no 1 article 12016

[31] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[32] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 481 no 1-2 pp 1ndash28 2009

[33] B Pontecorvo ldquoNeutrino experiments and the problem ofconservation of leptonic chargerdquo Soviet Physics-JETP vol 26no 5 pp 984ndash988 1968

[34] A Merle ldquokeV neutrino model buildingrdquo International Journalof Modern Physics D vol 22 no 10 Article ID 1330020 2013

[35] J DVergados andT S Kosmas ldquoSearching for cold darkmatterA case of coexistence of supersymmetry and nuclear physicsrdquoPhysics of Atomic Nuclei vol 61 p 1066 1998

[36] E Holmlund M Kortelainen T S Kosmas J Suhonen and JToivanen ldquoMicroscopic calculation of the LSP detection ratesfor the 71Ga 73Ge and 127I dark-matter detectorsrdquoPhysics LettersB vol 584 no 1-2 pp 31ndash39 2004

[37] R Adhikari M Agostini N Anh Ky et al ldquoA white paper onkeV sterile neutrino dark matterrdquo httpsarxivorgabs160204816

[38] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[39] MARE collaboration httpmaredfmuninsubriaitfrontendexecphp

[40] A Nucciotti ldquoNeutrino mass calorimetric searches in theMARE experimentrdquo httpsarxivorgabs10122290

[41] A Nucciotti ldquoOn behalf of the MARE collaboration lecture attheWorkshop ChalongeMeudonrdquo 2011 httpchalongeobspmfr

[42] KATRIN Collaboration httpwwwkatrinkitedu

[43] C Weinheimer ldquoDirect determination of neutrino mass fromtritium beta spectrumrdquo httpsarxivorgabs09121619

[44] C Weinheimer lecture at the 20th Paris Cosmology Chalonge-Hector de Vega Colloquium 2016 httpchalongeobspmfr

[45] G Drexlin lecture at the 19th Paris Cosmology Colloquium2015

[46] A Huber lecture at the Chalonge-de Vega Meudon Workshop2016 httpchalongeobspmfr

[47] C Tully lecture at the 19th Paris Cosmology Colloquium 2015[48] S Betts W R Blanchard R H Carnevale et al ldquoDevelop-

ment of a relic neutrino detection experiment at PTOLEMYprinceton tritiumobservatory for light early-universemassive-neutrino yieldrdquo httpsarxivorgabs13074738

[49] D M Asner R F Bradley L de Viveiros et al ldquoSingle-electrondetection and spectroscopy via relativistic cyclotron radiationrdquoPhysical Review Letters vol 114 no 16 Article ID 162501 2015

[50] P C-O Ranitzsch J-P Porst S Kempf et al ldquoDevelopmentof metallic magnetic calorimeters for high precision measure-ments of calorimetric 187Re and 163Ho spectrardquo Journal of LowTemperature Physics vol 167 no 5-6 pp 1004ndash1014 2012

[51] C Hassel ldquoLecture at the International School of AstrophysicsDaniel ChalongemdashHector de Vegardquo 2015

[52] L Gastaldo K Blaum A Doerr et al ldquoThe electron capture163Ho experiment ECHordquo Journal of Low Temperature Physicsvol 176 no 5 pp 876ndash884 2014

[53] B Alpert M Balata D Bennett et al ldquoThe electron capturedecay of 163Ho tomeasure the electron neutrino mass with sub-eV sensitivityrdquo The European Physical Journal C vol 75 article112 2015

[54] E Fermi ldquoRadioattivita prodotta da bombardamento di neu-tronirdquo Il Nuovo Cimento vol 11 no 7 pp 429ndash441 1934

[55] E Fermi ldquoVersuch einer theorie der 120573-strahlen Irdquo Zeitschriftfur Physik vol 88 no 3 pp 161ndash177 1934

[56] R B Firestone and V S Shirley Eds Table of Isotopes JohnWiley amp Sons New York NY USA 8th edition 1999

[57] W Bambynek H Behrens M H Chen et al ldquoOrbital electroncapture by the nucleusrdquo Reviews of Modern Physics vol 49 no1 pp 77ndash221 1977

[58] J L Campbell and T Papp ldquoWidths of the atomic KndashN7 levelsrdquoAtomic Data and Nuclear Data Tables vol 77 no 1 pp 1ndash562001

[59] J A Bearden and A F Burr ldquoReevaluation of X-ray atomicenergy levelsrdquo Reviews of Modern Physics vol 39 no 1 pp 125ndash142 1967

[60] M Cardona and L Ley Eds Photoemission in Solids I GeneralPrinciples Springer Berlin Germany 1978

[61] J C Fuggle and N Martensson ldquoCore-level binding energies inmetalsrdquo Journal of Electron Spectroscopy and Related Phenom-ena vol 21 no 3 pp 275ndash281 1980

[62] T Kosmas H Ejiri and A Hatzikoutelis ldquoNeutrino physics inthe frontiers of intensities and very high sensitivitiesrdquo Advancesin High Energy Physics vol 2015 Article ID 806067 3 pages2015

[63] A Giuliani and A Poves ldquoNeutrinoless double-beta decayrdquoAdvances in High Energy Physics vol 2012 Article ID 85701638 pages 2012

[64] J-U Nabi and H V Klapdor-Kleingrothaus ldquoWeak interactionrates of sd-shell nuclei in stellar environments calculated in theproton-neutron quasiparticle random-phase approximationrdquoAtomic Data andNuclear Data Tables vol 71 no 2 pp 149ndash3451999

12 Advances in High Energy Physics

[65] J-U Nabi and H V Klapdor-Kleingrothaus ldquoMicroscopic cal-culations of stellar weak interaction rates and energy losses forfp- and fpg-shell nucleirdquo Atomic Data and Nuclear Data Tablesvol 88 no 2 pp 237ndash476 2004

[66] Y Fukuda T Hayakawa E Ichihara et al ldquoEvidence foroscillation of atmospheric neutrinosrdquo Physical Review Lettersvol 81 no 8 pp 1562ndash1567 1998

[67] Q R Ahmad and SNOCollaboration ldquoMeasurement of the rateof V119890+119889 rarr 119901+119901+119890minus interactions produced by 8B solar neutrinosat the Sudbury Neutrino Observatoryrdquo Physical Review Lettersvol 87 no 7 Article ID 071301 6 pages 2001

[68] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 2002

[69] G J Feldman J Hartnell and T Kobayashi ldquoLong-baselineneutrino oscillation experimentsrdquo Advances in High EnergyPhysics vol 2013 Article ID 475749 30 pages 2013

[70] S T Petcov ldquoThenature ofmassive neutrinosrdquoAdvances inHighEnergy Physics vol 2013 Article ID 852987 20 pages 2013

[71] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[72] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 48 no 1-2 pp 1ndash28 2009

[73] F Munyaneza and P L Biermann ldquoDegenerate sterile neutrinodark matter in the cores of galaxiesrdquo Astronomy amp Astrophysicsvol 458 no 2 pp L9ndashL12 2006

[74] D Boyanovsky and C M Ho ldquoSterile neutrino production viaactive-sterile oscillations the quantum Zeno effectrdquo Journal ofHigh Energy Physics vol 2007 no 7 p 30 2007

[75] R Shrock ldquoNew tests for and bounds on neutrino masses andlepton mixingrdquo Physics Letters B vol 96 no 1-2 pp 159ndash1641980

[76] R E Shrock ldquoGeneral theory of weak processes involving neu-trinos I Leptonic pseudoscalar-meson decays with associatedtests for and bounds on neutrino masses and lepton mixingrdquoPhysical Review D vol 24 no 5 pp 1232ndash1274 1981

[77] R E Shrock ldquoImplications of neutrino masses and mixing forweak processesrdquo in Proceedings of the Weak Interactions asProbes of Unification vol 72 of AIP Conference Proceedings p368 Blacksburg Va USA 1981

[78] J M Conrad C M Ignarra G Karagiorgi M H Shaevitzand J Spitz ldquoSterile neutrino fits to short-baseline neutrinooscillation measurementsrdquo Advances in High Energy Physicsvol 2013 Article ID 163897 2013

[79] A Nucciotti ldquoThe use of low temperature detectors for directmeasurements of themass of the electron neutrinordquoAdvances inHigh Energy Physics vol 2016 Article ID 9153024 41 pages2016

[80] R G H Robertson ldquoExamination of the calorimetric spectrumto determine the neutrino mass in low-energy electron capturedecayrdquo Physical Review C vol 91 no 3 Article ID 035504 2015

[81] A De Rujula ldquoTwo old ways to measure the electron-neutrinomassrdquo httpsarxivorgabs13054857

[82] A De Rujula ldquoAn upper bound on the exceptional character-istics for Lusztigrsquos character formulardquo httpsarxivorgabs08111674v2

[83] A Faessler and F Simkovic ldquoCan electron capture tell us themass of the neutrinordquo Physica Scripta vol 91 no 4 Article ID043007 2016

[84] H J de Vega O Moreno E Moya de Guerra M RamonMedrano and N G Sanchez ldquoRole of sterile neutrino warmdark matter in rhenium and tritium beta decaysrdquo NuclearPhysics B vol 866 no 2 pp 177ndash195 2013

[85] E M de Guerra O Moreno P Sarriguren and M RamonMedrano ldquoTopics on nuclear structure with electroweakprobesrdquo Journal of Physics Conference Series vol 366 ArticleID 012011 2012

[86] J M Boillos O Moreno and E Moya de ldquoActive and sterileneutrino mass effects on beta decay spectrardquo in Proceedingsof the La Rabida International Scientific Meeting on NuclearPhysics Basic Concepts in Nuclear Physics Theory Experimentsand Applications vol 1541 ofAIP Conference Proceedings p 167La Rabida Spain September 2012

[87] P E Filianin K Blaum S A Eliseev et al ldquoOn the keVsterile neutrino search in electron capturerdquo Journal of PhysicsG Nuclear and Particle Physics vol 41 no 9 Article ID 0950042014

[88] S Eliseev K BlaumM Block et al ldquoDirect measurement of themass difference of 163Ho and 163Dy solves theQ-value puzzle forthe neutrino mass determinationrdquo Physical Review Letters vol115 no 6 Article ID 062501 2015

[89] A De Rujula ldquoA new way tomeasure neutrinomassesrdquoNuclearPhysics Section B vol 188 no 3 pp 414ndash458 1981

[90] M Lusignoli and M Vignati ldquoRelic antineutrino capture on163Hodecaying nucleirdquoPhysics Letters B vol 697 no 1 pp 11ndash142011

[91] A Faessler L Gastaldo and F Simkovic ldquoElectron capture in163Ho overlap plus exchange corrections and neutrino massrdquoJournal of Physics G Nuclear and Particle Physics vol 42 no 1Article ID 015108 2015

[92] G Audi F G Kondev M Wang et al ldquoThe Nubase2012evaluation of nuclear propertiesrdquo Chinese Physics C vol 36 no12 pp 1157ndash1286 2012

[93] S Mertens ldquoSensitivity of next-generation tritium beta-decayexperiments for keV-scale sterile neutrinosrdquo JCAP vol 1502 no2 p 20 2015

Submit your manuscripts athttpwwwhindawicom

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ThermodynamicsJournal of

4 Advances in High Energy Physics

An alternative capture process to the one in (6) involvesthe instantaneous formation of a second hole 1198671015840 due to themismatch between the spectator electron wave functions inthe parent and in the daughter atom Whereas the hole 119867 isleft by the captured electron and therefore fulfils the energyand angular momentum conditions for such process theextra hole 1198671015840 has a different origin namely the abovementioned incomplete overlap of electron wave functionswhichmay occur inmany electron shellsThe electron leavingthe extra hole may have been ldquoshaken uprdquo (excited) to anunoccupied orbital in the daughter atom

(119883) 997888rarr (119884)1198671198671015840 + ]119894 (9)

or it may have been ldquoshaken offrdquo to the continuum (ejected)

(119883) 997888rarr (119884+)1198671198671015840 + 119890minus + ]119894 (10)

In a shake-up process the deexcitation of the final atomcontributes to the calorimeter energywith a peak at an energyapproximately equal to the sum of the binding energies of the119867 and 1198671015840 electrons 1198641198671198671015840 asymp |119861(119884)119867 | + |119861(119883)1198671015840 | minus |119861(119883)out | Thussatellite peaks show up located after the single-hole peak at119864119867 In the case of a shake-off the calorimeter energy is thecombination of 1198641198671198671015840 and the electron kinetic energy thelatter showing a continuous distribution as corresponds to thethree-body decay of (10)

It is important to notice that some electron emissionprocesses of the type

(119883) 997888rarr (119884+)1198671015840 11986710158401015840 + 119890minus + ]119894 (11)

where none of the holes in the daughter atom correspondto the captured electron 119867 are quantum mechanicallyindistinguishable [81 82] from the three-step process in (6)and (8) and therefore they contribute to the same one-hole peaks at 119864119867 The contributions to the spectrum of thetwo-hole excitations that are actually distinct from the one-hole excitations ((9) and (10)) have been a subject of recentstudies devoted to search for the electron neutrino mass andare therefore focused on the endpoint of the spectrum (seeeg [80ndash83]) For heavy neutrino searches a similar analysisshould be performed at other regions of the energy spectrumnot just at the endpoint In this paper we restrict thecalculations to the one-hole states

The density of final states of the emitted neutrino inelectron capture is proportional to

120588 = 120588] (119864]) prop 119901]119864] = (1198642] minus 1198982])12 119864] (12)

where 119864] 119901] and 119898] are the neutrino total energy momen-tum and mass respectively On the other hand the proba-bility of capture of a bound electron follows a Breit-Wignerdistribution of width Γ119909 and peaks at the energy 119864119909

119875 (119864) = Γ1199092120587(119864 minus 119864119909)2 + Γ21199094 (13)

Using Fermirsquos Golden Rule the differential reaction ratewith respect to the neutrino energy yields

119889120582119889119864] = 119870EC (1198642] minus 1198982])12 119864]sum119867

119882(])119867sdot Γ1198672120587[119864] minus (119876 minus 119864119867)]2 + Γ21198674

(14)

where 119870EC contains among other factors the weak interac-tion coupling constant and the nuclear matrix element Thefactor119882(])119867 is the squared leptonic matrix element for a givenhole state119867

We write 119882(])119867 = 119862119867119878(])119867 to have a general expressionfor allowed and forbidden transitions The factor 119862119867 is thesquared amplitude of the bound-state electron radial wavefunction at the nuclear interior containing also the squaredoverlap between the initial and the final atom orbital wavefunctions and the effect of electron exchange (as defined inAppendix F-2 of [56]) The cases discussed in the nextsections involve allowed Gamow-Teller transitions (163Ho (72minus)rarr 163Dy (52minus)) and first forbiddenGamow-Teller transi-tions (202Pb (0+) rarr 202Tl (2minus) and 205Pb (52minus) rarr 205Tl (12+)) In the first case the factor 119878(])119867 = 1 for all 119867 states Inthe second case the factor 119878(])119867 = 1 for electrons captured withorbital angularmomentum 119897 = 1 (and neutrinos emitted with119897 = 0) and 119878(])119867 = 1199012] for electrons captured with orbitalangular momentum 119897 = 0 (and neutrinos emitted with 119897 =1) The reason for this is that due to conservation of totalangularmomentum first forbidden transitions have 119871 = 1 forthe leptonic pair Hence there is a linear momentum depen-dence in the leptonic matrix element corresponding to thelepton that carries the angular momentum in each specificcaptureThe corresponding 1199012119890 factor for the 119897 = 1 electrons isembedded in the 119862119867 factor [56]

Equation (14) includes every possible orbital electroncapture the probability of each peaking at 119864] = 119876 minus 119864119867 asdictated by energy conservation where 119864119867 is the excitationenergy of the final atom due to the electron hole119867 resultingfrom capture The energy collected by a calorimeter from theatomic deexcitations is 119864119888 = 119876 minus 119864] namely all the availableenergy except for the one carried away by the neutrino Thedifferential rate can be written in terms of the calorimeterenergy as

119889120582119889119864119888 = 119870EC [(119876 minus 119864119888)2 minus 1198982]]12 (119876 minus 119864119888)sum119867

119862119867119878(])119867sdot Γ1198672120587(119864119888 minus 119864119867)2 + Γ21198674

(15)

In the next section we shall also consider integrated ratesover particular energy ranges which can be calculatednumerically from (15) A convenient compact analyticalexpression of the integrated rate can be obtained from (15)by replacing the Breit-Wigner distributions by Dirac deltas

Advances in High Energy Physics 5

With this approximation the integral of the differential ratecan be written as

120582 = 119870ECsum119867

119862119867119878(])119867 [(119876 minus 119864119867)2 minus 1198982]]12 (119876 minus 119864119867) (16)

In this theoretical model we have neglected two-holepeaks deexcitations through virtual intermediate states andinterferences between deexcitation channels The theoreticalcalorimeter spectrum is thus a single-hole approximationthat assumes full collection of de-excitation energy by thecalorimeter and no pile-up

4 Sterile Neutrino Effect in the Spectrum

Due to the smallness of the mixing angle it is useful toconsider ratios between rates that emphasize the contributionof the heavy neutrino In [84ndash86] for the case of beta decay aratio was considered between the contributions of the heavyand the light neutrino to the differential rates as we shall seein Section 6 For the case of electron capture the differentialdecay rates contain many peaks and it is more convenient toconsider integrated decay rates over specific energy rangesas discussed in [87] In either case (beta decay and electroncapture) this amounts to consider that the detector collectsevents within energy ranges 119864119894 plusmn Δ and 119864119895 plusmn Δ in two regionsof the spectrum such that the mass of the hypothetical heavyneutrino lies in between namely 119864119894 + Δ lt 119898ℎ lt 119864119895 minus Δ with119864119895 lt 119876minusΔ A ratio 119877 between the number of collected eventsin both regions is then performed which corresponds to thetheoretical expression

119877 = Λ 119894Λ 119895 =120581119897119894 + tan2120577120581ℎ119894120581119897119895 (17)

where Λ 119894 = cos2120577120581119897119894 + sin2120577120581ℎ119894 (in the region where both 119897and ℎ mass eigenstates contribute) and Λ 119895 = cos2120577120581119897119894 (in theregion where ℎ is not energetically allowed) The integrals 120581are defined as

120581]119894119895 = int119864119894119895+Δ119864119894119895minusΔ

119889120582119889119864119888 (119898]) 119889119864119888 (18)

In electron capture the energies 119894 and 119895 can be selected asthe energies of two peaks and the integration intervals canbe chosen as the width of the capture peaks If the peaks areapproximated by delta functions the integrals (using119864119867 = 119864119894and Δ rarr 0) can be written as

120581]EC119894119903 = 119870EC119862119894119903119878(])119894119903 (119876119903 minus 119864119894)2 [1 minus ( 119898]119876119903 minus 119864119894)2]12 (19)

where 119876119903 is the atomic mass difference in a given isotope 119903119864119894 is the energy position of a given peak 119894 in the calorimeterspectrum 119898] is 119898119897 asymp 0 the mass of a light neutrino or119898ℎ the mass of a heavy neutrino and 119878(])119894119903 contains theneutrino momentum dependence for the peak 119894 comingfrom the leptonic matrix element squared As explained inthe previous section for allowed transitions 119878(])119894119903 = 1 For

first forbidden transitions some of the capture peaks have119878(])119894119903 = 1199012]119894119903 for the 119894 peaks corresponding to 11990412 shells (aswell as for those corresponding to 11990112 shells that contributethrough their admixtures with the 119897 = 0 orbital due torelativistic corrections) The other peaks have 119878(])119894119903 = 1where 119894 corresponds to 11990132 shells (and 11988932 shells throughtheir admixtures with the 119897 = 1 orbital due to relativisticcorrections)

For electron capture it is convenient to define a ratiosimilar to the one in (17) but where the numerator andthe denominator are themselves ratios of numbers of eventswithin the same peak but for different isotopes of the sameelement 119903 and 119904 so that some atomic corrections cancel out[87]

1198771015840 = Λ 119894119903Λ 119894119904Λ 119895119903Λ 119895119904 = (119877119897119894119895119903119904)2(120574+1)(1 + 1205962120574+1119894119903 tan2120577

1 + 1205962120574+1119894119904 tan2120577) (20)

where one should notice that the factors 119862119894 and 119862119895 cancel outin addition to the factor119870EC which cancels out in both ratios(119877 and 1198771015840) The factors in (20) have very simple analyticalforms when one uses (19)

119877119897119894119895119903119904 = (119876119903 minus 119864119894) (119876119904 minus 119864119895)(119876119903 minus 119864119895) (119876119904 minus 119864119894)

120596119894119903(119904) = [1 minus ( 119898ℎ119876119903(119904) minus 119864119894)2]12

(21)

where 120574 depends on the angular momenta of each lepton in agivenΔ119869120587 nuclear transition For instance for allowed decays120574 = 0 whereas for first forbidden decays 120574 = 1 for 11990412 and11990112 peaks or 120574 = 0 for 11990132 and 11988932 peaks In the expressionsabove we have assumed that both peaks 119894 and 119895 are of thesame type namely 120574119894 = 120574119895 = 120574

By measuring the ratio in (20) the mixing angle can thenbe obtained as

120577 = arctan[[

(119877119897119894119895119903119904)2(120574+1) minus 1198771015840exp1198771015840exp1205962120574+1119894119904 minus (119877119897119894119895119903119904)2(120574+1) 1205962120574+1119894119903

]]12

(22)

where the theoretical values of 119877119897119894119895119903119904 120596119894119903 and 120596119894119904 requirean accurate experimental knowledge of the position of theselected peaks and more importantly of the atomic massdifferences 119876119903 and 119876119904 In addition one has to consider fixedinitially unknown values of the mass of the heavy neutrino119898ℎ that is searched for or otherwise be content with thegeneration of exclusion plots in case of no effect observation

In summary we consider here two types of ratios that canbe useful in the search for a signal of keV sterile neutrinos inelectron capture (1) the ratio between the intensity of twopeaks in the electron capture spectrum of a given nucleusas for the case of 163Ho discussed in Section 5 (2) The casein which two isotopes of a given element undergo electroncapture where one can use the ratio defined in (20) as isthe case of Lead isotopes discussed in Section 5 For the

6 Advances in High Energy Physics

10minus5

Neutrino phase spaceAtomic deexcitationTotal

M1M2

N1

N2O1

O2

0 05 1 15 2 25 3Ec (keV)

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

(a)

m = 25 keVm = 2 keVm = 15 keV

m = 1 keVm = 05 keVm = 0

0 05 1 15 2 25 3Ec (keV)

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

10minus5

(b)

0 05 1 15 2 25 3

No mixingMax mixing

Ec (keV)

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

(c)

Figure 1 Calorimeter spectrum after electron capture in 163Ho (a) Full spectrum (solid curve) and contributions emitted neutrino phasespace (dotted curve) and daughter atom deexcitation (dashed curve) (b) Full spectrum for different neutrino masses 0 05 1 15 2 and25 keV (c) full spectrum for light (119898119897 asymp 0 keV) - heavy (119898ℎ = 2 keV) neutrino mixing using a maximal mixing angle 120577 = 45∘ for illustrationwith (solid curve) and without (dotted curve) the heavy neutrino contribution For comparison the spectrumwithout heavy neutrinomixing(dashed curve) is also shownWe use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of the figure Realistic valuesof 120577 lt 001∘ result in a reduction of the difference between the dashed and the solid lines in (c) by more than seven orders of magnitude

beta decay case in Section 6 we consider the ratio betweenthe differential rates but the ratios between integrated ratesare also useful and may be more realistic for comparison toexperiments

5 Results for Electron Capture Spectra

Let us first consider the capture of an atomic electron by thenucleus Holmium 163 (163Ho 119885 = 67 119873 = 96) to turn intoDysprosium 163 (163Dy 119885 = 66119873 = 97) with ground statespins and parities 119869120587 = 72minus and 119869120587 = 52minus respectively Itis an allowed unique Gamow-Teller transition (Δ119869 = 1 andno parity change the leptons carry no orbital angularmomentum Δ119871 = 0 and couple to spin 119878 = 1) with atomicmass difference 119876 = 2833 plusmn 0030 plusmn 0015 keV [88] which isbeing currently measured at the Electron Capture Holmium(ECHo) experiment [50ndash52] Only electrons with principalquantum number 119899 larger than 2 have a binding energylower than the 119876-value and can therefore be captured Inaddition being an allowed decay only electrons from 11990412 and

11990112 orbitals can be captured the latter through admixtureswith the 119897 = 0 orbital due to relativistic corrections Thuscapture from the orbitals M1 M2 N1 N2 O1 O2 and P1 (inspectroscopic notation) is possible [80 89ndash91]

In Figure 1 we show the differential electron capture ratein 163Ho as a function of the calorimeter energy 119864119888 = 119876minus119864]In Table 1 we give theoretical values of the strengths andwidths of each capture peak and experimental values ofthe electron binding energy again for each capture peakThe strengths 119862119867 are obtained in [56 57] from relativisticHartree-Fock calculations of the electron wave functions atthe nuclear site accounting for exchange and overlap contri-butions In the upper plot the emitted neutrino phase spacecontribution and the daughter atom deexcitation contribu-tion are plotted separately together with the product of bothwhich is the full differential electron capture rate In themiddle plot we show the full differential rate for differentmasses of the emitted neutrino 119898] asymp 0 (realistic) and 119898] asymp05 1 15 2 and 25 keV (unrealistic) As can be seen in theplot each curve ends at 119876minus119898] Finally the lower plot showsthe spectrum resulting from the mixing of a light neutrino

Advances in High Energy Physics 7

Table 1 Atomic parameters for electron capture in Holmium from shells M1 to O2The values of 119862119867 [56 57] correspond to the atomic shellsin Holmium while the orbital electron widths Γ119867 [58] and binding energies 119861119867 (ltQ) [59ndash61] correspond to the atomic shells in Dysprosium

M1 M2 N1 N2 O1 O2119862119867 005377 0002605 001373 00005891 0001708 00001Γ119867 [keV] 0013 0006 0006 0005 0005 0002119861119867 [keV] 2047 1842 0416 0332 0063 0026

Table 2 Same as Table 1 (data from the same references) but for electron capture in Lead going toThalliumThe upper table summarizes thevalues of the relevant parameters for 11990412 and 11990112 atomic shells while the lower table shows the values for 11990132 and 11988932 atomic shells In thelatter case 119862119867 includes an extra factor 1199012119890

L1 L2 M1 M2 N1 N2 O1 O2119862119867 08781 006647 02125 001759 005849 0004431 001016 00008052Γ119867 [keV] 0011 0006 0015 0010 0009 0007 mdash mdash119864119867 [keV] 15346 14697 3703 3415 0845 0720 0136 0099L3 M3 M4 N3 N4 O3 O4119862119867 [keV2] 18978193 5366014 45383 1363307 12607 237280 1500Γ119867 [keV] 0006 0009 0002 0006 0004 0001 0001119861119867 [keV] 12657 2956 2484 0608 0406 0072 0015

mass eigenstate 119898119897 asymp 0 with a heavy mass eigenstate 119898ℎ asymp2 keV (solid curve) as in (5) Although the spectrum withmixing is realistic in the sense that it is the result expected if aheavy mass eigenstate exists it is unrealistic in the degree ofmixing shown in the figure which has been maximized herefor the sake of visibility of the ldquokinkrdquo in the scale of the figure50 light neutrino and 50 heavy neutrino correspondingto amixing angle 120577 = 45∘The ldquokinkrdquo can be seen in this curveat 119864119888 = 119876 minus 119898ℎ = 0833 keV Above this calorimeter energythe heavy mass eigenstate cannot be produced due to energyconservation For comparison the spectrum for119898] asymp 0 withno mixing with heavy states is shown in the dashed curve

Another example of electron capture under study here isthat of the Lead isotopes 202 (202Pb 119885 = 82 119873 = 120)and 205 (205Pb 119885 = 82 119873 = 123) going to the Thalliumisotopes 202 (202Tl 119885 = 81 119873 = 121) and 205 (205Tl119885 = 81 119873 = 124) respectively where the process is in bothcases a first forbidden Gamow-Teller unique transition with0+ rarr 2minus for 202Pb and 52minus rarr 12+ for 205Pb (Δ119869 = 2 andparity change the leptons carry Δ119871 = 1 and couple to 119878 = 1)The atomic mass differences obtained from the data in [92]are 119876 = 46 plusmn 14 keV and 119876 = 506 plusmn 18 keV respectivelyAccording to the explanations given in Section 3 capturesfrom orbitals L1 L2 M1 M2 N1 N2 O1 and O2 containan extra neutrino momentum dependence 119878(]) = 1199012] thatmodifies the calorimeter spectrum and corresponds to 120574 =1 On the other hand captures from orbitals L3 M3 M4N3 N4 O3 and O4 which correspond to 120574 = 0 containan extra electron momentum dependence 1199012119890 which beingfixed in bound electrons is included in the values of 119862119867They are given in Table 2 together with the widths and theexperimental electron binding energies in the daughter atomThallium

Figure 2 shows the capture differential rate in 205Pb (darkcurves) and in 202Pb (light curves) Position width and

strength of the capture peaks are assumed to be the same inboth isotopes but the spectra are different because of the dif-ferent 119876-values Solid curves are for light neutrino emission119898] asymp 0 and dashed lines are for heavy neutrino emissionwith119898] = 40 keV (results are given separately for each massbeforemixing) Due to the large119876-values electron capture inLead isotopes allows us to explore this mass value althoughit may be somewhat larger than the expected value fromcosmological reasons The analysis of the ratio in (20) canbe computed for example using the capture peaks L3 at119864119888 = 12657 keV and M3 at 119864119888 = 2956 keV both beingof the 120574 = 0 type For 119898ℎ = 40 keV one would obtain119877l1198723 1198713 202 205 = 1037 1205961198723202 = 0369 1205961198723205 = 0543These results should be introduced in (22) together with theexperimental ratio 1198771015840exp to obtain the value of the mixingangle 120577

It is important to remark that the theoretical quantities 119877119897and 120596 computed as described above contain several approxi-mations They correspond to Dirac-delta peaks ignoring theactual shapes and widths and thus they just refer to onesingle capture peak neglecting the possible effect of the tailsof nearby one-hole peaks Moreover the influence of two-hole peaks close to the main ones has also been neglectedOther effects that have not been taken into account butwhose influence is expected to be very small are multihole(more than two) peaks virtual intermediate states (influenceof transitions through atomic shells not energetically acces-sible) or interference between atomic transitions resultingfrom addition of amplitudes instead of intensities [81 82]Theaccurate experimental determination of the ratio 1198771015840exp entailsits own difficulties among them the possible existence ofmetastable atomic states whose delayed de-excitations failto contribute to the collected spectrum and the variety ofchemical environments resulting in a complex mixture of 119876-values

8 Advances in High Energy Physics

L1

L2

L3

M1M2

M3

M4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

6 162 8 10 12 144 18 200Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(a)

N1

N2

N3

O1

O2

O3

N4O4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

05 0603 0401 02 07 08 09 1000Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(b)

Figure 2 Calorimeter spectrum after electron capture in 202Pb (light curves) and in 205Pb (black curves) with emission of a light neutrino119898119897 = 0 (solid curves) and a heavy neutrino119898ℎ = 40 keV (dashed curves) (a) Full spectrum showing L and M capture peak labels (b) Lowenergy region (119864119888 = 0-1 keV) showing N and O capture peak labels

6 Results for Beta Decay Spectra

In addition to electron capture we also show here the effectof heavy neutrino emission in the electron spectrum of betadecays for one of the cases of most interest Tritium It hasbeen studied in depth in previous works [84ndash86 93] togetherwith another process that has also drawn considerable theo-retical end experimental attention the beta decay of Rhenium187 [37 39ndash41 84ndash86] The beta decay of Tritium (3H 119885 = 1119873 = 2) going to Helium 3 (3He 119885 = 2119873 = 1) is

3H 997888rarr 3He + 119890minus + ]119890 (23)

and has a 119876-value of 1859 keV [92] Both the initial and thefinal nuclear ground states have spin-parity 12+ and thetransition is allowed with the electron and the antineutrinoemitted in 119904-wave The Karlsruhe Tritium Neutrino Experi-ment (KATRIN) [42ndash46] is currently studying this decay inorder to determine the active neutrino mass and could alsostudy the production of a heavy neutrino of amass lower than18 keV

The differential decay rate with respect to the electronenergy is given by

119889120582119889119864119890 = 119870120573 (1198642119890 minus 1198982119890)12 [(119876 + 119898119890 minus 119864119890)2 minus 1198982]]12

sdot 119864119890 (119876 + 119898119890 minus 119864119890) (24)

where 119870120573 contains among others the weak interactioncoupling the nuclearmatrix element and the Fermi function

As an illustration of the heavy neutrino effect in the Tri-tium beta decay we plot in Figure 3 the differential decay ratefrom (24) and (5) using a maximal mixing with an unrealisticvalue of the mixing angle 120577 = 45∘ to show the effect more

distinctly in the scale of the figure We have used heavy masscomponents with 119898ℎ = 2 keV [26] (Figure 3(a)) and with119898ℎ = 7 keV [29 30] (Figure 3(b)) For comparison thespectra without the heavy neutrino contribution (120582ℎ = 0) andwithout mixing (120577 = 0∘) are also shown in this plot The kinkin the spectrum at 119864119890 minus 119898119890 = 119876 minus 119898ℎ can be observed if theexperimental relative error is lower than the size of the stepthe latter being very small in realistic situations

The effect of a heavy neutrino emission can be analyzedthrough the ratio between the heavy and the light neutrinocontributions to the spectrum

R equiv 119889120582ℎ119889119864119890119889120582119897119889119864119890 tan2120577 (25)

The full differential decay rate is also related to the ratio Rthrough

119889120582119889119864119890 =119889120582119897119889119864119890 [1 +R] cos2120577 (26)

In Figure 4 we plot R the ratio of the heavy neutrinocontribution over the light neutrino contribution to the decayrate as a function of the momentum of the emitted electronfor a heavy neutrino mass 119898ℎ = 2 keV and different mixingangles 120577 = 001∘ 0005∘ and 0001∘ The size of this ratio(lt10minus7) gives an idea of the difficulty of finding the kink in thespectrum due to the production of the heavymass eigenstateAs can be seen in the figure the ratio is different from zeroand almost constant in the range 0 le 119901119890 lt (119901119890)max wherethe maximum electron momentum is given by (119901119890)max =[(119876 minus 119898ℎ)(119876 minus 119898ℎ + 2119898119890)]12 For the heavy mass used inthe figure (119901119890)max ≃ 1313 keV The ratio R decreases as themixing angle decreases being approximately proportional to

Advances in High Energy Physics 9

Mixing but no heavy neutrino contribution

104

105

106

107

108d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 2 keV

(a)

104

105

106

107

108

d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

Mixing but no heavy neutrino contribution

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 7 keV

(b)

Figure 3 Electron spectrum of Tritium beta decay for a light-heavy neutrino mixing angle 120577 = 45∘ shown just for illustration and a heavyneutrino mass (solid curve)119898ℎ = 2 keV (a) and119898ℎ = 7 keV (b) The spectrum without heavy neutrino mass contribution (dotted curve) andwithout mixing (dashed curve) are also shown We use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of thefigure Realistic values of 120577 lt 001∘ result in a reduction of the difference between the dotted and the solid lines by more than seven orders ofmagnitude

10minus11

10minus10

10minus9

10minus8

10minus7

10minus12

120577 = 001∘

120577 = 0005∘

120577 = 0001∘

15090 100 110 120 130 1408070pe (keV)

ℛ

Figure 4 Ratio R defined in (25) for the Tritium beta decay as afunction of the electronmomentum for a heavy neutrinomass119898ℎ =2 keV and different values of the mixing angle 120577 = 001∘ (solid line)0005∘ (dashed line) and 0001∘ (dotted line)

1205772 It also decreases for larger heavy neutrino masses 119898ℎ Adifferent ratio between the spectra with mixing and without

mixing (Rlowast) has also been considered in [84ndash86 93] relatedwithR in (25) as

Rlowast = minussin2120577 +R cos2120577 = 119889120582119889119864119890119889120582119897119889119864119890 minus 1 (27)

7 Conclusions

Signatures of hypothetical keV sterile neutrinos which couldbe warm dark matter candidates (WDM) may be found inthe spectra of ordinary weak nuclear decays The electronspectrum in beta decay is cleaner for this purpose becauseit is a smooth curve while the deexcitation spectrum afterelectron capture shows many peaks On the other hand inbeta decay the possible excitation of the atoms or moleculesis not disentangled while in electron capture the calorimetercollects all the energy independently of the excitation ofthe atom and of the deexcitation path and one has higherstatistics around the capture peaks

Figures of electron spectrum for beta decay in 3H aswell as of calorimeter spectrum for electron capture in 163Hoand 202Pb and 205Pb are given considering various valuesof the heavy neutrino mass (2 keV 7 keV and 40 keV) thatmay be experimentally probed In both weak processes thesmall value of the light-heavy neutrino mixing angle requiresextremely high experimental precision and the use of sourceswith large stability to reduce systematic errors This is why it

10 Advances in High Energy Physics

is useful to consider relevant ratios between transition rates asfirst introduced in [84ndash87] to analyze the data We considertwo cases the case of a single isotope where one may use theratio of accumulated number of events in different regionsof the spectrum which allows us to remove uncertaintiesrelated to the nuclear matrix element and to the valuesof overall constants And in the case of two isotopes weconsider ratios of the above mentioned ratios that allow usto reduce uncertainties from atomic parameters We give aswell analytical expressions that can be used to obtain a goodtheoretical approximation to the experimental ratios ((19) to(21))

Both the experimental measurement and the theoreticalmodel must be accurate enough to detect differences in theexpected versus themeasured ratios of the order of themixingangle squared 1205772 ≲ 10minus8 This is also the size of the kinkexpected in the electron spectrumof beta decay located at thelimit of the regionwhere the production of a heavy neutrino isenergetically allowed In order to identify this kink among thestatistical fluctuations of themeasured spectrum the numberof collected events must be larger than the inverse of thesquared ratioR in (25)

Our results particularly on the electron capture spectraof Lead isotopes that include for the first time all thepossible peaks may help in designing and analyzing futureexperiments to search for sterile neutrinos in the keV massrangeThe use of the different types of ratios between numberof events discussed here for both electron capture and betadecay processes may also help in planning the experimentsby establishing the threshold of statistical and systematicuncertainties required for the detection of sterile neutrinosof given mass and mixing or to properly extract exclusionplots

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

E Moya de Guerra and O Moreno acknowledge MINECOfor partial financial support (FIS2014-51971-P) and IEM-CSIC for hospitality O Moreno acknowledges supportfrom a Marie Curie International Outgoing Fellowshipwithin the European Union Seventh Framework Pro-gramme under Grant Agreement PIOF-GA-2011-298364(ELECTROWEAK) M R M thanks N G Sanchez forcorrespondence and conversations

References

[1] F Zwicky ldquoDieRotverschiebung von extragalaktischenNebelnrdquoHelvetica Physica Acta vol 6 pp 110ndash127 1933

[2] M Roos ldquoAstrophysical and cosmological probes of darkmatterrdquo Journal of Modern Physics vol 3 pp 1152ndash1171 2012

[3] S Dodelson Modern Cosmology Academic Press New YorkNY USA 2003

[4] S Weinberg Cosmology Oxford University Press Oxford UK2008

[5] J Lesgourgues and S Pastor ldquoNeutrino mass from cosmologyrdquoAdvances in High Energy Physics vol 2012 Article ID 60851534 pages 2012

[6] D Boyanovsky H J de Vega and N G Sanchez ldquoConstraintson dark matter particles from theory galaxy observations andN-body simulationsrdquo Physical Review D vol 77 no 4 ArticleID 043518 2008

[7] D Boyanovsky H J de Vega and N G Sanchez ldquoDarkmatter transfer function free streaming particle statistics andmemory of gravitational clusteringrdquo Physical Review D vol 78no 6 Article ID 063546 2008

[8] C Destri H J de Vega and N G Sanchez ldquoWarm dark matterprimordial spectra and the onset of structure formation atredshift zrdquo Physical Review D vol 88 no 8 Article ID 0835122013

[9] H J De Vega P Salucci and N G Sanchez ldquoThe mass ofthe dark matter particle theory and galaxy observationsrdquo NewAstronomy vol 17 no 7 pp 653ndash666 2012

[10] H J de Vega P Salucci and N G Sanchez ldquoObservationalrotation curves and density profiles versus the Thomas-Fermigalaxy structure theoryrdquoMonthlyNotices of the Royal Astronom-ical Society vol 442 no 3 pp 2717ndash2727 2014

[11] H J de Vega and N G Sanchez ldquoModel-independent analysisof dark matter points to a particle mass at the keV scalerdquoMonthly Notices of the Royal Astronomical Society vol 404 no2 pp 885ndash894 2010

[12] H J de Vega and N G Sanchez ldquoConstant surface gravity anddensity profile of dark matterrdquo International Journal of ModernPhysics A vol 26 no 6 pp 1057ndash1072 2011

[13] H J de Vega and N G Sanchez ldquoCosmological evolutionof warm dark matter fluctuations I Efficient computationalframework with Volterra integral equationsrdquo Physical ReviewDvol 85 no 4 Article ID 043516 20 pages 2012

[14] H J de Vega and N G Sanchez ldquoThe dark matter distributionfunction and halo thermalization from the Eddington equationin galaxiesrdquo International Journal of Modern Physics A vol 31no 13 Article ID 1650073 2016

[15] NMenci S Paduroiu P Salucci N Sanchez and C RWatsonldquolectures at the 20th Paris Cosmology Colloquium Chalonge-Hector de Vegardquo 2016 httpchalongeobspmfr

[16] E W Kolb and M S Turner The Early Universe AddisonWesley Boston Mass USA 1990

[17] J F Navarro C S Frenk and S D M White ldquoA universal den-sity profile from hierarchical clusteringrdquo Astrophysical JournalLetters vol 490 no 2 pp 493ndash508 1997

[18] W J G de Blok ldquoThe core-cusp problemrdquo Advances in Astron-omy vol 2010 Article ID 789293 14 pages 2010

[19] G Gilmore M I Wilkinson R F GWyse et al ldquoThe observedproperties of dark matter on small spatial scalesrdquo AstrophysicalJournal vol 663 no 2 pp 948ndash959 2007

[20] P Salucci and Ch Frigerio Martins ldquoThe mass distribution inSpiral galaxiesrdquo EAS Publications Series vol 36 pp 133ndash1402009

[21] J van Eymeren C Trachternach B S Koribalski and R-JDettmar ldquoNon-circular motions and the cusp-core discrepancyin dwarf galaxiesrdquo Astronomy amp Astrophysics vol 505 no 1 pp1ndash20 2009

[22] M Walker and J Penarrubia ldquoA method for measuring (slopesof) the mass profiles of dwarf spheroidal galaxiesrdquo The Astro-physical Journal vol 742 no 1 p 20 2011

Advances in High Energy Physics 11

[23] R F G Wyse and G Gilmore ldquoObserved properties of darkmatter on small spatial scalesrdquo in Proceedings of the Proceedingsof the International Astronomical Union Symposium (IUA rsquo07)vol 3 no 244 pp 44ndash52 Cardiff UK June 2007

[24] A Burkert ldquoThe structure of dark matter halos in dwarfgalaxiesrdquo Astrophysical Journal vol 447 no 1 pp L25ndashL281995

[25] C Destri H J de Vega and N G Sanchez ldquoQuantum WDMfermions and gravitation determine the observed galaxy struc-turesrdquo Astroparticle Physics vol 46 pp 14ndash22 2013

[26] N Menci N G Sanchez M Castellano and A GrazianldquoConstraining the warm dark matter particle mass throughultra-deep UV luminosity functions at Z = 2rdquoTheAstrophysicalJournal vol 818 no 1 article 90 2016

[27] E Bulbul M Markevitch A Foster R K Smith M Loewen-stein and SW Randall ldquoDetection of an unidentified emissionline in the stackedX-ray spectrumof galaxy clustersrdquoAstrophys-ical Journal vol 789 article 13 2014

[28] A Boyarsky O Ruchayskiy D Iakubovskyi and J FranseldquoUnidentified line inX-ray spectra of the andromeda galaxy andperseus galaxy clusterrdquo Physical Review Letters vol 113 no 25Article ID 251301 2014

[29] K N Abazajian ldquoResonantly produced 7 keV sterile neutrinodark matter models and the properties of Milky Way satellitesrdquoPhysical Review Letters vol 112 no 16 Article ID 161303 5pages 2014

[30] N Menci A Grazian M Castellano and N G SanchezldquoShock connectivity in the 2010 August and 2012 July solarenergetic particle events inferred from observations and ENLILmodelingrdquo Astrophysical Journal Letters vol 825 no 1 article 12016

[31] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[32] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 481 no 1-2 pp 1ndash28 2009

[33] B Pontecorvo ldquoNeutrino experiments and the problem ofconservation of leptonic chargerdquo Soviet Physics-JETP vol 26no 5 pp 984ndash988 1968

[34] A Merle ldquokeV neutrino model buildingrdquo International Journalof Modern Physics D vol 22 no 10 Article ID 1330020 2013

[35] J DVergados andT S Kosmas ldquoSearching for cold darkmatterA case of coexistence of supersymmetry and nuclear physicsrdquoPhysics of Atomic Nuclei vol 61 p 1066 1998

[36] E Holmlund M Kortelainen T S Kosmas J Suhonen and JToivanen ldquoMicroscopic calculation of the LSP detection ratesfor the 71Ga 73Ge and 127I dark-matter detectorsrdquoPhysics LettersB vol 584 no 1-2 pp 31ndash39 2004

[37] R Adhikari M Agostini N Anh Ky et al ldquoA white paper onkeV sterile neutrino dark matterrdquo httpsarxivorgabs160204816

[38] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[39] MARE collaboration httpmaredfmuninsubriaitfrontendexecphp

[40] A Nucciotti ldquoNeutrino mass calorimetric searches in theMARE experimentrdquo httpsarxivorgabs10122290

[41] A Nucciotti ldquoOn behalf of the MARE collaboration lecture attheWorkshop ChalongeMeudonrdquo 2011 httpchalongeobspmfr

[42] KATRIN Collaboration httpwwwkatrinkitedu

[43] C Weinheimer ldquoDirect determination of neutrino mass fromtritium beta spectrumrdquo httpsarxivorgabs09121619

[44] C Weinheimer lecture at the 20th Paris Cosmology Chalonge-Hector de Vega Colloquium 2016 httpchalongeobspmfr

[45] G Drexlin lecture at the 19th Paris Cosmology Colloquium2015

[46] A Huber lecture at the Chalonge-de Vega Meudon Workshop2016 httpchalongeobspmfr

[47] C Tully lecture at the 19th Paris Cosmology Colloquium 2015[48] S Betts W R Blanchard R H Carnevale et al ldquoDevelop-

ment of a relic neutrino detection experiment at PTOLEMYprinceton tritiumobservatory for light early-universemassive-neutrino yieldrdquo httpsarxivorgabs13074738

[49] D M Asner R F Bradley L de Viveiros et al ldquoSingle-electrondetection and spectroscopy via relativistic cyclotron radiationrdquoPhysical Review Letters vol 114 no 16 Article ID 162501 2015

[50] P C-O Ranitzsch J-P Porst S Kempf et al ldquoDevelopmentof metallic magnetic calorimeters for high precision measure-ments of calorimetric 187Re and 163Ho spectrardquo Journal of LowTemperature Physics vol 167 no 5-6 pp 1004ndash1014 2012

[51] C Hassel ldquoLecture at the International School of AstrophysicsDaniel ChalongemdashHector de Vegardquo 2015

[52] L Gastaldo K Blaum A Doerr et al ldquoThe electron capture163Ho experiment ECHordquo Journal of Low Temperature Physicsvol 176 no 5 pp 876ndash884 2014

[53] B Alpert M Balata D Bennett et al ldquoThe electron capturedecay of 163Ho tomeasure the electron neutrino mass with sub-eV sensitivityrdquo The European Physical Journal C vol 75 article112 2015

[54] E Fermi ldquoRadioattivita prodotta da bombardamento di neu-tronirdquo Il Nuovo Cimento vol 11 no 7 pp 429ndash441 1934

[55] E Fermi ldquoVersuch einer theorie der 120573-strahlen Irdquo Zeitschriftfur Physik vol 88 no 3 pp 161ndash177 1934

[56] R B Firestone and V S Shirley Eds Table of Isotopes JohnWiley amp Sons New York NY USA 8th edition 1999

[57] W Bambynek H Behrens M H Chen et al ldquoOrbital electroncapture by the nucleusrdquo Reviews of Modern Physics vol 49 no1 pp 77ndash221 1977

[58] J L Campbell and T Papp ldquoWidths of the atomic KndashN7 levelsrdquoAtomic Data and Nuclear Data Tables vol 77 no 1 pp 1ndash562001

[59] J A Bearden and A F Burr ldquoReevaluation of X-ray atomicenergy levelsrdquo Reviews of Modern Physics vol 39 no 1 pp 125ndash142 1967

[60] M Cardona and L Ley Eds Photoemission in Solids I GeneralPrinciples Springer Berlin Germany 1978

[61] J C Fuggle and N Martensson ldquoCore-level binding energies inmetalsrdquo Journal of Electron Spectroscopy and Related Phenom-ena vol 21 no 3 pp 275ndash281 1980

[62] T Kosmas H Ejiri and A Hatzikoutelis ldquoNeutrino physics inthe frontiers of intensities and very high sensitivitiesrdquo Advancesin High Energy Physics vol 2015 Article ID 806067 3 pages2015

[63] A Giuliani and A Poves ldquoNeutrinoless double-beta decayrdquoAdvances in High Energy Physics vol 2012 Article ID 85701638 pages 2012

[64] J-U Nabi and H V Klapdor-Kleingrothaus ldquoWeak interactionrates of sd-shell nuclei in stellar environments calculated in theproton-neutron quasiparticle random-phase approximationrdquoAtomic Data andNuclear Data Tables vol 71 no 2 pp 149ndash3451999

12 Advances in High Energy Physics

[65] J-U Nabi and H V Klapdor-Kleingrothaus ldquoMicroscopic cal-culations of stellar weak interaction rates and energy losses forfp- and fpg-shell nucleirdquo Atomic Data and Nuclear Data Tablesvol 88 no 2 pp 237ndash476 2004

[66] Y Fukuda T Hayakawa E Ichihara et al ldquoEvidence foroscillation of atmospheric neutrinosrdquo Physical Review Lettersvol 81 no 8 pp 1562ndash1567 1998

[67] Q R Ahmad and SNOCollaboration ldquoMeasurement of the rateof V119890+119889 rarr 119901+119901+119890minus interactions produced by 8B solar neutrinosat the Sudbury Neutrino Observatoryrdquo Physical Review Lettersvol 87 no 7 Article ID 071301 6 pages 2001

[68] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 2002

[69] G J Feldman J Hartnell and T Kobayashi ldquoLong-baselineneutrino oscillation experimentsrdquo Advances in High EnergyPhysics vol 2013 Article ID 475749 30 pages 2013

[70] S T Petcov ldquoThenature ofmassive neutrinosrdquoAdvances inHighEnergy Physics vol 2013 Article ID 852987 20 pages 2013

[71] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[72] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 48 no 1-2 pp 1ndash28 2009

[73] F Munyaneza and P L Biermann ldquoDegenerate sterile neutrinodark matter in the cores of galaxiesrdquo Astronomy amp Astrophysicsvol 458 no 2 pp L9ndashL12 2006

[74] D Boyanovsky and C M Ho ldquoSterile neutrino production viaactive-sterile oscillations the quantum Zeno effectrdquo Journal ofHigh Energy Physics vol 2007 no 7 p 30 2007

[75] R Shrock ldquoNew tests for and bounds on neutrino masses andlepton mixingrdquo Physics Letters B vol 96 no 1-2 pp 159ndash1641980

[76] R E Shrock ldquoGeneral theory of weak processes involving neu-trinos I Leptonic pseudoscalar-meson decays with associatedtests for and bounds on neutrino masses and lepton mixingrdquoPhysical Review D vol 24 no 5 pp 1232ndash1274 1981

[77] R E Shrock ldquoImplications of neutrino masses and mixing forweak processesrdquo in Proceedings of the Weak Interactions asProbes of Unification vol 72 of AIP Conference Proceedings p368 Blacksburg Va USA 1981

[78] J M Conrad C M Ignarra G Karagiorgi M H Shaevitzand J Spitz ldquoSterile neutrino fits to short-baseline neutrinooscillation measurementsrdquo Advances in High Energy Physicsvol 2013 Article ID 163897 2013

[79] A Nucciotti ldquoThe use of low temperature detectors for directmeasurements of themass of the electron neutrinordquoAdvances inHigh Energy Physics vol 2016 Article ID 9153024 41 pages2016

[80] R G H Robertson ldquoExamination of the calorimetric spectrumto determine the neutrino mass in low-energy electron capturedecayrdquo Physical Review C vol 91 no 3 Article ID 035504 2015

[81] A De Rujula ldquoTwo old ways to measure the electron-neutrinomassrdquo httpsarxivorgabs13054857

[82] A De Rujula ldquoAn upper bound on the exceptional character-istics for Lusztigrsquos character formulardquo httpsarxivorgabs08111674v2

[83] A Faessler and F Simkovic ldquoCan electron capture tell us themass of the neutrinordquo Physica Scripta vol 91 no 4 Article ID043007 2016

[84] H J de Vega O Moreno E Moya de Guerra M RamonMedrano and N G Sanchez ldquoRole of sterile neutrino warmdark matter in rhenium and tritium beta decaysrdquo NuclearPhysics B vol 866 no 2 pp 177ndash195 2013

[85] E M de Guerra O Moreno P Sarriguren and M RamonMedrano ldquoTopics on nuclear structure with electroweakprobesrdquo Journal of Physics Conference Series vol 366 ArticleID 012011 2012

[86] J M Boillos O Moreno and E Moya de ldquoActive and sterileneutrino mass effects on beta decay spectrardquo in Proceedingsof the La Rabida International Scientific Meeting on NuclearPhysics Basic Concepts in Nuclear Physics Theory Experimentsand Applications vol 1541 ofAIP Conference Proceedings p 167La Rabida Spain September 2012

[87] P E Filianin K Blaum S A Eliseev et al ldquoOn the keVsterile neutrino search in electron capturerdquo Journal of PhysicsG Nuclear and Particle Physics vol 41 no 9 Article ID 0950042014

[88] S Eliseev K BlaumM Block et al ldquoDirect measurement of themass difference of 163Ho and 163Dy solves theQ-value puzzle forthe neutrino mass determinationrdquo Physical Review Letters vol115 no 6 Article ID 062501 2015

[89] A De Rujula ldquoA new way tomeasure neutrinomassesrdquoNuclearPhysics Section B vol 188 no 3 pp 414ndash458 1981

[90] M Lusignoli and M Vignati ldquoRelic antineutrino capture on163Hodecaying nucleirdquoPhysics Letters B vol 697 no 1 pp 11ndash142011

[91] A Faessler L Gastaldo and F Simkovic ldquoElectron capture in163Ho overlap plus exchange corrections and neutrino massrdquoJournal of Physics G Nuclear and Particle Physics vol 42 no 1Article ID 015108 2015

[92] G Audi F G Kondev M Wang et al ldquoThe Nubase2012evaluation of nuclear propertiesrdquo Chinese Physics C vol 36 no12 pp 1157ndash1286 2012

[93] S Mertens ldquoSensitivity of next-generation tritium beta-decayexperiments for keV-scale sterile neutrinosrdquo JCAP vol 1502 no2 p 20 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

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Superconductivity

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Statistical MechanicsInternational Journal of

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AstrophysicsJournal of

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Solid State PhysicsJournal of

 Computational  Methods in Physics

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Soft MatterJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 5

With this approximation the integral of the differential ratecan be written as

120582 = 119870ECsum119867

119862119867119878(])119867 [(119876 minus 119864119867)2 minus 1198982]]12 (119876 minus 119864119867) (16)

In this theoretical model we have neglected two-holepeaks deexcitations through virtual intermediate states andinterferences between deexcitation channels The theoreticalcalorimeter spectrum is thus a single-hole approximationthat assumes full collection of de-excitation energy by thecalorimeter and no pile-up

4 Sterile Neutrino Effect in the Spectrum

Due to the smallness of the mixing angle it is useful toconsider ratios between rates that emphasize the contributionof the heavy neutrino In [84ndash86] for the case of beta decay aratio was considered between the contributions of the heavyand the light neutrino to the differential rates as we shall seein Section 6 For the case of electron capture the differentialdecay rates contain many peaks and it is more convenient toconsider integrated decay rates over specific energy rangesas discussed in [87] In either case (beta decay and electroncapture) this amounts to consider that the detector collectsevents within energy ranges 119864119894 plusmn Δ and 119864119895 plusmn Δ in two regionsof the spectrum such that the mass of the hypothetical heavyneutrino lies in between namely 119864119894 + Δ lt 119898ℎ lt 119864119895 minus Δ with119864119895 lt 119876minusΔ A ratio 119877 between the number of collected eventsin both regions is then performed which corresponds to thetheoretical expression

119877 = Λ 119894Λ 119895 =120581119897119894 + tan2120577120581ℎ119894120581119897119895 (17)

where Λ 119894 = cos2120577120581119897119894 + sin2120577120581ℎ119894 (in the region where both 119897and ℎ mass eigenstates contribute) and Λ 119895 = cos2120577120581119897119894 (in theregion where ℎ is not energetically allowed) The integrals 120581are defined as

120581]119894119895 = int119864119894119895+Δ119864119894119895minusΔ

119889120582119889119864119888 (119898]) 119889119864119888 (18)

In electron capture the energies 119894 and 119895 can be selected asthe energies of two peaks and the integration intervals canbe chosen as the width of the capture peaks If the peaks areapproximated by delta functions the integrals (using119864119867 = 119864119894and Δ rarr 0) can be written as

120581]EC119894119903 = 119870EC119862119894119903119878(])119894119903 (119876119903 minus 119864119894)2 [1 minus ( 119898]119876119903 minus 119864119894)2]12 (19)

where 119876119903 is the atomic mass difference in a given isotope 119903119864119894 is the energy position of a given peak 119894 in the calorimeterspectrum 119898] is 119898119897 asymp 0 the mass of a light neutrino or119898ℎ the mass of a heavy neutrino and 119878(])119894119903 contains theneutrino momentum dependence for the peak 119894 comingfrom the leptonic matrix element squared As explained inthe previous section for allowed transitions 119878(])119894119903 = 1 For

first forbidden transitions some of the capture peaks have119878(])119894119903 = 1199012]119894119903 for the 119894 peaks corresponding to 11990412 shells (aswell as for those corresponding to 11990112 shells that contributethrough their admixtures with the 119897 = 0 orbital due torelativistic corrections) The other peaks have 119878(])119894119903 = 1where 119894 corresponds to 11990132 shells (and 11988932 shells throughtheir admixtures with the 119897 = 1 orbital due to relativisticcorrections)

For electron capture it is convenient to define a ratiosimilar to the one in (17) but where the numerator andthe denominator are themselves ratios of numbers of eventswithin the same peak but for different isotopes of the sameelement 119903 and 119904 so that some atomic corrections cancel out[87]

1198771015840 = Λ 119894119903Λ 119894119904Λ 119895119903Λ 119895119904 = (119877119897119894119895119903119904)2(120574+1)(1 + 1205962120574+1119894119903 tan2120577

1 + 1205962120574+1119894119904 tan2120577) (20)

where one should notice that the factors 119862119894 and 119862119895 cancel outin addition to the factor119870EC which cancels out in both ratios(119877 and 1198771015840) The factors in (20) have very simple analyticalforms when one uses (19)

119877119897119894119895119903119904 = (119876119903 minus 119864119894) (119876119904 minus 119864119895)(119876119903 minus 119864119895) (119876119904 minus 119864119894)

120596119894119903(119904) = [1 minus ( 119898ℎ119876119903(119904) minus 119864119894)2]12

(21)

where 120574 depends on the angular momenta of each lepton in agivenΔ119869120587 nuclear transition For instance for allowed decays120574 = 0 whereas for first forbidden decays 120574 = 1 for 11990412 and11990112 peaks or 120574 = 0 for 11990132 and 11988932 peaks In the expressionsabove we have assumed that both peaks 119894 and 119895 are of thesame type namely 120574119894 = 120574119895 = 120574

By measuring the ratio in (20) the mixing angle can thenbe obtained as

120577 = arctan[[

(119877119897119894119895119903119904)2(120574+1) minus 1198771015840exp1198771015840exp1205962120574+1119894119904 minus (119877119897119894119895119903119904)2(120574+1) 1205962120574+1119894119903

]]12

(22)

where the theoretical values of 119877119897119894119895119903119904 120596119894119903 and 120596119894119904 requirean accurate experimental knowledge of the position of theselected peaks and more importantly of the atomic massdifferences 119876119903 and 119876119904 In addition one has to consider fixedinitially unknown values of the mass of the heavy neutrino119898ℎ that is searched for or otherwise be content with thegeneration of exclusion plots in case of no effect observation

In summary we consider here two types of ratios that canbe useful in the search for a signal of keV sterile neutrinos inelectron capture (1) the ratio between the intensity of twopeaks in the electron capture spectrum of a given nucleusas for the case of 163Ho discussed in Section 5 (2) The casein which two isotopes of a given element undergo electroncapture where one can use the ratio defined in (20) as isthe case of Lead isotopes discussed in Section 5 For the

6 Advances in High Energy Physics

10minus5

Neutrino phase spaceAtomic deexcitationTotal

M1M2

N1

N2O1

O2

0 05 1 15 2 25 3Ec (keV)

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

(a)

m = 25 keVm = 2 keVm = 15 keV

m = 1 keVm = 05 keVm = 0

0 05 1 15 2 25 3Ec (keV)

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

10minus5

(b)

0 05 1 15 2 25 3

No mixingMax mixing

Ec (keV)

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

(c)

Figure 1 Calorimeter spectrum after electron capture in 163Ho (a) Full spectrum (solid curve) and contributions emitted neutrino phasespace (dotted curve) and daughter atom deexcitation (dashed curve) (b) Full spectrum for different neutrino masses 0 05 1 15 2 and25 keV (c) full spectrum for light (119898119897 asymp 0 keV) - heavy (119898ℎ = 2 keV) neutrino mixing using a maximal mixing angle 120577 = 45∘ for illustrationwith (solid curve) and without (dotted curve) the heavy neutrino contribution For comparison the spectrumwithout heavy neutrinomixing(dashed curve) is also shownWe use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of the figure Realistic valuesof 120577 lt 001∘ result in a reduction of the difference between the dashed and the solid lines in (c) by more than seven orders of magnitude

beta decay case in Section 6 we consider the ratio betweenthe differential rates but the ratios between integrated ratesare also useful and may be more realistic for comparison toexperiments

5 Results for Electron Capture Spectra

Let us first consider the capture of an atomic electron by thenucleus Holmium 163 (163Ho 119885 = 67 119873 = 96) to turn intoDysprosium 163 (163Dy 119885 = 66119873 = 97) with ground statespins and parities 119869120587 = 72minus and 119869120587 = 52minus respectively Itis an allowed unique Gamow-Teller transition (Δ119869 = 1 andno parity change the leptons carry no orbital angularmomentum Δ119871 = 0 and couple to spin 119878 = 1) with atomicmass difference 119876 = 2833 plusmn 0030 plusmn 0015 keV [88] which isbeing currently measured at the Electron Capture Holmium(ECHo) experiment [50ndash52] Only electrons with principalquantum number 119899 larger than 2 have a binding energylower than the 119876-value and can therefore be captured Inaddition being an allowed decay only electrons from 11990412 and

11990112 orbitals can be captured the latter through admixtureswith the 119897 = 0 orbital due to relativistic corrections Thuscapture from the orbitals M1 M2 N1 N2 O1 O2 and P1 (inspectroscopic notation) is possible [80 89ndash91]

In Figure 1 we show the differential electron capture ratein 163Ho as a function of the calorimeter energy 119864119888 = 119876minus119864]In Table 1 we give theoretical values of the strengths andwidths of each capture peak and experimental values ofthe electron binding energy again for each capture peakThe strengths 119862119867 are obtained in [56 57] from relativisticHartree-Fock calculations of the electron wave functions atthe nuclear site accounting for exchange and overlap contri-butions In the upper plot the emitted neutrino phase spacecontribution and the daughter atom deexcitation contribu-tion are plotted separately together with the product of bothwhich is the full differential electron capture rate In themiddle plot we show the full differential rate for differentmasses of the emitted neutrino 119898] asymp 0 (realistic) and 119898] asymp05 1 15 2 and 25 keV (unrealistic) As can be seen in theplot each curve ends at 119876minus119898] Finally the lower plot showsthe spectrum resulting from the mixing of a light neutrino

Advances in High Energy Physics 7

Table 1 Atomic parameters for electron capture in Holmium from shells M1 to O2The values of 119862119867 [56 57] correspond to the atomic shellsin Holmium while the orbital electron widths Γ119867 [58] and binding energies 119861119867 (ltQ) [59ndash61] correspond to the atomic shells in Dysprosium

M1 M2 N1 N2 O1 O2119862119867 005377 0002605 001373 00005891 0001708 00001Γ119867 [keV] 0013 0006 0006 0005 0005 0002119861119867 [keV] 2047 1842 0416 0332 0063 0026

Table 2 Same as Table 1 (data from the same references) but for electron capture in Lead going toThalliumThe upper table summarizes thevalues of the relevant parameters for 11990412 and 11990112 atomic shells while the lower table shows the values for 11990132 and 11988932 atomic shells In thelatter case 119862119867 includes an extra factor 1199012119890

L1 L2 M1 M2 N1 N2 O1 O2119862119867 08781 006647 02125 001759 005849 0004431 001016 00008052Γ119867 [keV] 0011 0006 0015 0010 0009 0007 mdash mdash119864119867 [keV] 15346 14697 3703 3415 0845 0720 0136 0099L3 M3 M4 N3 N4 O3 O4119862119867 [keV2] 18978193 5366014 45383 1363307 12607 237280 1500Γ119867 [keV] 0006 0009 0002 0006 0004 0001 0001119861119867 [keV] 12657 2956 2484 0608 0406 0072 0015

mass eigenstate 119898119897 asymp 0 with a heavy mass eigenstate 119898ℎ asymp2 keV (solid curve) as in (5) Although the spectrum withmixing is realistic in the sense that it is the result expected if aheavy mass eigenstate exists it is unrealistic in the degree ofmixing shown in the figure which has been maximized herefor the sake of visibility of the ldquokinkrdquo in the scale of the figure50 light neutrino and 50 heavy neutrino correspondingto amixing angle 120577 = 45∘The ldquokinkrdquo can be seen in this curveat 119864119888 = 119876 minus 119898ℎ = 0833 keV Above this calorimeter energythe heavy mass eigenstate cannot be produced due to energyconservation For comparison the spectrum for119898] asymp 0 withno mixing with heavy states is shown in the dashed curve

Another example of electron capture under study here isthat of the Lead isotopes 202 (202Pb 119885 = 82 119873 = 120)and 205 (205Pb 119885 = 82 119873 = 123) going to the Thalliumisotopes 202 (202Tl 119885 = 81 119873 = 121) and 205 (205Tl119885 = 81 119873 = 124) respectively where the process is in bothcases a first forbidden Gamow-Teller unique transition with0+ rarr 2minus for 202Pb and 52minus rarr 12+ for 205Pb (Δ119869 = 2 andparity change the leptons carry Δ119871 = 1 and couple to 119878 = 1)The atomic mass differences obtained from the data in [92]are 119876 = 46 plusmn 14 keV and 119876 = 506 plusmn 18 keV respectivelyAccording to the explanations given in Section 3 capturesfrom orbitals L1 L2 M1 M2 N1 N2 O1 and O2 containan extra neutrino momentum dependence 119878(]) = 1199012] thatmodifies the calorimeter spectrum and corresponds to 120574 =1 On the other hand captures from orbitals L3 M3 M4N3 N4 O3 and O4 which correspond to 120574 = 0 containan extra electron momentum dependence 1199012119890 which beingfixed in bound electrons is included in the values of 119862119867They are given in Table 2 together with the widths and theexperimental electron binding energies in the daughter atomThallium

Figure 2 shows the capture differential rate in 205Pb (darkcurves) and in 202Pb (light curves) Position width and

strength of the capture peaks are assumed to be the same inboth isotopes but the spectra are different because of the dif-ferent 119876-values Solid curves are for light neutrino emission119898] asymp 0 and dashed lines are for heavy neutrino emissionwith119898] = 40 keV (results are given separately for each massbeforemixing) Due to the large119876-values electron capture inLead isotopes allows us to explore this mass value althoughit may be somewhat larger than the expected value fromcosmological reasons The analysis of the ratio in (20) canbe computed for example using the capture peaks L3 at119864119888 = 12657 keV and M3 at 119864119888 = 2956 keV both beingof the 120574 = 0 type For 119898ℎ = 40 keV one would obtain119877l1198723 1198713 202 205 = 1037 1205961198723202 = 0369 1205961198723205 = 0543These results should be introduced in (22) together with theexperimental ratio 1198771015840exp to obtain the value of the mixingangle 120577

It is important to remark that the theoretical quantities 119877119897and 120596 computed as described above contain several approxi-mations They correspond to Dirac-delta peaks ignoring theactual shapes and widths and thus they just refer to onesingle capture peak neglecting the possible effect of the tailsof nearby one-hole peaks Moreover the influence of two-hole peaks close to the main ones has also been neglectedOther effects that have not been taken into account butwhose influence is expected to be very small are multihole(more than two) peaks virtual intermediate states (influenceof transitions through atomic shells not energetically acces-sible) or interference between atomic transitions resultingfrom addition of amplitudes instead of intensities [81 82]Theaccurate experimental determination of the ratio 1198771015840exp entailsits own difficulties among them the possible existence ofmetastable atomic states whose delayed de-excitations failto contribute to the collected spectrum and the variety ofchemical environments resulting in a complex mixture of 119876-values

8 Advances in High Energy Physics

L1

L2

L3

M1M2

M3

M4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

6 162 8 10 12 144 18 200Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(a)

N1

N2

N3

O1

O2

O3

N4O4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

05 0603 0401 02 07 08 09 1000Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(b)

Figure 2 Calorimeter spectrum after electron capture in 202Pb (light curves) and in 205Pb (black curves) with emission of a light neutrino119898119897 = 0 (solid curves) and a heavy neutrino119898ℎ = 40 keV (dashed curves) (a) Full spectrum showing L and M capture peak labels (b) Lowenergy region (119864119888 = 0-1 keV) showing N and O capture peak labels

6 Results for Beta Decay Spectra

In addition to electron capture we also show here the effectof heavy neutrino emission in the electron spectrum of betadecays for one of the cases of most interest Tritium It hasbeen studied in depth in previous works [84ndash86 93] togetherwith another process that has also drawn considerable theo-retical end experimental attention the beta decay of Rhenium187 [37 39ndash41 84ndash86] The beta decay of Tritium (3H 119885 = 1119873 = 2) going to Helium 3 (3He 119885 = 2119873 = 1) is

3H 997888rarr 3He + 119890minus + ]119890 (23)

and has a 119876-value of 1859 keV [92] Both the initial and thefinal nuclear ground states have spin-parity 12+ and thetransition is allowed with the electron and the antineutrinoemitted in 119904-wave The Karlsruhe Tritium Neutrino Experi-ment (KATRIN) [42ndash46] is currently studying this decay inorder to determine the active neutrino mass and could alsostudy the production of a heavy neutrino of amass lower than18 keV

The differential decay rate with respect to the electronenergy is given by

119889120582119889119864119890 = 119870120573 (1198642119890 minus 1198982119890)12 [(119876 + 119898119890 minus 119864119890)2 minus 1198982]]12

sdot 119864119890 (119876 + 119898119890 minus 119864119890) (24)

where 119870120573 contains among others the weak interactioncoupling the nuclearmatrix element and the Fermi function

As an illustration of the heavy neutrino effect in the Tri-tium beta decay we plot in Figure 3 the differential decay ratefrom (24) and (5) using a maximal mixing with an unrealisticvalue of the mixing angle 120577 = 45∘ to show the effect more

distinctly in the scale of the figure We have used heavy masscomponents with 119898ℎ = 2 keV [26] (Figure 3(a)) and with119898ℎ = 7 keV [29 30] (Figure 3(b)) For comparison thespectra without the heavy neutrino contribution (120582ℎ = 0) andwithout mixing (120577 = 0∘) are also shown in this plot The kinkin the spectrum at 119864119890 minus 119898119890 = 119876 minus 119898ℎ can be observed if theexperimental relative error is lower than the size of the stepthe latter being very small in realistic situations

The effect of a heavy neutrino emission can be analyzedthrough the ratio between the heavy and the light neutrinocontributions to the spectrum

R equiv 119889120582ℎ119889119864119890119889120582119897119889119864119890 tan2120577 (25)

The full differential decay rate is also related to the ratio Rthrough

119889120582119889119864119890 =119889120582119897119889119864119890 [1 +R] cos2120577 (26)

In Figure 4 we plot R the ratio of the heavy neutrinocontribution over the light neutrino contribution to the decayrate as a function of the momentum of the emitted electronfor a heavy neutrino mass 119898ℎ = 2 keV and different mixingangles 120577 = 001∘ 0005∘ and 0001∘ The size of this ratio(lt10minus7) gives an idea of the difficulty of finding the kink in thespectrum due to the production of the heavymass eigenstateAs can be seen in the figure the ratio is different from zeroand almost constant in the range 0 le 119901119890 lt (119901119890)max wherethe maximum electron momentum is given by (119901119890)max =[(119876 minus 119898ℎ)(119876 minus 119898ℎ + 2119898119890)]12 For the heavy mass used inthe figure (119901119890)max ≃ 1313 keV The ratio R decreases as themixing angle decreases being approximately proportional to

Advances in High Energy Physics 9

Mixing but no heavy neutrino contribution

104

105

106

107

108d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 2 keV

(a)

104

105

106

107

108

d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

Mixing but no heavy neutrino contribution

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 7 keV

(b)

Figure 3 Electron spectrum of Tritium beta decay for a light-heavy neutrino mixing angle 120577 = 45∘ shown just for illustration and a heavyneutrino mass (solid curve)119898ℎ = 2 keV (a) and119898ℎ = 7 keV (b) The spectrum without heavy neutrino mass contribution (dotted curve) andwithout mixing (dashed curve) are also shown We use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of thefigure Realistic values of 120577 lt 001∘ result in a reduction of the difference between the dotted and the solid lines by more than seven orders ofmagnitude

10minus11

10minus10

10minus9

10minus8

10minus7

10minus12

120577 = 001∘

120577 = 0005∘

120577 = 0001∘

15090 100 110 120 130 1408070pe (keV)

ℛ

Figure 4 Ratio R defined in (25) for the Tritium beta decay as afunction of the electronmomentum for a heavy neutrinomass119898ℎ =2 keV and different values of the mixing angle 120577 = 001∘ (solid line)0005∘ (dashed line) and 0001∘ (dotted line)

1205772 It also decreases for larger heavy neutrino masses 119898ℎ Adifferent ratio between the spectra with mixing and without

mixing (Rlowast) has also been considered in [84ndash86 93] relatedwithR in (25) as

Rlowast = minussin2120577 +R cos2120577 = 119889120582119889119864119890119889120582119897119889119864119890 minus 1 (27)

7 Conclusions

Signatures of hypothetical keV sterile neutrinos which couldbe warm dark matter candidates (WDM) may be found inthe spectra of ordinary weak nuclear decays The electronspectrum in beta decay is cleaner for this purpose becauseit is a smooth curve while the deexcitation spectrum afterelectron capture shows many peaks On the other hand inbeta decay the possible excitation of the atoms or moleculesis not disentangled while in electron capture the calorimetercollects all the energy independently of the excitation ofthe atom and of the deexcitation path and one has higherstatistics around the capture peaks

Figures of electron spectrum for beta decay in 3H aswell as of calorimeter spectrum for electron capture in 163Hoand 202Pb and 205Pb are given considering various valuesof the heavy neutrino mass (2 keV 7 keV and 40 keV) thatmay be experimentally probed In both weak processes thesmall value of the light-heavy neutrino mixing angle requiresextremely high experimental precision and the use of sourceswith large stability to reduce systematic errors This is why it

10 Advances in High Energy Physics

is useful to consider relevant ratios between transition rates asfirst introduced in [84ndash87] to analyze the data We considertwo cases the case of a single isotope where one may use theratio of accumulated number of events in different regionsof the spectrum which allows us to remove uncertaintiesrelated to the nuclear matrix element and to the valuesof overall constants And in the case of two isotopes weconsider ratios of the above mentioned ratios that allow usto reduce uncertainties from atomic parameters We give aswell analytical expressions that can be used to obtain a goodtheoretical approximation to the experimental ratios ((19) to(21))

Both the experimental measurement and the theoreticalmodel must be accurate enough to detect differences in theexpected versus themeasured ratios of the order of themixingangle squared 1205772 ≲ 10minus8 This is also the size of the kinkexpected in the electron spectrumof beta decay located at thelimit of the regionwhere the production of a heavy neutrino isenergetically allowed In order to identify this kink among thestatistical fluctuations of themeasured spectrum the numberof collected events must be larger than the inverse of thesquared ratioR in (25)

Our results particularly on the electron capture spectraof Lead isotopes that include for the first time all thepossible peaks may help in designing and analyzing futureexperiments to search for sterile neutrinos in the keV massrangeThe use of the different types of ratios between numberof events discussed here for both electron capture and betadecay processes may also help in planning the experimentsby establishing the threshold of statistical and systematicuncertainties required for the detection of sterile neutrinosof given mass and mixing or to properly extract exclusionplots

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

E Moya de Guerra and O Moreno acknowledge MINECOfor partial financial support (FIS2014-51971-P) and IEM-CSIC for hospitality O Moreno acknowledges supportfrom a Marie Curie International Outgoing Fellowshipwithin the European Union Seventh Framework Pro-gramme under Grant Agreement PIOF-GA-2011-298364(ELECTROWEAK) M R M thanks N G Sanchez forcorrespondence and conversations

References

[1] F Zwicky ldquoDieRotverschiebung von extragalaktischenNebelnrdquoHelvetica Physica Acta vol 6 pp 110ndash127 1933

[2] M Roos ldquoAstrophysical and cosmological probes of darkmatterrdquo Journal of Modern Physics vol 3 pp 1152ndash1171 2012

[3] S Dodelson Modern Cosmology Academic Press New YorkNY USA 2003

[4] S Weinberg Cosmology Oxford University Press Oxford UK2008

[5] J Lesgourgues and S Pastor ldquoNeutrino mass from cosmologyrdquoAdvances in High Energy Physics vol 2012 Article ID 60851534 pages 2012

[6] D Boyanovsky H J de Vega and N G Sanchez ldquoConstraintson dark matter particles from theory galaxy observations andN-body simulationsrdquo Physical Review D vol 77 no 4 ArticleID 043518 2008

[7] D Boyanovsky H J de Vega and N G Sanchez ldquoDarkmatter transfer function free streaming particle statistics andmemory of gravitational clusteringrdquo Physical Review D vol 78no 6 Article ID 063546 2008

[8] C Destri H J de Vega and N G Sanchez ldquoWarm dark matterprimordial spectra and the onset of structure formation atredshift zrdquo Physical Review D vol 88 no 8 Article ID 0835122013

[9] H J De Vega P Salucci and N G Sanchez ldquoThe mass ofthe dark matter particle theory and galaxy observationsrdquo NewAstronomy vol 17 no 7 pp 653ndash666 2012

[10] H J de Vega P Salucci and N G Sanchez ldquoObservationalrotation curves and density profiles versus the Thomas-Fermigalaxy structure theoryrdquoMonthlyNotices of the Royal Astronom-ical Society vol 442 no 3 pp 2717ndash2727 2014

[11] H J de Vega and N G Sanchez ldquoModel-independent analysisof dark matter points to a particle mass at the keV scalerdquoMonthly Notices of the Royal Astronomical Society vol 404 no2 pp 885ndash894 2010

[12] H J de Vega and N G Sanchez ldquoConstant surface gravity anddensity profile of dark matterrdquo International Journal of ModernPhysics A vol 26 no 6 pp 1057ndash1072 2011

[13] H J de Vega and N G Sanchez ldquoCosmological evolutionof warm dark matter fluctuations I Efficient computationalframework with Volterra integral equationsrdquo Physical ReviewDvol 85 no 4 Article ID 043516 20 pages 2012

[14] H J de Vega and N G Sanchez ldquoThe dark matter distributionfunction and halo thermalization from the Eddington equationin galaxiesrdquo International Journal of Modern Physics A vol 31no 13 Article ID 1650073 2016

[15] NMenci S Paduroiu P Salucci N Sanchez and C RWatsonldquolectures at the 20th Paris Cosmology Colloquium Chalonge-Hector de Vegardquo 2016 httpchalongeobspmfr

[16] E W Kolb and M S Turner The Early Universe AddisonWesley Boston Mass USA 1990

[17] J F Navarro C S Frenk and S D M White ldquoA universal den-sity profile from hierarchical clusteringrdquo Astrophysical JournalLetters vol 490 no 2 pp 493ndash508 1997

[18] W J G de Blok ldquoThe core-cusp problemrdquo Advances in Astron-omy vol 2010 Article ID 789293 14 pages 2010

[19] G Gilmore M I Wilkinson R F GWyse et al ldquoThe observedproperties of dark matter on small spatial scalesrdquo AstrophysicalJournal vol 663 no 2 pp 948ndash959 2007

[20] P Salucci and Ch Frigerio Martins ldquoThe mass distribution inSpiral galaxiesrdquo EAS Publications Series vol 36 pp 133ndash1402009

[21] J van Eymeren C Trachternach B S Koribalski and R-JDettmar ldquoNon-circular motions and the cusp-core discrepancyin dwarf galaxiesrdquo Astronomy amp Astrophysics vol 505 no 1 pp1ndash20 2009

[22] M Walker and J Penarrubia ldquoA method for measuring (slopesof) the mass profiles of dwarf spheroidal galaxiesrdquo The Astro-physical Journal vol 742 no 1 p 20 2011

Advances in High Energy Physics 11

[23] R F G Wyse and G Gilmore ldquoObserved properties of darkmatter on small spatial scalesrdquo in Proceedings of the Proceedingsof the International Astronomical Union Symposium (IUA rsquo07)vol 3 no 244 pp 44ndash52 Cardiff UK June 2007

[24] A Burkert ldquoThe structure of dark matter halos in dwarfgalaxiesrdquo Astrophysical Journal vol 447 no 1 pp L25ndashL281995

[25] C Destri H J de Vega and N G Sanchez ldquoQuantum WDMfermions and gravitation determine the observed galaxy struc-turesrdquo Astroparticle Physics vol 46 pp 14ndash22 2013

[26] N Menci N G Sanchez M Castellano and A GrazianldquoConstraining the warm dark matter particle mass throughultra-deep UV luminosity functions at Z = 2rdquoTheAstrophysicalJournal vol 818 no 1 article 90 2016

[27] E Bulbul M Markevitch A Foster R K Smith M Loewen-stein and SW Randall ldquoDetection of an unidentified emissionline in the stackedX-ray spectrumof galaxy clustersrdquoAstrophys-ical Journal vol 789 article 13 2014

[28] A Boyarsky O Ruchayskiy D Iakubovskyi and J FranseldquoUnidentified line inX-ray spectra of the andromeda galaxy andperseus galaxy clusterrdquo Physical Review Letters vol 113 no 25Article ID 251301 2014

[29] K N Abazajian ldquoResonantly produced 7 keV sterile neutrinodark matter models and the properties of Milky Way satellitesrdquoPhysical Review Letters vol 112 no 16 Article ID 161303 5pages 2014

[30] N Menci A Grazian M Castellano and N G SanchezldquoShock connectivity in the 2010 August and 2012 July solarenergetic particle events inferred from observations and ENLILmodelingrdquo Astrophysical Journal Letters vol 825 no 1 article 12016

[31] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[32] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 481 no 1-2 pp 1ndash28 2009

[33] B Pontecorvo ldquoNeutrino experiments and the problem ofconservation of leptonic chargerdquo Soviet Physics-JETP vol 26no 5 pp 984ndash988 1968

[34] A Merle ldquokeV neutrino model buildingrdquo International Journalof Modern Physics D vol 22 no 10 Article ID 1330020 2013

[35] J DVergados andT S Kosmas ldquoSearching for cold darkmatterA case of coexistence of supersymmetry and nuclear physicsrdquoPhysics of Atomic Nuclei vol 61 p 1066 1998

[36] E Holmlund M Kortelainen T S Kosmas J Suhonen and JToivanen ldquoMicroscopic calculation of the LSP detection ratesfor the 71Ga 73Ge and 127I dark-matter detectorsrdquoPhysics LettersB vol 584 no 1-2 pp 31ndash39 2004

[37] R Adhikari M Agostini N Anh Ky et al ldquoA white paper onkeV sterile neutrino dark matterrdquo httpsarxivorgabs160204816

[38] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[39] MARE collaboration httpmaredfmuninsubriaitfrontendexecphp

[40] A Nucciotti ldquoNeutrino mass calorimetric searches in theMARE experimentrdquo httpsarxivorgabs10122290

[41] A Nucciotti ldquoOn behalf of the MARE collaboration lecture attheWorkshop ChalongeMeudonrdquo 2011 httpchalongeobspmfr

[42] KATRIN Collaboration httpwwwkatrinkitedu

[43] C Weinheimer ldquoDirect determination of neutrino mass fromtritium beta spectrumrdquo httpsarxivorgabs09121619

[44] C Weinheimer lecture at the 20th Paris Cosmology Chalonge-Hector de Vega Colloquium 2016 httpchalongeobspmfr

[45] G Drexlin lecture at the 19th Paris Cosmology Colloquium2015

[46] A Huber lecture at the Chalonge-de Vega Meudon Workshop2016 httpchalongeobspmfr

[47] C Tully lecture at the 19th Paris Cosmology Colloquium 2015[48] S Betts W R Blanchard R H Carnevale et al ldquoDevelop-

ment of a relic neutrino detection experiment at PTOLEMYprinceton tritiumobservatory for light early-universemassive-neutrino yieldrdquo httpsarxivorgabs13074738

[49] D M Asner R F Bradley L de Viveiros et al ldquoSingle-electrondetection and spectroscopy via relativistic cyclotron radiationrdquoPhysical Review Letters vol 114 no 16 Article ID 162501 2015

[50] P C-O Ranitzsch J-P Porst S Kempf et al ldquoDevelopmentof metallic magnetic calorimeters for high precision measure-ments of calorimetric 187Re and 163Ho spectrardquo Journal of LowTemperature Physics vol 167 no 5-6 pp 1004ndash1014 2012

[51] C Hassel ldquoLecture at the International School of AstrophysicsDaniel ChalongemdashHector de Vegardquo 2015

[52] L Gastaldo K Blaum A Doerr et al ldquoThe electron capture163Ho experiment ECHordquo Journal of Low Temperature Physicsvol 176 no 5 pp 876ndash884 2014

[53] B Alpert M Balata D Bennett et al ldquoThe electron capturedecay of 163Ho tomeasure the electron neutrino mass with sub-eV sensitivityrdquo The European Physical Journal C vol 75 article112 2015

[54] E Fermi ldquoRadioattivita prodotta da bombardamento di neu-tronirdquo Il Nuovo Cimento vol 11 no 7 pp 429ndash441 1934

[55] E Fermi ldquoVersuch einer theorie der 120573-strahlen Irdquo Zeitschriftfur Physik vol 88 no 3 pp 161ndash177 1934

[56] R B Firestone and V S Shirley Eds Table of Isotopes JohnWiley amp Sons New York NY USA 8th edition 1999

[57] W Bambynek H Behrens M H Chen et al ldquoOrbital electroncapture by the nucleusrdquo Reviews of Modern Physics vol 49 no1 pp 77ndash221 1977

[58] J L Campbell and T Papp ldquoWidths of the atomic KndashN7 levelsrdquoAtomic Data and Nuclear Data Tables vol 77 no 1 pp 1ndash562001

[59] J A Bearden and A F Burr ldquoReevaluation of X-ray atomicenergy levelsrdquo Reviews of Modern Physics vol 39 no 1 pp 125ndash142 1967

[60] M Cardona and L Ley Eds Photoemission in Solids I GeneralPrinciples Springer Berlin Germany 1978

[61] J C Fuggle and N Martensson ldquoCore-level binding energies inmetalsrdquo Journal of Electron Spectroscopy and Related Phenom-ena vol 21 no 3 pp 275ndash281 1980

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[64] J-U Nabi and H V Klapdor-Kleingrothaus ldquoWeak interactionrates of sd-shell nuclei in stellar environments calculated in theproton-neutron quasiparticle random-phase approximationrdquoAtomic Data andNuclear Data Tables vol 71 no 2 pp 149ndash3451999

12 Advances in High Energy Physics

[65] J-U Nabi and H V Klapdor-Kleingrothaus ldquoMicroscopic cal-culations of stellar weak interaction rates and energy losses forfp- and fpg-shell nucleirdquo Atomic Data and Nuclear Data Tablesvol 88 no 2 pp 237ndash476 2004

[66] Y Fukuda T Hayakawa E Ichihara et al ldquoEvidence foroscillation of atmospheric neutrinosrdquo Physical Review Lettersvol 81 no 8 pp 1562ndash1567 1998

[67] Q R Ahmad and SNOCollaboration ldquoMeasurement of the rateof V119890+119889 rarr 119901+119901+119890minus interactions produced by 8B solar neutrinosat the Sudbury Neutrino Observatoryrdquo Physical Review Lettersvol 87 no 7 Article ID 071301 6 pages 2001

[68] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 2002

[69] G J Feldman J Hartnell and T Kobayashi ldquoLong-baselineneutrino oscillation experimentsrdquo Advances in High EnergyPhysics vol 2013 Article ID 475749 30 pages 2013

[70] S T Petcov ldquoThenature ofmassive neutrinosrdquoAdvances inHighEnergy Physics vol 2013 Article ID 852987 20 pages 2013

[71] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[72] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 48 no 1-2 pp 1ndash28 2009

[73] F Munyaneza and P L Biermann ldquoDegenerate sterile neutrinodark matter in the cores of galaxiesrdquo Astronomy amp Astrophysicsvol 458 no 2 pp L9ndashL12 2006

[74] D Boyanovsky and C M Ho ldquoSterile neutrino production viaactive-sterile oscillations the quantum Zeno effectrdquo Journal ofHigh Energy Physics vol 2007 no 7 p 30 2007

[75] R Shrock ldquoNew tests for and bounds on neutrino masses andlepton mixingrdquo Physics Letters B vol 96 no 1-2 pp 159ndash1641980

[76] R E Shrock ldquoGeneral theory of weak processes involving neu-trinos I Leptonic pseudoscalar-meson decays with associatedtests for and bounds on neutrino masses and lepton mixingrdquoPhysical Review D vol 24 no 5 pp 1232ndash1274 1981

[77] R E Shrock ldquoImplications of neutrino masses and mixing forweak processesrdquo in Proceedings of the Weak Interactions asProbes of Unification vol 72 of AIP Conference Proceedings p368 Blacksburg Va USA 1981

[78] J M Conrad C M Ignarra G Karagiorgi M H Shaevitzand J Spitz ldquoSterile neutrino fits to short-baseline neutrinooscillation measurementsrdquo Advances in High Energy Physicsvol 2013 Article ID 163897 2013

[79] A Nucciotti ldquoThe use of low temperature detectors for directmeasurements of themass of the electron neutrinordquoAdvances inHigh Energy Physics vol 2016 Article ID 9153024 41 pages2016

[80] R G H Robertson ldquoExamination of the calorimetric spectrumto determine the neutrino mass in low-energy electron capturedecayrdquo Physical Review C vol 91 no 3 Article ID 035504 2015

[81] A De Rujula ldquoTwo old ways to measure the electron-neutrinomassrdquo httpsarxivorgabs13054857

[82] A De Rujula ldquoAn upper bound on the exceptional character-istics for Lusztigrsquos character formulardquo httpsarxivorgabs08111674v2

[83] A Faessler and F Simkovic ldquoCan electron capture tell us themass of the neutrinordquo Physica Scripta vol 91 no 4 Article ID043007 2016

[84] H J de Vega O Moreno E Moya de Guerra M RamonMedrano and N G Sanchez ldquoRole of sterile neutrino warmdark matter in rhenium and tritium beta decaysrdquo NuclearPhysics B vol 866 no 2 pp 177ndash195 2013

[85] E M de Guerra O Moreno P Sarriguren and M RamonMedrano ldquoTopics on nuclear structure with electroweakprobesrdquo Journal of Physics Conference Series vol 366 ArticleID 012011 2012

[86] J M Boillos O Moreno and E Moya de ldquoActive and sterileneutrino mass effects on beta decay spectrardquo in Proceedingsof the La Rabida International Scientific Meeting on NuclearPhysics Basic Concepts in Nuclear Physics Theory Experimentsand Applications vol 1541 ofAIP Conference Proceedings p 167La Rabida Spain September 2012

[87] P E Filianin K Blaum S A Eliseev et al ldquoOn the keVsterile neutrino search in electron capturerdquo Journal of PhysicsG Nuclear and Particle Physics vol 41 no 9 Article ID 0950042014

[88] S Eliseev K BlaumM Block et al ldquoDirect measurement of themass difference of 163Ho and 163Dy solves theQ-value puzzle forthe neutrino mass determinationrdquo Physical Review Letters vol115 no 6 Article ID 062501 2015

[89] A De Rujula ldquoA new way tomeasure neutrinomassesrdquoNuclearPhysics Section B vol 188 no 3 pp 414ndash458 1981

[90] M Lusignoli and M Vignati ldquoRelic antineutrino capture on163Hodecaying nucleirdquoPhysics Letters B vol 697 no 1 pp 11ndash142011

[91] A Faessler L Gastaldo and F Simkovic ldquoElectron capture in163Ho overlap plus exchange corrections and neutrino massrdquoJournal of Physics G Nuclear and Particle Physics vol 42 no 1Article ID 015108 2015

[92] G Audi F G Kondev M Wang et al ldquoThe Nubase2012evaluation of nuclear propertiesrdquo Chinese Physics C vol 36 no12 pp 1157ndash1286 2012

[93] S Mertens ldquoSensitivity of next-generation tritium beta-decayexperiments for keV-scale sterile neutrinosrdquo JCAP vol 1502 no2 p 20 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

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Superconductivity

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Statistical MechanicsInternational Journal of

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AstrophysicsJournal of

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Solid State PhysicsJournal of

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Soft MatterJournal of

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PhotonicsJournal of

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ThermodynamicsJournal of

6 Advances in High Energy Physics

10minus5

Neutrino phase spaceAtomic deexcitationTotal

M1M2

N1

N2O1

O2

0 05 1 15 2 25 3Ec (keV)

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

(a)

m = 25 keVm = 2 keVm = 15 keV

m = 1 keVm = 05 keVm = 0

0 05 1 15 2 25 3Ec (keV)

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

10minus5

(b)

0 05 1 15 2 25 3

No mixingMax mixing

Ec (keV)

10minus5

10minus4

10minus3

10minus2

10minus1

100

101

d120582d

Ec

(au

)

(c)

Figure 1 Calorimeter spectrum after electron capture in 163Ho (a) Full spectrum (solid curve) and contributions emitted neutrino phasespace (dotted curve) and daughter atom deexcitation (dashed curve) (b) Full spectrum for different neutrino masses 0 05 1 15 2 and25 keV (c) full spectrum for light (119898119897 asymp 0 keV) - heavy (119898ℎ = 2 keV) neutrino mixing using a maximal mixing angle 120577 = 45∘ for illustrationwith (solid curve) and without (dotted curve) the heavy neutrino contribution For comparison the spectrumwithout heavy neutrinomixing(dashed curve) is also shownWe use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of the figure Realistic valuesof 120577 lt 001∘ result in a reduction of the difference between the dashed and the solid lines in (c) by more than seven orders of magnitude

beta decay case in Section 6 we consider the ratio betweenthe differential rates but the ratios between integrated ratesare also useful and may be more realistic for comparison toexperiments

5 Results for Electron Capture Spectra

Let us first consider the capture of an atomic electron by thenucleus Holmium 163 (163Ho 119885 = 67 119873 = 96) to turn intoDysprosium 163 (163Dy 119885 = 66119873 = 97) with ground statespins and parities 119869120587 = 72minus and 119869120587 = 52minus respectively Itis an allowed unique Gamow-Teller transition (Δ119869 = 1 andno parity change the leptons carry no orbital angularmomentum Δ119871 = 0 and couple to spin 119878 = 1) with atomicmass difference 119876 = 2833 plusmn 0030 plusmn 0015 keV [88] which isbeing currently measured at the Electron Capture Holmium(ECHo) experiment [50ndash52] Only electrons with principalquantum number 119899 larger than 2 have a binding energylower than the 119876-value and can therefore be captured Inaddition being an allowed decay only electrons from 11990412 and

11990112 orbitals can be captured the latter through admixtureswith the 119897 = 0 orbital due to relativistic corrections Thuscapture from the orbitals M1 M2 N1 N2 O1 O2 and P1 (inspectroscopic notation) is possible [80 89ndash91]

In Figure 1 we show the differential electron capture ratein 163Ho as a function of the calorimeter energy 119864119888 = 119876minus119864]In Table 1 we give theoretical values of the strengths andwidths of each capture peak and experimental values ofthe electron binding energy again for each capture peakThe strengths 119862119867 are obtained in [56 57] from relativisticHartree-Fock calculations of the electron wave functions atthe nuclear site accounting for exchange and overlap contri-butions In the upper plot the emitted neutrino phase spacecontribution and the daughter atom deexcitation contribu-tion are plotted separately together with the product of bothwhich is the full differential electron capture rate In themiddle plot we show the full differential rate for differentmasses of the emitted neutrino 119898] asymp 0 (realistic) and 119898] asymp05 1 15 2 and 25 keV (unrealistic) As can be seen in theplot each curve ends at 119876minus119898] Finally the lower plot showsthe spectrum resulting from the mixing of a light neutrino

Advances in High Energy Physics 7

Table 1 Atomic parameters for electron capture in Holmium from shells M1 to O2The values of 119862119867 [56 57] correspond to the atomic shellsin Holmium while the orbital electron widths Γ119867 [58] and binding energies 119861119867 (ltQ) [59ndash61] correspond to the atomic shells in Dysprosium

M1 M2 N1 N2 O1 O2119862119867 005377 0002605 001373 00005891 0001708 00001Γ119867 [keV] 0013 0006 0006 0005 0005 0002119861119867 [keV] 2047 1842 0416 0332 0063 0026

Table 2 Same as Table 1 (data from the same references) but for electron capture in Lead going toThalliumThe upper table summarizes thevalues of the relevant parameters for 11990412 and 11990112 atomic shells while the lower table shows the values for 11990132 and 11988932 atomic shells In thelatter case 119862119867 includes an extra factor 1199012119890

L1 L2 M1 M2 N1 N2 O1 O2119862119867 08781 006647 02125 001759 005849 0004431 001016 00008052Γ119867 [keV] 0011 0006 0015 0010 0009 0007 mdash mdash119864119867 [keV] 15346 14697 3703 3415 0845 0720 0136 0099L3 M3 M4 N3 N4 O3 O4119862119867 [keV2] 18978193 5366014 45383 1363307 12607 237280 1500Γ119867 [keV] 0006 0009 0002 0006 0004 0001 0001119861119867 [keV] 12657 2956 2484 0608 0406 0072 0015

mass eigenstate 119898119897 asymp 0 with a heavy mass eigenstate 119898ℎ asymp2 keV (solid curve) as in (5) Although the spectrum withmixing is realistic in the sense that it is the result expected if aheavy mass eigenstate exists it is unrealistic in the degree ofmixing shown in the figure which has been maximized herefor the sake of visibility of the ldquokinkrdquo in the scale of the figure50 light neutrino and 50 heavy neutrino correspondingto amixing angle 120577 = 45∘The ldquokinkrdquo can be seen in this curveat 119864119888 = 119876 minus 119898ℎ = 0833 keV Above this calorimeter energythe heavy mass eigenstate cannot be produced due to energyconservation For comparison the spectrum for119898] asymp 0 withno mixing with heavy states is shown in the dashed curve

Another example of electron capture under study here isthat of the Lead isotopes 202 (202Pb 119885 = 82 119873 = 120)and 205 (205Pb 119885 = 82 119873 = 123) going to the Thalliumisotopes 202 (202Tl 119885 = 81 119873 = 121) and 205 (205Tl119885 = 81 119873 = 124) respectively where the process is in bothcases a first forbidden Gamow-Teller unique transition with0+ rarr 2minus for 202Pb and 52minus rarr 12+ for 205Pb (Δ119869 = 2 andparity change the leptons carry Δ119871 = 1 and couple to 119878 = 1)The atomic mass differences obtained from the data in [92]are 119876 = 46 plusmn 14 keV and 119876 = 506 plusmn 18 keV respectivelyAccording to the explanations given in Section 3 capturesfrom orbitals L1 L2 M1 M2 N1 N2 O1 and O2 containan extra neutrino momentum dependence 119878(]) = 1199012] thatmodifies the calorimeter spectrum and corresponds to 120574 =1 On the other hand captures from orbitals L3 M3 M4N3 N4 O3 and O4 which correspond to 120574 = 0 containan extra electron momentum dependence 1199012119890 which beingfixed in bound electrons is included in the values of 119862119867They are given in Table 2 together with the widths and theexperimental electron binding energies in the daughter atomThallium

Figure 2 shows the capture differential rate in 205Pb (darkcurves) and in 202Pb (light curves) Position width and

strength of the capture peaks are assumed to be the same inboth isotopes but the spectra are different because of the dif-ferent 119876-values Solid curves are for light neutrino emission119898] asymp 0 and dashed lines are for heavy neutrino emissionwith119898] = 40 keV (results are given separately for each massbeforemixing) Due to the large119876-values electron capture inLead isotopes allows us to explore this mass value althoughit may be somewhat larger than the expected value fromcosmological reasons The analysis of the ratio in (20) canbe computed for example using the capture peaks L3 at119864119888 = 12657 keV and M3 at 119864119888 = 2956 keV both beingof the 120574 = 0 type For 119898ℎ = 40 keV one would obtain119877l1198723 1198713 202 205 = 1037 1205961198723202 = 0369 1205961198723205 = 0543These results should be introduced in (22) together with theexperimental ratio 1198771015840exp to obtain the value of the mixingangle 120577

It is important to remark that the theoretical quantities 119877119897and 120596 computed as described above contain several approxi-mations They correspond to Dirac-delta peaks ignoring theactual shapes and widths and thus they just refer to onesingle capture peak neglecting the possible effect of the tailsof nearby one-hole peaks Moreover the influence of two-hole peaks close to the main ones has also been neglectedOther effects that have not been taken into account butwhose influence is expected to be very small are multihole(more than two) peaks virtual intermediate states (influenceof transitions through atomic shells not energetically acces-sible) or interference between atomic transitions resultingfrom addition of amplitudes instead of intensities [81 82]Theaccurate experimental determination of the ratio 1198771015840exp entailsits own difficulties among them the possible existence ofmetastable atomic states whose delayed de-excitations failto contribute to the collected spectrum and the variety ofchemical environments resulting in a complex mixture of 119876-values

8 Advances in High Energy Physics

L1

L2

L3

M1M2

M3

M4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

6 162 8 10 12 144 18 200Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(a)

N1

N2

N3

O1

O2

O3

N4O4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

05 0603 0401 02 07 08 09 1000Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(b)

Figure 2 Calorimeter spectrum after electron capture in 202Pb (light curves) and in 205Pb (black curves) with emission of a light neutrino119898119897 = 0 (solid curves) and a heavy neutrino119898ℎ = 40 keV (dashed curves) (a) Full spectrum showing L and M capture peak labels (b) Lowenergy region (119864119888 = 0-1 keV) showing N and O capture peak labels

6 Results for Beta Decay Spectra

In addition to electron capture we also show here the effectof heavy neutrino emission in the electron spectrum of betadecays for one of the cases of most interest Tritium It hasbeen studied in depth in previous works [84ndash86 93] togetherwith another process that has also drawn considerable theo-retical end experimental attention the beta decay of Rhenium187 [37 39ndash41 84ndash86] The beta decay of Tritium (3H 119885 = 1119873 = 2) going to Helium 3 (3He 119885 = 2119873 = 1) is

3H 997888rarr 3He + 119890minus + ]119890 (23)

and has a 119876-value of 1859 keV [92] Both the initial and thefinal nuclear ground states have spin-parity 12+ and thetransition is allowed with the electron and the antineutrinoemitted in 119904-wave The Karlsruhe Tritium Neutrino Experi-ment (KATRIN) [42ndash46] is currently studying this decay inorder to determine the active neutrino mass and could alsostudy the production of a heavy neutrino of amass lower than18 keV

The differential decay rate with respect to the electronenergy is given by

119889120582119889119864119890 = 119870120573 (1198642119890 minus 1198982119890)12 [(119876 + 119898119890 minus 119864119890)2 minus 1198982]]12

sdot 119864119890 (119876 + 119898119890 minus 119864119890) (24)

where 119870120573 contains among others the weak interactioncoupling the nuclearmatrix element and the Fermi function

As an illustration of the heavy neutrino effect in the Tri-tium beta decay we plot in Figure 3 the differential decay ratefrom (24) and (5) using a maximal mixing with an unrealisticvalue of the mixing angle 120577 = 45∘ to show the effect more

distinctly in the scale of the figure We have used heavy masscomponents with 119898ℎ = 2 keV [26] (Figure 3(a)) and with119898ℎ = 7 keV [29 30] (Figure 3(b)) For comparison thespectra without the heavy neutrino contribution (120582ℎ = 0) andwithout mixing (120577 = 0∘) are also shown in this plot The kinkin the spectrum at 119864119890 minus 119898119890 = 119876 minus 119898ℎ can be observed if theexperimental relative error is lower than the size of the stepthe latter being very small in realistic situations

The effect of a heavy neutrino emission can be analyzedthrough the ratio between the heavy and the light neutrinocontributions to the spectrum

R equiv 119889120582ℎ119889119864119890119889120582119897119889119864119890 tan2120577 (25)

The full differential decay rate is also related to the ratio Rthrough

119889120582119889119864119890 =119889120582119897119889119864119890 [1 +R] cos2120577 (26)

In Figure 4 we plot R the ratio of the heavy neutrinocontribution over the light neutrino contribution to the decayrate as a function of the momentum of the emitted electronfor a heavy neutrino mass 119898ℎ = 2 keV and different mixingangles 120577 = 001∘ 0005∘ and 0001∘ The size of this ratio(lt10minus7) gives an idea of the difficulty of finding the kink in thespectrum due to the production of the heavymass eigenstateAs can be seen in the figure the ratio is different from zeroand almost constant in the range 0 le 119901119890 lt (119901119890)max wherethe maximum electron momentum is given by (119901119890)max =[(119876 minus 119898ℎ)(119876 minus 119898ℎ + 2119898119890)]12 For the heavy mass used inthe figure (119901119890)max ≃ 1313 keV The ratio R decreases as themixing angle decreases being approximately proportional to

Advances in High Energy Physics 9

Mixing but no heavy neutrino contribution

104

105

106

107

108d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 2 keV

(a)

104

105

106

107

108

d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

Mixing but no heavy neutrino contribution

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 7 keV

(b)

Figure 3 Electron spectrum of Tritium beta decay for a light-heavy neutrino mixing angle 120577 = 45∘ shown just for illustration and a heavyneutrino mass (solid curve)119898ℎ = 2 keV (a) and119898ℎ = 7 keV (b) The spectrum without heavy neutrino mass contribution (dotted curve) andwithout mixing (dashed curve) are also shown We use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of thefigure Realistic values of 120577 lt 001∘ result in a reduction of the difference between the dotted and the solid lines by more than seven orders ofmagnitude

10minus11

10minus10

10minus9

10minus8

10minus7

10minus12

120577 = 001∘

120577 = 0005∘

120577 = 0001∘

15090 100 110 120 130 1408070pe (keV)

ℛ

Figure 4 Ratio R defined in (25) for the Tritium beta decay as afunction of the electronmomentum for a heavy neutrinomass119898ℎ =2 keV and different values of the mixing angle 120577 = 001∘ (solid line)0005∘ (dashed line) and 0001∘ (dotted line)

1205772 It also decreases for larger heavy neutrino masses 119898ℎ Adifferent ratio between the spectra with mixing and without

mixing (Rlowast) has also been considered in [84ndash86 93] relatedwithR in (25) as

Rlowast = minussin2120577 +R cos2120577 = 119889120582119889119864119890119889120582119897119889119864119890 minus 1 (27)

7 Conclusions

Signatures of hypothetical keV sterile neutrinos which couldbe warm dark matter candidates (WDM) may be found inthe spectra of ordinary weak nuclear decays The electronspectrum in beta decay is cleaner for this purpose becauseit is a smooth curve while the deexcitation spectrum afterelectron capture shows many peaks On the other hand inbeta decay the possible excitation of the atoms or moleculesis not disentangled while in electron capture the calorimetercollects all the energy independently of the excitation ofthe atom and of the deexcitation path and one has higherstatistics around the capture peaks

Figures of electron spectrum for beta decay in 3H aswell as of calorimeter spectrum for electron capture in 163Hoand 202Pb and 205Pb are given considering various valuesof the heavy neutrino mass (2 keV 7 keV and 40 keV) thatmay be experimentally probed In both weak processes thesmall value of the light-heavy neutrino mixing angle requiresextremely high experimental precision and the use of sourceswith large stability to reduce systematic errors This is why it

10 Advances in High Energy Physics

is useful to consider relevant ratios between transition rates asfirst introduced in [84ndash87] to analyze the data We considertwo cases the case of a single isotope where one may use theratio of accumulated number of events in different regionsof the spectrum which allows us to remove uncertaintiesrelated to the nuclear matrix element and to the valuesof overall constants And in the case of two isotopes weconsider ratios of the above mentioned ratios that allow usto reduce uncertainties from atomic parameters We give aswell analytical expressions that can be used to obtain a goodtheoretical approximation to the experimental ratios ((19) to(21))

Both the experimental measurement and the theoreticalmodel must be accurate enough to detect differences in theexpected versus themeasured ratios of the order of themixingangle squared 1205772 ≲ 10minus8 This is also the size of the kinkexpected in the electron spectrumof beta decay located at thelimit of the regionwhere the production of a heavy neutrino isenergetically allowed In order to identify this kink among thestatistical fluctuations of themeasured spectrum the numberof collected events must be larger than the inverse of thesquared ratioR in (25)

Our results particularly on the electron capture spectraof Lead isotopes that include for the first time all thepossible peaks may help in designing and analyzing futureexperiments to search for sterile neutrinos in the keV massrangeThe use of the different types of ratios between numberof events discussed here for both electron capture and betadecay processes may also help in planning the experimentsby establishing the threshold of statistical and systematicuncertainties required for the detection of sterile neutrinosof given mass and mixing or to properly extract exclusionplots

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

E Moya de Guerra and O Moreno acknowledge MINECOfor partial financial support (FIS2014-51971-P) and IEM-CSIC for hospitality O Moreno acknowledges supportfrom a Marie Curie International Outgoing Fellowshipwithin the European Union Seventh Framework Pro-gramme under Grant Agreement PIOF-GA-2011-298364(ELECTROWEAK) M R M thanks N G Sanchez forcorrespondence and conversations

References

[1] F Zwicky ldquoDieRotverschiebung von extragalaktischenNebelnrdquoHelvetica Physica Acta vol 6 pp 110ndash127 1933

[2] M Roos ldquoAstrophysical and cosmological probes of darkmatterrdquo Journal of Modern Physics vol 3 pp 1152ndash1171 2012

[3] S Dodelson Modern Cosmology Academic Press New YorkNY USA 2003

[4] S Weinberg Cosmology Oxford University Press Oxford UK2008

[5] J Lesgourgues and S Pastor ldquoNeutrino mass from cosmologyrdquoAdvances in High Energy Physics vol 2012 Article ID 60851534 pages 2012

[6] D Boyanovsky H J de Vega and N G Sanchez ldquoConstraintson dark matter particles from theory galaxy observations andN-body simulationsrdquo Physical Review D vol 77 no 4 ArticleID 043518 2008

[7] D Boyanovsky H J de Vega and N G Sanchez ldquoDarkmatter transfer function free streaming particle statistics andmemory of gravitational clusteringrdquo Physical Review D vol 78no 6 Article ID 063546 2008

[8] C Destri H J de Vega and N G Sanchez ldquoWarm dark matterprimordial spectra and the onset of structure formation atredshift zrdquo Physical Review D vol 88 no 8 Article ID 0835122013

[9] H J De Vega P Salucci and N G Sanchez ldquoThe mass ofthe dark matter particle theory and galaxy observationsrdquo NewAstronomy vol 17 no 7 pp 653ndash666 2012

[10] H J de Vega P Salucci and N G Sanchez ldquoObservationalrotation curves and density profiles versus the Thomas-Fermigalaxy structure theoryrdquoMonthlyNotices of the Royal Astronom-ical Society vol 442 no 3 pp 2717ndash2727 2014

[11] H J de Vega and N G Sanchez ldquoModel-independent analysisof dark matter points to a particle mass at the keV scalerdquoMonthly Notices of the Royal Astronomical Society vol 404 no2 pp 885ndash894 2010

[12] H J de Vega and N G Sanchez ldquoConstant surface gravity anddensity profile of dark matterrdquo International Journal of ModernPhysics A vol 26 no 6 pp 1057ndash1072 2011

[13] H J de Vega and N G Sanchez ldquoCosmological evolutionof warm dark matter fluctuations I Efficient computationalframework with Volterra integral equationsrdquo Physical ReviewDvol 85 no 4 Article ID 043516 20 pages 2012

[14] H J de Vega and N G Sanchez ldquoThe dark matter distributionfunction and halo thermalization from the Eddington equationin galaxiesrdquo International Journal of Modern Physics A vol 31no 13 Article ID 1650073 2016

[15] NMenci S Paduroiu P Salucci N Sanchez and C RWatsonldquolectures at the 20th Paris Cosmology Colloquium Chalonge-Hector de Vegardquo 2016 httpchalongeobspmfr

[16] E W Kolb and M S Turner The Early Universe AddisonWesley Boston Mass USA 1990

[17] J F Navarro C S Frenk and S D M White ldquoA universal den-sity profile from hierarchical clusteringrdquo Astrophysical JournalLetters vol 490 no 2 pp 493ndash508 1997

[18] W J G de Blok ldquoThe core-cusp problemrdquo Advances in Astron-omy vol 2010 Article ID 789293 14 pages 2010

[19] G Gilmore M I Wilkinson R F GWyse et al ldquoThe observedproperties of dark matter on small spatial scalesrdquo AstrophysicalJournal vol 663 no 2 pp 948ndash959 2007

[20] P Salucci and Ch Frigerio Martins ldquoThe mass distribution inSpiral galaxiesrdquo EAS Publications Series vol 36 pp 133ndash1402009

[21] J van Eymeren C Trachternach B S Koribalski and R-JDettmar ldquoNon-circular motions and the cusp-core discrepancyin dwarf galaxiesrdquo Astronomy amp Astrophysics vol 505 no 1 pp1ndash20 2009

[22] M Walker and J Penarrubia ldquoA method for measuring (slopesof) the mass profiles of dwarf spheroidal galaxiesrdquo The Astro-physical Journal vol 742 no 1 p 20 2011

Advances in High Energy Physics 11

[23] R F G Wyse and G Gilmore ldquoObserved properties of darkmatter on small spatial scalesrdquo in Proceedings of the Proceedingsof the International Astronomical Union Symposium (IUA rsquo07)vol 3 no 244 pp 44ndash52 Cardiff UK June 2007

[24] A Burkert ldquoThe structure of dark matter halos in dwarfgalaxiesrdquo Astrophysical Journal vol 447 no 1 pp L25ndashL281995

[25] C Destri H J de Vega and N G Sanchez ldquoQuantum WDMfermions and gravitation determine the observed galaxy struc-turesrdquo Astroparticle Physics vol 46 pp 14ndash22 2013

[26] N Menci N G Sanchez M Castellano and A GrazianldquoConstraining the warm dark matter particle mass throughultra-deep UV luminosity functions at Z = 2rdquoTheAstrophysicalJournal vol 818 no 1 article 90 2016

[27] E Bulbul M Markevitch A Foster R K Smith M Loewen-stein and SW Randall ldquoDetection of an unidentified emissionline in the stackedX-ray spectrumof galaxy clustersrdquoAstrophys-ical Journal vol 789 article 13 2014

[28] A Boyarsky O Ruchayskiy D Iakubovskyi and J FranseldquoUnidentified line inX-ray spectra of the andromeda galaxy andperseus galaxy clusterrdquo Physical Review Letters vol 113 no 25Article ID 251301 2014

[29] K N Abazajian ldquoResonantly produced 7 keV sterile neutrinodark matter models and the properties of Milky Way satellitesrdquoPhysical Review Letters vol 112 no 16 Article ID 161303 5pages 2014

[30] N Menci A Grazian M Castellano and N G SanchezldquoShock connectivity in the 2010 August and 2012 July solarenergetic particle events inferred from observations and ENLILmodelingrdquo Astrophysical Journal Letters vol 825 no 1 article 12016

[31] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[32] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 481 no 1-2 pp 1ndash28 2009

[33] B Pontecorvo ldquoNeutrino experiments and the problem ofconservation of leptonic chargerdquo Soviet Physics-JETP vol 26no 5 pp 984ndash988 1968

[34] A Merle ldquokeV neutrino model buildingrdquo International Journalof Modern Physics D vol 22 no 10 Article ID 1330020 2013

[35] J DVergados andT S Kosmas ldquoSearching for cold darkmatterA case of coexistence of supersymmetry and nuclear physicsrdquoPhysics of Atomic Nuclei vol 61 p 1066 1998

[36] E Holmlund M Kortelainen T S Kosmas J Suhonen and JToivanen ldquoMicroscopic calculation of the LSP detection ratesfor the 71Ga 73Ge and 127I dark-matter detectorsrdquoPhysics LettersB vol 584 no 1-2 pp 31ndash39 2004

[37] R Adhikari M Agostini N Anh Ky et al ldquoA white paper onkeV sterile neutrino dark matterrdquo httpsarxivorgabs160204816

[38] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[39] MARE collaboration httpmaredfmuninsubriaitfrontendexecphp

[40] A Nucciotti ldquoNeutrino mass calorimetric searches in theMARE experimentrdquo httpsarxivorgabs10122290

[41] A Nucciotti ldquoOn behalf of the MARE collaboration lecture attheWorkshop ChalongeMeudonrdquo 2011 httpchalongeobspmfr

[42] KATRIN Collaboration httpwwwkatrinkitedu

[43] C Weinheimer ldquoDirect determination of neutrino mass fromtritium beta spectrumrdquo httpsarxivorgabs09121619

[44] C Weinheimer lecture at the 20th Paris Cosmology Chalonge-Hector de Vega Colloquium 2016 httpchalongeobspmfr

[45] G Drexlin lecture at the 19th Paris Cosmology Colloquium2015

[46] A Huber lecture at the Chalonge-de Vega Meudon Workshop2016 httpchalongeobspmfr

[47] C Tully lecture at the 19th Paris Cosmology Colloquium 2015[48] S Betts W R Blanchard R H Carnevale et al ldquoDevelop-

ment of a relic neutrino detection experiment at PTOLEMYprinceton tritiumobservatory for light early-universemassive-neutrino yieldrdquo httpsarxivorgabs13074738

[49] D M Asner R F Bradley L de Viveiros et al ldquoSingle-electrondetection and spectroscopy via relativistic cyclotron radiationrdquoPhysical Review Letters vol 114 no 16 Article ID 162501 2015

[50] P C-O Ranitzsch J-P Porst S Kempf et al ldquoDevelopmentof metallic magnetic calorimeters for high precision measure-ments of calorimetric 187Re and 163Ho spectrardquo Journal of LowTemperature Physics vol 167 no 5-6 pp 1004ndash1014 2012

[51] C Hassel ldquoLecture at the International School of AstrophysicsDaniel ChalongemdashHector de Vegardquo 2015

[52] L Gastaldo K Blaum A Doerr et al ldquoThe electron capture163Ho experiment ECHordquo Journal of Low Temperature Physicsvol 176 no 5 pp 876ndash884 2014

[53] B Alpert M Balata D Bennett et al ldquoThe electron capturedecay of 163Ho tomeasure the electron neutrino mass with sub-eV sensitivityrdquo The European Physical Journal C vol 75 article112 2015

[54] E Fermi ldquoRadioattivita prodotta da bombardamento di neu-tronirdquo Il Nuovo Cimento vol 11 no 7 pp 429ndash441 1934

[55] E Fermi ldquoVersuch einer theorie der 120573-strahlen Irdquo Zeitschriftfur Physik vol 88 no 3 pp 161ndash177 1934

[56] R B Firestone and V S Shirley Eds Table of Isotopes JohnWiley amp Sons New York NY USA 8th edition 1999

[57] W Bambynek H Behrens M H Chen et al ldquoOrbital electroncapture by the nucleusrdquo Reviews of Modern Physics vol 49 no1 pp 77ndash221 1977

[58] J L Campbell and T Papp ldquoWidths of the atomic KndashN7 levelsrdquoAtomic Data and Nuclear Data Tables vol 77 no 1 pp 1ndash562001

[59] J A Bearden and A F Burr ldquoReevaluation of X-ray atomicenergy levelsrdquo Reviews of Modern Physics vol 39 no 1 pp 125ndash142 1967

[60] M Cardona and L Ley Eds Photoemission in Solids I GeneralPrinciples Springer Berlin Germany 1978

[61] J C Fuggle and N Martensson ldquoCore-level binding energies inmetalsrdquo Journal of Electron Spectroscopy and Related Phenom-ena vol 21 no 3 pp 275ndash281 1980

[62] T Kosmas H Ejiri and A Hatzikoutelis ldquoNeutrino physics inthe frontiers of intensities and very high sensitivitiesrdquo Advancesin High Energy Physics vol 2015 Article ID 806067 3 pages2015

[63] A Giuliani and A Poves ldquoNeutrinoless double-beta decayrdquoAdvances in High Energy Physics vol 2012 Article ID 85701638 pages 2012

[64] J-U Nabi and H V Klapdor-Kleingrothaus ldquoWeak interactionrates of sd-shell nuclei in stellar environments calculated in theproton-neutron quasiparticle random-phase approximationrdquoAtomic Data andNuclear Data Tables vol 71 no 2 pp 149ndash3451999

12 Advances in High Energy Physics

[65] J-U Nabi and H V Klapdor-Kleingrothaus ldquoMicroscopic cal-culations of stellar weak interaction rates and energy losses forfp- and fpg-shell nucleirdquo Atomic Data and Nuclear Data Tablesvol 88 no 2 pp 237ndash476 2004

[66] Y Fukuda T Hayakawa E Ichihara et al ldquoEvidence foroscillation of atmospheric neutrinosrdquo Physical Review Lettersvol 81 no 8 pp 1562ndash1567 1998

[67] Q R Ahmad and SNOCollaboration ldquoMeasurement of the rateof V119890+119889 rarr 119901+119901+119890minus interactions produced by 8B solar neutrinosat the Sudbury Neutrino Observatoryrdquo Physical Review Lettersvol 87 no 7 Article ID 071301 6 pages 2001

[68] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 2002

[69] G J Feldman J Hartnell and T Kobayashi ldquoLong-baselineneutrino oscillation experimentsrdquo Advances in High EnergyPhysics vol 2013 Article ID 475749 30 pages 2013

[70] S T Petcov ldquoThenature ofmassive neutrinosrdquoAdvances inHighEnergy Physics vol 2013 Article ID 852987 20 pages 2013

[71] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[72] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 48 no 1-2 pp 1ndash28 2009

[73] F Munyaneza and P L Biermann ldquoDegenerate sterile neutrinodark matter in the cores of galaxiesrdquo Astronomy amp Astrophysicsvol 458 no 2 pp L9ndashL12 2006

[74] D Boyanovsky and C M Ho ldquoSterile neutrino production viaactive-sterile oscillations the quantum Zeno effectrdquo Journal ofHigh Energy Physics vol 2007 no 7 p 30 2007

[75] R Shrock ldquoNew tests for and bounds on neutrino masses andlepton mixingrdquo Physics Letters B vol 96 no 1-2 pp 159ndash1641980

[76] R E Shrock ldquoGeneral theory of weak processes involving neu-trinos I Leptonic pseudoscalar-meson decays with associatedtests for and bounds on neutrino masses and lepton mixingrdquoPhysical Review D vol 24 no 5 pp 1232ndash1274 1981

[77] R E Shrock ldquoImplications of neutrino masses and mixing forweak processesrdquo in Proceedings of the Weak Interactions asProbes of Unification vol 72 of AIP Conference Proceedings p368 Blacksburg Va USA 1981

[78] J M Conrad C M Ignarra G Karagiorgi M H Shaevitzand J Spitz ldquoSterile neutrino fits to short-baseline neutrinooscillation measurementsrdquo Advances in High Energy Physicsvol 2013 Article ID 163897 2013

[79] A Nucciotti ldquoThe use of low temperature detectors for directmeasurements of themass of the electron neutrinordquoAdvances inHigh Energy Physics vol 2016 Article ID 9153024 41 pages2016

[80] R G H Robertson ldquoExamination of the calorimetric spectrumto determine the neutrino mass in low-energy electron capturedecayrdquo Physical Review C vol 91 no 3 Article ID 035504 2015

[81] A De Rujula ldquoTwo old ways to measure the electron-neutrinomassrdquo httpsarxivorgabs13054857

[82] A De Rujula ldquoAn upper bound on the exceptional character-istics for Lusztigrsquos character formulardquo httpsarxivorgabs08111674v2

[83] A Faessler and F Simkovic ldquoCan electron capture tell us themass of the neutrinordquo Physica Scripta vol 91 no 4 Article ID043007 2016

[84] H J de Vega O Moreno E Moya de Guerra M RamonMedrano and N G Sanchez ldquoRole of sterile neutrino warmdark matter in rhenium and tritium beta decaysrdquo NuclearPhysics B vol 866 no 2 pp 177ndash195 2013

[85] E M de Guerra O Moreno P Sarriguren and M RamonMedrano ldquoTopics on nuclear structure with electroweakprobesrdquo Journal of Physics Conference Series vol 366 ArticleID 012011 2012

[86] J M Boillos O Moreno and E Moya de ldquoActive and sterileneutrino mass effects on beta decay spectrardquo in Proceedingsof the La Rabida International Scientific Meeting on NuclearPhysics Basic Concepts in Nuclear Physics Theory Experimentsand Applications vol 1541 ofAIP Conference Proceedings p 167La Rabida Spain September 2012

[87] P E Filianin K Blaum S A Eliseev et al ldquoOn the keVsterile neutrino search in electron capturerdquo Journal of PhysicsG Nuclear and Particle Physics vol 41 no 9 Article ID 0950042014

[88] S Eliseev K BlaumM Block et al ldquoDirect measurement of themass difference of 163Ho and 163Dy solves theQ-value puzzle forthe neutrino mass determinationrdquo Physical Review Letters vol115 no 6 Article ID 062501 2015

[89] A De Rujula ldquoA new way tomeasure neutrinomassesrdquoNuclearPhysics Section B vol 188 no 3 pp 414ndash458 1981

[90] M Lusignoli and M Vignati ldquoRelic antineutrino capture on163Hodecaying nucleirdquoPhysics Letters B vol 697 no 1 pp 11ndash142011

[91] A Faessler L Gastaldo and F Simkovic ldquoElectron capture in163Ho overlap plus exchange corrections and neutrino massrdquoJournal of Physics G Nuclear and Particle Physics vol 42 no 1Article ID 015108 2015

[92] G Audi F G Kondev M Wang et al ldquoThe Nubase2012evaluation of nuclear propertiesrdquo Chinese Physics C vol 36 no12 pp 1157ndash1286 2012

[93] S Mertens ldquoSensitivity of next-generation tritium beta-decayexperiments for keV-scale sterile neutrinosrdquo JCAP vol 1502 no2 p 20 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

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Advances in Condensed Matter Physics

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Superconductivity

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ThermodynamicsJournal of

Advances in High Energy Physics 7

Table 1 Atomic parameters for electron capture in Holmium from shells M1 to O2The values of 119862119867 [56 57] correspond to the atomic shellsin Holmium while the orbital electron widths Γ119867 [58] and binding energies 119861119867 (ltQ) [59ndash61] correspond to the atomic shells in Dysprosium

M1 M2 N1 N2 O1 O2119862119867 005377 0002605 001373 00005891 0001708 00001Γ119867 [keV] 0013 0006 0006 0005 0005 0002119861119867 [keV] 2047 1842 0416 0332 0063 0026

Table 2 Same as Table 1 (data from the same references) but for electron capture in Lead going toThalliumThe upper table summarizes thevalues of the relevant parameters for 11990412 and 11990112 atomic shells while the lower table shows the values for 11990132 and 11988932 atomic shells In thelatter case 119862119867 includes an extra factor 1199012119890

L1 L2 M1 M2 N1 N2 O1 O2119862119867 08781 006647 02125 001759 005849 0004431 001016 00008052Γ119867 [keV] 0011 0006 0015 0010 0009 0007 mdash mdash119864119867 [keV] 15346 14697 3703 3415 0845 0720 0136 0099L3 M3 M4 N3 N4 O3 O4119862119867 [keV2] 18978193 5366014 45383 1363307 12607 237280 1500Γ119867 [keV] 0006 0009 0002 0006 0004 0001 0001119861119867 [keV] 12657 2956 2484 0608 0406 0072 0015

mass eigenstate 119898119897 asymp 0 with a heavy mass eigenstate 119898ℎ asymp2 keV (solid curve) as in (5) Although the spectrum withmixing is realistic in the sense that it is the result expected if aheavy mass eigenstate exists it is unrealistic in the degree ofmixing shown in the figure which has been maximized herefor the sake of visibility of the ldquokinkrdquo in the scale of the figure50 light neutrino and 50 heavy neutrino correspondingto amixing angle 120577 = 45∘The ldquokinkrdquo can be seen in this curveat 119864119888 = 119876 minus 119898ℎ = 0833 keV Above this calorimeter energythe heavy mass eigenstate cannot be produced due to energyconservation For comparison the spectrum for119898] asymp 0 withno mixing with heavy states is shown in the dashed curve

Another example of electron capture under study here isthat of the Lead isotopes 202 (202Pb 119885 = 82 119873 = 120)and 205 (205Pb 119885 = 82 119873 = 123) going to the Thalliumisotopes 202 (202Tl 119885 = 81 119873 = 121) and 205 (205Tl119885 = 81 119873 = 124) respectively where the process is in bothcases a first forbidden Gamow-Teller unique transition with0+ rarr 2minus for 202Pb and 52minus rarr 12+ for 205Pb (Δ119869 = 2 andparity change the leptons carry Δ119871 = 1 and couple to 119878 = 1)The atomic mass differences obtained from the data in [92]are 119876 = 46 plusmn 14 keV and 119876 = 506 plusmn 18 keV respectivelyAccording to the explanations given in Section 3 capturesfrom orbitals L1 L2 M1 M2 N1 N2 O1 and O2 containan extra neutrino momentum dependence 119878(]) = 1199012] thatmodifies the calorimeter spectrum and corresponds to 120574 =1 On the other hand captures from orbitals L3 M3 M4N3 N4 O3 and O4 which correspond to 120574 = 0 containan extra electron momentum dependence 1199012119890 which beingfixed in bound electrons is included in the values of 119862119867They are given in Table 2 together with the widths and theexperimental electron binding energies in the daughter atomThallium

Figure 2 shows the capture differential rate in 205Pb (darkcurves) and in 202Pb (light curves) Position width and

strength of the capture peaks are assumed to be the same inboth isotopes but the spectra are different because of the dif-ferent 119876-values Solid curves are for light neutrino emission119898] asymp 0 and dashed lines are for heavy neutrino emissionwith119898] = 40 keV (results are given separately for each massbeforemixing) Due to the large119876-values electron capture inLead isotopes allows us to explore this mass value althoughit may be somewhat larger than the expected value fromcosmological reasons The analysis of the ratio in (20) canbe computed for example using the capture peaks L3 at119864119888 = 12657 keV and M3 at 119864119888 = 2956 keV both beingof the 120574 = 0 type For 119898ℎ = 40 keV one would obtain119877l1198723 1198713 202 205 = 1037 1205961198723202 = 0369 1205961198723205 = 0543These results should be introduced in (22) together with theexperimental ratio 1198771015840exp to obtain the value of the mixingangle 120577

It is important to remark that the theoretical quantities 119877119897and 120596 computed as described above contain several approxi-mations They correspond to Dirac-delta peaks ignoring theactual shapes and widths and thus they just refer to onesingle capture peak neglecting the possible effect of the tailsof nearby one-hole peaks Moreover the influence of two-hole peaks close to the main ones has also been neglectedOther effects that have not been taken into account butwhose influence is expected to be very small are multihole(more than two) peaks virtual intermediate states (influenceof transitions through atomic shells not energetically acces-sible) or interference between atomic transitions resultingfrom addition of amplitudes instead of intensities [81 82]Theaccurate experimental determination of the ratio 1198771015840exp entailsits own difficulties among them the possible existence ofmetastable atomic states whose delayed de-excitations failto contribute to the collected spectrum and the variety ofchemical environments resulting in a complex mixture of 119876-values

8 Advances in High Energy Physics

L1

L2

L3

M1M2

M3

M4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

6 162 8 10 12 144 18 200Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(a)

N1

N2

N3

O1

O2

O3

N4O4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

05 0603 0401 02 07 08 09 1000Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(b)

Figure 2 Calorimeter spectrum after electron capture in 202Pb (light curves) and in 205Pb (black curves) with emission of a light neutrino119898119897 = 0 (solid curves) and a heavy neutrino119898ℎ = 40 keV (dashed curves) (a) Full spectrum showing L and M capture peak labels (b) Lowenergy region (119864119888 = 0-1 keV) showing N and O capture peak labels

6 Results for Beta Decay Spectra

In addition to electron capture we also show here the effectof heavy neutrino emission in the electron spectrum of betadecays for one of the cases of most interest Tritium It hasbeen studied in depth in previous works [84ndash86 93] togetherwith another process that has also drawn considerable theo-retical end experimental attention the beta decay of Rhenium187 [37 39ndash41 84ndash86] The beta decay of Tritium (3H 119885 = 1119873 = 2) going to Helium 3 (3He 119885 = 2119873 = 1) is

3H 997888rarr 3He + 119890minus + ]119890 (23)

and has a 119876-value of 1859 keV [92] Both the initial and thefinal nuclear ground states have spin-parity 12+ and thetransition is allowed with the electron and the antineutrinoemitted in 119904-wave The Karlsruhe Tritium Neutrino Experi-ment (KATRIN) [42ndash46] is currently studying this decay inorder to determine the active neutrino mass and could alsostudy the production of a heavy neutrino of amass lower than18 keV

The differential decay rate with respect to the electronenergy is given by

119889120582119889119864119890 = 119870120573 (1198642119890 minus 1198982119890)12 [(119876 + 119898119890 minus 119864119890)2 minus 1198982]]12

sdot 119864119890 (119876 + 119898119890 minus 119864119890) (24)

where 119870120573 contains among others the weak interactioncoupling the nuclearmatrix element and the Fermi function

As an illustration of the heavy neutrino effect in the Tri-tium beta decay we plot in Figure 3 the differential decay ratefrom (24) and (5) using a maximal mixing with an unrealisticvalue of the mixing angle 120577 = 45∘ to show the effect more

distinctly in the scale of the figure We have used heavy masscomponents with 119898ℎ = 2 keV [26] (Figure 3(a)) and with119898ℎ = 7 keV [29 30] (Figure 3(b)) For comparison thespectra without the heavy neutrino contribution (120582ℎ = 0) andwithout mixing (120577 = 0∘) are also shown in this plot The kinkin the spectrum at 119864119890 minus 119898119890 = 119876 minus 119898ℎ can be observed if theexperimental relative error is lower than the size of the stepthe latter being very small in realistic situations

The effect of a heavy neutrino emission can be analyzedthrough the ratio between the heavy and the light neutrinocontributions to the spectrum

R equiv 119889120582ℎ119889119864119890119889120582119897119889119864119890 tan2120577 (25)

The full differential decay rate is also related to the ratio Rthrough

119889120582119889119864119890 =119889120582119897119889119864119890 [1 +R] cos2120577 (26)

In Figure 4 we plot R the ratio of the heavy neutrinocontribution over the light neutrino contribution to the decayrate as a function of the momentum of the emitted electronfor a heavy neutrino mass 119898ℎ = 2 keV and different mixingangles 120577 = 001∘ 0005∘ and 0001∘ The size of this ratio(lt10minus7) gives an idea of the difficulty of finding the kink in thespectrum due to the production of the heavymass eigenstateAs can be seen in the figure the ratio is different from zeroand almost constant in the range 0 le 119901119890 lt (119901119890)max wherethe maximum electron momentum is given by (119901119890)max =[(119876 minus 119898ℎ)(119876 minus 119898ℎ + 2119898119890)]12 For the heavy mass used inthe figure (119901119890)max ≃ 1313 keV The ratio R decreases as themixing angle decreases being approximately proportional to

Advances in High Energy Physics 9

Mixing but no heavy neutrino contribution

104

105

106

107

108d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 2 keV

(a)

104

105

106

107

108

d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

Mixing but no heavy neutrino contribution

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 7 keV

(b)

Figure 3 Electron spectrum of Tritium beta decay for a light-heavy neutrino mixing angle 120577 = 45∘ shown just for illustration and a heavyneutrino mass (solid curve)119898ℎ = 2 keV (a) and119898ℎ = 7 keV (b) The spectrum without heavy neutrino mass contribution (dotted curve) andwithout mixing (dashed curve) are also shown We use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of thefigure Realistic values of 120577 lt 001∘ result in a reduction of the difference between the dotted and the solid lines by more than seven orders ofmagnitude

10minus11

10minus10

10minus9

10minus8

10minus7

10minus12

120577 = 001∘

120577 = 0005∘

120577 = 0001∘

15090 100 110 120 130 1408070pe (keV)

ℛ

Figure 4 Ratio R defined in (25) for the Tritium beta decay as afunction of the electronmomentum for a heavy neutrinomass119898ℎ =2 keV and different values of the mixing angle 120577 = 001∘ (solid line)0005∘ (dashed line) and 0001∘ (dotted line)

1205772 It also decreases for larger heavy neutrino masses 119898ℎ Adifferent ratio between the spectra with mixing and without

mixing (Rlowast) has also been considered in [84ndash86 93] relatedwithR in (25) as

Rlowast = minussin2120577 +R cos2120577 = 119889120582119889119864119890119889120582119897119889119864119890 minus 1 (27)

7 Conclusions

Signatures of hypothetical keV sterile neutrinos which couldbe warm dark matter candidates (WDM) may be found inthe spectra of ordinary weak nuclear decays The electronspectrum in beta decay is cleaner for this purpose becauseit is a smooth curve while the deexcitation spectrum afterelectron capture shows many peaks On the other hand inbeta decay the possible excitation of the atoms or moleculesis not disentangled while in electron capture the calorimetercollects all the energy independently of the excitation ofthe atom and of the deexcitation path and one has higherstatistics around the capture peaks

Figures of electron spectrum for beta decay in 3H aswell as of calorimeter spectrum for electron capture in 163Hoand 202Pb and 205Pb are given considering various valuesof the heavy neutrino mass (2 keV 7 keV and 40 keV) thatmay be experimentally probed In both weak processes thesmall value of the light-heavy neutrino mixing angle requiresextremely high experimental precision and the use of sourceswith large stability to reduce systematic errors This is why it

10 Advances in High Energy Physics

is useful to consider relevant ratios between transition rates asfirst introduced in [84ndash87] to analyze the data We considertwo cases the case of a single isotope where one may use theratio of accumulated number of events in different regionsof the spectrum which allows us to remove uncertaintiesrelated to the nuclear matrix element and to the valuesof overall constants And in the case of two isotopes weconsider ratios of the above mentioned ratios that allow usto reduce uncertainties from atomic parameters We give aswell analytical expressions that can be used to obtain a goodtheoretical approximation to the experimental ratios ((19) to(21))

Both the experimental measurement and the theoreticalmodel must be accurate enough to detect differences in theexpected versus themeasured ratios of the order of themixingangle squared 1205772 ≲ 10minus8 This is also the size of the kinkexpected in the electron spectrumof beta decay located at thelimit of the regionwhere the production of a heavy neutrino isenergetically allowed In order to identify this kink among thestatistical fluctuations of themeasured spectrum the numberof collected events must be larger than the inverse of thesquared ratioR in (25)

Our results particularly on the electron capture spectraof Lead isotopes that include for the first time all thepossible peaks may help in designing and analyzing futureexperiments to search for sterile neutrinos in the keV massrangeThe use of the different types of ratios between numberof events discussed here for both electron capture and betadecay processes may also help in planning the experimentsby establishing the threshold of statistical and systematicuncertainties required for the detection of sterile neutrinosof given mass and mixing or to properly extract exclusionplots

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

E Moya de Guerra and O Moreno acknowledge MINECOfor partial financial support (FIS2014-51971-P) and IEM-CSIC for hospitality O Moreno acknowledges supportfrom a Marie Curie International Outgoing Fellowshipwithin the European Union Seventh Framework Pro-gramme under Grant Agreement PIOF-GA-2011-298364(ELECTROWEAK) M R M thanks N G Sanchez forcorrespondence and conversations

References

[1] F Zwicky ldquoDieRotverschiebung von extragalaktischenNebelnrdquoHelvetica Physica Acta vol 6 pp 110ndash127 1933

[2] M Roos ldquoAstrophysical and cosmological probes of darkmatterrdquo Journal of Modern Physics vol 3 pp 1152ndash1171 2012

[3] S Dodelson Modern Cosmology Academic Press New YorkNY USA 2003

[4] S Weinberg Cosmology Oxford University Press Oxford UK2008

[5] J Lesgourgues and S Pastor ldquoNeutrino mass from cosmologyrdquoAdvances in High Energy Physics vol 2012 Article ID 60851534 pages 2012

[6] D Boyanovsky H J de Vega and N G Sanchez ldquoConstraintson dark matter particles from theory galaxy observations andN-body simulationsrdquo Physical Review D vol 77 no 4 ArticleID 043518 2008

[7] D Boyanovsky H J de Vega and N G Sanchez ldquoDarkmatter transfer function free streaming particle statistics andmemory of gravitational clusteringrdquo Physical Review D vol 78no 6 Article ID 063546 2008

[8] C Destri H J de Vega and N G Sanchez ldquoWarm dark matterprimordial spectra and the onset of structure formation atredshift zrdquo Physical Review D vol 88 no 8 Article ID 0835122013

[9] H J De Vega P Salucci and N G Sanchez ldquoThe mass ofthe dark matter particle theory and galaxy observationsrdquo NewAstronomy vol 17 no 7 pp 653ndash666 2012

[10] H J de Vega P Salucci and N G Sanchez ldquoObservationalrotation curves and density profiles versus the Thomas-Fermigalaxy structure theoryrdquoMonthlyNotices of the Royal Astronom-ical Society vol 442 no 3 pp 2717ndash2727 2014

[11] H J de Vega and N G Sanchez ldquoModel-independent analysisof dark matter points to a particle mass at the keV scalerdquoMonthly Notices of the Royal Astronomical Society vol 404 no2 pp 885ndash894 2010

[12] H J de Vega and N G Sanchez ldquoConstant surface gravity anddensity profile of dark matterrdquo International Journal of ModernPhysics A vol 26 no 6 pp 1057ndash1072 2011

[13] H J de Vega and N G Sanchez ldquoCosmological evolutionof warm dark matter fluctuations I Efficient computationalframework with Volterra integral equationsrdquo Physical ReviewDvol 85 no 4 Article ID 043516 20 pages 2012

[14] H J de Vega and N G Sanchez ldquoThe dark matter distributionfunction and halo thermalization from the Eddington equationin galaxiesrdquo International Journal of Modern Physics A vol 31no 13 Article ID 1650073 2016

[15] NMenci S Paduroiu P Salucci N Sanchez and C RWatsonldquolectures at the 20th Paris Cosmology Colloquium Chalonge-Hector de Vegardquo 2016 httpchalongeobspmfr

[16] E W Kolb and M S Turner The Early Universe AddisonWesley Boston Mass USA 1990

[17] J F Navarro C S Frenk and S D M White ldquoA universal den-sity profile from hierarchical clusteringrdquo Astrophysical JournalLetters vol 490 no 2 pp 493ndash508 1997

[18] W J G de Blok ldquoThe core-cusp problemrdquo Advances in Astron-omy vol 2010 Article ID 789293 14 pages 2010

[19] G Gilmore M I Wilkinson R F GWyse et al ldquoThe observedproperties of dark matter on small spatial scalesrdquo AstrophysicalJournal vol 663 no 2 pp 948ndash959 2007

[20] P Salucci and Ch Frigerio Martins ldquoThe mass distribution inSpiral galaxiesrdquo EAS Publications Series vol 36 pp 133ndash1402009

[21] J van Eymeren C Trachternach B S Koribalski and R-JDettmar ldquoNon-circular motions and the cusp-core discrepancyin dwarf galaxiesrdquo Astronomy amp Astrophysics vol 505 no 1 pp1ndash20 2009

[22] M Walker and J Penarrubia ldquoA method for measuring (slopesof) the mass profiles of dwarf spheroidal galaxiesrdquo The Astro-physical Journal vol 742 no 1 p 20 2011

Advances in High Energy Physics 11

[23] R F G Wyse and G Gilmore ldquoObserved properties of darkmatter on small spatial scalesrdquo in Proceedings of the Proceedingsof the International Astronomical Union Symposium (IUA rsquo07)vol 3 no 244 pp 44ndash52 Cardiff UK June 2007

[24] A Burkert ldquoThe structure of dark matter halos in dwarfgalaxiesrdquo Astrophysical Journal vol 447 no 1 pp L25ndashL281995

[25] C Destri H J de Vega and N G Sanchez ldquoQuantum WDMfermions and gravitation determine the observed galaxy struc-turesrdquo Astroparticle Physics vol 46 pp 14ndash22 2013

[26] N Menci N G Sanchez M Castellano and A GrazianldquoConstraining the warm dark matter particle mass throughultra-deep UV luminosity functions at Z = 2rdquoTheAstrophysicalJournal vol 818 no 1 article 90 2016

[27] E Bulbul M Markevitch A Foster R K Smith M Loewen-stein and SW Randall ldquoDetection of an unidentified emissionline in the stackedX-ray spectrumof galaxy clustersrdquoAstrophys-ical Journal vol 789 article 13 2014

[28] A Boyarsky O Ruchayskiy D Iakubovskyi and J FranseldquoUnidentified line inX-ray spectra of the andromeda galaxy andperseus galaxy clusterrdquo Physical Review Letters vol 113 no 25Article ID 251301 2014

[29] K N Abazajian ldquoResonantly produced 7 keV sterile neutrinodark matter models and the properties of Milky Way satellitesrdquoPhysical Review Letters vol 112 no 16 Article ID 161303 5pages 2014

[30] N Menci A Grazian M Castellano and N G SanchezldquoShock connectivity in the 2010 August and 2012 July solarenergetic particle events inferred from observations and ENLILmodelingrdquo Astrophysical Journal Letters vol 825 no 1 article 12016

[31] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[32] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 481 no 1-2 pp 1ndash28 2009

[33] B Pontecorvo ldquoNeutrino experiments and the problem ofconservation of leptonic chargerdquo Soviet Physics-JETP vol 26no 5 pp 984ndash988 1968

[34] A Merle ldquokeV neutrino model buildingrdquo International Journalof Modern Physics D vol 22 no 10 Article ID 1330020 2013

[35] J DVergados andT S Kosmas ldquoSearching for cold darkmatterA case of coexistence of supersymmetry and nuclear physicsrdquoPhysics of Atomic Nuclei vol 61 p 1066 1998

[36] E Holmlund M Kortelainen T S Kosmas J Suhonen and JToivanen ldquoMicroscopic calculation of the LSP detection ratesfor the 71Ga 73Ge and 127I dark-matter detectorsrdquoPhysics LettersB vol 584 no 1-2 pp 31ndash39 2004

[37] R Adhikari M Agostini N Anh Ky et al ldquoA white paper onkeV sterile neutrino dark matterrdquo httpsarxivorgabs160204816

[38] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[39] MARE collaboration httpmaredfmuninsubriaitfrontendexecphp

[40] A Nucciotti ldquoNeutrino mass calorimetric searches in theMARE experimentrdquo httpsarxivorgabs10122290

[41] A Nucciotti ldquoOn behalf of the MARE collaboration lecture attheWorkshop ChalongeMeudonrdquo 2011 httpchalongeobspmfr

[42] KATRIN Collaboration httpwwwkatrinkitedu

[43] C Weinheimer ldquoDirect determination of neutrino mass fromtritium beta spectrumrdquo httpsarxivorgabs09121619

[44] C Weinheimer lecture at the 20th Paris Cosmology Chalonge-Hector de Vega Colloquium 2016 httpchalongeobspmfr

[45] G Drexlin lecture at the 19th Paris Cosmology Colloquium2015

[46] A Huber lecture at the Chalonge-de Vega Meudon Workshop2016 httpchalongeobspmfr

[47] C Tully lecture at the 19th Paris Cosmology Colloquium 2015[48] S Betts W R Blanchard R H Carnevale et al ldquoDevelop-

ment of a relic neutrino detection experiment at PTOLEMYprinceton tritiumobservatory for light early-universemassive-neutrino yieldrdquo httpsarxivorgabs13074738

[49] D M Asner R F Bradley L de Viveiros et al ldquoSingle-electrondetection and spectroscopy via relativistic cyclotron radiationrdquoPhysical Review Letters vol 114 no 16 Article ID 162501 2015

[50] P C-O Ranitzsch J-P Porst S Kempf et al ldquoDevelopmentof metallic magnetic calorimeters for high precision measure-ments of calorimetric 187Re and 163Ho spectrardquo Journal of LowTemperature Physics vol 167 no 5-6 pp 1004ndash1014 2012

[51] C Hassel ldquoLecture at the International School of AstrophysicsDaniel ChalongemdashHector de Vegardquo 2015

[52] L Gastaldo K Blaum A Doerr et al ldquoThe electron capture163Ho experiment ECHordquo Journal of Low Temperature Physicsvol 176 no 5 pp 876ndash884 2014

[53] B Alpert M Balata D Bennett et al ldquoThe electron capturedecay of 163Ho tomeasure the electron neutrino mass with sub-eV sensitivityrdquo The European Physical Journal C vol 75 article112 2015

[54] E Fermi ldquoRadioattivita prodotta da bombardamento di neu-tronirdquo Il Nuovo Cimento vol 11 no 7 pp 429ndash441 1934

[55] E Fermi ldquoVersuch einer theorie der 120573-strahlen Irdquo Zeitschriftfur Physik vol 88 no 3 pp 161ndash177 1934

[56] R B Firestone and V S Shirley Eds Table of Isotopes JohnWiley amp Sons New York NY USA 8th edition 1999

[57] W Bambynek H Behrens M H Chen et al ldquoOrbital electroncapture by the nucleusrdquo Reviews of Modern Physics vol 49 no1 pp 77ndash221 1977

[58] J L Campbell and T Papp ldquoWidths of the atomic KndashN7 levelsrdquoAtomic Data and Nuclear Data Tables vol 77 no 1 pp 1ndash562001

[59] J A Bearden and A F Burr ldquoReevaluation of X-ray atomicenergy levelsrdquo Reviews of Modern Physics vol 39 no 1 pp 125ndash142 1967

[60] M Cardona and L Ley Eds Photoemission in Solids I GeneralPrinciples Springer Berlin Germany 1978

[61] J C Fuggle and N Martensson ldquoCore-level binding energies inmetalsrdquo Journal of Electron Spectroscopy and Related Phenom-ena vol 21 no 3 pp 275ndash281 1980

[62] T Kosmas H Ejiri and A Hatzikoutelis ldquoNeutrino physics inthe frontiers of intensities and very high sensitivitiesrdquo Advancesin High Energy Physics vol 2015 Article ID 806067 3 pages2015

[63] A Giuliani and A Poves ldquoNeutrinoless double-beta decayrdquoAdvances in High Energy Physics vol 2012 Article ID 85701638 pages 2012

[64] J-U Nabi and H V Klapdor-Kleingrothaus ldquoWeak interactionrates of sd-shell nuclei in stellar environments calculated in theproton-neutron quasiparticle random-phase approximationrdquoAtomic Data andNuclear Data Tables vol 71 no 2 pp 149ndash3451999

12 Advances in High Energy Physics

[65] J-U Nabi and H V Klapdor-Kleingrothaus ldquoMicroscopic cal-culations of stellar weak interaction rates and energy losses forfp- and fpg-shell nucleirdquo Atomic Data and Nuclear Data Tablesvol 88 no 2 pp 237ndash476 2004

[66] Y Fukuda T Hayakawa E Ichihara et al ldquoEvidence foroscillation of atmospheric neutrinosrdquo Physical Review Lettersvol 81 no 8 pp 1562ndash1567 1998

[67] Q R Ahmad and SNOCollaboration ldquoMeasurement of the rateof V119890+119889 rarr 119901+119901+119890minus interactions produced by 8B solar neutrinosat the Sudbury Neutrino Observatoryrdquo Physical Review Lettersvol 87 no 7 Article ID 071301 6 pages 2001

[68] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 2002

[69] G J Feldman J Hartnell and T Kobayashi ldquoLong-baselineneutrino oscillation experimentsrdquo Advances in High EnergyPhysics vol 2013 Article ID 475749 30 pages 2013

[70] S T Petcov ldquoThenature ofmassive neutrinosrdquoAdvances inHighEnergy Physics vol 2013 Article ID 852987 20 pages 2013

[71] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[72] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 48 no 1-2 pp 1ndash28 2009

[73] F Munyaneza and P L Biermann ldquoDegenerate sterile neutrinodark matter in the cores of galaxiesrdquo Astronomy amp Astrophysicsvol 458 no 2 pp L9ndashL12 2006

[74] D Boyanovsky and C M Ho ldquoSterile neutrino production viaactive-sterile oscillations the quantum Zeno effectrdquo Journal ofHigh Energy Physics vol 2007 no 7 p 30 2007

[75] R Shrock ldquoNew tests for and bounds on neutrino masses andlepton mixingrdquo Physics Letters B vol 96 no 1-2 pp 159ndash1641980

[76] R E Shrock ldquoGeneral theory of weak processes involving neu-trinos I Leptonic pseudoscalar-meson decays with associatedtests for and bounds on neutrino masses and lepton mixingrdquoPhysical Review D vol 24 no 5 pp 1232ndash1274 1981

[77] R E Shrock ldquoImplications of neutrino masses and mixing forweak processesrdquo in Proceedings of the Weak Interactions asProbes of Unification vol 72 of AIP Conference Proceedings p368 Blacksburg Va USA 1981

[78] J M Conrad C M Ignarra G Karagiorgi M H Shaevitzand J Spitz ldquoSterile neutrino fits to short-baseline neutrinooscillation measurementsrdquo Advances in High Energy Physicsvol 2013 Article ID 163897 2013

[79] A Nucciotti ldquoThe use of low temperature detectors for directmeasurements of themass of the electron neutrinordquoAdvances inHigh Energy Physics vol 2016 Article ID 9153024 41 pages2016

[80] R G H Robertson ldquoExamination of the calorimetric spectrumto determine the neutrino mass in low-energy electron capturedecayrdquo Physical Review C vol 91 no 3 Article ID 035504 2015

[81] A De Rujula ldquoTwo old ways to measure the electron-neutrinomassrdquo httpsarxivorgabs13054857

[82] A De Rujula ldquoAn upper bound on the exceptional character-istics for Lusztigrsquos character formulardquo httpsarxivorgabs08111674v2

[83] A Faessler and F Simkovic ldquoCan electron capture tell us themass of the neutrinordquo Physica Scripta vol 91 no 4 Article ID043007 2016

[84] H J de Vega O Moreno E Moya de Guerra M RamonMedrano and N G Sanchez ldquoRole of sterile neutrino warmdark matter in rhenium and tritium beta decaysrdquo NuclearPhysics B vol 866 no 2 pp 177ndash195 2013

[85] E M de Guerra O Moreno P Sarriguren and M RamonMedrano ldquoTopics on nuclear structure with electroweakprobesrdquo Journal of Physics Conference Series vol 366 ArticleID 012011 2012

[86] J M Boillos O Moreno and E Moya de ldquoActive and sterileneutrino mass effects on beta decay spectrardquo in Proceedingsof the La Rabida International Scientific Meeting on NuclearPhysics Basic Concepts in Nuclear Physics Theory Experimentsand Applications vol 1541 ofAIP Conference Proceedings p 167La Rabida Spain September 2012

[87] P E Filianin K Blaum S A Eliseev et al ldquoOn the keVsterile neutrino search in electron capturerdquo Journal of PhysicsG Nuclear and Particle Physics vol 41 no 9 Article ID 0950042014

[88] S Eliseev K BlaumM Block et al ldquoDirect measurement of themass difference of 163Ho and 163Dy solves theQ-value puzzle forthe neutrino mass determinationrdquo Physical Review Letters vol115 no 6 Article ID 062501 2015

[89] A De Rujula ldquoA new way tomeasure neutrinomassesrdquoNuclearPhysics Section B vol 188 no 3 pp 414ndash458 1981

[90] M Lusignoli and M Vignati ldquoRelic antineutrino capture on163Hodecaying nucleirdquoPhysics Letters B vol 697 no 1 pp 11ndash142011

[91] A Faessler L Gastaldo and F Simkovic ldquoElectron capture in163Ho overlap plus exchange corrections and neutrino massrdquoJournal of Physics G Nuclear and Particle Physics vol 42 no 1Article ID 015108 2015

[92] G Audi F G Kondev M Wang et al ldquoThe Nubase2012evaluation of nuclear propertiesrdquo Chinese Physics C vol 36 no12 pp 1157ndash1286 2012

[93] S Mertens ldquoSensitivity of next-generation tritium beta-decayexperiments for keV-scale sterile neutrinosrdquo JCAP vol 1502 no2 p 20 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

L1

L2

L3

M1M2

M3

M4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

6 162 8 10 12 144 18 200Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(a)

N1

N2

N3

O1

O2

O3

N4O4

102

103

104

105

106

107

108

109

1010

d120582d

Ec

(au

)

05 0603 0401 02 07 08 09 1000Ec (keV)

m = 40 keV m = 0

m = 40 keV m = 0

202Pb202Pb

205Pb205Pb

(b)

Figure 2 Calorimeter spectrum after electron capture in 202Pb (light curves) and in 205Pb (black curves) with emission of a light neutrino119898119897 = 0 (solid curves) and a heavy neutrino119898ℎ = 40 keV (dashed curves) (a) Full spectrum showing L and M capture peak labels (b) Lowenergy region (119864119888 = 0-1 keV) showing N and O capture peak labels

6 Results for Beta Decay Spectra

In addition to electron capture we also show here the effectof heavy neutrino emission in the electron spectrum of betadecays for one of the cases of most interest Tritium It hasbeen studied in depth in previous works [84ndash86 93] togetherwith another process that has also drawn considerable theo-retical end experimental attention the beta decay of Rhenium187 [37 39ndash41 84ndash86] The beta decay of Tritium (3H 119885 = 1119873 = 2) going to Helium 3 (3He 119885 = 2119873 = 1) is

3H 997888rarr 3He + 119890minus + ]119890 (23)

and has a 119876-value of 1859 keV [92] Both the initial and thefinal nuclear ground states have spin-parity 12+ and thetransition is allowed with the electron and the antineutrinoemitted in 119904-wave The Karlsruhe Tritium Neutrino Experi-ment (KATRIN) [42ndash46] is currently studying this decay inorder to determine the active neutrino mass and could alsostudy the production of a heavy neutrino of amass lower than18 keV

The differential decay rate with respect to the electronenergy is given by

119889120582119889119864119890 = 119870120573 (1198642119890 minus 1198982119890)12 [(119876 + 119898119890 minus 119864119890)2 minus 1198982]]12

sdot 119864119890 (119876 + 119898119890 minus 119864119890) (24)

where 119870120573 contains among others the weak interactioncoupling the nuclearmatrix element and the Fermi function

As an illustration of the heavy neutrino effect in the Tri-tium beta decay we plot in Figure 3 the differential decay ratefrom (24) and (5) using a maximal mixing with an unrealisticvalue of the mixing angle 120577 = 45∘ to show the effect more

distinctly in the scale of the figure We have used heavy masscomponents with 119898ℎ = 2 keV [26] (Figure 3(a)) and with119898ℎ = 7 keV [29 30] (Figure 3(b)) For comparison thespectra without the heavy neutrino contribution (120582ℎ = 0) andwithout mixing (120577 = 0∘) are also shown in this plot The kinkin the spectrum at 119864119890 minus 119898119890 = 119876 minus 119898ℎ can be observed if theexperimental relative error is lower than the size of the stepthe latter being very small in realistic situations

The effect of a heavy neutrino emission can be analyzedthrough the ratio between the heavy and the light neutrinocontributions to the spectrum

R equiv 119889120582ℎ119889119864119890119889120582119897119889119864119890 tan2120577 (25)

The full differential decay rate is also related to the ratio Rthrough

119889120582119889119864119890 =119889120582119897119889119864119890 [1 +R] cos2120577 (26)

In Figure 4 we plot R the ratio of the heavy neutrinocontribution over the light neutrino contribution to the decayrate as a function of the momentum of the emitted electronfor a heavy neutrino mass 119898ℎ = 2 keV and different mixingangles 120577 = 001∘ 0005∘ and 0001∘ The size of this ratio(lt10minus7) gives an idea of the difficulty of finding the kink in thespectrum due to the production of the heavymass eigenstateAs can be seen in the figure the ratio is different from zeroand almost constant in the range 0 le 119901119890 lt (119901119890)max wherethe maximum electron momentum is given by (119901119890)max =[(119876 minus 119898ℎ)(119876 minus 119898ℎ + 2119898119890)]12 For the heavy mass used inthe figure (119901119890)max ≃ 1313 keV The ratio R decreases as themixing angle decreases being approximately proportional to

Advances in High Energy Physics 9

Mixing but no heavy neutrino contribution

104

105

106

107

108d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 2 keV

(a)

104

105

106

107

108

d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

Mixing but no heavy neutrino contribution

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 7 keV

(b)

Figure 3 Electron spectrum of Tritium beta decay for a light-heavy neutrino mixing angle 120577 = 45∘ shown just for illustration and a heavyneutrino mass (solid curve)119898ℎ = 2 keV (a) and119898ℎ = 7 keV (b) The spectrum without heavy neutrino mass contribution (dotted curve) andwithout mixing (dashed curve) are also shown We use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of thefigure Realistic values of 120577 lt 001∘ result in a reduction of the difference between the dotted and the solid lines by more than seven orders ofmagnitude

10minus11

10minus10

10minus9

10minus8

10minus7

10minus12

120577 = 001∘

120577 = 0005∘

120577 = 0001∘

15090 100 110 120 130 1408070pe (keV)

ℛ

Figure 4 Ratio R defined in (25) for the Tritium beta decay as afunction of the electronmomentum for a heavy neutrinomass119898ℎ =2 keV and different values of the mixing angle 120577 = 001∘ (solid line)0005∘ (dashed line) and 0001∘ (dotted line)

1205772 It also decreases for larger heavy neutrino masses 119898ℎ Adifferent ratio between the spectra with mixing and without

mixing (Rlowast) has also been considered in [84ndash86 93] relatedwithR in (25) as

Rlowast = minussin2120577 +R cos2120577 = 119889120582119889119864119890119889120582119897119889119864119890 minus 1 (27)

7 Conclusions

Signatures of hypothetical keV sterile neutrinos which couldbe warm dark matter candidates (WDM) may be found inthe spectra of ordinary weak nuclear decays The electronspectrum in beta decay is cleaner for this purpose becauseit is a smooth curve while the deexcitation spectrum afterelectron capture shows many peaks On the other hand inbeta decay the possible excitation of the atoms or moleculesis not disentangled while in electron capture the calorimetercollects all the energy independently of the excitation ofthe atom and of the deexcitation path and one has higherstatistics around the capture peaks

Figures of electron spectrum for beta decay in 3H aswell as of calorimeter spectrum for electron capture in 163Hoand 202Pb and 205Pb are given considering various valuesof the heavy neutrino mass (2 keV 7 keV and 40 keV) thatmay be experimentally probed In both weak processes thesmall value of the light-heavy neutrino mixing angle requiresextremely high experimental precision and the use of sourceswith large stability to reduce systematic errors This is why it

10 Advances in High Energy Physics

is useful to consider relevant ratios between transition rates asfirst introduced in [84ndash87] to analyze the data We considertwo cases the case of a single isotope where one may use theratio of accumulated number of events in different regionsof the spectrum which allows us to remove uncertaintiesrelated to the nuclear matrix element and to the valuesof overall constants And in the case of two isotopes weconsider ratios of the above mentioned ratios that allow usto reduce uncertainties from atomic parameters We give aswell analytical expressions that can be used to obtain a goodtheoretical approximation to the experimental ratios ((19) to(21))

Both the experimental measurement and the theoreticalmodel must be accurate enough to detect differences in theexpected versus themeasured ratios of the order of themixingangle squared 1205772 ≲ 10minus8 This is also the size of the kinkexpected in the electron spectrumof beta decay located at thelimit of the regionwhere the production of a heavy neutrino isenergetically allowed In order to identify this kink among thestatistical fluctuations of themeasured spectrum the numberof collected events must be larger than the inverse of thesquared ratioR in (25)

Our results particularly on the electron capture spectraof Lead isotopes that include for the first time all thepossible peaks may help in designing and analyzing futureexperiments to search for sterile neutrinos in the keV massrangeThe use of the different types of ratios between numberof events discussed here for both electron capture and betadecay processes may also help in planning the experimentsby establishing the threshold of statistical and systematicuncertainties required for the detection of sterile neutrinosof given mass and mixing or to properly extract exclusionplots

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

E Moya de Guerra and O Moreno acknowledge MINECOfor partial financial support (FIS2014-51971-P) and IEM-CSIC for hospitality O Moreno acknowledges supportfrom a Marie Curie International Outgoing Fellowshipwithin the European Union Seventh Framework Pro-gramme under Grant Agreement PIOF-GA-2011-298364(ELECTROWEAK) M R M thanks N G Sanchez forcorrespondence and conversations

References

[1] F Zwicky ldquoDieRotverschiebung von extragalaktischenNebelnrdquoHelvetica Physica Acta vol 6 pp 110ndash127 1933

[2] M Roos ldquoAstrophysical and cosmological probes of darkmatterrdquo Journal of Modern Physics vol 3 pp 1152ndash1171 2012

[3] S Dodelson Modern Cosmology Academic Press New YorkNY USA 2003

[4] S Weinberg Cosmology Oxford University Press Oxford UK2008

[5] J Lesgourgues and S Pastor ldquoNeutrino mass from cosmologyrdquoAdvances in High Energy Physics vol 2012 Article ID 60851534 pages 2012

[6] D Boyanovsky H J de Vega and N G Sanchez ldquoConstraintson dark matter particles from theory galaxy observations andN-body simulationsrdquo Physical Review D vol 77 no 4 ArticleID 043518 2008

[7] D Boyanovsky H J de Vega and N G Sanchez ldquoDarkmatter transfer function free streaming particle statistics andmemory of gravitational clusteringrdquo Physical Review D vol 78no 6 Article ID 063546 2008

[8] C Destri H J de Vega and N G Sanchez ldquoWarm dark matterprimordial spectra and the onset of structure formation atredshift zrdquo Physical Review D vol 88 no 8 Article ID 0835122013

[9] H J De Vega P Salucci and N G Sanchez ldquoThe mass ofthe dark matter particle theory and galaxy observationsrdquo NewAstronomy vol 17 no 7 pp 653ndash666 2012

[10] H J de Vega P Salucci and N G Sanchez ldquoObservationalrotation curves and density profiles versus the Thomas-Fermigalaxy structure theoryrdquoMonthlyNotices of the Royal Astronom-ical Society vol 442 no 3 pp 2717ndash2727 2014

[11] H J de Vega and N G Sanchez ldquoModel-independent analysisof dark matter points to a particle mass at the keV scalerdquoMonthly Notices of the Royal Astronomical Society vol 404 no2 pp 885ndash894 2010

[12] H J de Vega and N G Sanchez ldquoConstant surface gravity anddensity profile of dark matterrdquo International Journal of ModernPhysics A vol 26 no 6 pp 1057ndash1072 2011

[13] H J de Vega and N G Sanchez ldquoCosmological evolutionof warm dark matter fluctuations I Efficient computationalframework with Volterra integral equationsrdquo Physical ReviewDvol 85 no 4 Article ID 043516 20 pages 2012

[14] H J de Vega and N G Sanchez ldquoThe dark matter distributionfunction and halo thermalization from the Eddington equationin galaxiesrdquo International Journal of Modern Physics A vol 31no 13 Article ID 1650073 2016

[15] NMenci S Paduroiu P Salucci N Sanchez and C RWatsonldquolectures at the 20th Paris Cosmology Colloquium Chalonge-Hector de Vegardquo 2016 httpchalongeobspmfr

[16] E W Kolb and M S Turner The Early Universe AddisonWesley Boston Mass USA 1990

[17] J F Navarro C S Frenk and S D M White ldquoA universal den-sity profile from hierarchical clusteringrdquo Astrophysical JournalLetters vol 490 no 2 pp 493ndash508 1997

[18] W J G de Blok ldquoThe core-cusp problemrdquo Advances in Astron-omy vol 2010 Article ID 789293 14 pages 2010

[19] G Gilmore M I Wilkinson R F GWyse et al ldquoThe observedproperties of dark matter on small spatial scalesrdquo AstrophysicalJournal vol 663 no 2 pp 948ndash959 2007

[20] P Salucci and Ch Frigerio Martins ldquoThe mass distribution inSpiral galaxiesrdquo EAS Publications Series vol 36 pp 133ndash1402009

[21] J van Eymeren C Trachternach B S Koribalski and R-JDettmar ldquoNon-circular motions and the cusp-core discrepancyin dwarf galaxiesrdquo Astronomy amp Astrophysics vol 505 no 1 pp1ndash20 2009

[22] M Walker and J Penarrubia ldquoA method for measuring (slopesof) the mass profiles of dwarf spheroidal galaxiesrdquo The Astro-physical Journal vol 742 no 1 p 20 2011

Advances in High Energy Physics 11

[23] R F G Wyse and G Gilmore ldquoObserved properties of darkmatter on small spatial scalesrdquo in Proceedings of the Proceedingsof the International Astronomical Union Symposium (IUA rsquo07)vol 3 no 244 pp 44ndash52 Cardiff UK June 2007

[24] A Burkert ldquoThe structure of dark matter halos in dwarfgalaxiesrdquo Astrophysical Journal vol 447 no 1 pp L25ndashL281995

[25] C Destri H J de Vega and N G Sanchez ldquoQuantum WDMfermions and gravitation determine the observed galaxy struc-turesrdquo Astroparticle Physics vol 46 pp 14ndash22 2013

[26] N Menci N G Sanchez M Castellano and A GrazianldquoConstraining the warm dark matter particle mass throughultra-deep UV luminosity functions at Z = 2rdquoTheAstrophysicalJournal vol 818 no 1 article 90 2016

[27] E Bulbul M Markevitch A Foster R K Smith M Loewen-stein and SW Randall ldquoDetection of an unidentified emissionline in the stackedX-ray spectrumof galaxy clustersrdquoAstrophys-ical Journal vol 789 article 13 2014

[28] A Boyarsky O Ruchayskiy D Iakubovskyi and J FranseldquoUnidentified line inX-ray spectra of the andromeda galaxy andperseus galaxy clusterrdquo Physical Review Letters vol 113 no 25Article ID 251301 2014

[29] K N Abazajian ldquoResonantly produced 7 keV sterile neutrinodark matter models and the properties of Milky Way satellitesrdquoPhysical Review Letters vol 112 no 16 Article ID 161303 5pages 2014

[30] N Menci A Grazian M Castellano and N G SanchezldquoShock connectivity in the 2010 August and 2012 July solarenergetic particle events inferred from observations and ENLILmodelingrdquo Astrophysical Journal Letters vol 825 no 1 article 12016

[31] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[32] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 481 no 1-2 pp 1ndash28 2009

[33] B Pontecorvo ldquoNeutrino experiments and the problem ofconservation of leptonic chargerdquo Soviet Physics-JETP vol 26no 5 pp 984ndash988 1968

[34] A Merle ldquokeV neutrino model buildingrdquo International Journalof Modern Physics D vol 22 no 10 Article ID 1330020 2013

[35] J DVergados andT S Kosmas ldquoSearching for cold darkmatterA case of coexistence of supersymmetry and nuclear physicsrdquoPhysics of Atomic Nuclei vol 61 p 1066 1998

[36] E Holmlund M Kortelainen T S Kosmas J Suhonen and JToivanen ldquoMicroscopic calculation of the LSP detection ratesfor the 71Ga 73Ge and 127I dark-matter detectorsrdquoPhysics LettersB vol 584 no 1-2 pp 31ndash39 2004

[37] R Adhikari M Agostini N Anh Ky et al ldquoA white paper onkeV sterile neutrino dark matterrdquo httpsarxivorgabs160204816

[38] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[39] MARE collaboration httpmaredfmuninsubriaitfrontendexecphp

[40] A Nucciotti ldquoNeutrino mass calorimetric searches in theMARE experimentrdquo httpsarxivorgabs10122290

[41] A Nucciotti ldquoOn behalf of the MARE collaboration lecture attheWorkshop ChalongeMeudonrdquo 2011 httpchalongeobspmfr

[42] KATRIN Collaboration httpwwwkatrinkitedu

[43] C Weinheimer ldquoDirect determination of neutrino mass fromtritium beta spectrumrdquo httpsarxivorgabs09121619

[44] C Weinheimer lecture at the 20th Paris Cosmology Chalonge-Hector de Vega Colloquium 2016 httpchalongeobspmfr

[45] G Drexlin lecture at the 19th Paris Cosmology Colloquium2015

[46] A Huber lecture at the Chalonge-de Vega Meudon Workshop2016 httpchalongeobspmfr

[47] C Tully lecture at the 19th Paris Cosmology Colloquium 2015[48] S Betts W R Blanchard R H Carnevale et al ldquoDevelop-

ment of a relic neutrino detection experiment at PTOLEMYprinceton tritiumobservatory for light early-universemassive-neutrino yieldrdquo httpsarxivorgabs13074738

[49] D M Asner R F Bradley L de Viveiros et al ldquoSingle-electrondetection and spectroscopy via relativistic cyclotron radiationrdquoPhysical Review Letters vol 114 no 16 Article ID 162501 2015

[50] P C-O Ranitzsch J-P Porst S Kempf et al ldquoDevelopmentof metallic magnetic calorimeters for high precision measure-ments of calorimetric 187Re and 163Ho spectrardquo Journal of LowTemperature Physics vol 167 no 5-6 pp 1004ndash1014 2012

[51] C Hassel ldquoLecture at the International School of AstrophysicsDaniel ChalongemdashHector de Vegardquo 2015

[52] L Gastaldo K Blaum A Doerr et al ldquoThe electron capture163Ho experiment ECHordquo Journal of Low Temperature Physicsvol 176 no 5 pp 876ndash884 2014

[53] B Alpert M Balata D Bennett et al ldquoThe electron capturedecay of 163Ho tomeasure the electron neutrino mass with sub-eV sensitivityrdquo The European Physical Journal C vol 75 article112 2015

[54] E Fermi ldquoRadioattivita prodotta da bombardamento di neu-tronirdquo Il Nuovo Cimento vol 11 no 7 pp 429ndash441 1934

[55] E Fermi ldquoVersuch einer theorie der 120573-strahlen Irdquo Zeitschriftfur Physik vol 88 no 3 pp 161ndash177 1934

[56] R B Firestone and V S Shirley Eds Table of Isotopes JohnWiley amp Sons New York NY USA 8th edition 1999

[57] W Bambynek H Behrens M H Chen et al ldquoOrbital electroncapture by the nucleusrdquo Reviews of Modern Physics vol 49 no1 pp 77ndash221 1977

[58] J L Campbell and T Papp ldquoWidths of the atomic KndashN7 levelsrdquoAtomic Data and Nuclear Data Tables vol 77 no 1 pp 1ndash562001

[59] J A Bearden and A F Burr ldquoReevaluation of X-ray atomicenergy levelsrdquo Reviews of Modern Physics vol 39 no 1 pp 125ndash142 1967

[60] M Cardona and L Ley Eds Photoemission in Solids I GeneralPrinciples Springer Berlin Germany 1978

[61] J C Fuggle and N Martensson ldquoCore-level binding energies inmetalsrdquo Journal of Electron Spectroscopy and Related Phenom-ena vol 21 no 3 pp 275ndash281 1980

[62] T Kosmas H Ejiri and A Hatzikoutelis ldquoNeutrino physics inthe frontiers of intensities and very high sensitivitiesrdquo Advancesin High Energy Physics vol 2015 Article ID 806067 3 pages2015

[63] A Giuliani and A Poves ldquoNeutrinoless double-beta decayrdquoAdvances in High Energy Physics vol 2012 Article ID 85701638 pages 2012

[64] J-U Nabi and H V Klapdor-Kleingrothaus ldquoWeak interactionrates of sd-shell nuclei in stellar environments calculated in theproton-neutron quasiparticle random-phase approximationrdquoAtomic Data andNuclear Data Tables vol 71 no 2 pp 149ndash3451999

12 Advances in High Energy Physics

[65] J-U Nabi and H V Klapdor-Kleingrothaus ldquoMicroscopic cal-culations of stellar weak interaction rates and energy losses forfp- and fpg-shell nucleirdquo Atomic Data and Nuclear Data Tablesvol 88 no 2 pp 237ndash476 2004

[66] Y Fukuda T Hayakawa E Ichihara et al ldquoEvidence foroscillation of atmospheric neutrinosrdquo Physical Review Lettersvol 81 no 8 pp 1562ndash1567 1998

[67] Q R Ahmad and SNOCollaboration ldquoMeasurement of the rateof V119890+119889 rarr 119901+119901+119890minus interactions produced by 8B solar neutrinosat the Sudbury Neutrino Observatoryrdquo Physical Review Lettersvol 87 no 7 Article ID 071301 6 pages 2001

[68] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 2002

[69] G J Feldman J Hartnell and T Kobayashi ldquoLong-baselineneutrino oscillation experimentsrdquo Advances in High EnergyPhysics vol 2013 Article ID 475749 30 pages 2013

[70] S T Petcov ldquoThenature ofmassive neutrinosrdquoAdvances inHighEnergy Physics vol 2013 Article ID 852987 20 pages 2013

[71] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[72] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 48 no 1-2 pp 1ndash28 2009

[73] F Munyaneza and P L Biermann ldquoDegenerate sterile neutrinodark matter in the cores of galaxiesrdquo Astronomy amp Astrophysicsvol 458 no 2 pp L9ndashL12 2006

[74] D Boyanovsky and C M Ho ldquoSterile neutrino production viaactive-sterile oscillations the quantum Zeno effectrdquo Journal ofHigh Energy Physics vol 2007 no 7 p 30 2007

[75] R Shrock ldquoNew tests for and bounds on neutrino masses andlepton mixingrdquo Physics Letters B vol 96 no 1-2 pp 159ndash1641980

[76] R E Shrock ldquoGeneral theory of weak processes involving neu-trinos I Leptonic pseudoscalar-meson decays with associatedtests for and bounds on neutrino masses and lepton mixingrdquoPhysical Review D vol 24 no 5 pp 1232ndash1274 1981

[77] R E Shrock ldquoImplications of neutrino masses and mixing forweak processesrdquo in Proceedings of the Weak Interactions asProbes of Unification vol 72 of AIP Conference Proceedings p368 Blacksburg Va USA 1981

[78] J M Conrad C M Ignarra G Karagiorgi M H Shaevitzand J Spitz ldquoSterile neutrino fits to short-baseline neutrinooscillation measurementsrdquo Advances in High Energy Physicsvol 2013 Article ID 163897 2013

[79] A Nucciotti ldquoThe use of low temperature detectors for directmeasurements of themass of the electron neutrinordquoAdvances inHigh Energy Physics vol 2016 Article ID 9153024 41 pages2016

[80] R G H Robertson ldquoExamination of the calorimetric spectrumto determine the neutrino mass in low-energy electron capturedecayrdquo Physical Review C vol 91 no 3 Article ID 035504 2015

[81] A De Rujula ldquoTwo old ways to measure the electron-neutrinomassrdquo httpsarxivorgabs13054857

[82] A De Rujula ldquoAn upper bound on the exceptional character-istics for Lusztigrsquos character formulardquo httpsarxivorgabs08111674v2

[83] A Faessler and F Simkovic ldquoCan electron capture tell us themass of the neutrinordquo Physica Scripta vol 91 no 4 Article ID043007 2016

[84] H J de Vega O Moreno E Moya de Guerra M RamonMedrano and N G Sanchez ldquoRole of sterile neutrino warmdark matter in rhenium and tritium beta decaysrdquo NuclearPhysics B vol 866 no 2 pp 177ndash195 2013

[85] E M de Guerra O Moreno P Sarriguren and M RamonMedrano ldquoTopics on nuclear structure with electroweakprobesrdquo Journal of Physics Conference Series vol 366 ArticleID 012011 2012

[86] J M Boillos O Moreno and E Moya de ldquoActive and sterileneutrino mass effects on beta decay spectrardquo in Proceedingsof the La Rabida International Scientific Meeting on NuclearPhysics Basic Concepts in Nuclear Physics Theory Experimentsand Applications vol 1541 ofAIP Conference Proceedings p 167La Rabida Spain September 2012

[87] P E Filianin K Blaum S A Eliseev et al ldquoOn the keVsterile neutrino search in electron capturerdquo Journal of PhysicsG Nuclear and Particle Physics vol 41 no 9 Article ID 0950042014

[88] S Eliseev K BlaumM Block et al ldquoDirect measurement of themass difference of 163Ho and 163Dy solves theQ-value puzzle forthe neutrino mass determinationrdquo Physical Review Letters vol115 no 6 Article ID 062501 2015

[89] A De Rujula ldquoA new way tomeasure neutrinomassesrdquoNuclearPhysics Section B vol 188 no 3 pp 414ndash458 1981

[90] M Lusignoli and M Vignati ldquoRelic antineutrino capture on163Hodecaying nucleirdquoPhysics Letters B vol 697 no 1 pp 11ndash142011

[91] A Faessler L Gastaldo and F Simkovic ldquoElectron capture in163Ho overlap plus exchange corrections and neutrino massrdquoJournal of Physics G Nuclear and Particle Physics vol 42 no 1Article ID 015108 2015

[92] G Audi F G Kondev M Wang et al ldquoThe Nubase2012evaluation of nuclear propertiesrdquo Chinese Physics C vol 36 no12 pp 1157ndash1286 2012

[93] S Mertens ldquoSensitivity of next-generation tritium beta-decayexperiments for keV-scale sterile neutrinosrdquo JCAP vol 1502 no2 p 20 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 9

Mixing but no heavy neutrino contribution

104

105

106

107

108d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 2 keV

(a)

104

105

106

107

108

d120582d

Ee

6 162 8 10 12 144 18 200Ee minus me (keV)

Mixing but no heavy neutrino contribution

No mixing (120577 = 0)Maximum (unphysical) mixing with mℎ = 7 keV

(b)

Figure 3 Electron spectrum of Tritium beta decay for a light-heavy neutrino mixing angle 120577 = 45∘ shown just for illustration and a heavyneutrino mass (solid curve)119898ℎ = 2 keV (a) and119898ℎ = 7 keV (b) The spectrum without heavy neutrino mass contribution (dotted curve) andwithout mixing (dashed curve) are also shown We use the maximal unrealistic value 120577 = 45∘ to make the effect visible in the scale of thefigure Realistic values of 120577 lt 001∘ result in a reduction of the difference between the dotted and the solid lines by more than seven orders ofmagnitude

10minus11

10minus10

10minus9

10minus8

10minus7

10minus12

120577 = 001∘

120577 = 0005∘

120577 = 0001∘

15090 100 110 120 130 1408070pe (keV)

ℛ

Figure 4 Ratio R defined in (25) for the Tritium beta decay as afunction of the electronmomentum for a heavy neutrinomass119898ℎ =2 keV and different values of the mixing angle 120577 = 001∘ (solid line)0005∘ (dashed line) and 0001∘ (dotted line)

1205772 It also decreases for larger heavy neutrino masses 119898ℎ Adifferent ratio between the spectra with mixing and without

mixing (Rlowast) has also been considered in [84ndash86 93] relatedwithR in (25) as

Rlowast = minussin2120577 +R cos2120577 = 119889120582119889119864119890119889120582119897119889119864119890 minus 1 (27)

7 Conclusions

Signatures of hypothetical keV sterile neutrinos which couldbe warm dark matter candidates (WDM) may be found inthe spectra of ordinary weak nuclear decays The electronspectrum in beta decay is cleaner for this purpose becauseit is a smooth curve while the deexcitation spectrum afterelectron capture shows many peaks On the other hand inbeta decay the possible excitation of the atoms or moleculesis not disentangled while in electron capture the calorimetercollects all the energy independently of the excitation ofthe atom and of the deexcitation path and one has higherstatistics around the capture peaks

Figures of electron spectrum for beta decay in 3H aswell as of calorimeter spectrum for electron capture in 163Hoand 202Pb and 205Pb are given considering various valuesof the heavy neutrino mass (2 keV 7 keV and 40 keV) thatmay be experimentally probed In both weak processes thesmall value of the light-heavy neutrino mixing angle requiresextremely high experimental precision and the use of sourceswith large stability to reduce systematic errors This is why it

10 Advances in High Energy Physics

is useful to consider relevant ratios between transition rates asfirst introduced in [84ndash87] to analyze the data We considertwo cases the case of a single isotope where one may use theratio of accumulated number of events in different regionsof the spectrum which allows us to remove uncertaintiesrelated to the nuclear matrix element and to the valuesof overall constants And in the case of two isotopes weconsider ratios of the above mentioned ratios that allow usto reduce uncertainties from atomic parameters We give aswell analytical expressions that can be used to obtain a goodtheoretical approximation to the experimental ratios ((19) to(21))

Both the experimental measurement and the theoreticalmodel must be accurate enough to detect differences in theexpected versus themeasured ratios of the order of themixingangle squared 1205772 ≲ 10minus8 This is also the size of the kinkexpected in the electron spectrumof beta decay located at thelimit of the regionwhere the production of a heavy neutrino isenergetically allowed In order to identify this kink among thestatistical fluctuations of themeasured spectrum the numberof collected events must be larger than the inverse of thesquared ratioR in (25)

Our results particularly on the electron capture spectraof Lead isotopes that include for the first time all thepossible peaks may help in designing and analyzing futureexperiments to search for sterile neutrinos in the keV massrangeThe use of the different types of ratios between numberof events discussed here for both electron capture and betadecay processes may also help in planning the experimentsby establishing the threshold of statistical and systematicuncertainties required for the detection of sterile neutrinosof given mass and mixing or to properly extract exclusionplots

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

E Moya de Guerra and O Moreno acknowledge MINECOfor partial financial support (FIS2014-51971-P) and IEM-CSIC for hospitality O Moreno acknowledges supportfrom a Marie Curie International Outgoing Fellowshipwithin the European Union Seventh Framework Pro-gramme under Grant Agreement PIOF-GA-2011-298364(ELECTROWEAK) M R M thanks N G Sanchez forcorrespondence and conversations

References

[1] F Zwicky ldquoDieRotverschiebung von extragalaktischenNebelnrdquoHelvetica Physica Acta vol 6 pp 110ndash127 1933

[2] M Roos ldquoAstrophysical and cosmological probes of darkmatterrdquo Journal of Modern Physics vol 3 pp 1152ndash1171 2012

[3] S Dodelson Modern Cosmology Academic Press New YorkNY USA 2003

[4] S Weinberg Cosmology Oxford University Press Oxford UK2008

[5] J Lesgourgues and S Pastor ldquoNeutrino mass from cosmologyrdquoAdvances in High Energy Physics vol 2012 Article ID 60851534 pages 2012

[6] D Boyanovsky H J de Vega and N G Sanchez ldquoConstraintson dark matter particles from theory galaxy observations andN-body simulationsrdquo Physical Review D vol 77 no 4 ArticleID 043518 2008

[7] D Boyanovsky H J de Vega and N G Sanchez ldquoDarkmatter transfer function free streaming particle statistics andmemory of gravitational clusteringrdquo Physical Review D vol 78no 6 Article ID 063546 2008

[8] C Destri H J de Vega and N G Sanchez ldquoWarm dark matterprimordial spectra and the onset of structure formation atredshift zrdquo Physical Review D vol 88 no 8 Article ID 0835122013

[9] H J De Vega P Salucci and N G Sanchez ldquoThe mass ofthe dark matter particle theory and galaxy observationsrdquo NewAstronomy vol 17 no 7 pp 653ndash666 2012

[10] H J de Vega P Salucci and N G Sanchez ldquoObservationalrotation curves and density profiles versus the Thomas-Fermigalaxy structure theoryrdquoMonthlyNotices of the Royal Astronom-ical Society vol 442 no 3 pp 2717ndash2727 2014

[11] H J de Vega and N G Sanchez ldquoModel-independent analysisof dark matter points to a particle mass at the keV scalerdquoMonthly Notices of the Royal Astronomical Society vol 404 no2 pp 885ndash894 2010

[12] H J de Vega and N G Sanchez ldquoConstant surface gravity anddensity profile of dark matterrdquo International Journal of ModernPhysics A vol 26 no 6 pp 1057ndash1072 2011

[13] H J de Vega and N G Sanchez ldquoCosmological evolutionof warm dark matter fluctuations I Efficient computationalframework with Volterra integral equationsrdquo Physical ReviewDvol 85 no 4 Article ID 043516 20 pages 2012

[14] H J de Vega and N G Sanchez ldquoThe dark matter distributionfunction and halo thermalization from the Eddington equationin galaxiesrdquo International Journal of Modern Physics A vol 31no 13 Article ID 1650073 2016

[15] NMenci S Paduroiu P Salucci N Sanchez and C RWatsonldquolectures at the 20th Paris Cosmology Colloquium Chalonge-Hector de Vegardquo 2016 httpchalongeobspmfr

[16] E W Kolb and M S Turner The Early Universe AddisonWesley Boston Mass USA 1990

[17] J F Navarro C S Frenk and S D M White ldquoA universal den-sity profile from hierarchical clusteringrdquo Astrophysical JournalLetters vol 490 no 2 pp 493ndash508 1997

[18] W J G de Blok ldquoThe core-cusp problemrdquo Advances in Astron-omy vol 2010 Article ID 789293 14 pages 2010

[19] G Gilmore M I Wilkinson R F GWyse et al ldquoThe observedproperties of dark matter on small spatial scalesrdquo AstrophysicalJournal vol 663 no 2 pp 948ndash959 2007

[20] P Salucci and Ch Frigerio Martins ldquoThe mass distribution inSpiral galaxiesrdquo EAS Publications Series vol 36 pp 133ndash1402009

[21] J van Eymeren C Trachternach B S Koribalski and R-JDettmar ldquoNon-circular motions and the cusp-core discrepancyin dwarf galaxiesrdquo Astronomy amp Astrophysics vol 505 no 1 pp1ndash20 2009

[22] M Walker and J Penarrubia ldquoA method for measuring (slopesof) the mass profiles of dwarf spheroidal galaxiesrdquo The Astro-physical Journal vol 742 no 1 p 20 2011

Advances in High Energy Physics 11

[23] R F G Wyse and G Gilmore ldquoObserved properties of darkmatter on small spatial scalesrdquo in Proceedings of the Proceedingsof the International Astronomical Union Symposium (IUA rsquo07)vol 3 no 244 pp 44ndash52 Cardiff UK June 2007

[24] A Burkert ldquoThe structure of dark matter halos in dwarfgalaxiesrdquo Astrophysical Journal vol 447 no 1 pp L25ndashL281995

[25] C Destri H J de Vega and N G Sanchez ldquoQuantum WDMfermions and gravitation determine the observed galaxy struc-turesrdquo Astroparticle Physics vol 46 pp 14ndash22 2013

[26] N Menci N G Sanchez M Castellano and A GrazianldquoConstraining the warm dark matter particle mass throughultra-deep UV luminosity functions at Z = 2rdquoTheAstrophysicalJournal vol 818 no 1 article 90 2016

[27] E Bulbul M Markevitch A Foster R K Smith M Loewen-stein and SW Randall ldquoDetection of an unidentified emissionline in the stackedX-ray spectrumof galaxy clustersrdquoAstrophys-ical Journal vol 789 article 13 2014

[28] A Boyarsky O Ruchayskiy D Iakubovskyi and J FranseldquoUnidentified line inX-ray spectra of the andromeda galaxy andperseus galaxy clusterrdquo Physical Review Letters vol 113 no 25Article ID 251301 2014

[29] K N Abazajian ldquoResonantly produced 7 keV sterile neutrinodark matter models and the properties of Milky Way satellitesrdquoPhysical Review Letters vol 112 no 16 Article ID 161303 5pages 2014

[30] N Menci A Grazian M Castellano and N G SanchezldquoShock connectivity in the 2010 August and 2012 July solarenergetic particle events inferred from observations and ENLILmodelingrdquo Astrophysical Journal Letters vol 825 no 1 article 12016

[31] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[32] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 481 no 1-2 pp 1ndash28 2009

[33] B Pontecorvo ldquoNeutrino experiments and the problem ofconservation of leptonic chargerdquo Soviet Physics-JETP vol 26no 5 pp 984ndash988 1968

[34] A Merle ldquokeV neutrino model buildingrdquo International Journalof Modern Physics D vol 22 no 10 Article ID 1330020 2013

[35] J DVergados andT S Kosmas ldquoSearching for cold darkmatterA case of coexistence of supersymmetry and nuclear physicsrdquoPhysics of Atomic Nuclei vol 61 p 1066 1998

[36] E Holmlund M Kortelainen T S Kosmas J Suhonen and JToivanen ldquoMicroscopic calculation of the LSP detection ratesfor the 71Ga 73Ge and 127I dark-matter detectorsrdquoPhysics LettersB vol 584 no 1-2 pp 31ndash39 2004

[37] R Adhikari M Agostini N Anh Ky et al ldquoA white paper onkeV sterile neutrino dark matterrdquo httpsarxivorgabs160204816

[38] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[39] MARE collaboration httpmaredfmuninsubriaitfrontendexecphp

[40] A Nucciotti ldquoNeutrino mass calorimetric searches in theMARE experimentrdquo httpsarxivorgabs10122290

[41] A Nucciotti ldquoOn behalf of the MARE collaboration lecture attheWorkshop ChalongeMeudonrdquo 2011 httpchalongeobspmfr

[42] KATRIN Collaboration httpwwwkatrinkitedu

[43] C Weinheimer ldquoDirect determination of neutrino mass fromtritium beta spectrumrdquo httpsarxivorgabs09121619

[44] C Weinheimer lecture at the 20th Paris Cosmology Chalonge-Hector de Vega Colloquium 2016 httpchalongeobspmfr

[45] G Drexlin lecture at the 19th Paris Cosmology Colloquium2015

[46] A Huber lecture at the Chalonge-de Vega Meudon Workshop2016 httpchalongeobspmfr

[47] C Tully lecture at the 19th Paris Cosmology Colloquium 2015[48] S Betts W R Blanchard R H Carnevale et al ldquoDevelop-

ment of a relic neutrino detection experiment at PTOLEMYprinceton tritiumobservatory for light early-universemassive-neutrino yieldrdquo httpsarxivorgabs13074738

[49] D M Asner R F Bradley L de Viveiros et al ldquoSingle-electrondetection and spectroscopy via relativistic cyclotron radiationrdquoPhysical Review Letters vol 114 no 16 Article ID 162501 2015

[50] P C-O Ranitzsch J-P Porst S Kempf et al ldquoDevelopmentof metallic magnetic calorimeters for high precision measure-ments of calorimetric 187Re and 163Ho spectrardquo Journal of LowTemperature Physics vol 167 no 5-6 pp 1004ndash1014 2012

[51] C Hassel ldquoLecture at the International School of AstrophysicsDaniel ChalongemdashHector de Vegardquo 2015

[52] L Gastaldo K Blaum A Doerr et al ldquoThe electron capture163Ho experiment ECHordquo Journal of Low Temperature Physicsvol 176 no 5 pp 876ndash884 2014

[53] B Alpert M Balata D Bennett et al ldquoThe electron capturedecay of 163Ho tomeasure the electron neutrino mass with sub-eV sensitivityrdquo The European Physical Journal C vol 75 article112 2015

[54] E Fermi ldquoRadioattivita prodotta da bombardamento di neu-tronirdquo Il Nuovo Cimento vol 11 no 7 pp 429ndash441 1934

[55] E Fermi ldquoVersuch einer theorie der 120573-strahlen Irdquo Zeitschriftfur Physik vol 88 no 3 pp 161ndash177 1934

[56] R B Firestone and V S Shirley Eds Table of Isotopes JohnWiley amp Sons New York NY USA 8th edition 1999

[57] W Bambynek H Behrens M H Chen et al ldquoOrbital electroncapture by the nucleusrdquo Reviews of Modern Physics vol 49 no1 pp 77ndash221 1977

[58] J L Campbell and T Papp ldquoWidths of the atomic KndashN7 levelsrdquoAtomic Data and Nuclear Data Tables vol 77 no 1 pp 1ndash562001

[59] J A Bearden and A F Burr ldquoReevaluation of X-ray atomicenergy levelsrdquo Reviews of Modern Physics vol 39 no 1 pp 125ndash142 1967

[60] M Cardona and L Ley Eds Photoemission in Solids I GeneralPrinciples Springer Berlin Germany 1978

[61] J C Fuggle and N Martensson ldquoCore-level binding energies inmetalsrdquo Journal of Electron Spectroscopy and Related Phenom-ena vol 21 no 3 pp 275ndash281 1980

[62] T Kosmas H Ejiri and A Hatzikoutelis ldquoNeutrino physics inthe frontiers of intensities and very high sensitivitiesrdquo Advancesin High Energy Physics vol 2015 Article ID 806067 3 pages2015

[63] A Giuliani and A Poves ldquoNeutrinoless double-beta decayrdquoAdvances in High Energy Physics vol 2012 Article ID 85701638 pages 2012

[64] J-U Nabi and H V Klapdor-Kleingrothaus ldquoWeak interactionrates of sd-shell nuclei in stellar environments calculated in theproton-neutron quasiparticle random-phase approximationrdquoAtomic Data andNuclear Data Tables vol 71 no 2 pp 149ndash3451999

12 Advances in High Energy Physics

[65] J-U Nabi and H V Klapdor-Kleingrothaus ldquoMicroscopic cal-culations of stellar weak interaction rates and energy losses forfp- and fpg-shell nucleirdquo Atomic Data and Nuclear Data Tablesvol 88 no 2 pp 237ndash476 2004

[66] Y Fukuda T Hayakawa E Ichihara et al ldquoEvidence foroscillation of atmospheric neutrinosrdquo Physical Review Lettersvol 81 no 8 pp 1562ndash1567 1998

[67] Q R Ahmad and SNOCollaboration ldquoMeasurement of the rateof V119890+119889 rarr 119901+119901+119890minus interactions produced by 8B solar neutrinosat the Sudbury Neutrino Observatoryrdquo Physical Review Lettersvol 87 no 7 Article ID 071301 6 pages 2001

[68] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 2002

[69] G J Feldman J Hartnell and T Kobayashi ldquoLong-baselineneutrino oscillation experimentsrdquo Advances in High EnergyPhysics vol 2013 Article ID 475749 30 pages 2013

[70] S T Petcov ldquoThenature ofmassive neutrinosrdquoAdvances inHighEnergy Physics vol 2013 Article ID 852987 20 pages 2013

[71] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[72] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 48 no 1-2 pp 1ndash28 2009

[73] F Munyaneza and P L Biermann ldquoDegenerate sterile neutrinodark matter in the cores of galaxiesrdquo Astronomy amp Astrophysicsvol 458 no 2 pp L9ndashL12 2006

[74] D Boyanovsky and C M Ho ldquoSterile neutrino production viaactive-sterile oscillations the quantum Zeno effectrdquo Journal ofHigh Energy Physics vol 2007 no 7 p 30 2007

[75] R Shrock ldquoNew tests for and bounds on neutrino masses andlepton mixingrdquo Physics Letters B vol 96 no 1-2 pp 159ndash1641980

[76] R E Shrock ldquoGeneral theory of weak processes involving neu-trinos I Leptonic pseudoscalar-meson decays with associatedtests for and bounds on neutrino masses and lepton mixingrdquoPhysical Review D vol 24 no 5 pp 1232ndash1274 1981

[77] R E Shrock ldquoImplications of neutrino masses and mixing forweak processesrdquo in Proceedings of the Weak Interactions asProbes of Unification vol 72 of AIP Conference Proceedings p368 Blacksburg Va USA 1981

[78] J M Conrad C M Ignarra G Karagiorgi M H Shaevitzand J Spitz ldquoSterile neutrino fits to short-baseline neutrinooscillation measurementsrdquo Advances in High Energy Physicsvol 2013 Article ID 163897 2013

[79] A Nucciotti ldquoThe use of low temperature detectors for directmeasurements of themass of the electron neutrinordquoAdvances inHigh Energy Physics vol 2016 Article ID 9153024 41 pages2016

[80] R G H Robertson ldquoExamination of the calorimetric spectrumto determine the neutrino mass in low-energy electron capturedecayrdquo Physical Review C vol 91 no 3 Article ID 035504 2015

[81] A De Rujula ldquoTwo old ways to measure the electron-neutrinomassrdquo httpsarxivorgabs13054857

[82] A De Rujula ldquoAn upper bound on the exceptional character-istics for Lusztigrsquos character formulardquo httpsarxivorgabs08111674v2

[83] A Faessler and F Simkovic ldquoCan electron capture tell us themass of the neutrinordquo Physica Scripta vol 91 no 4 Article ID043007 2016

[84] H J de Vega O Moreno E Moya de Guerra M RamonMedrano and N G Sanchez ldquoRole of sterile neutrino warmdark matter in rhenium and tritium beta decaysrdquo NuclearPhysics B vol 866 no 2 pp 177ndash195 2013

[85] E M de Guerra O Moreno P Sarriguren and M RamonMedrano ldquoTopics on nuclear structure with electroweakprobesrdquo Journal of Physics Conference Series vol 366 ArticleID 012011 2012

[86] J M Boillos O Moreno and E Moya de ldquoActive and sterileneutrino mass effects on beta decay spectrardquo in Proceedingsof the La Rabida International Scientific Meeting on NuclearPhysics Basic Concepts in Nuclear Physics Theory Experimentsand Applications vol 1541 ofAIP Conference Proceedings p 167La Rabida Spain September 2012

[87] P E Filianin K Blaum S A Eliseev et al ldquoOn the keVsterile neutrino search in electron capturerdquo Journal of PhysicsG Nuclear and Particle Physics vol 41 no 9 Article ID 0950042014

[88] S Eliseev K BlaumM Block et al ldquoDirect measurement of themass difference of 163Ho and 163Dy solves theQ-value puzzle forthe neutrino mass determinationrdquo Physical Review Letters vol115 no 6 Article ID 062501 2015

[89] A De Rujula ldquoA new way tomeasure neutrinomassesrdquoNuclearPhysics Section B vol 188 no 3 pp 414ndash458 1981

[90] M Lusignoli and M Vignati ldquoRelic antineutrino capture on163Hodecaying nucleirdquoPhysics Letters B vol 697 no 1 pp 11ndash142011

[91] A Faessler L Gastaldo and F Simkovic ldquoElectron capture in163Ho overlap plus exchange corrections and neutrino massrdquoJournal of Physics G Nuclear and Particle Physics vol 42 no 1Article ID 015108 2015

[92] G Audi F G Kondev M Wang et al ldquoThe Nubase2012evaluation of nuclear propertiesrdquo Chinese Physics C vol 36 no12 pp 1157ndash1286 2012

[93] S Mertens ldquoSensitivity of next-generation tritium beta-decayexperiments for keV-scale sterile neutrinosrdquo JCAP vol 1502 no2 p 20 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Advances in High Energy Physics

is useful to consider relevant ratios between transition rates asfirst introduced in [84ndash87] to analyze the data We considertwo cases the case of a single isotope where one may use theratio of accumulated number of events in different regionsof the spectrum which allows us to remove uncertaintiesrelated to the nuclear matrix element and to the valuesof overall constants And in the case of two isotopes weconsider ratios of the above mentioned ratios that allow usto reduce uncertainties from atomic parameters We give aswell analytical expressions that can be used to obtain a goodtheoretical approximation to the experimental ratios ((19) to(21))

Both the experimental measurement and the theoreticalmodel must be accurate enough to detect differences in theexpected versus themeasured ratios of the order of themixingangle squared 1205772 ≲ 10minus8 This is also the size of the kinkexpected in the electron spectrumof beta decay located at thelimit of the regionwhere the production of a heavy neutrino isenergetically allowed In order to identify this kink among thestatistical fluctuations of themeasured spectrum the numberof collected events must be larger than the inverse of thesquared ratioR in (25)

Our results particularly on the electron capture spectraof Lead isotopes that include for the first time all thepossible peaks may help in designing and analyzing futureexperiments to search for sterile neutrinos in the keV massrangeThe use of the different types of ratios between numberof events discussed here for both electron capture and betadecay processes may also help in planning the experimentsby establishing the threshold of statistical and systematicuncertainties required for the detection of sterile neutrinosof given mass and mixing or to properly extract exclusionplots

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

E Moya de Guerra and O Moreno acknowledge MINECOfor partial financial support (FIS2014-51971-P) and IEM-CSIC for hospitality O Moreno acknowledges supportfrom a Marie Curie International Outgoing Fellowshipwithin the European Union Seventh Framework Pro-gramme under Grant Agreement PIOF-GA-2011-298364(ELECTROWEAK) M R M thanks N G Sanchez forcorrespondence and conversations

References

[1] F Zwicky ldquoDieRotverschiebung von extragalaktischenNebelnrdquoHelvetica Physica Acta vol 6 pp 110ndash127 1933

[2] M Roos ldquoAstrophysical and cosmological probes of darkmatterrdquo Journal of Modern Physics vol 3 pp 1152ndash1171 2012

[3] S Dodelson Modern Cosmology Academic Press New YorkNY USA 2003

[4] S Weinberg Cosmology Oxford University Press Oxford UK2008

[5] J Lesgourgues and S Pastor ldquoNeutrino mass from cosmologyrdquoAdvances in High Energy Physics vol 2012 Article ID 60851534 pages 2012

[6] D Boyanovsky H J de Vega and N G Sanchez ldquoConstraintson dark matter particles from theory galaxy observations andN-body simulationsrdquo Physical Review D vol 77 no 4 ArticleID 043518 2008

[7] D Boyanovsky H J de Vega and N G Sanchez ldquoDarkmatter transfer function free streaming particle statistics andmemory of gravitational clusteringrdquo Physical Review D vol 78no 6 Article ID 063546 2008

[8] C Destri H J de Vega and N G Sanchez ldquoWarm dark matterprimordial spectra and the onset of structure formation atredshift zrdquo Physical Review D vol 88 no 8 Article ID 0835122013

[9] H J De Vega P Salucci and N G Sanchez ldquoThe mass ofthe dark matter particle theory and galaxy observationsrdquo NewAstronomy vol 17 no 7 pp 653ndash666 2012

[10] H J de Vega P Salucci and N G Sanchez ldquoObservationalrotation curves and density profiles versus the Thomas-Fermigalaxy structure theoryrdquoMonthlyNotices of the Royal Astronom-ical Society vol 442 no 3 pp 2717ndash2727 2014

[11] H J de Vega and N G Sanchez ldquoModel-independent analysisof dark matter points to a particle mass at the keV scalerdquoMonthly Notices of the Royal Astronomical Society vol 404 no2 pp 885ndash894 2010

[12] H J de Vega and N G Sanchez ldquoConstant surface gravity anddensity profile of dark matterrdquo International Journal of ModernPhysics A vol 26 no 6 pp 1057ndash1072 2011

[13] H J de Vega and N G Sanchez ldquoCosmological evolutionof warm dark matter fluctuations I Efficient computationalframework with Volterra integral equationsrdquo Physical ReviewDvol 85 no 4 Article ID 043516 20 pages 2012

[14] H J de Vega and N G Sanchez ldquoThe dark matter distributionfunction and halo thermalization from the Eddington equationin galaxiesrdquo International Journal of Modern Physics A vol 31no 13 Article ID 1650073 2016

[15] NMenci S Paduroiu P Salucci N Sanchez and C RWatsonldquolectures at the 20th Paris Cosmology Colloquium Chalonge-Hector de Vegardquo 2016 httpchalongeobspmfr

[16] E W Kolb and M S Turner The Early Universe AddisonWesley Boston Mass USA 1990

[17] J F Navarro C S Frenk and S D M White ldquoA universal den-sity profile from hierarchical clusteringrdquo Astrophysical JournalLetters vol 490 no 2 pp 493ndash508 1997

[18] W J G de Blok ldquoThe core-cusp problemrdquo Advances in Astron-omy vol 2010 Article ID 789293 14 pages 2010

[19] G Gilmore M I Wilkinson R F GWyse et al ldquoThe observedproperties of dark matter on small spatial scalesrdquo AstrophysicalJournal vol 663 no 2 pp 948ndash959 2007

[20] P Salucci and Ch Frigerio Martins ldquoThe mass distribution inSpiral galaxiesrdquo EAS Publications Series vol 36 pp 133ndash1402009

[21] J van Eymeren C Trachternach B S Koribalski and R-JDettmar ldquoNon-circular motions and the cusp-core discrepancyin dwarf galaxiesrdquo Astronomy amp Astrophysics vol 505 no 1 pp1ndash20 2009

[22] M Walker and J Penarrubia ldquoA method for measuring (slopesof) the mass profiles of dwarf spheroidal galaxiesrdquo The Astro-physical Journal vol 742 no 1 p 20 2011

Advances in High Energy Physics 11

[23] R F G Wyse and G Gilmore ldquoObserved properties of darkmatter on small spatial scalesrdquo in Proceedings of the Proceedingsof the International Astronomical Union Symposium (IUA rsquo07)vol 3 no 244 pp 44ndash52 Cardiff UK June 2007

[24] A Burkert ldquoThe structure of dark matter halos in dwarfgalaxiesrdquo Astrophysical Journal vol 447 no 1 pp L25ndashL281995

[25] C Destri H J de Vega and N G Sanchez ldquoQuantum WDMfermions and gravitation determine the observed galaxy struc-turesrdquo Astroparticle Physics vol 46 pp 14ndash22 2013

[26] N Menci N G Sanchez M Castellano and A GrazianldquoConstraining the warm dark matter particle mass throughultra-deep UV luminosity functions at Z = 2rdquoTheAstrophysicalJournal vol 818 no 1 article 90 2016

[27] E Bulbul M Markevitch A Foster R K Smith M Loewen-stein and SW Randall ldquoDetection of an unidentified emissionline in the stackedX-ray spectrumof galaxy clustersrdquoAstrophys-ical Journal vol 789 article 13 2014

[28] A Boyarsky O Ruchayskiy D Iakubovskyi and J FranseldquoUnidentified line inX-ray spectra of the andromeda galaxy andperseus galaxy clusterrdquo Physical Review Letters vol 113 no 25Article ID 251301 2014

[29] K N Abazajian ldquoResonantly produced 7 keV sterile neutrinodark matter models and the properties of Milky Way satellitesrdquoPhysical Review Letters vol 112 no 16 Article ID 161303 5pages 2014

[30] N Menci A Grazian M Castellano and N G SanchezldquoShock connectivity in the 2010 August and 2012 July solarenergetic particle events inferred from observations and ENLILmodelingrdquo Astrophysical Journal Letters vol 825 no 1 article 12016

[31] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[32] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 481 no 1-2 pp 1ndash28 2009

[33] B Pontecorvo ldquoNeutrino experiments and the problem ofconservation of leptonic chargerdquo Soviet Physics-JETP vol 26no 5 pp 984ndash988 1968

[34] A Merle ldquokeV neutrino model buildingrdquo International Journalof Modern Physics D vol 22 no 10 Article ID 1330020 2013

[35] J DVergados andT S Kosmas ldquoSearching for cold darkmatterA case of coexistence of supersymmetry and nuclear physicsrdquoPhysics of Atomic Nuclei vol 61 p 1066 1998

[36] E Holmlund M Kortelainen T S Kosmas J Suhonen and JToivanen ldquoMicroscopic calculation of the LSP detection ratesfor the 71Ga 73Ge and 127I dark-matter detectorsrdquoPhysics LettersB vol 584 no 1-2 pp 31ndash39 2004

[37] R Adhikari M Agostini N Anh Ky et al ldquoA white paper onkeV sterile neutrino dark matterrdquo httpsarxivorgabs160204816

[38] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[39] MARE collaboration httpmaredfmuninsubriaitfrontendexecphp

[40] A Nucciotti ldquoNeutrino mass calorimetric searches in theMARE experimentrdquo httpsarxivorgabs10122290

[41] A Nucciotti ldquoOn behalf of the MARE collaboration lecture attheWorkshop ChalongeMeudonrdquo 2011 httpchalongeobspmfr

[42] KATRIN Collaboration httpwwwkatrinkitedu

[43] C Weinheimer ldquoDirect determination of neutrino mass fromtritium beta spectrumrdquo httpsarxivorgabs09121619

[44] C Weinheimer lecture at the 20th Paris Cosmology Chalonge-Hector de Vega Colloquium 2016 httpchalongeobspmfr

[45] G Drexlin lecture at the 19th Paris Cosmology Colloquium2015

[46] A Huber lecture at the Chalonge-de Vega Meudon Workshop2016 httpchalongeobspmfr

[47] C Tully lecture at the 19th Paris Cosmology Colloquium 2015[48] S Betts W R Blanchard R H Carnevale et al ldquoDevelop-

ment of a relic neutrino detection experiment at PTOLEMYprinceton tritiumobservatory for light early-universemassive-neutrino yieldrdquo httpsarxivorgabs13074738

[49] D M Asner R F Bradley L de Viveiros et al ldquoSingle-electrondetection and spectroscopy via relativistic cyclotron radiationrdquoPhysical Review Letters vol 114 no 16 Article ID 162501 2015

[50] P C-O Ranitzsch J-P Porst S Kempf et al ldquoDevelopmentof metallic magnetic calorimeters for high precision measure-ments of calorimetric 187Re and 163Ho spectrardquo Journal of LowTemperature Physics vol 167 no 5-6 pp 1004ndash1014 2012

[51] C Hassel ldquoLecture at the International School of AstrophysicsDaniel ChalongemdashHector de Vegardquo 2015

[52] L Gastaldo K Blaum A Doerr et al ldquoThe electron capture163Ho experiment ECHordquo Journal of Low Temperature Physicsvol 176 no 5 pp 876ndash884 2014

[53] B Alpert M Balata D Bennett et al ldquoThe electron capturedecay of 163Ho tomeasure the electron neutrino mass with sub-eV sensitivityrdquo The European Physical Journal C vol 75 article112 2015

[54] E Fermi ldquoRadioattivita prodotta da bombardamento di neu-tronirdquo Il Nuovo Cimento vol 11 no 7 pp 429ndash441 1934

[55] E Fermi ldquoVersuch einer theorie der 120573-strahlen Irdquo Zeitschriftfur Physik vol 88 no 3 pp 161ndash177 1934

[56] R B Firestone and V S Shirley Eds Table of Isotopes JohnWiley amp Sons New York NY USA 8th edition 1999

[57] W Bambynek H Behrens M H Chen et al ldquoOrbital electroncapture by the nucleusrdquo Reviews of Modern Physics vol 49 no1 pp 77ndash221 1977

[58] J L Campbell and T Papp ldquoWidths of the atomic KndashN7 levelsrdquoAtomic Data and Nuclear Data Tables vol 77 no 1 pp 1ndash562001

[59] J A Bearden and A F Burr ldquoReevaluation of X-ray atomicenergy levelsrdquo Reviews of Modern Physics vol 39 no 1 pp 125ndash142 1967

[60] M Cardona and L Ley Eds Photoemission in Solids I GeneralPrinciples Springer Berlin Germany 1978

[61] J C Fuggle and N Martensson ldquoCore-level binding energies inmetalsrdquo Journal of Electron Spectroscopy and Related Phenom-ena vol 21 no 3 pp 275ndash281 1980

[62] T Kosmas H Ejiri and A Hatzikoutelis ldquoNeutrino physics inthe frontiers of intensities and very high sensitivitiesrdquo Advancesin High Energy Physics vol 2015 Article ID 806067 3 pages2015

[63] A Giuliani and A Poves ldquoNeutrinoless double-beta decayrdquoAdvances in High Energy Physics vol 2012 Article ID 85701638 pages 2012

[64] J-U Nabi and H V Klapdor-Kleingrothaus ldquoWeak interactionrates of sd-shell nuclei in stellar environments calculated in theproton-neutron quasiparticle random-phase approximationrdquoAtomic Data andNuclear Data Tables vol 71 no 2 pp 149ndash3451999

12 Advances in High Energy Physics

[65] J-U Nabi and H V Klapdor-Kleingrothaus ldquoMicroscopic cal-culations of stellar weak interaction rates and energy losses forfp- and fpg-shell nucleirdquo Atomic Data and Nuclear Data Tablesvol 88 no 2 pp 237ndash476 2004

[66] Y Fukuda T Hayakawa E Ichihara et al ldquoEvidence foroscillation of atmospheric neutrinosrdquo Physical Review Lettersvol 81 no 8 pp 1562ndash1567 1998

[67] Q R Ahmad and SNOCollaboration ldquoMeasurement of the rateof V119890+119889 rarr 119901+119901+119890minus interactions produced by 8B solar neutrinosat the Sudbury Neutrino Observatoryrdquo Physical Review Lettersvol 87 no 7 Article ID 071301 6 pages 2001

[68] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 2002

[69] G J Feldman J Hartnell and T Kobayashi ldquoLong-baselineneutrino oscillation experimentsrdquo Advances in High EnergyPhysics vol 2013 Article ID 475749 30 pages 2013

[70] S T Petcov ldquoThenature ofmassive neutrinosrdquoAdvances inHighEnergy Physics vol 2013 Article ID 852987 20 pages 2013

[71] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[72] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 48 no 1-2 pp 1ndash28 2009

[73] F Munyaneza and P L Biermann ldquoDegenerate sterile neutrinodark matter in the cores of galaxiesrdquo Astronomy amp Astrophysicsvol 458 no 2 pp L9ndashL12 2006

[74] D Boyanovsky and C M Ho ldquoSterile neutrino production viaactive-sterile oscillations the quantum Zeno effectrdquo Journal ofHigh Energy Physics vol 2007 no 7 p 30 2007

[75] R Shrock ldquoNew tests for and bounds on neutrino masses andlepton mixingrdquo Physics Letters B vol 96 no 1-2 pp 159ndash1641980

[76] R E Shrock ldquoGeneral theory of weak processes involving neu-trinos I Leptonic pseudoscalar-meson decays with associatedtests for and bounds on neutrino masses and lepton mixingrdquoPhysical Review D vol 24 no 5 pp 1232ndash1274 1981

[77] R E Shrock ldquoImplications of neutrino masses and mixing forweak processesrdquo in Proceedings of the Weak Interactions asProbes of Unification vol 72 of AIP Conference Proceedings p368 Blacksburg Va USA 1981

[78] J M Conrad C M Ignarra G Karagiorgi M H Shaevitzand J Spitz ldquoSterile neutrino fits to short-baseline neutrinooscillation measurementsrdquo Advances in High Energy Physicsvol 2013 Article ID 163897 2013

[79] A Nucciotti ldquoThe use of low temperature detectors for directmeasurements of themass of the electron neutrinordquoAdvances inHigh Energy Physics vol 2016 Article ID 9153024 41 pages2016

[80] R G H Robertson ldquoExamination of the calorimetric spectrumto determine the neutrino mass in low-energy electron capturedecayrdquo Physical Review C vol 91 no 3 Article ID 035504 2015

[81] A De Rujula ldquoTwo old ways to measure the electron-neutrinomassrdquo httpsarxivorgabs13054857

[82] A De Rujula ldquoAn upper bound on the exceptional character-istics for Lusztigrsquos character formulardquo httpsarxivorgabs08111674v2

[83] A Faessler and F Simkovic ldquoCan electron capture tell us themass of the neutrinordquo Physica Scripta vol 91 no 4 Article ID043007 2016

[84] H J de Vega O Moreno E Moya de Guerra M RamonMedrano and N G Sanchez ldquoRole of sterile neutrino warmdark matter in rhenium and tritium beta decaysrdquo NuclearPhysics B vol 866 no 2 pp 177ndash195 2013

[85] E M de Guerra O Moreno P Sarriguren and M RamonMedrano ldquoTopics on nuclear structure with electroweakprobesrdquo Journal of Physics Conference Series vol 366 ArticleID 012011 2012

[86] J M Boillos O Moreno and E Moya de ldquoActive and sterileneutrino mass effects on beta decay spectrardquo in Proceedingsof the La Rabida International Scientific Meeting on NuclearPhysics Basic Concepts in Nuclear Physics Theory Experimentsand Applications vol 1541 ofAIP Conference Proceedings p 167La Rabida Spain September 2012

[87] P E Filianin K Blaum S A Eliseev et al ldquoOn the keVsterile neutrino search in electron capturerdquo Journal of PhysicsG Nuclear and Particle Physics vol 41 no 9 Article ID 0950042014

[88] S Eliseev K BlaumM Block et al ldquoDirect measurement of themass difference of 163Ho and 163Dy solves theQ-value puzzle forthe neutrino mass determinationrdquo Physical Review Letters vol115 no 6 Article ID 062501 2015

[89] A De Rujula ldquoA new way tomeasure neutrinomassesrdquoNuclearPhysics Section B vol 188 no 3 pp 414ndash458 1981

[90] M Lusignoli and M Vignati ldquoRelic antineutrino capture on163Hodecaying nucleirdquoPhysics Letters B vol 697 no 1 pp 11ndash142011

[91] A Faessler L Gastaldo and F Simkovic ldquoElectron capture in163Ho overlap plus exchange corrections and neutrino massrdquoJournal of Physics G Nuclear and Particle Physics vol 42 no 1Article ID 015108 2015

[92] G Audi F G Kondev M Wang et al ldquoThe Nubase2012evaluation of nuclear propertiesrdquo Chinese Physics C vol 36 no12 pp 1157ndash1286 2012

[93] S Mertens ldquoSensitivity of next-generation tritium beta-decayexperiments for keV-scale sterile neutrinosrdquo JCAP vol 1502 no2 p 20 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 11

[23] R F G Wyse and G Gilmore ldquoObserved properties of darkmatter on small spatial scalesrdquo in Proceedings of the Proceedingsof the International Astronomical Union Symposium (IUA rsquo07)vol 3 no 244 pp 44ndash52 Cardiff UK June 2007

[24] A Burkert ldquoThe structure of dark matter halos in dwarfgalaxiesrdquo Astrophysical Journal vol 447 no 1 pp L25ndashL281995

[25] C Destri H J de Vega and N G Sanchez ldquoQuantum WDMfermions and gravitation determine the observed galaxy struc-turesrdquo Astroparticle Physics vol 46 pp 14ndash22 2013

[26] N Menci N G Sanchez M Castellano and A GrazianldquoConstraining the warm dark matter particle mass throughultra-deep UV luminosity functions at Z = 2rdquoTheAstrophysicalJournal vol 818 no 1 article 90 2016

[27] E Bulbul M Markevitch A Foster R K Smith M Loewen-stein and SW Randall ldquoDetection of an unidentified emissionline in the stackedX-ray spectrumof galaxy clustersrdquoAstrophys-ical Journal vol 789 article 13 2014

[28] A Boyarsky O Ruchayskiy D Iakubovskyi and J FranseldquoUnidentified line inX-ray spectra of the andromeda galaxy andperseus galaxy clusterrdquo Physical Review Letters vol 113 no 25Article ID 251301 2014

[29] K N Abazajian ldquoResonantly produced 7 keV sterile neutrinodark matter models and the properties of Milky Way satellitesrdquoPhysical Review Letters vol 112 no 16 Article ID 161303 5pages 2014

[30] N Menci A Grazian M Castellano and N G SanchezldquoShock connectivity in the 2010 August and 2012 July solarenergetic particle events inferred from observations and ENLILmodelingrdquo Astrophysical Journal Letters vol 825 no 1 article 12016

[31] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[32] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 481 no 1-2 pp 1ndash28 2009

[33] B Pontecorvo ldquoNeutrino experiments and the problem ofconservation of leptonic chargerdquo Soviet Physics-JETP vol 26no 5 pp 984ndash988 1968

[34] A Merle ldquokeV neutrino model buildingrdquo International Journalof Modern Physics D vol 22 no 10 Article ID 1330020 2013

[35] J DVergados andT S Kosmas ldquoSearching for cold darkmatterA case of coexistence of supersymmetry and nuclear physicsrdquoPhysics of Atomic Nuclei vol 61 p 1066 1998

[36] E Holmlund M Kortelainen T S Kosmas J Suhonen and JToivanen ldquoMicroscopic calculation of the LSP detection ratesfor the 71Ga 73Ge and 127I dark-matter detectorsrdquoPhysics LettersB vol 584 no 1-2 pp 31ndash39 2004

[37] R Adhikari M Agostini N Anh Ky et al ldquoA white paper onkeV sterile neutrino dark matterrdquo httpsarxivorgabs160204816

[38] G Drexlin V Hannen S Mertens and C WeinheimerldquoCurrent direct neutrino mass experimentsrdquo Advances in HighEnergy Physics vol 2013 Article ID 293986 39 pages 2013

[39] MARE collaboration httpmaredfmuninsubriaitfrontendexecphp

[40] A Nucciotti ldquoNeutrino mass calorimetric searches in theMARE experimentrdquo httpsarxivorgabs10122290

[41] A Nucciotti ldquoOn behalf of the MARE collaboration lecture attheWorkshop ChalongeMeudonrdquo 2011 httpchalongeobspmfr

[42] KATRIN Collaboration httpwwwkatrinkitedu

[43] C Weinheimer ldquoDirect determination of neutrino mass fromtritium beta spectrumrdquo httpsarxivorgabs09121619

[44] C Weinheimer lecture at the 20th Paris Cosmology Chalonge-Hector de Vega Colloquium 2016 httpchalongeobspmfr

[45] G Drexlin lecture at the 19th Paris Cosmology Colloquium2015

[46] A Huber lecture at the Chalonge-de Vega Meudon Workshop2016 httpchalongeobspmfr

[47] C Tully lecture at the 19th Paris Cosmology Colloquium 2015[48] S Betts W R Blanchard R H Carnevale et al ldquoDevelop-

ment of a relic neutrino detection experiment at PTOLEMYprinceton tritiumobservatory for light early-universemassive-neutrino yieldrdquo httpsarxivorgabs13074738

[49] D M Asner R F Bradley L de Viveiros et al ldquoSingle-electrondetection and spectroscopy via relativistic cyclotron radiationrdquoPhysical Review Letters vol 114 no 16 Article ID 162501 2015

[50] P C-O Ranitzsch J-P Porst S Kempf et al ldquoDevelopmentof metallic magnetic calorimeters for high precision measure-ments of calorimetric 187Re and 163Ho spectrardquo Journal of LowTemperature Physics vol 167 no 5-6 pp 1004ndash1014 2012

[51] C Hassel ldquoLecture at the International School of AstrophysicsDaniel ChalongemdashHector de Vegardquo 2015

[52] L Gastaldo K Blaum A Doerr et al ldquoThe electron capture163Ho experiment ECHordquo Journal of Low Temperature Physicsvol 176 no 5 pp 876ndash884 2014

[53] B Alpert M Balata D Bennett et al ldquoThe electron capturedecay of 163Ho tomeasure the electron neutrino mass with sub-eV sensitivityrdquo The European Physical Journal C vol 75 article112 2015

[54] E Fermi ldquoRadioattivita prodotta da bombardamento di neu-tronirdquo Il Nuovo Cimento vol 11 no 7 pp 429ndash441 1934

[55] E Fermi ldquoVersuch einer theorie der 120573-strahlen Irdquo Zeitschriftfur Physik vol 88 no 3 pp 161ndash177 1934

[56] R B Firestone and V S Shirley Eds Table of Isotopes JohnWiley amp Sons New York NY USA 8th edition 1999

[57] W Bambynek H Behrens M H Chen et al ldquoOrbital electroncapture by the nucleusrdquo Reviews of Modern Physics vol 49 no1 pp 77ndash221 1977

[58] J L Campbell and T Papp ldquoWidths of the atomic KndashN7 levelsrdquoAtomic Data and Nuclear Data Tables vol 77 no 1 pp 1ndash562001

[59] J A Bearden and A F Burr ldquoReevaluation of X-ray atomicenergy levelsrdquo Reviews of Modern Physics vol 39 no 1 pp 125ndash142 1967

[60] M Cardona and L Ley Eds Photoemission in Solids I GeneralPrinciples Springer Berlin Germany 1978

[61] J C Fuggle and N Martensson ldquoCore-level binding energies inmetalsrdquo Journal of Electron Spectroscopy and Related Phenom-ena vol 21 no 3 pp 275ndash281 1980

[62] T Kosmas H Ejiri and A Hatzikoutelis ldquoNeutrino physics inthe frontiers of intensities and very high sensitivitiesrdquo Advancesin High Energy Physics vol 2015 Article ID 806067 3 pages2015

[63] A Giuliani and A Poves ldquoNeutrinoless double-beta decayrdquoAdvances in High Energy Physics vol 2012 Article ID 85701638 pages 2012

[64] J-U Nabi and H V Klapdor-Kleingrothaus ldquoWeak interactionrates of sd-shell nuclei in stellar environments calculated in theproton-neutron quasiparticle random-phase approximationrdquoAtomic Data andNuclear Data Tables vol 71 no 2 pp 149ndash3451999

12 Advances in High Energy Physics

[65] J-U Nabi and H V Klapdor-Kleingrothaus ldquoMicroscopic cal-culations of stellar weak interaction rates and energy losses forfp- and fpg-shell nucleirdquo Atomic Data and Nuclear Data Tablesvol 88 no 2 pp 237ndash476 2004

[66] Y Fukuda T Hayakawa E Ichihara et al ldquoEvidence foroscillation of atmospheric neutrinosrdquo Physical Review Lettersvol 81 no 8 pp 1562ndash1567 1998

[67] Q R Ahmad and SNOCollaboration ldquoMeasurement of the rateof V119890+119889 rarr 119901+119901+119890minus interactions produced by 8B solar neutrinosat the Sudbury Neutrino Observatoryrdquo Physical Review Lettersvol 87 no 7 Article ID 071301 6 pages 2001

[68] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 2002

[69] G J Feldman J Hartnell and T Kobayashi ldquoLong-baselineneutrino oscillation experimentsrdquo Advances in High EnergyPhysics vol 2013 Article ID 475749 30 pages 2013

[70] S T Petcov ldquoThenature ofmassive neutrinosrdquoAdvances inHighEnergy Physics vol 2013 Article ID 852987 20 pages 2013

[71] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[72] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 48 no 1-2 pp 1ndash28 2009

[73] F Munyaneza and P L Biermann ldquoDegenerate sterile neutrinodark matter in the cores of galaxiesrdquo Astronomy amp Astrophysicsvol 458 no 2 pp L9ndashL12 2006

[74] D Boyanovsky and C M Ho ldquoSterile neutrino production viaactive-sterile oscillations the quantum Zeno effectrdquo Journal ofHigh Energy Physics vol 2007 no 7 p 30 2007

[75] R Shrock ldquoNew tests for and bounds on neutrino masses andlepton mixingrdquo Physics Letters B vol 96 no 1-2 pp 159ndash1641980

[76] R E Shrock ldquoGeneral theory of weak processes involving neu-trinos I Leptonic pseudoscalar-meson decays with associatedtests for and bounds on neutrino masses and lepton mixingrdquoPhysical Review D vol 24 no 5 pp 1232ndash1274 1981

[77] R E Shrock ldquoImplications of neutrino masses and mixing forweak processesrdquo in Proceedings of the Weak Interactions asProbes of Unification vol 72 of AIP Conference Proceedings p368 Blacksburg Va USA 1981

[78] J M Conrad C M Ignarra G Karagiorgi M H Shaevitzand J Spitz ldquoSterile neutrino fits to short-baseline neutrinooscillation measurementsrdquo Advances in High Energy Physicsvol 2013 Article ID 163897 2013

[79] A Nucciotti ldquoThe use of low temperature detectors for directmeasurements of themass of the electron neutrinordquoAdvances inHigh Energy Physics vol 2016 Article ID 9153024 41 pages2016

[80] R G H Robertson ldquoExamination of the calorimetric spectrumto determine the neutrino mass in low-energy electron capturedecayrdquo Physical Review C vol 91 no 3 Article ID 035504 2015

[81] A De Rujula ldquoTwo old ways to measure the electron-neutrinomassrdquo httpsarxivorgabs13054857

[82] A De Rujula ldquoAn upper bound on the exceptional character-istics for Lusztigrsquos character formulardquo httpsarxivorgabs08111674v2

[83] A Faessler and F Simkovic ldquoCan electron capture tell us themass of the neutrinordquo Physica Scripta vol 91 no 4 Article ID043007 2016

[84] H J de Vega O Moreno E Moya de Guerra M RamonMedrano and N G Sanchez ldquoRole of sterile neutrino warmdark matter in rhenium and tritium beta decaysrdquo NuclearPhysics B vol 866 no 2 pp 177ndash195 2013

[85] E M de Guerra O Moreno P Sarriguren and M RamonMedrano ldquoTopics on nuclear structure with electroweakprobesrdquo Journal of Physics Conference Series vol 366 ArticleID 012011 2012

[86] J M Boillos O Moreno and E Moya de ldquoActive and sterileneutrino mass effects on beta decay spectrardquo in Proceedingsof the La Rabida International Scientific Meeting on NuclearPhysics Basic Concepts in Nuclear Physics Theory Experimentsand Applications vol 1541 ofAIP Conference Proceedings p 167La Rabida Spain September 2012

[87] P E Filianin K Blaum S A Eliseev et al ldquoOn the keVsterile neutrino search in electron capturerdquo Journal of PhysicsG Nuclear and Particle Physics vol 41 no 9 Article ID 0950042014

[88] S Eliseev K BlaumM Block et al ldquoDirect measurement of themass difference of 163Ho and 163Dy solves theQ-value puzzle forthe neutrino mass determinationrdquo Physical Review Letters vol115 no 6 Article ID 062501 2015

[89] A De Rujula ldquoA new way tomeasure neutrinomassesrdquoNuclearPhysics Section B vol 188 no 3 pp 414ndash458 1981

[90] M Lusignoli and M Vignati ldquoRelic antineutrino capture on163Hodecaying nucleirdquoPhysics Letters B vol 697 no 1 pp 11ndash142011

[91] A Faessler L Gastaldo and F Simkovic ldquoElectron capture in163Ho overlap plus exchange corrections and neutrino massrdquoJournal of Physics G Nuclear and Particle Physics vol 42 no 1Article ID 015108 2015

[92] G Audi F G Kondev M Wang et al ldquoThe Nubase2012evaluation of nuclear propertiesrdquo Chinese Physics C vol 36 no12 pp 1157ndash1286 2012

[93] S Mertens ldquoSensitivity of next-generation tritium beta-decayexperiments for keV-scale sterile neutrinosrdquo JCAP vol 1502 no2 p 20 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

12 Advances in High Energy Physics

[65] J-U Nabi and H V Klapdor-Kleingrothaus ldquoMicroscopic cal-culations of stellar weak interaction rates and energy losses forfp- and fpg-shell nucleirdquo Atomic Data and Nuclear Data Tablesvol 88 no 2 pp 237ndash476 2004

[66] Y Fukuda T Hayakawa E Ichihara et al ldquoEvidence foroscillation of atmospheric neutrinosrdquo Physical Review Lettersvol 81 no 8 pp 1562ndash1567 1998

[67] Q R Ahmad and SNOCollaboration ldquoMeasurement of the rateof V119890+119889 rarr 119901+119901+119890minus interactions produced by 8B solar neutrinosat the Sudbury Neutrino Observatoryrdquo Physical Review Lettersvol 87 no 7 Article ID 071301 6 pages 2001

[68] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 2002

[69] G J Feldman J Hartnell and T Kobayashi ldquoLong-baselineneutrino oscillation experimentsrdquo Advances in High EnergyPhysics vol 2013 Article ID 475749 30 pages 2013

[70] S T Petcov ldquoThenature ofmassive neutrinosrdquoAdvances inHighEnergy Physics vol 2013 Article ID 852987 20 pages 2013

[71] ADDolgov ldquoNeutrinos in cosmologyrdquoPhysics Report vol 370no 4-5 pp 333ndash535 2002

[72] A Kusenko ldquoSterile neutrinos the dark side of the lightfermionsrdquo Physics Reports vol 48 no 1-2 pp 1ndash28 2009

[73] F Munyaneza and P L Biermann ldquoDegenerate sterile neutrinodark matter in the cores of galaxiesrdquo Astronomy amp Astrophysicsvol 458 no 2 pp L9ndashL12 2006

[74] D Boyanovsky and C M Ho ldquoSterile neutrino production viaactive-sterile oscillations the quantum Zeno effectrdquo Journal ofHigh Energy Physics vol 2007 no 7 p 30 2007

[75] R Shrock ldquoNew tests for and bounds on neutrino masses andlepton mixingrdquo Physics Letters B vol 96 no 1-2 pp 159ndash1641980

[76] R E Shrock ldquoGeneral theory of weak processes involving neu-trinos I Leptonic pseudoscalar-meson decays with associatedtests for and bounds on neutrino masses and lepton mixingrdquoPhysical Review D vol 24 no 5 pp 1232ndash1274 1981

[77] R E Shrock ldquoImplications of neutrino masses and mixing forweak processesrdquo in Proceedings of the Weak Interactions asProbes of Unification vol 72 of AIP Conference Proceedings p368 Blacksburg Va USA 1981

[78] J M Conrad C M Ignarra G Karagiorgi M H Shaevitzand J Spitz ldquoSterile neutrino fits to short-baseline neutrinooscillation measurementsrdquo Advances in High Energy Physicsvol 2013 Article ID 163897 2013

[79] A Nucciotti ldquoThe use of low temperature detectors for directmeasurements of themass of the electron neutrinordquoAdvances inHigh Energy Physics vol 2016 Article ID 9153024 41 pages2016

[80] R G H Robertson ldquoExamination of the calorimetric spectrumto determine the neutrino mass in low-energy electron capturedecayrdquo Physical Review C vol 91 no 3 Article ID 035504 2015

[81] A De Rujula ldquoTwo old ways to measure the electron-neutrinomassrdquo httpsarxivorgabs13054857

[82] A De Rujula ldquoAn upper bound on the exceptional character-istics for Lusztigrsquos character formulardquo httpsarxivorgabs08111674v2

[83] A Faessler and F Simkovic ldquoCan electron capture tell us themass of the neutrinordquo Physica Scripta vol 91 no 4 Article ID043007 2016

[84] H J de Vega O Moreno E Moya de Guerra M RamonMedrano and N G Sanchez ldquoRole of sterile neutrino warmdark matter in rhenium and tritium beta decaysrdquo NuclearPhysics B vol 866 no 2 pp 177ndash195 2013

[85] E M de Guerra O Moreno P Sarriguren and M RamonMedrano ldquoTopics on nuclear structure with electroweakprobesrdquo Journal of Physics Conference Series vol 366 ArticleID 012011 2012

[86] J M Boillos O Moreno and E Moya de ldquoActive and sterileneutrino mass effects on beta decay spectrardquo in Proceedingsof the La Rabida International Scientific Meeting on NuclearPhysics Basic Concepts in Nuclear Physics Theory Experimentsand Applications vol 1541 ofAIP Conference Proceedings p 167La Rabida Spain September 2012

[87] P E Filianin K Blaum S A Eliseev et al ldquoOn the keVsterile neutrino search in electron capturerdquo Journal of PhysicsG Nuclear and Particle Physics vol 41 no 9 Article ID 0950042014

[88] S Eliseev K BlaumM Block et al ldquoDirect measurement of themass difference of 163Ho and 163Dy solves theQ-value puzzle forthe neutrino mass determinationrdquo Physical Review Letters vol115 no 6 Article ID 062501 2015

[89] A De Rujula ldquoA new way tomeasure neutrinomassesrdquoNuclearPhysics Section B vol 188 no 3 pp 414ndash458 1981

[90] M Lusignoli and M Vignati ldquoRelic antineutrino capture on163Hodecaying nucleirdquoPhysics Letters B vol 697 no 1 pp 11ndash142011

[91] A Faessler L Gastaldo and F Simkovic ldquoElectron capture in163Ho overlap plus exchange corrections and neutrino massrdquoJournal of Physics G Nuclear and Particle Physics vol 42 no 1Article ID 015108 2015

[92] G Audi F G Kondev M Wang et al ldquoThe Nubase2012evaluation of nuclear propertiesrdquo Chinese Physics C vol 36 no12 pp 1157ndash1286 2012

[93] S Mertens ldquoSensitivity of next-generation tritium beta-decayexperiments for keV-scale sterile neutrinosrdquo JCAP vol 1502 no2 p 20 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of