Matrix Elements for Neutrinoless Double Beta Decay

Fedor Simkovica,b

aBogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region, RussiabDepartment of Nuclear Physics and Biophysics, Comenius University, Mlynska dolina F1, SK–842 15 Bratislava, Slovakia

Abstract

The status of calculation of the neutrinoless double beta decay (0νββ-decay) nuclear matrix elements (NMEs) isreviewed. The 0νββ-decay NMEs obtained in the framework of the Quasiparticle Random Phase Approximation(QRPA), Large Scale Shell Model (LSSM), Interacting Boson Model (IBM), Projected Hartree-Fock Bogoliubovapproach (P-HFB) and Energy Density Functional Method (EDF) are presented and discussed. The problem of re-liable determination of the 0νββ-decay NMEs is addressed. It is manifested that the uncertainty associated with thecalculation of the 0νββ-decay NMEs can be diminished by suitable chosen nuclear probes. Further, the recent devel-opment in the field of double beta decay showed that the search for the neutrinoless double electron capture process(0νECEC) may establish the Majorana nature of neutrinos. This possibility should be considered as alternative andcomplementary to searches for the 0νββ-decay.

Keywords: neutrinoless double beta decay, nuclear structure

1. Introduction

The fundamental importance of the search for the0νββ-decay,

(A, Z) → (A, Z + 2) + 2e−, (1)

is widely accepted. After 70 years the brilliant hypoth-esis of Ettore Majorana is still valid and is strongly sup-ported by the discovery of neutrino oscillations and bythe construction of the Grand Unified Theories. The0νββ-decay is currently the most powerful tool to clar-ify if the neutrino is a Dirac or a Majorana particle. Thisissue is intimately related with the origin of neutrinomasses having a strong impact also on astrophysics andcosmology.

The main aim of the experiments on the search for0νββ-decay is the measurement of the effective Majo-rana neutrino mass mββ. Under the assumption of themixing of three massive Majorana neutrinos the effec-tive Majorana neutrino mass mββ takes the form

mββ = U2e1 m1 + U2

e2 m2 + U2e3 m3. (2)

Here, Uei and mi (i = 1, 2, 3) are elements ofPontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrinomixing matrix and masses of neutrinos, respectively.

Experimental searches for the 0νββ-decay, of everincreasing sensitivity, are being pursued worldwide.However, interpreting existing results as a measurementof the Majorana neutrino effective mass and planningnew experiments, depends crucially on the knowledgeof the corresponding nuclear matrix elements that gov-ern the decay rate. Accurate determination of the nu-clear matrix elements, and a realistic estimate of theiruncertainty, is of great importance.

2. Current status of the 0νββ-decay NMEs

The inverse value of the 0νββ-decay half-life for agiven isotope (A,Z) is a product of the effective mass ofMajorana neutrinos mββ, the known phase-space factorG0ν(Qββ, Z) (depending on nuclear charge Z and the en-ergy release Qββ of the reaction) and the nuclear matrixelement M0ν, which depends on the nuclear structure of

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www.elsevier.com/locate/npbps

doi:10.1016/j.nuclphysbps.2012.09.026

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

M0ν

LSSMEDFPHFBIBM-2QRPA

76Ge 82Se 96Zr 100Mo116Cd124Sn 130Te 136Xe48Ca 150Nd128Te

Figure 1: The 0νββ-decay NMEs calculated within different nuclearstructure approaches: Large Scale Shell Model (LSSM) [5], En-ergy Density Functional Method (EDF) [9], Projected Hartree-FockBogoliubov approach (P-HFB) [7], Interacting Boson Model (IBM)[8] and (Renormalized) Quasiparticle Random Phase Approximation(R)QRPA [4]. The Miller-Spencer Jastrow two-nucleon short-rangecorrelations are taken into account. gA = 1.25 is considered. Thenuclear radius is R = 1.2 fm A1/3.

the particular isotope under study [1]:

(T 0ν1/2)−1 = G0ν(Qββ, Z) |M0ν|2 |mββ|2, (3)

where M0ν consists of Fermi, Gamow-Teller and tensorparts as

M0ν = −MF

g2A

+ MGT + MT . (4)

From the measurement of half-life of the 0νββ-decayonly the product |mββ| |M0ν(A, Z)| of effective neutrinomass and nuclear matrix element can be determined.Clearly, the accuracy of the determination of |mββ| fromthe measured 0νββ-decay half-life is mainly given byour knowledge of nuclear matrix elements. Without ac-curate calculation of the 0νββ-decay NMEs, it is notpossible to reach qualitative conclusions about neutrinomasses, the type of neutrino mass spectrum and CP vi-olation.

The nuclear matrix elements for 0νββ-decay must beevaluated using tools of nuclear structure theory. Unfor-tunately, there are no observables that could be directlylinked to the magnitude of 0νββ-decay nuclear matrixelements and that could be used to determine them inan essentially model independent way. The calculationof the 0νββ-decay matrix elements is a difficult prob-lem because ground and many excited states of open-shell nuclei with complicated nuclear structure have tobe considered.

The main two basic approaches used for evaluationof double beta decay NMEs are the Quasiparticle Ran-dom Phase Approximation (QRPA) [2, 3, 4] and theLarge Scale Shell Model (LSSM) [5]. Both methodshave the same starting point, namely a Slater determi-nant of independent particles. However, there are sub-stantial differences between both approaches, namelythe kind of correlations they include are complemen-tary. The QRPA treats a large single particle modelspace, but truncates heavily the included configurations[2]. The LSSM, by a contrast, treats a small fraction ofthis model space, but allows the nucleons to correlate inarbitrary ways [6].

Due to its simplicity the QRPA is a popular tech-nique to calculate the 0νββ-decay NMEs. One of themost important factors of the QRPA calculation of the0νββ-decay NMEs is the way how the particle-particlestrength of the nuclear Hamiltonian gpp is fixed. It hasbeen shown that by adjusting gpp to the 2νββ-decayrates the uncertainty associated with variations in QRPAcalculations of the 0νββ-decay NMEs can be signifi-cantly eliminated [2]. In particular, the results obtainedin this way are essentially independent of the size ofthe basis, the form of different realistic nucleon-nucleonpotentials, or on whether QRPA or renormalized QRPA(take into account Pauli exclusion principle) is used.

Matrix elements for the double beta decay arecalculated also by angular momentum projected(with real quasi-particle transformation) Hartree-Fock-Bogoliubov (P-HFB) wave functions [7] the InteractingBoson Model (IBM) [8] and by Energy Density Func-tional Method (EDF) [9]. The P-HFB allows only, thatneutron pairs with angular momenta 0+, 2+, 4+, · · · aretransformed into two protons in the 0νββ-decay. Inaddition the pairs different from 0+ are strongly sup-pressed compared to the results of the LSSM and theQRPA. The approaches LSSM and QRPA show also,that other neutron pairs contribute strongly, which cannot be included into real P-HFB. One would need toextend the P-HFB approach to complex quasi-particletransformations and probably also to several orthogonalP-HFB configurations. The IBM is also restrictive: Itallows only that 0+ and 2+ neutron pairs are changedinto proton pairs.

The calculated 0νββ-decay NMEs within these ap-proaches are presented in Fig. 1. It is surprising that theIBM results agree well with the QRPA ones. Results ofthese approaches exhibit some dependence on A unlikethe LSSM values, which are practically the same exceptfor 48Ca. The value of the 0νββ-decay NME for thisisotope is suppressed as 48Ca is a magic nucleus.

F. Šimkovic / Nuclear Physics B (Proc. Suppl.) 229–232 (2012) 160–164 161

1025

1026

1027

1028

1029

T0ν 1/2[y]

LSSMEDFPHFBIBM-2(R)QRPA

76Ge 82Se 96Zr 100Mo116Cd124Sn 130Te 136Xe48Ca 150Nd128Te

Figure 2: The 0νββ-decay half-lives for mββ = 50 meV and nuclearmatrix elements of different many-body approaches. Notation as inFig. 1.

Comparing plots of the 0νββ-decay nuclear matrix el-ements calculated using different methods gives someinsight in the advantages or disadvantages of differentcandidate nuclei. However, matrix elements are notquite the relevant quantities. Experimentally, half-livesare measured or constrained, and the effective Majorananeutrino mass mββ is the ultimate goal. In the case of theinverted hierarchy of neutrino masses mββ is about 20-50 meV. In Fig. 2 the calculated half-lives for doubleβ-decaying nuclei of interest are showed for mββ = 50meV and nuclear matrix elements presented in Fig. 1.

3. Reducing the uncertainty in NMEs

The improvement of the calculation of double betadecay nuclear matrix elements is a very important andchallenging problem. The uncertainty associated withthe calculation of the 0νββ-decay NMEs can be di-minished by suitably chosen nuclear probes. A com-plementary experimental information from related pro-cesses like charge-exchange reactions, muon captureand charged current (anti)neutrino-nucleus reactions ishighly required. A direct confrontation of nuclear struc-ture models with data from these processes might im-prove quality of nuclear structure models. The con-strained parameter space of nuclear models is a promis-ing way to reduce uncertainty in the calculated 0νββ-decay NMEs.

In practice, knowledge of the 2νββ-decay rate and ofthe ordinary decay f t values were used to constrain thenuclear model parameters, in particular when the quasi-particle random phase approximation (QRPA) was em-ployed [3]. Clearly, when other relevant data become

available, and the nuclear model is constrained to re-produce them, confidence in the deduced 0νββ-decayNMEs is increased.

The occupancies of valence neutron and proton orbitsdetermined experimentally in Refs. [10], represent im-portant constraints for nuclear models used in the eval-uation of the 0νββ-decay NME for transition 76Ge →76

S e transition. In Ref. [11] the input mean field hasbeen modified in such a way that the valence orbits inthe model obey these constraints. Within QRPA and itsgeneralizations it was found that it is important to alsochoose the variant of the basic method that makes suchcomparison meaningful by conserving the average par-ticle number in the correlated ground state. When fol-lowing this procedure, but otherwise keeping the samesteps as in evaluation of M0ν within QRPA before, theconclusion was that for the 76Ge →76 S e transition thematrix element is smaller by 25%, reducing the previ-ously bothersome difference with the shell model pre-diction noticeably. Clearly, having the experimental or-bit occupancies available and adjusting the input to ful-fill the corresponding constraint makes a difference. Itwould be very useful to have similar constraints avail-able also in other systems, in particular for 130Te and/or136Xe.

Charge-exchange reactions of (p, n) and (n, p) type atintermediate energies and at forward angles, i.e., lowmomentum transfers (qtr ∼ 0 and ΔL = 0), selec-tively excite Gamow-Teller (GT) transitions owing tothe dominance of the Vστ component of the effective in-teraction. However, experiments which employ the el-ementary (p, n) and (n, p) reactions have rather limitedresolution and alternatives to them have now success-fully been established through the (n, p)-type (d, 2He)or (t, 3He) reactions and the (p, n)-type (3He, t) reac-tion. Resolutions on the order of 100 keV in the caseof (d, 2He), 190 keV for (t, 3He) and 30 keV for (3He, t)have routinely been achieved [12].

The connection between the two-neutrino doublebeta decay (2νββ-decay) half-life and the GT transitionstrength B(GT) is as follows:

(T 2ν

1/2

)−1= G2ν(Qββ, Z)

∣∣∣M2νDGT

∣∣∣2 , (5)

where G2ν(Q, Z) is a phase-space factor. It contains theweak interaction coupling constant gA at fourth orderlike G0ν(Q, Z). The 2νββ-decay matrix element can bededuced by combining GT+ and GT− distributions inthe following way:

M2νDGT =

∑

m

MGT+m · MGT−

m

Qββ/2 + me + Ex(1+m) − E0,

F. Šimkovic / Nuclear Physics B (Proc. Suppl.) 229–232 (2012) 160–164162

B(GT±) =1

2Ji + 1

∣∣∣MGT± ∣∣∣2 . (6)

Here, (Ex(1+m) − E0) is the energy difference betweenthe mth intermediate 1+ state and the initial ground state.The∑

m runs over all states of the intermediate nucleus.In this approach the effect of a destructive interferenceamong contributions of different states of intermediatenucleus to M2ν

DGT is neglected.The results of the charge-exchange reaction exper-

iments therefore furnish important information aboutthe nuclear physics relevant for double β-decay [12].This information directly feeds into a model calcula-tions, which are aimed at describing reliably the nuclearphysics around both decay variants, the 2νββ-decay andthe 0νββ-decay. A high energy resolution of the orderof 30 keV, which can presently only be obtained at theRCNP facility in Osaka, allows a precise determinationof the GT strength distribution. The high resolution cangive significant insight into the details of the nuclearstructure. It may be important to understand if the con-centration of the low-energy B(GT) strength within asingle strong transition, as was observed in the case of96Zr and 100Mo, is a somewhat general feature of nucleiwith masses A ∼ 100 or above [12]. An open questionis the 2νββ-decay half-life of 136Xe, which has not beenmeasured yet. The reason of suppression of this processis not known. Clearly, explanation of these effects hassignificant bearing on the double β-decay rate.

Till now, Miller-Spencer Jastrow short-range correla-tions (SRC) have been introduced into the involved two-body transition matrix elements, changing two neutronsinto two protons, to achieve healing of the correlatedwave functions. These correlations, which have originin the short-range repulsion of the realistic NN inter-action, are neglected in the mean-field (e.g., the LSSMand the QRPA descriptions). The effect of these SRCwas considered as an uncertainty [3].

In Ref. [4] the coupled cluster method (CCM) short-range correlations has been considered. They were ob-tained as a solution of the CCM with realistic CD-Bonn and Argonne V18 interactions. An analysis ofthe squared correlation functions, represented by a ra-tio of correlated and uncorrelated neutrino potentials,has showed a principal differences among the Miller-Spencer and the CCM SRC. In addition, the importanceof the effect of the finite nucleon size (FNS) was stud-ied. It was found that both CCM SRC and the FNSeffect reduces the 0νββ-decay NMEs by a comparableamount for a considered choice of form-factor masses.The suppression due to Miller-Spencer SRC is abouttwice larger when compared to results without SRC and

the FNS effect.By performing a consistent calculation of the 0νββ-

decay NMEs in which pairing, ground-state correlationsand the short-range correlations originate from the samerealistic NN interaction, namely from the CD-Bonn andArgonne potentials, matrix elements for the 0νββ decayare obtained, which are larger in magnitude when com-pared with the traditional approach of using the Miller-Spencer Jastrow correlations.

4. Neutrinoless double electron capture

Recently, a new theoretical background for the neu-trinoless double electron capture (0νECEC) transitionsto final atomic state with a minimal mass differencewith original atom has been proposed[13]. A new phe-nomenon is an oscillation plus deexcitations of atoms

(A,Z) ↔ (A,Z + 2)∗∗,(A,Z) ↔ (A,Z− 2)∗∗, (7)

which has origin in a mixing of a pair of neutral atoms,(A,Z) and (A,Z ± 2)∗∗ differing by two units in lep-ton numbers. The first atom (A,Z) is in a groundstate and the second atom (A,Z ± 2)∗ might be inan excited state both in respect of atomic and nuclearstructure. The underlying mechanism is a transition oftwo protons and two bound electrons to two neutronsp + p + e−b + e−b ↔ n + n. A signature of the os-cillations might be an electromagnetic deexcitation ofthe involved unstable nucleus and atomic shell with theelectron holes.

A phenomenological analysis of the oscillations plusdeexcitations of atoms pointed out on a resonant en-hancement of the 0νECEC decay rate, that has a Breit-Wigner form. It was manifested that it is reasonableto hope that a search for oscillation plus deexcitationof atoms, which are sufficiently long lived to conduct apractical experiment, may uncover processes with lep-ton number violation. For that purpose systems of twoatoms with the smallest mass difference were looked forin the periodic table. The favorable atomic systems areas follows [14]: 106Cd → 106Pd∗∗, 124Xe → 124Te∗∗,136Ce → 136Ba∗∗, 152Gd → 152Sm∗∗, 156Dy →156Gd∗∗, 164Er → 164Dy∗∗, 168Yb → 168Er∗∗, 180W →180Hf∗∗, 184Os → 184W∗∗, 190Pt → 190Os∗∗ etc. Itis worth mentioning that the lepton number conservingdouble electron capture with emission of two neutrinos,

(A,Z) ↔ (A,Z + 2)∗∗ + 2ν, (8)

is strongly suppressed due to a very small phase space.

F. Šimkovic / Nuclear Physics B (Proc. Suppl.) 229–232 (2012) 160–164 163

Within the formalism of oscillations plus deexcitationof atoms the decay width of the 0νECEC process in-volving 0+ → 0+ nuclear transition is given by [14]

ΓνECECab =

|Vab|2(QECEC − Bab)2 + 1

4Γ2ab

Γab, (9)

where QECEC stands for a difference between the ini-tial and final masses of atoms in ground state, Bab =

Ea + Eb + EC is the energy of two electron holes,which quantum numbers are denoted by indices a andb (a ≡ (na, ja, la) and b ≡ (nb, jb, lb). EC is an inter-action energy of two holes. Γab is a width of excitedfinal atom with two electron holes. The lepton numberviolating potential takes form

Vab =1

4πG2βmββ

g2A

R< Fab > M0ν. (10)

Here, R is a nuclear radius, < Fαβ > is a combination ofaveraged upper and lower bispinor components of theatomic electron wave functions and M0ν is the nuclearmatrix element [14].

Assuming an effective mass for the Majorana neu-trino of 1 eV, some 0νECEC half-lives are predictedto be as low as 1022 years in the unitary limit [14]. Itis argued that, in order to obtain more accurate predic-tions for the 0νECEC half-lives, precision mass mea-surements of the atoms involved are necessary, whichcan readily be accomplished by today’s high precisionpenning traps. Further advancements also require a bet-ter understanding of high-lying excited states of the fi-nal nuclei (i.e. excitation energy, angular momentumand parity) and the calculation of the nuclear matrix el-ements.

5. Conclusion and outlook

Many new projects for measurements of 0νββ-decayhave been proposed, which hope to probe effective neu-trino mass mββ down to 10-50 meV. Nuclear matrix ele-ments need to be evaluated with uncertainty of less than30% to establish the neutrino mass spectrum and CPviolating phases. The improvement of the calculationof the nuclear matrix elements is a very important andchallenging problem.

Recently, there has been significant progress in un-derstanding the source of the spread of calculatedNMEs. Nevertheless, there is no consensus amongnuclear theorists about their correct values, and corre-sponding uncertainty. However, a recent developmentin the field is encouraging. There is a reason to be hope-ful that the uncertainty will be reduced.

An important cross-check for nuclear models wouldbe to explore the structure of the intermediate odd-oddnuclei by the charge exchange reactions. There are pos-sibilities for improving the QRPA calculation of NMEs,e.g., by taking into account the deformation of parentand daughter nuclei. Further progress in the NSM calcu-lation will be possible due to increasing computer speedand memory. This will allow to extend the consideredmodel spaces. The exactly solvable models can alsohelp to find the ultimate solution of this important prob-lem. It is also clear that in order to have confidence incalculated NMEs multiple 0νββ-decay experimental re-sults are required.

Finally, it is reasonable to hope that the search for theneutrinoless double electron capture process of atoms,which are sufficiently long lived to conduct a practi-cal experiment, may established the Majorana nature ofneutrinos. This possibility should be considered as al-ternative and complementary to searches for the 0νββ-decay.

The presented NMEs obtained within the QRPA-likeapproaches were obtained in collaboration with AmandFaessler, Vadim Rodin and Petr Vogel. The author ac-knowledges the support by the VEGA Grant agency un-der the contract No. 1/0249/03.

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