Neutrinoless double beta decay.Introduction,

what its observation would prove,and its nuclear matrix elements.

Petr Vogel Caltech

ECT,Trento, 7/31/2009

ββ decay can exist in two modes. The two-neutrino (2νββ) decay is an allowed but slow process, while the neutrinoless (0νββ) mode would violate the total lepton number conservation law and thuswould be a sign of new physics

virtual state of the intermediate nucleusvirtual state of the intermediate nucleus and of the exchanged neutrino

(so far)

ν masses are much smaller than the masses of other fermions

Is that a possible “Hint of” a new mass-generating mechanism?

DESERT

The most popular theory of why neutrinos areso light is the —

See-Saw Mechanism

ν

NRVeryheavyneutrino

Familiarlightneutrino

}{

(Gell-Mann, Ramond, Slansky (1979), Yanagida(1979), Mohapatra, Senjanovic(1980))

It assumes that the very heavy neutrinos NR exist. Their large massexplains why the observed neutrinos are so light, mν ~ v2/MN

(v is the Higgs vacuum expectation value, v ~ 250 GeV). Both the light and heavy neutrinos are then Majorana fermions.

APS Joint Study on the Future ofNeutrino Physics (2004)

(physics/0411216)

We recommend, as a high priority, a phased programof sensitive searches for neutrinoless double betadecay (first on the list of recommendations)

The answer to the question whether neutrinos aretheir own antiparticles is of central importance, notonly to our understanding of neutrinos, but also toour understanding of the origin of mass.

Study of the 0νββ-decay is one of the highestpriority issues in particle and nuclear physics

5.63.367150Nd→150Sm8.92.479136Xe→136Ba34.52.533130Te→130Xe5.642.228124Sn→124Te7.52.802116Cd→116Sn11.82.013110Pd→110Cd9.63.034100Mo→100Ru2.83.35096Zr→96Mo9.22.99582Se→82Kr7.82.04076Ge →76Se0.1874.27148Ca→48Ti

Candidate Nuclei for Double Beta DecayQ (MeV) Abund.(%)

All candidatenuclei on thislist have Q > 2MeV.The nuclei with anarrow are usedin the presentor planned largemass experiments.For most of thenuclei in this listthe 2νββ decayhas been observed

If (or when) the 0νββ decay is observed twotheoretical problems must be resolved:

a)What is the mechanism of the decay, i.e., what kind of virtual particle is exchanged between the affected nucleons (or quarks)?b) How to relate the observed decay rate to the fundamental parameters, i.e., what is the value of the corresponding nuclear matrix elements?

Two basic categories are long-range andshort-range contributions to the 0νββ decay.

The long-range category involves two pointlike verticesand the exchange of a light Majorana neutrino betweenthem. The standard (plain vanilla) type of that categoryis when1/T1/2

0ν = G0ν(Q,Z) |M0ν|2 |<mββ>|2, <mββ>=ΣiUei2 mi ,

which represents simple relation between the decay rateand the parameters of the neutrino mass matrix.

The short-range category involves only a single pointlikevertex (six fermions, four hadrons and two leptons),i.e. a dimension 9 operator. The relation between thedecay rate and neutrino mass is not simple in that case.

The relative size of the heavy (AH) vs. light particle (AL) exchange to the decay amplitude is (a crude estimate, due originaly to Mohapatra)

AL ~ GF2 mββ/<q2>, AH ~ GF

2 MW4/Λ5 ,

where Λ is the heavy scale and q ~ 100 MeV is the virtualneutrino momentum.

For Λ ~ 1 TeV and mββ ~ 0.1 – 0.5 eV AL/AH ~ 1, hence bothmechanisms contribute equally.

It is well known that the amplitude for the light neutrinoexchange scales as <mββ>. On the other hand, if heavy

particles of scale Λ are involved the amplitude scales as 1/Λ5

(dimension 9 operator).

Note in passing that less attention has been devoted in the past tothe evaluation of the nuclear matrix elements for the case of heavy particle exchange (short-range contribution to 0νββ decay).Proper treatment of the nucleon-nucleon repulsion in that case is obviously crucial; it is traditionally treated crudely using nucleonform factors.Including pion exchange avoids this problem and seems to lead tolarger and more consistently evaluated matrix elements. (Vergados 82, Faessler et al. 97, Prezeau et al. 03)

0νββ amplitude is contained in the ππee vertex

Lets restrict our further considerations to the simplest long-rangemechanism involving the virtual exchange of a light Majorana neutrino.

As long as the mass eigenstates νi that are the components of theflavor neutrinos νe, νµ, and ντ are really Majorana neutrinos, the 0νββdecay will occur, with the rate

1/T1/2= G(Etot,Z) (M0ν)2 <mββ>2,

where G(Etot,Z) is easily calculable phase space factor, M0ν is the nuclearmatrix element, calculable with difficulties (and discussed later), and

<mββ> = | Σi |Uei|2 exp(iαi) mi |,

where αi are unknown Majorana phases (only two of them are relevant).Using the formula above we can relate <mββ> to otherobservables related to the absolute neutrino mass.

Usual representation of that relation. It shows that the <mββ>axis can be divided into three distinct regions as indicated. However, it creates the impression (false) that determining <mββ> would decide between the two competing hierarchies.

inverted

normal

degenerate

In double beta decay two neutrons bound in theground state of an initial even-even nucleus aresimultaneously transformed into two protons thatagain are bound in the ground state of the finalnucleus.It is therefore necessary to evaluate, with a sufficientaccuracy, the ground state wave functions of bothnuclei, and evaluate the matrix element of the 0νββ-decay operator connecting them.

This cannot be done exactly; some approximationand/or truncation is always needed. Moreover,unfortunately, there is no other analogous observablethat can be used to judge the quality of the result.

Nuclear Matrix Elements:

Can one use the 2νββ-decay matris elements for that?What are the similarities and differences?

Both 2νββ and 0νββ operators connect the same states.Both change two neutrons into two protons.

However, in 2νββ the momentum transfer q < few MeV;thus eiqr ~ 1, long wavelength approximation is valid, only the GT operator στ need to be considered.

In 0νββ q ~ 100-200 MeV, eiqr = 1 + many terms, thereis no natural cutoff in that expansion.

Explaining 2νββ-decay rate is necessary but not sufficient

To obtain the 0νββ operator, we need to integrate neutrinopropagator first over dq0 (the energy of the virtual neutrino), and then Fourier transform the corresponding second order perturbation expression (this is now the integral over the three-momentum of the virtual neutrino):

This is the `neutrino potential’ (Am is the total energy in the virtualnuclear state with respect to (Mi + Mf)/2). The constants are addedfor future convenience. r is the distance between the two neutronsthat are transformed into protons.

This H(r, E), together with spin and isospin operators will appear inthe nuclear matrix elements, In compact form

H(r ) = R/r Φ(Εr)

where Φ (Εr) ≤ 1 is a slowly varying function. Since r < R thepotential is ≥ 1 (but less than 5-10).

Without short range correlations

With short range correlations

76Ge

76Ge

100Mo

100Mo

130Te

130Te

How good is theclosure approximation?

Comparison betweenthe QRPA M0ν with theproper energies of the virtual intermediate states (symbols with arrows) and the closureapproximation (lines)with different <En - Ei>.

Note the mild dependenceon <En - Ei> and the factthat the exact resultsare reasonably closeto the closure approximationresults for <En - Ei> < 20 MeV.

Graph by F. Simkovic

Two complementary procedures are commonly used:a) Nuclear shell model (NSM)b) Quasiparticle random phase approximation (QRPA)

In NSM a limited valence space is used but all configurations of valence nucleons are included.Describes well properties of low-lying nuclear states.Technically difficult, thus only few 0νββ calculations.

In QRPA a large valence space is used, but only a classof configurations is included. Describes collectivestates, but not details of dominantly few-particle states.Relatively simple, thus more 0νββ calculations.

Evaluation of M0ν involves transformation to the relative coordinatesof the nucleons (the operators OK depend on rij)

unsymmetrized two-bodyradial integral involves`neutrino potentials’

From QRPA forfinal nucleus

From QRPA forinitial nucleus

overlap

Note the two separate multipole decompositions. Jπ refers to thevirtual state in odd-odd nucleus, while J refers to the angularmomentum of the neutron pair transformed into proton pair.

The vectors X and Y are obtained by solving the equations of motion:

with

Eigenvalue equation forω2, unphysical solutions with ω2 < 0 possible

particle-hole

particle-particle

Evaluate the M2ν is relatively simple

But the two `vacua’ |0+QRPA> are not identical, hence the

Overlap is included (this is an approximation).But more importantly, how does one choose gpp?

The usual practice in QRPA is to give up on the predictability of the 2νββ decay, instead to choose gpp such that the M2ν has the correct value ( ~ ±20% deviation from the nominal gpp = 1)

overlap

2νββ matrix elements for 76Ge as a function of gpp in QRPA and RQRPA,calculation performed with 9,12, and 21 orbits. Note the crossing of zeroand approach to collapse (infinite slope)

99

1212

21 21

experimentalM2ν

Important bonus: Fixing gpp in order to reproduce M2ν correctly alsoessentially removes the dependence on the size of the s.p. basis.

M0ν full lines,M2ν dashed lines.

By fixing gpp to theexp. value of M2ν

we get the same M0ν

with 9 and 21 levels,but with different gppfor the two cases, 1.05 vs. 0.85

experimentalvalue of M2ν

82Se

130Te

Why it is difficult to calculatethe matrix elements accurately?

Contributions of differentangular momenta J of theneutron pair that is transformed in the decay into the proton pair with the same J.

Note the opposite signs, and thus tendency to cancel, between the J = 0 (pairing) and the J≠ 0(ground state correlations) parts.

The same restricted s.p. space is used for QRPA and NSM. There is a reasonablequalitative agreement between these two methods

J

from isovectorp-p and n-n pairinginteraction

from isoscalar n-p interaction

The opposite signs, and similar magnitudes of the J = 0 and J ≠ 0 parts is universal. Here forthree nuclei with coupling constant gpp adjusted so that the 2νββ rate is correctly reproduced.Now two oscillator shells are included.

Contributions to the 0νββ-decayNME in 100Mo from different Jπ

multipole states in the virtualintermediate odd-odd 100Tc.The strength gpp of the particle-particle neutron-proton interactionis varied by ± 5% with respectto the value gpp = 1.096 that givesgood agreement with the known2νββ decay rate.Note the rapid variation with gppof the 1+ multipole. The othermultipoles (as well as the negativeparity ones not shown) changeonly slowly with gpp.It is therefore reasonable toadjust gpp so that the contributionof the 1+ is fixed to the known2νββ where only that multipolecontributes.This is what we do.

J+

J-

How good is QRPA? Can we check its validity?

To do that (approximately) we use a two-level model that can besolved exactly using the algebra based on SO(5)xSO(5).It has many features analogous to real nuclei. The hamiltonian is

number operator pair operators

distance between the two shells

strength of n-p interaction

From Engel & Vogel, PRC69,034304(2004)

Comparison of exact (solid lines) and QRPA (dashed) M2ν and M0ν,for different level spacings ε. In this model QRPA works perfectly.

Dependence of the M0ν on the distance r between the two neutrons that are transformed into the two protons.

The “neutrino potential” is H(r)= R/r Φ(ωr) where Φ(ωr) is rather slowly varying function. This is a longrange potential, more or less like a Coulomb potential.Thus, naively, one expect that the matrix elementwill get its main contribution from r ~ R, i.e. themean distance between the nucleons in a nucleus.

This is not so. Due to the “pairing” and “broken pairs”competition, only distances r < 2-3 fm contribute, i.e., only nearest neighbors.

Full matrix element

The radial dependence of M0ν for the three indicatednuclei. The contributionssummed over all componentsis shown in the upper panel.The `pairing’ J = 0 and`broken pairs’ J ≠ 0 partsare shown separately below.Note that these two partsessentially cancel each otherfor r > 2-3 fm. This is ageneric behavior. Hencethe treatment of small values of r or large valuesof q are quite important.

C(r)

CJ(r)

M0ν = ∫C(r)dr

pairing part

broken pairs part

total

The radial dependence of M0ν for the indicated nuclei, evaluated inthe nuclear shell model. (Menendes et al, arXiv:0801.3760).Note the similarity to the QRPA evaluation of the same function.

The radial dependence of M0ν evaluated in the exactly solvable model described earlier. Note that the cancellation at r > 2-3 fm is most pronounced near gpp = 1.

The finding that the relative distances r < 2- 3 fm, and correspondinglythat the momentum transfer q > ~100 MeV means that one needs toconsider a number of effects that typically play a minor role in thestructure of nuclear ground states:a) Short range repulsionb) Nucleon finite sizec) Induced weak currents (Pseudoscalar and weak magnetism)

Each of these, with the present treatment, causes correction(or uncertainty) of ~20% in the 0νββ matrix element.

There is a consensus now that these effects must be included butonly emerging consensus how to treat them, in particular a).

Two-nucleon probability distribution, with and without correlations,MC with realistic interaction. O. Benhar et al. RMP65,817(1993)

= nuclear matter, saturation density

= nuclear matter, halfof the saturation density

no s.r.c.

only protons

See also Bisconti et al., Phys. Rev. C73, 054304(2006) for the more modern version of this

(5.3)(4.0)

(4.1)(5.0)

Dependence on the distance between the two transformed nucleons andthe effect of different treatments of short range correlations. Thiscauses changes of M0ν by ~ 20%.

Graph by F. Simkovic

C(r)

Figure from Engel & Hagen, arXiv:0904.1709. The GT part of M0ν is shown.Full line for the bare operator, s.r.c not included. Dashed line is obtained whenthe effective operator generated by summing over high energy ladderdiagrams is used. The dotted line is from Simkovic et al. (0902.0331), obtainedby fits to the coupled-cluster equations at the one and two-body level.

These results,together withthe consistenttreatment ofthe UCOMmethod suggestthat the effectof s.r.c isquite small.

Dependence of the 0νββ matrix element on the ΜΑ = MV = Λcut parameter inthe usual dipole nucleon form factor . When correction for short rangecorrelations is included the M0ν changes little for Λcut ≥ 1000 MeV.

Graph by F. Simkovic

f(q2)

~1/(1+q2/Λ2)2

Hadronic current expressed in terms of nucleon fields Ψ:

Vector gV(q2) = gV/(1 + q2/MV2)2, gV = 1, MV = 0.85 GeV

Axial vector gA(q2) = gA/(1 + q2/MA2)2, gA = 1.25, MA = 1.09 GeV

Weak Magnetism gM(q2) = (µp - µn) gV(q2)

Induced pseudoscalar gP(q2) = 2mpgA(q2)/(q2 + mπ2)

After the nonrelativistic reduction the space part of the current is

Contributions of different parts of the nucleon current.Note that the AP (axial-pseudoscalar interference) contains q2/(q2 + mπ

2), and MM contains q2/4Mp2.

76Ge→76Se

Full estimated range of M0ν within QRPA framework and comparison with NSM (higher order currents now included in NSM)

The 2ν matrix elements, unlike the 0ν ones, exhibit pronounced shelleffects. They vary fast as a function of Z or A.

0νββ nuclear matrix elements calculated very recently with the Interacting Boson Model-2, see Barea and Iachello, Phys. Rev. C79, 044301(2009).

Why are the QRPA and NSM matrix elements different?

Various possible explanations:a) Assumed occupancies of individual valence orbits might be differentb) In QRPA more single particle states are includedc) In NSM all configurations (seniorities) are includedd) In NSM the deformation effects are includede) All of the above

pf5/2

g9/2

Pf5/2g9/2

P 0.5f5/2 0.8g9/2 0.7

Neutron orbit occupancies, original Woods-Saxon vs.adjusted effective mean field. For 76Ge -> 76Se experiment

Assumed occupancies of individual valence orbits might be different

Experiment from J.P.Schiffer et al, Phys.Rev.Lett. 100, 1120501(2008),used (d,p),(p,d),(3He,α),(α,3He) to derive occupancies of neutron orbits

P 1.8 ± 0.15f5/2 2.0 ± 0.25g9/2 0.2 ± 0.25

P 2.1 ± 0.15f5/2 3.2 ± 0.25g9/2 0.8 ± 0.25

P 0.3f5/2 1.2g9/2 0.6

Proton orbit occupancies, original Woods-Saxon vs.adjusted effective mean field. experiment

Experiment from B.P.Kay et al, Phys.Rev.C79,021301(2009), based on (d,3He)

Full estimated range of M0ν within QRPA framework and comparison with NSM (higher order currents now included in NSM)

♥

New QRPA value with adjusted mean field so that experimentaloccupancies are reproduced

♠ New NSM value with adjusted mean field (monopole) where experimentaloccupancies are better reproduced

In QRPA more single particle states are included

Contribution of initial neutron orbit pairs against the final proton pairs.The nonvalence orbits are labeled as r. Adding all parts with r-type orbits gives +2.83 - 3.22 = -0.39 which is only ~12% of the total matrix element 3.27(The total matrix element is made of 9.74 - 6.46 = 3.27.)In the figure all entries are, however, normalized so that their sum is unity..

It appears, therefore, that all of these effects, possible differencesin the assumed occupancies of valence orbits, additional single particle states included in QRPA but not in NSM, inclusion ofcomplicated configurations (higher seniority and/or deformation)in NSM but only crudely in QRPA, can and probably do affectthe resulting nuclear matrix elements, and might explain thedifferent outcomes of the two methods. In particular, the difference in deformation of the initial andfinal nuclei makes the evaluation of the matrix element for150Nd -> 150Sm very difficult.

Summary on matrix elements

1) There is, as of now, agreement of all practitioners on what needs to beincluded in the evaluation of the 0νββ nuclear matrix elements, eventhough there is no complete agreement how to do it (e.g. for

the short range correlations).2) The NSM and QRPA have both many basic features in common, in particular the (sometimes severe) cancellation between the effect of pairing and `broken pairs’ configurations and in the radial distance dependence.3) There are still noticeable differences between these two methods, and several possible causes have been identified.4) Both methods predict that the 0νββ nuclear matrix elements should vary slowly and rather smoothly with A and Z, unlike the 2νββ matrix

elements. That makes the comparison of experiments with different sources easier.

There is a steady progressin the sensitivity of thesearches for 0νββ decay.Several experiments thatare funded and almostready to go will reachsensitivity to ~0.1 eV.There is one (so far unconfirmed) claim thatthe 0νββ decay of 76Gewas actually observed.The deduced mass <mββ>would be then 0.3-0.7 eV.

Moore’s law of 0νββ decay:

0νββ half-lives for <mββ> = 50 meV basedon the matrix elements of Rodin et al.

76Ge (2.1 - 2.6) x 1027 y82Se (6.0 - 8.7) x 1026 y100Mo (1.1 - 1.7) x 1027 y130Te (0.7 - 1.7) x 1027 y136Xe (1.5 - 5.6) x 1027 Y (no 2ν observed yet)

Note: Calculated matrix elements decrease withincreasing A, but the phase-space factors usuallyincrease, particularly the Coulomb factor, hencerelatively little variation of T1/2 with A.

Note: The sensitivity to <mββ> scales as 1/(T1/2)1/2

Summary and/or Conclusions

Study of 0νββ decay entered a new era. No longer is the aim just to push the sensitivity higher and the background lower, but to explore specific regions of the <mββ> values.In agreement with the `phased’ program the plan is to explore the`degenerate’ region (0.1-1 eV) first, with ~100 kg sources, andprepare the study of `inverted hierarchy’ (0.01-0.1eV) regionwith ~ ton sources that should follow later.In this context it is important to keep in mind the questions I discussed:a) Relation of <mββ> and the absolute mass (rather clear already,

becoming less uncertain with better oscillation results).b) Mechanism of the decay (exploring LFV, models of LNV, running of LHC to explore the ~TeV mass particles).c) Nuclear matrix elements (exploring better, and agreeing on, the reasons for the spread of calculated values, and deciding on the optimum way of performing the calculations, while pursuing vigorously also the application of the shell model).