Research Article Dark Atoms and the Positron-Annihilation-Line Excess in …downloads.hindawi.com/journals/ahep/2014/869425.pdf · 2019-07-31 · Research Article Dark Atoms and the

Research ArticleDark Atoms and the Positron-Annihilation-Line Excessin the Galactic Bulge

J-R Cudell1 M Yu Khlopov234 and Q Wallemacq1

1 IFPA ldquoDepartement drsquoAGOrdquo Universite de Liege Sart Tilman 4000 Liege Belgium2National Research Nuclear University ldquoMoscow Engineering Physics Instituterdquo Moscow 115409 Russia3 Centre for Cosmoparticle Physics ldquoCosmionrdquo 115409 Moscow Russia4APC Laboratory 10 rue Alice Domon et Leonie Duquet 75205 Paris Cedex 13 France

Correspondence should be addressed to Q Wallemacq quentinwallemacqulgacbe

Received 25 November 2013 Accepted 13 January 2014 Published 25 February 2014

Academic Editor Chris Kouvaris

Copyright copy 2014 J-R Cudell 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

It was recently proposed that stable particles of charge minus2 Ominusminus can exist and constitute dark matter after they bind with primordialhelium in O-helium (OHe) atoms We study here in detail the possibility that this model provides an explanation for the excessof gamma radiation in the positron-annihilation line from the galactic bulge observed by INTEGRAL This explanation assumesthat OHe excited to a 2s state through collisions in the central part of the Galaxy deexcites to its ground state via an 1198640 transitionemitting an electron-positron pair The cross-section for OHe collisions with excitation to 2s level is calculated and it is shownthat the rate of such excitations in the galactic bulge strongly depends not only on the mass of O-helium which is determined bythe mass of Ominusminus but also on the density and velocity distribution of dark matter Given the astrophysical uncertainties on thesedistributions this mechanism constrains the Ominusminus mass to lie in two possible regions One of these is reachable in the experimentalsearches for stable multicharged particles at the LHC

1 Introduction

According to modern cosmology dark matter correspondsto 25 of the total cosmological density is nonbaryonicand consists of new stable particles Such particles (see [1ndash6] for reviews and references) should be stable providethe measured dark-matter density and be decoupled fromplasma and radiation at least before the beginning of thematter-dominated era It was recently shown that heavy stableparticles of charge minus2 Ominusminus bound to primordial heliumin OHe atoms can provide an interesting explanation forcosmological dark matter [6 7] It should also be notedthat the nuclear cross-section of the O-helium interactionwith matter escapes the severe constraints [8ndash10] on stronglyinteracting dark-matter particles (SIMPs) [8ndash16] imposed bythe XQC experiment [17 18]

The hypothesis of composite O-helium dark matter firstconsidered to provide a solution to the puzzles of direct dark-matter searches can offer an explanation for another puzzle

of modern astrophysics [6 7 19] this composite dark-mattermodel can explain the excess of gamma radiation in theelectron-positron-annihilation line observed by INTEGRALin the galactic bulge (see [20] for a review and references)The explanation assumes that OHe provides all the galacticdark matter and that its collisions in the central part ofthe Galaxy result in 2s-level excitations of OHe which aredeexcited to the ground state by an 1198640 transition in which anelectron-positron pair is emitted If the 2s level is excited pairproduction dominates over the two-photon channel in thedeexcitation because electrons are much lighter than heliumnuclei and positron production is not accompanied by astrong gamma-ray signal

According to [21] the rate of positron production 3 sdot

10

42 sminus1 is sufficient to explain the excess in the positron-annihilation line from the bulge measured by INTEGRAL Inthe present paper we study the process of 2s-level excitationof OHe from collisions in the galactic bulge and determinethe conditions under which such collisions can provide

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 869425 5 pageshttpdxdoiorg1011552014869425

2 Advances in High Energy Physics

the observed excess Inelastic interactions of O-helium withmatter in interstellar space and subsequent deexcitation cangive rise to radiation in the range from a few keV to afew MeV In the galactic bulge with radius 119903

119887

sim 1 kpcthe number density of O-helium can be of the order of119899

119900

asymp 3 sdot 10

minus3

119878

3

cmminus3 or larger and the collision rate of O-helium in this central region was estimated in [19] 119889119873119889119905 =

119899

2

119900

120590Vℎ

4120587119903

3

119887

3 asymp 3 sdot 10

42

119878

minus2

3

sminus1 with 119878

3

= 119898OHe1TeV At thevelocity of V

ℎ

sim 3 sdot 10

7 cms energy transfer in such collisionsis Δ119864 sim 1MeV119878

3

These collisions can lead to excitation ofO-helium If OHe levels with nonzero angular momentumare excited gamma lines should be observed from transitions(119899 gt 119898) 119864

119899119898

= 1598MeV(1119898

2

minus 1119899

2

) (or from similartransitions corresponding to the case 119868

119900

= 1287MeV) at thelevel 3 sdot 10

minus4

119878

minus2

3

(cm2 sMeV ster)minus1

2 Collisional Excitation Cross-Section

The studied reaction is the collision between two OHe atomsboth being initially in their ground state |1s⟩ giving rise tothe excitation of one of them to a |119899s⟩ state while the otherremains in its ground state

OHe (1s) +OHe (1s) 997888rarr OHe (1s) +OHe (119899s) (1)

If we work in the rest frame of the OHe that gets excitedand if we neglect its recoil after the collision the differentialcross-section of the process is given by

119889120590 (1s 997888rarr 119899s) = 2120587

1003816

1003816

1003816

1003816

1003816

⟨119899s

119901

1015840

|119880|1s

119901⟩

1003816

1003816

1003816

1003816

1003816

2

times 120575(

119901

10158402

2119872

+ 119864

119899s minus119901

2

2119872

minus 119864

1s)119889

3

119901

1015840

(2120587)

3

(2)

where 119872 is the mass of OHe

119901 and

119901

1015840 are the momenta ofthe incident OHe before and after the collision 119864

1s and 119864

119899sare the ground-state and excited-state energies of the targetOHe and 119880 is the interaction potential between the incidentand the target OHersquos

Wewill neglect the internal structure of the incidentOHeso that its wave functions are plane waves120595

119901

is normalized toobtain a unit incident current density and the normalisationof 120595

119901

1015840 is chosen for it to be pointlike that is the Fouriertransform of 120575(3)( 119903) [22]

120595

119901

= radic

119872

119901

119890

119894

119901sdot 119903

120595

119901

1015840 = 119890

119894

119901

1015840sdot 119903

(3)

where 119903 is the position vector of the incidentOHe and119901 = |

119901|In the following we will be led to consider Ominusminus masses

which are much larger than themass of helium or the bound-state energies Therefore the origin of the rest frame of thetarget OHe coincides with the position of its Ominusminus componentand its reduced mass 120583 can be taken as the mass of helium119872He

The OHe that gets excited is described as a hydrogen-likeatom with energy levels 119864

119899s = minus05119872He(119885He119885O120572)2

119899

2 andinitial and final bound-state wave functions 120595

1s and 120595

119899s of ahydrogenoid atom with a Bohr radius 119886

0

= (119872He119885He119885O120572)minus1

The incident OHe interacts with the Ominusminus and heliumcomponents in the target OHe so that the interactionpotential 119880 is the sum of the two contributions 119880O and 119880He

119880 (119903) = 119880O (

119903) + 119880He ( 119903 minus 119903He) (4)

where 119903He is the position vector of the helium componentThe first term119880O gives a zero contribution to the integral

of expression (2) since the states 1205951s and 120595

119899s are orthogonalFor the second term we treat the incident OHe as a heavyneutron colliding on a helium nucleus through short-rangenuclear forces The interaction potential can then be writtenin the form of a contact term

119880He ( 119903 minus 119903He) = minus

2120587

119872He119886

0

120575 ( 119903 minus 119903He) (5)

where we have normalised the delta function to obtain anOHe-helium elastic cross-section equal to 4120587119886

2

0

Going to spherical coordinates for

119901

1015840 and integrating over119901

1015840

= |

119901

1015840

| in the differential cross-section (2) together withthe previous expressions (3) (4) and (5) we get

119889120590 (1s 997888rarr 119899s) = (

119872

119872He)

2

119886

2

0

(

119901

1015840

119901

)

times

1003816

1003816

1003816

1003816

1003816

1003816

1003816

int 119890

minus119894 119902 119903He120595

lowast

119899s1205951s1198893

119903He1003816

1003816

1003816

1003816

1003816

1003816

1003816

2

119889Ω

(6)

where 119902 =

119901

1015840

minus

119901 is the transferred momentum and 119889Ω is thesolid angle From the integration over the delta function in(2) we have obtained the conservation of energy during theprocess

119901

10158402

= 119901

2

+ 2119872(119864

1s minus 119864

119899s) (7)

It leads to the threshold energy corresponding to 119901

10158402

= 0 andto aminimum incident velocity Vmin =

radic2(119864

119899s minus 119864

1s)119872Theprevious expression for 119901

1015840 allows us to express the squaredmodulus of 119902 as

119902

2

= 2 (119901

2

+ 119872(119864

1s minus 119864

119899s)

minus119901

radic

119901

2

+ 2119872(119864

1s minus 119864

119899s) cos 120579)

(8)

where 120579 is the deviation angle of the incident OHe withrespect to the collision axis in the rest frame of the targetOHe

119890

+

119890

minus pairs will be dominantly produced if OHe is excitedto a 2s state since the only deexcitation channel is in this casefrom 2s to 1s As 119890

+

119890

minus pair production is the only possiblechannel the differential pair-production cross-section 119889120590

119890119890

isequal to the differential collisional excitation cross-sectionBy particularizing expression (6) to the case 119899 = 2 one finallygets

119889120590

119890119890

119889 cos 120579= 512

2

(

2120587119872

2

119872

2

He)119886

6

0

(

119901

1015840

119901

)

119902

4

2(4119886

2

0

119902

2

+ 9)

6

(9)

Advances in High Energy Physics 3

3 The 119890

+

119890

minus Pair-Production Rate in theGalactic Bulge

The total 119890+119890minus pair-production rate in the galactic bulge isgiven by

119889119873

119889119905

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119890119890

= int

119881

119887

120588

2

DM (

119877)

119872

2

⟨120590

119890119890

V⟩ (

119877) 119889

119877

(10)

where119881119887

is the volume of the galactic bulge which is a sphereof radius 119877

119887

= 15 kpc 120588DM is the energy density distributionof dark matter in the galactic halo and ⟨120590

119890119890

V⟩ is the pair-production cross-section 120590

119890119890

times relative velocity V aver-aged over the velocity distribution of dark-matter particlesThe total pair-production cross-section 120590

119890119890

is obtained byintegrating (9) over the diffusion angle Its dependence on therelative velocity V is contained in1199011199011015840 and 119902 through119901 = 119872Vand the expressions (7) and (8) of 1199011015840 and 119902 in terms of 119901

We use a Burkert [23 24] flat cored dark-matter densityprofile known to reproduce well the kinematics of disksystems in massive spiral galaxies and supported by recentsimulations including supernova feedback and radiationpressure of massive stars [25] in response to the cuspy haloproblem

120588DM (119877) = 120588

0

119877

3

0

(119877 + 119877

0

) (119877

2

+ 119877

2

0

)

(11)

where 119877 is the distance from the galactic center The centraldark-matter density 120588

0

is left as a free parameter and 119877

0

isdetermined by requiring that the local dark-matter density at119877 = 119877

⊙

= 8 kpc is 120588⊙

= 03GeVcm3 The dark-matter massenclosed in a sphere of radius 119877 is therefore given by

119872DM (119877) = 120588

0

120587119877

3

0

log(119877

2

+ 119877

2

0

119877

2

0

)

+2 log(119877 + 119877

0

119877

0

) minus 2 arctan(

119877

119877

0

)

(12)

For the baryons in the bulge we use an exponential profile[26] of the form

120588

119887

(119877) =

119872bulge

8120587119877

3

119887

119890

minus119877119877

119887 (13)

where 119872bulge = 10

10

119872

⊙

[27] is the mass of the bulge Thisgives the baryonic mass distribution in the galactic bulge

119872

119887

(119877) = 119872bulge 1 minus 119890

minus119877119877

119887(1 +

119877

119877

119887

+

119877

2

119877

2

119887

) (14)

We assume a Maxwell-Boltzmann velocity distributionfor the dark-matter particles of the galactic halo with avelocity dispersion 119906(119877) and a cutoff at the galactic escapevelocity Vesc(119877)

119891 (119877 Vℎ

) =

1

119862 (119877)

119890

minusV2ℎ119906

2(119877)

(15)

where Vℎ

is the velocity of the dark-matter particles in theframe of the halo and 119862(119877) = 120587119906

2

(radic120587119906 erf(Vesc119906) minus

2Vesc119890minusV2esc119906

2

) is a normalization constant such thatint

Vesc(119877)0

119891(119877 Vℎ

)119889Vℎ

= 1The radial dependence of the velocity dispersion is

obtained via the virial theorem

119906 (119877) =

radic

119866119872tot (119877)

119877

(16)

where119872tot = 119872DM + 119872

119887

while Vesc =

radic

2119906Using the velocity distribution (15) going to center-of-

mass and relative velocities VCM and V and performing theintegrals over VCM we obtain for the mean pair-productioncross-section times relative velocity

⟨120590

119890119890

V⟩ =

1

119906

2

radic

2120587119906 erf (radic2Vesc119906) minus 4Vesc119890minus2V2esc119906

2

(radic120587119906 erf (Vesc119906) minus 2Vesc119890minusV2

esc1199062

)

2

times int

2Vesc

0

120590

119890119890

(V) V3119890minusV22119906

2

119889V

(17)

which is also a function of 119877 through 119906 and Vesc Putting (9)(11) (12) (14) (16) and (17) together allows us to compute thepair-production rate in the galactic bulge defined in (10) as afunction of 120588

0

and119872

4 Results

The rate of excessive 119890+119890minus pairs to be generated in the galacticbulge was estimated in [21] to be 119889119873119889119905|obs = 3 times 10

42 sminus1We computed 119889119873119889119905|

119890119890

for a large range of central dark-matter densities going from 03GeVcm3 to an ultimateupper limit of 10

4 GeVcm3 [28] For each value of 120588

0

wesearched for themass119872 ofOHe that reproduces the observedrate The results are shown in Figure 1

The observed rate can be reproduced from a value of120588

0

≃ 115GeVcm3 corresponding to an OHe mass of 119872 ≃

125TeV As 120588

0

gets larger two values of 119872 are possiblewith the lower one going from 125TeV to 130GeV and theupper one going from 125 to 130TeV as 120588

0

goes from 115 to10

4 GeVcm3

5 Conclusion

The existence of heavy stable particles is one of the mostpopular solutions for the dark- matter problem Usually theyare considered to be electrically neutral But dark mattercan potentially be made of stable heavy charged particlesbound in neutral atom-like states by Coulomb attractionAn analysis of the cosmological data and of the atomiccomposition of theUniverse forces the particle to have chargeminus2 Ominusminus is then trapped by primordial helium in neutral O-helium states and this avoids the problem of overproductionof anomalous isotopes which are severely constrained byobservations Here we have shown that the cosmologicalmodel of O-helium dark matter can explain the puzzle ofpositron line emission from the center of our Galaxy


1000100101

10000

1000

100001 01

1205880(GeV

cm

3)

M (TeV)

Figure 1 Values of the central dark-matter density 120588

0

(GeVcm3)and of the OHe mass 119872 (TeV) reproducing the excess of 119890+119890minus pairsproduction in the galactic bulge Below the red curve the predictedrate is too low

The proposed explanation is based on the assumptionthat OHe dominates the dark-matter sector Its collisionscan lead to 1198640 deexcitations of the 2s states excited by thecollisionsThe estimated luminosity in the electron-positron-annihilation line strongly depends not only on the mass ofOminusminus but also on the density profile and velocity distribution ofdarkmatter in the galactic bulge Note that the density profilewe considered is used only to obtain a reasonable estimatefor the uncertainties on the density in the bulge It indeedunderestimates the mass of the Galaxy but it shows thatthe uncertainties on the astrophysical parameters are largeenough to reproduce the observed excess for a rather widerange of masses of Ominusminus For a fixed density profile and a fixedvelocity distribution only two values of the Ominusminus mass leadto the necessary rate of positron production The lower valueof this mass which does not exceed 125TeV is within thereach of experimental searches for multicharged stable heavyparticles at the LHC

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors express their gratitude to A S Romaniouk fordiscussions

References

[1] M Yu Khlopov Cosmoparticle Physics World Scientific Singa-pore 1999

[2] M Yu Khlopov ldquoCosmoarcheology Direct and indirect astro-physical effects of hypothetical particles and fieldsrdquo inCosmion-94 M Yu Khlopov M E Prokhorov A A Starobinsky and J

Tran Thanh Van Eds pp 67ndash76 Editions Frontieres QuebecCanada 1996

[3] M Y Khlopov ldquoProceedings to the 9th workshop lsquowhat comesbeyond the standard modelsrsquordquo Bled Workshops in Physics vol 7no 2 p 51 2006


[5] M Yu Khlopov Fundamentals of Cosmoparticle Physics CISP-Springer Cambridge UK 2012

[6] M Yu Khlopov ldquoFundamental particle structure in the cosmo-logical dark matterrdquo International Journal of Modern Physics Avol 28 no 29 Article ID 1330042 60 pages 2013

[7] M Yu Khlopov ldquoPhysics of dark matter in the light of darkatomsrdquoModern Physics Letters A vol 26 no 38 Article ID 28232011

[8] B D Wandelt R Dave G R Farrar P C McGuire D NSpergel and P J Steinhardt ldquoSelf-interacting dark matterrdquohttparxivorgabsastro-ph0006344

[9] P C McGuire and P J Steinhardt ldquoCracking open the windowfor strongly interacting massive particles as the halo darkmatterrdquo httparxivorgabsastro-ph0105567

[10] G Zaharijas and G R Farrar ldquoWindow in the dark matterexclusion limitsrdquo Physical Review D vol 72 no 8 Article ID083502 11 pages 2005

[11] C B Dover et al ldquoCosmological constraints on new stablehadronsrdquo Physical Review Letters vol 42 no 17 pp 1117ndash11201979

[12] S Wolfram ldquoAbundances of new stable particles produced inthe early universerdquo Physics Letters B vol 82 no 1 pp 65ndash681979

[13] G D Starkman A Gould R Esmailzadeh and S DimopoulosldquoOpening the window on strongly interacting dark matterrdquoPhysical Review D vol 41 no 12 pp 3594ndash3603 1990

[14] D Javorsek D Elmore E Fischbach et al ldquoNew experimentallimits on strongly interactingmassive particles at the TeV scalerdquoPhysical Review Letters vol 87 no 23 Article ID 231804 2001

[15] S Mitra ldquoUranusrsquos anomalously low excess heat constrainsstrongly interacting dark matterrdquo Physical Review D vol 70 no10 Article ID 103517 2004

[16] G D Mack J F Beacom and G Bertone ldquoTowards closingthe window on strongly interacting dark matter far-reachingconstraints from Earthrsquos heat flowrdquo Physical Review D vol 76no 4 Article ID 043523 2007

[17] D McCammon R Almy S Deiker et al ldquoA soundingrocket payload for X-ray astronomy employing high-resolutionmicrocalorimetersrdquoNuclear Instruments andMethods in PhysicsResearch Section A vol 370 no 1 pp 266ndash268 1996

[18] D McCammon R Almy E Apodaca et al ldquoA high spectralresolution observation of the soft X-ray diffuse backgroundwith thermal detectors rdquoThe Astrophysical Journal vol 576 no1 p 188 2002

[19] M Yu Khlopov ldquoComposite dark matter from stable chargedconstituentsrdquo httparxivorgabs08063581

[20] B J Teegarden K Watanabe P Jean et al ldquoINTEGRAL SPIlimits on electron-positron annihilation radiation from thegalactic planerdquoThe Astrophysical Journal vol 621 no 1 p 2962005

[21] D P Finkbeiner and N Weiner ldquoExciting dark matter and theINTEGRALSPI 511 keV signalrdquo Physical Review D vol 76 no8 Article ID 083519 2007


[22] LD Landau andEM LifshitzQuantumMechanics PergamonPress Elmsford NY USA 1965

[23] A Burkert ldquoThe structure of dark matter haloes in dwarfgalaxiesrdquo IAU Symposia vol 171 p 175 1996

[24] A Burkert ldquoThe structure of dark matter haloes in dwarfgalaxiesrdquoThe Astrophysical Journal vol 447 no 1 p L25 1995

[25] A V Maccio G Stinson C B Brook et al ldquoHALO Expansionin cosmological hydro simulations toward a baryonic solutionof the cuspcore problem in massive spiralsrdquo The AstrophysicalJournal Letters vol 744 no 1 p L9 2012

[26] O Y Gnedin A V Kravtsov A A Klypin and D NagaildquoResponse of dark matter halos to condensation of Baryonscosmological simulations and improved adiabatic contractionmodelrdquoThe Astrophysical Journal vol 616 no 1 p 16 2004

[27] H Mo F van den Bosch and S White Galaxy Formation andEvolution Cambridge University Press Cambridge UK 2010

[28] X Hernandez and W H Lee ldquoAn upper limit to the centraldensity of dark matter haloes from consistency with the pres-ence ofmassive central black holesrdquoMonthlyNotices of the RoyalAstronomical Society vol 404 no 1 p L10 2010

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the observed excess Inelastic interactions of O-helium withmatter in interstellar space and subsequent deexcitation cangive rise to radiation in the range from a few keV to afew MeV In the galactic bulge with radius 119903

119887

sim 1 kpcthe number density of O-helium can be of the order of119899

119900

asymp 3 sdot 10

minus3

119878

3

cmminus3 or larger and the collision rate of O-helium in this central region was estimated in [19] 119889119873119889119905 =

119899

2

119900

120590Vℎ

4120587119903

3

119887

3 asymp 3 sdot 10

42

119878

minus2

3

sminus1 with 119878

3

= 119898OHe1TeV At thevelocity of V

ℎ

sim 3 sdot 10

7 cms energy transfer in such collisionsis Δ119864 sim 1MeV119878

3

These collisions can lead to excitation ofO-helium If OHe levels with nonzero angular momentumare excited gamma lines should be observed from transitions(119899 gt 119898) 119864

119899119898

= 1598MeV(1119898

2

minus 1119899

2

) (or from similartransitions corresponding to the case 119868

119900

= 1287MeV) at thelevel 3 sdot 10

minus4

119878

minus2

3

(cm2 sMeV ster)minus1

2 Collisional Excitation Cross-Section

The studied reaction is the collision between two OHe atomsboth being initially in their ground state |1s⟩ giving rise tothe excitation of one of them to a |119899s⟩ state while the otherremains in its ground state

OHe (1s) +OHe (1s) 997888rarr OHe (1s) +OHe (119899s) (1)

If we work in the rest frame of the OHe that gets excitedand if we neglect its recoil after the collision the differentialcross-section of the process is given by

119889120590 (1s 997888rarr 119899s) = 2120587

1003816

1003816

1003816

1003816

1003816

⟨119899s

119901

1015840

|119880|1s

119901⟩

1003816

1003816

1003816

1003816

1003816

2

times 120575(

119901

10158402

2119872

+ 119864

119899s minus119901

2

2119872

minus 119864

1s)119889

3

119901

1015840

(2120587)

3

(2)

where 119872 is the mass of OHe

119901 and

119901

1015840 are the momenta ofthe incident OHe before and after the collision 119864

1s and 119864

119899sare the ground-state and excited-state energies of the targetOHe and 119880 is the interaction potential between the incidentand the target OHersquos

Wewill neglect the internal structure of the incidentOHeso that its wave functions are plane waves120595

119901

is normalized toobtain a unit incident current density and the normalisationof 120595

119901

1015840 is chosen for it to be pointlike that is the Fouriertransform of 120575(3)( 119903) [22]

120595

119901

= radic

119872

119901

119890

119894

119901sdot 119903

120595

119901

1015840 = 119890

119894

119901

1015840sdot 119903

(3)

where 119903 is the position vector of the incidentOHe and119901 = |

119901|In the following we will be led to consider Ominusminus masses

which are much larger than themass of helium or the bound-state energies Therefore the origin of the rest frame of thetarget OHe coincides with the position of its Ominusminus componentand its reduced mass 120583 can be taken as the mass of helium119872He

The OHe that gets excited is described as a hydrogen-likeatom with energy levels 119864

119899s = minus05119872He(119885He119885O120572)2

119899

2 andinitial and final bound-state wave functions 120595

1s and 120595

119899s of ahydrogenoid atom with a Bohr radius 119886

0

= (119872He119885He119885O120572)minus1

The incident OHe interacts with the Ominusminus and heliumcomponents in the target OHe so that the interactionpotential 119880 is the sum of the two contributions 119880O and 119880He

119880 (119903) = 119880O (

119903) + 119880He ( 119903 minus 119903He) (4)

where 119903He is the position vector of the helium componentThe first term119880O gives a zero contribution to the integral

of expression (2) since the states 1205951s and 120595

119899s are orthogonalFor the second term we treat the incident OHe as a heavyneutron colliding on a helium nucleus through short-rangenuclear forces The interaction potential can then be writtenin the form of a contact term

119880He ( 119903 minus 119903He) = minus

2120587

119872He119886

0

120575 ( 119903 minus 119903He) (5)

where we have normalised the delta function to obtain anOHe-helium elastic cross-section equal to 4120587119886

2

0

Going to spherical coordinates for

119901

1015840 and integrating over119901

1015840

= |

119901

1015840

| in the differential cross-section (2) together withthe previous expressions (3) (4) and (5) we get

119889120590 (1s 997888rarr 119899s) = (

119872

119872He)

2

119886

2

0

(

119901

1015840

119901

)

times

1003816

1003816

1003816

1003816

1003816

1003816

1003816

int 119890

minus119894 119902 119903He120595

lowast

119899s1205951s1198893

119903He1003816

1003816

1003816

1003816

1003816

1003816

1003816

2

119889Ω

(6)

where 119902 =

119901

1015840

minus

119901 is the transferred momentum and 119889Ω is thesolid angle From the integration over the delta function in(2) we have obtained the conservation of energy during theprocess

119901

10158402

= 119901

2

+ 2119872(119864

1s minus 119864

119899s) (7)

It leads to the threshold energy corresponding to 119901

10158402

= 0 andto aminimum incident velocity Vmin =

radic2(119864

119899s minus 119864

1s)119872Theprevious expression for 119901

1015840 allows us to express the squaredmodulus of 119902 as

119902

2

= 2 (119901

2

+ 119872(119864

1s minus 119864

119899s)

minus119901

radic

119901

2

+ 2119872(119864

1s minus 119864

119899s) cos 120579)

(8)

where 120579 is the deviation angle of the incident OHe withrespect to the collision axis in the rest frame of the targetOHe

119890

+

119890

minus pairs will be dominantly produced if OHe is excitedto a 2s state since the only deexcitation channel is in this casefrom 2s to 1s As 119890

+

119890

minus pair production is the only possiblechannel the differential pair-production cross-section 119889120590

119890119890

isequal to the differential collisional excitation cross-sectionBy particularizing expression (6) to the case 119899 = 2 one finallygets

119889120590

119890119890

119889 cos 120579= 512

2

(

2120587119872

2

119872

2

He)119886

6

0

(

119901

1015840

119901

)

119902

4

2(4119886

2

0

119902

2

+ 9)

6

(9)


3 The 119890

+

119890



119889119873

119889119905

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119890119890

= int

119881

119887

120588

2

DM (

119877)

119872

2

⟨120590

119890119890

V⟩ (

119877) 119889

119877

(10)

where119881119887


119887


119890119890


119890119890


119890119890



120588DM (119877) = 120588

0

119877

3

0

(119877 + 119877

0

) (119877

2

+ 119877

2

0

)

(11)


0


0


⊙

= 8 kpc is 120588⊙


119872DM (119877) = 120588

0

120587119877

3

0

log(119877

2

+ 119877

2

0

119877

2

0

)

+2 log(119877 + 119877

0

119877

0

) minus 2 arctan(

119877

119877

0

)

(12)


120588

119887

(119877) =

119872bulge

8120587119877

3

119887

119890

minus119877119877

119887 (13)


10

119872

⊙


119872

119887

(119877) = 119872bulge 1 minus 119890

minus119877119877

119887(1 +

119877

119877

119887

+

119877

2

119877

2

119887

) (14)


119891 (119877 Vℎ

) =

1

119862 (119877)

119890

minusV2ℎ119906

2(119877)

(15)

where Vℎ


2



2


Vesc(119877)0

119891(119877 Vℎ

)119889Vℎ



119906 (119877) =

radic

119866119872tot (119877)

119877

(16)

where119872tot = 119872DM + 119872

119887

while Vesc =

radic



⟨120590

119890119890

V⟩ =

1

119906

2

radic


2


esc1199062

)

2

times int

2Vesc

0

120590

119890119890

(V) V3119890minusV22119906

2

119889V

(17)


0

and119872

4 Results



119890119890



0



0


125TeV As 120588

0


0

goes from 115 to10

4 GeVcm3

5 Conclusion



1000100101

10000

1000

100001 01

1205880(GeV

cm

3)

M (TeV)


0





Acknowledgment


References




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




3 The 119890

+

119890



119889119873

119889119905

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119890119890

= int

119881

119887

120588

2

DM (

119877)

119872

2

⟨120590

119890119890

V⟩ (

119877) 119889

119877

(10)

where119881119887


119887


119890119890


119890119890


119890119890



120588DM (119877) = 120588

0

119877

3

0

(119877 + 119877

0

) (119877

2

+ 119877

2

0

)

(11)


0


0


⊙

= 8 kpc is 120588⊙


119872DM (119877) = 120588

0

120587119877

3

0

log(119877

2

+ 119877

2

0

119877

2

0

)

+2 log(119877 + 119877

0

119877

0

) minus 2 arctan(

119877

119877

0

)

(12)


120588

119887

(119877) =

119872bulge

8120587119877

3

119887

119890

minus119877119877

119887 (13)


10

119872

⊙


119872

119887

(119877) = 119872bulge 1 minus 119890

minus119877119877

119887(1 +

119877

119877

119887

+

119877

2

119877

2

119887

) (14)


119891 (119877 Vℎ

) =

1

119862 (119877)

119890

minusV2ℎ119906

2(119877)

(15)

where Vℎ


2



2


Vesc(119877)0

119891(119877 Vℎ

)119889Vℎ



119906 (119877) =

radic

119866119872tot (119877)

119877

(16)

where119872tot = 119872DM + 119872

119887

while Vesc =

radic



⟨120590

119890119890

V⟩ =

1

119906

2

radic


2


esc1199062

)

2

times int

2Vesc

0

120590

119890119890

(V) V3119890minusV22119906

2

119889V

(17)


0

and119872

4 Results



119890119890



0



0


125TeV As 120588

0


0

goes from 115 to10

4 GeVcm3

5 Conclusion



1000100101

10000

1000

100001 01

1205880(GeV

cm

3)

M (TeV)


0





Acknowledgment


References




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1000100101

10000

1000

100001 01

1205880(GeV

cm

3)

M (TeV)


0





Acknowledgment


References




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics
















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article Dark Atoms and the Positron-Annihilation-Line Excess in …downloads.hindawi.com/journals/ahep/2014/869425.pdf · 2019-07-31 · Research Article Dark Atoms and the