nme

Nuclear matrix elements for neutrinoless double-beta decay and double-electron capture

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2012 J. Phys. G: Nucl. Part. Phys. 39 124006

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IOP PUBLISHING JOURNAL OF PHYSICS G: NUCLEAR AND PARTICLE PHYSICS

J. Phys. G: Nucl. Part. Phys. 39 (2012) 124006 (19pp) doi:10.1088/0954-3899/39/12/124006

Nuclear matrix elements for neutrinoless double-betadecay and double-electron capture

Amand Faessler1, Vadim Rodin1 and Fedor Simkovic2,3

1 Institute of Theoretical Physics, University of Tuebingen, Tuebingen 72076,Germany2 Department of Nuclear Physics and Biophysics, Comenius University, Mlynska dolina F1,Bratislava SK-842 15, Slovakia3 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research (JINR),Dubna 141980, Moscow, Russia

E-mail: [email protected]

Received 13 July 2012Published 19 November 2012Online at stacks.iop.org/JPhysG/39/124006

AbstractA new generation of neutrinoless double-beta decay (0-decay) experimentswith improved sensitivity is currently being designed and under construction.They will probe the inverted hierarchy region of the neutrino mass pattern.There is also a revived interest in the resonant neutrinoless double-electroncapture (0ECEC), which also has the potential to probe lepton numberconservation and to investigate the neutrino nature and mass scale. Theprimary concerns are the nuclear matrix elements. Clearly, the accuracy ofthe determination of the effective Majorana neutrino mass from the measured0-decay half-life is mainly determined by our knowledge of the nuclearmatrix elements. We review recent progress achieved in the calculation of 0and 0ECEC nuclear matrix elements within the quasiparticle random phaseapproximation. A considered self-consistent approach allows us to derive thepairing, residual interactions and the two-nucleon short-range correlations fromthe same modern realistic nucleonnucleon potentials. The effect of nucleardeformation is taken into account. The possibility of evaluating 0-decaymatrix elements phenomenologically is discussed.

1. Introduction

The physics community faces a challenging problem, discovering whether neutrinos are indeedMajorana particles (i.e. identical to their own antiparticles) as many particle models suggest orDirac particles (i.e. different from their antiparticles). The best sensitivity on small Majorananeutrino masses can be reached in the investigation of neutrinoless double-beta decay (0-decay) [1, 2],

(A, Z) (A, Z + 2) + e + e (1)

0954-3899/12/124006+19$33.00 2012 IOP Publishing Ltd Printed in the UK & the USA 1

J. Phys. G: Nucl. Part. Phys. 39 (2012) 124006 A Faessler et al

and the resonant neutrinoless double-electron capture (0ECEC) [3, 2],

eb + eb + (A, Z) (A, Z 2). (2)

A double asterisk in equation (2) means that, in general, the final atom (A, Z 2) is excitedwith respect to both the electron shell, due to the formation of two vacancies for the electrons,and the nucleus. Observing the 0-decay and/or 0ECEC would tell us that the total leptonnumber is not a conserved quantity and that neutrinos are massive Majorana fermions.

The search for the 0-decay represents the new frontier of neutrino physics, allowingus, in principle, to fix the neutrino mass scale, the neutrino nature and possibly CP violationeffects. There are a few tens of nuclear systems [4], which offer the opportunity to study the0-decay and the most favorable are those with a large Q-value.

Neutrinoless double-beta decay has not yet been found. The strongest limits on the half-lifeT 01/2 of the 0-decay were set in HeidelbergMoscow (76Ge, 1.91025 years) [5], NEMO3(100Mo, 1.0 1024 years) [6], CUORICINO (130Te, 3.0 1024 years) [7] and KamLAND-Zen (136Xe, 5.7 1024 years) [8] experiments. However, there is an unconfirmed, but notrefuted, claim of evidence for neutrinoless double decay in 76Ge by some participants of theHeidelbergMoscow Collaboration [9] with a half-life of T 01/2 = 2.23+0.440.31 1025 years. It isexpected that the GERDA experiment [10] in its first phase will check this result quite soon.

The main aim of experiments searching for 0-decay is the measurement of the effectiveMajorana neutrino mass m

m =

jU2e jm j, (3)

where Ue j is the element of PontecorvoMakiNakagawaSakata (PMNS) unitary mixingmatrix and mj is the mass of the neutrino. For the most discussed case, that of the mixing ofthree massive neutrinos ( j = 1,2,3), the PMNS matrix contains three charge parity (or CP)violating phases by assuming neutrinos to be Majorana particles.

The effective Majorana neutrino mass can be calculated by using neutrino oscillationparameters: an assumption about the mass of the lightest neutrino, by choosing a type ofspectrum (normal or inverted) and values of CP-violating phases. In future experiments,CUORE (130Te), EXO, KamLAND-Zen (136Xe), MAJORANA (76Ge), SuperNEMO (82Se),SNO+ (150Nd), and others [1, 2], a sensitivity

|m | a few tens of meV (4)

is planned to be reached. This is the region of the inverted hierarchy of neutrino masses. Inthe case of the normal mass hierarchy |m | is too small, a few meV, to be probed in the0-decay experiments of the next generation.

We note that it is reasonable to hope that the search for the 0ECEC atoms, whichare sufficiently long lived to conduct a practical experiment, may also establish the Majorananature of neutrinos. This possibility is considered as alternative and complementary to searchesfor the 0-decay.

To interpret the data from the 0-decay and the 0ECEC (neutrinoless double-electroncapture) accurately a better understanding of the nuclear structure effects important for thedescription of the nuclear matrix elements (NMEs) is needed. In this connection it is crucial todevelop and advance theoretical methods capable of reliably evaluating NMEs, and realisticallyassessing their uncertainties.

2


2. 0-decay NMEs: two-nucleon short-range correlations and uncertainties

The inverse value of the 0-decay half-life for a given isotope (A, Z) can be written as1

T 01/2=

mme2|M0 |2G0 (E0, Z). (5)

Here, G0 (E0, Z) and M0 are, respectively, the known phase-space factor (E0 is the energyrelease) and the nuclear matrix element, which depends on the nuclear structure of the particularisotopes (A, Z), (A, Z + 1) and (A, Z + 2) under study. The phase space factor G0 (E0, Z)includes the fourth power of unquenched axial-vector coupling constant gA and the inversesquare of the nuclear radius R2, compensated by the factor R in M0 . The assumed value ofthe nuclear radius is R = r0A1/3 with r0 = 1.2 f m.

The nuclear matrix element M0 is defined as

M0 =(

geffAgA

)2M0 . (6)

Here, geffA is the quenched axial-vector coupling constant. This definition of M0 [11] allow the

display of the effects of uncertainties in geffA and the use of the same phase factor G0 (E0, Z)when calculating the 0-decay rate.

The nuclear matrix element M0 consists of the Fermi (F), GamowTeller (GT) and tensor(T) parts as

M0 = M0F

(geffA )2+ M0GT M0T

= 0+i

kl+k

+l

[HF(rkl )

(geffA )2+ HGT(rkl )kl HT(rkl )Skl

] 0+f .(7)

Here

Skl = 3(k rkl )(l rkl ) kl, kl = k l . (8)The radial parts of the exchange potentials are

HF,GT,T(rkl ) = 2

R

0

j0,0,2(qrkl )hF,GT,T(q2)qq + E dq, (9)

where R is the nuclear radius and E is the average energy of the virtual intermediate states usedin the closure approximation. The closure approximation is adopted in all the calculations ofthe NMEs relevant for 0-decay with the exception of the QRPA. The functions hF,GT,T(q2)are given by [12]hF(q2) = f 2V (q2),hGT(q2) = 23 f

2V (q

2)(p n)2

(geffA )2q2

4m2p+ f 2A (q2)

(1 2

3q2

q2 + m2+ 1

3q4

(q2 + m2 )2)

,

hT(q2) = 13 f2V (q

2)(p n)2

(geffA )2q2

4m2p+ 1

3f 2A (q2)

(2

q2

(q2 + m2 ) q

4

(q2 + m2 )2)

. (10)

For the vector normalized to unity and axial-vector form factors the usual dipole approximationis adopted: fV (q2) = 1/(1 + q2/M2V )2, fA(q2) = 1/(1 + q2/M2A)2. MV = 850 MeV, and MA =1086 MeV. The difference in magnetic moments of proton and neutron is (p n) = 4.71,and gA = 1.254 is assumed.

3


The above definition of the M0 includes the contribution of the higher orderterms of the nucleon current, and the GoldbergerTreiman PCAC relation, gP(q2) =2mpgA(q2)/(q2 + m2 ) was employed for the induced pseudoscalar term.

The nuclear matrix elements M0 for the 0 decay must be evaluated using tools ofnuclear structure theory. Unfortunately, there are no observables that could be simply anddirectly linked to the magnitude of 0 nuclear matrix elements and that could be used todetermine them in an essentially model independent way.

Over many years two approaches have been used: the quasiparticle randomphase approximation (QRPA)[1113] and the interacting shell model (ISM)[14]. There aresubstantial differences between these approaches. The QRPA treats a large, single particlemodel space, but heavily truncates the included configurations. The ISM, in contrast, treatsa small fraction of this model space, but allows the nucleons to correlate in many differentways. In the last few years several new approaches have been used for the calculation ofthe 0-decay NMEs: the angular momentum projected HartreeFockBogoliubov method(PHFB) [15], the interacting boson model (IBM) [16], and the energy density functionalmethod (EDF) [17].

The standard QRPA method consists of two steps. First, the mean field corresponding tothe minimum energy is determined and the like-particle pairing interaction is taken into accountby employing the quasiparticle representation. In the second step the linearized equations ofmotion are solved in order to describe small amplitude vibrational-like modes around thatminimum. In the renormalized version of QRPA (RQRPA) the violation of the Pauli exclusionprinciple is partially corrected.

The drawback of QRPA is the fact that, unlike in BCS, the particle number is not conservedautomatically, even on average. For realistic Hamiltonians the differences between averagedparticle numbers on the RPA ground state and the exact particle numbers could be of the order ofunity (an extra or missing neutron or proton). The self-consistent renormalized QRPA method(SRQRPA) removes this drawback by treating the BCS and QRPA vacua simultaneously [18].For the neutron-proton systems the method was proposed and tested on the exactly solvablesimplified models in [19]. It is a generalization of the procedure proposed earlier in [20].

In the QRPA the phonon operators are defined asQ(k)J,M = pn

[Xk(pn)JA

(pn)J,M Y k(pn)JA(pn)J,M

], (11)

where Xk(pn)J and Y k(pn)J are the usual variational amplitudes, and A(pn)J,M is the angular

momentum coupled two-quasiparticle creation operator. p, n signify the quantum numbers ofthe proton, respectively neutron, orbits. The X and Y amplitudes, as well as the correspondingenergy eigenvalues k are determined by solving the QRPA eigenvalue equations for each J(

A BB A

)(XY

)=

(XY

). (12)

The matrices A and B above are determined by the Hamiltonian rewritten in terms of thequasiparticle operators:

AJpn,pn = O|(apan)(JM)H(apa

n )

(JM)|O (13)BJpn,pn = O|H(apan)(JM)(1)M(apan )(JM)|O

Here, |O is the BCS vacuum state.In the RQRPA and SRQRPA instead of |O the correlated QRPA ground state |0+QRPA

is considered. Then instead of the standard X and Y the renormalized amplitudes are usedeverywhere including the QRPA equations of motion:

Xm(pn,J ) = D1/2pn Xm(pn,J ), Ym

(pn,J ) = D1/2pn Y m(pn,J ), (14)4


where renormalization factors Dpn are given byDpn =

0+QRPA

[A(pn)J,M, A(pn)J,M]0+QRPA = 1 p n= 1 1

2 jp + 1nDpn

(J,k(2J + 1)

Y J,kpn 2)

12 jn + 1p

Dpn(J,k(2J + 1)

Y J,kpn2). (15)Here, n(p) is the expectation value of the number of quasiparticles in the orbit n(p),

n(p) 0+QRPA

[a+n(p)an(p)

]00

0+QRPA2 jn(p) + 1

. (16)

a+jn(p),m, a jn(p),m are the creation and annihilitation operators for the quasiparticle with quantumnumbers n(p), m. The renormalization coefficients Dpn and the quasiparticle occupationnumbers j can be obtained iteratively using the equations of motion of the (S)RQRPA.

In the correlated QRPA ground state the occupation numbers are no longer the pure BCSquantities. Instead, they depend, in addition, on the solutions of the QRPA equations of motionfor all multipoles J, and can be evaluated using

nQRPAn(p) =

0+QRPA

mc+n(p),mcn(p),m0+QRPA (2 jn(p) + 1)

[v2n(p) +

(u2n(p) v2n(p)

)n(p)

]. (17)

Here, c+j,m is the creation operator for a proton in the orbit jp or a neutron in the orbit jn andc j,m is the corresponding annihilation operator. The amplitudes v jp and v jn are obtained bysolving the gap equations.

In SRQRPA the BCS equations are reformulated. This is achieved by recalculating the uand v amplitudes from the minimum condition of the RQRPA ground-state energy. In SRQRPAthus the state around which the vibrational modes occur is no longer the quasiparticle vacuum,but instead the Bogoliubov transformation is chosen in such a way that provides the optimaland consistent basis while preserving the form of the phonon operator, equation (11).

In practice, the SRQRPA equations are solved double iteratively. One begins with thestandard BCS u, v amplitudes, solves the RQRPA equations of motion and calculates thefactors Dpn. The u, v amplitudes are recalculated and the procedure is repeated until self-consistency is achieved. Numerically, the double iteration procedure represents a challengingproblem. It was resolved in [21] where instead of the G-matrix based interaction the pairingpart (and only that part) of the problem was replaced by a pairing interaction that uses aconstant matrix element whose value was adjusted to reproduce the experimental oddevenmass differences.

In the QRPA, RQRPA and SRQRPA the M0 is written as the sum over the virtualintermediate states, labeled by their angular momentum and parity J and indices ki and k f :

MK =

J ,ki,k f ,J

pnpn

(1) jn+ jp+J+J2J + 1{ jp jn J

jn jp J}

p(1), p(2);J f (r12)OK f (r12) n(1), n(2);J 0+f [c+p cn]J

|Jk f Jk f |JkiJki||[c+p cn]J0+i .

(18)The operators OK, K = Fermi (F), GamowTeller (GT), and Tensor (T), contain neutrinopotentials, spin and isospin operators, and RPA energies Eki,k fJ . Two separate multipoledecompositions are built into equation (18). One is in terms the J of the virtual states in

5


the intermediate nucleus, the good quantum numbers of the QRPA and RQRPA. The otherdecomposition is based on the angular momenta and parities J of the pairs of neutrons thatare transformed into protons with the same J . The nucleon orbits are labeled in equation(18) by p, p, n, n.

The QRPA-like approaches do not allow the introduction of short-range correlations(SRCs) into the two-nucleon relative wavefunction. The traditional way is to introduce anexplicit Jastrow-type correlation function f (r12) into the involved two-body transition matrixelements (see equation (18)). In the parametrization of Miller and Spencer [22] we have

f (r12) = 1 cear2 (1 br2), a = 1.1 fm2, b = 0.68 fm2. (19)These two parameters (a and b) are correlated and chosen in the way that the norm of therelative wavefunction |J is conserved.

Recently, it was proposed [13] that instead of the Jastrow method the UCOM approachfor the description of the two-body correlated wavefunction [23] was adopted. The UCOMmethod produces good results for the binding energies of nuclei already at the HartreeFocklevel [24].

A self-consistent calculation of the 0-decay NMEs in the QRPA-like approaches wasperformed in [25]. The pairing and residual interactions as well as the two-nucleon short-rangecorrelations were for the first time derived from the same modern realistic nucleonnucleonpotentials, namely from the charge-dependent Bonn potential (CD-Bonn) and the ArgonneV18 potential. A method of choice was the coupled cluster method (CCM) [26]. For purposeof numerical calculation of the 0-decay NMEs the CCM short-range correlation functionswere presented in an analytic form of Jastrow-like function as [25]

fA,B(r12) = 1 cear2 (1 br2). (20)The set of parameters for Argonne and CD-Bonn NN interactions is given by

fA(r12) : a = 1.59 fm2, b = 1.45 fm2, c = 0.92,fB(r12) : a = 1.52 fm2, b = 1.88 fm2, c = 0.46. (21)

The calculated NMEs with these short-range correlation functions agree to within a few percentwith those obtained without this approximation. We note that the dependence of the SRC onthe value of oscillator length b is rather weak.

In table 1 the QRPA and RQRPA results are presented separately for the different typesof two-nucleon short-range correlations (SRC) considered: MilllerSpencer Jastrow SRCs(Jastrow) [11]; Fermi hypernetted chain SRCc (FHCh); unitary correlation operator methodSRCs (UCOM) [12]; the coupled cluster method SRCs derived from the Argonne and CD-Bonn potentials [25] based on an extension of the Brueckner theory (coupled cluster method =CCM). Two different values of the axial coupling constant, free nucleon geffA = gA = 1.254 andquenched geffA = 1.0, are taken into account. The strength of the particle-particle interactionis adjusted so the experimental value of the 2-decay nuclear matrix element is correctlyreproduced [11]. The NME calculated within this procedure, which includes three differentmodel spaces, is denoted as the averaged 0-decay NME M0. We note that the values ofNMEs become essentially independent of the size of the single-particle basis and rather stablewith respect to the possible quenching of the gA.

From table 1 it follows that the QRPA values are about 10-15% larger in comparison withthe RQRPA values. The largest NMEs are those calculated with the CCM CD-Bonn correlationfunction. In comparison the NMEs obtained with the CCM CD-Argonne correlation functionand the UCOM SRCs are about 10% smaller. This is explained by the fact that the CCMArgonne correlation function cuts out the small r12 part from the relative wavefunction of the

6


Table 1. Averaged 0 nuclear matrix elements M0 and their variance (in parentheses)calculated within the QRPA and the RQRPA. Different types of two-nucleon short-rangecorrelations (SRC) are considered: Miller-Spencer Jastrow SRCs (Jastrow) [11]; Fermi hypernettedchain SRCc (FHCh); unitary correlation operator method SRCs (UCOM) [12]; the coupled clustermethod SRCs derived from the Argonne and CD-Bonn potentials [25]. Three sets of single particlelevel schemes are used, ranging in size from 9 to 23 orbits. The strength of the particle-particleinteraction is adjusted so the experimental value of the 2-decay nuclear matrix element iscorrectly reproduced. Both free nucleon (geffA = gA = 1.254) and quenched (geffA = 1.0) valuesof axial-vector coupling constant are considered. We note that unlike in [11, 12, 25] r0 = 1.2 fminstead of r0 = 1.1 fm is assumed.

M0Nucleus SRC CCM SRC

transition geffA Method Jastrow FHCh UCOM Argonne CD-Bonn76Ge 1.25 QRPA 4.92(0.19) 5.15(0.17) 5.98(0.27) 6.34(0.29) 6.89(0.35)

RQRPA 4.28(0.13) 4.48(0.13) 5.17(0.20) 5.42(0.21) 5.93(0.25)1.00 QRPA 4.18(0.15) 4.36(0.15) 4.97(0.23) 5.20(0.22) 5.63(0.27)

RQRPA 3.77(0.14) 3.94(0.13) 4.47(0.20) 4.59(0.15) 5.04(0.24)82Se 1.25 QRPA 4.39(0.16) 4.57(0.16) 5.32(0.23) 5.66(0.26) 6.16(0.29)

RQRPA 3.81(0.14) 3.97(0.14) 4.59(0.17) 4.84(0.21) 5.30(0.22)1.00 QRPA 3.59(0.13) 3.74(0.13) 4.29(0.19) 4.57(0.20) 4.89(0.22)

RQRPA 3.17(0.10) 3.32(0.10) 3.79(0.13) 4.00(0.15) 4.29(0.16)96Zr 1.25 QRPA 1.22(0.03) 1.23(0.04) 1.77(0.02) 2.07(0.10) 2.28(0.03)

RQRPA 1.31(0.15) 1.33(0.15) 1.77(0.02) 2.01(0.17) 2.19(0.22)1.00 QRPA 1.32(0.08) 1.34(0.07) 1.73(0.10) 1.90(0.12) 2.11(0.12)

RQRPA 1.22(0.12) 1.24(0.12) 1.57(0.14) 1.69(0.13) 1.88(0.16)100Mo 1.25 QRPA 3.64(0.21) 3.73(0.21) 4.71(0.28) 5.18(0.36) 5.73(0.34)

RQRPA 3.03(0.21) 3.12(0.21) 3.88(0.26) 4.20(0.34) 4.67(0.31)1.00 QRPA 2.96(0.15) 3.02(0.15) 3.74(0.21) 4.03(0.27) 4.44(0.24)

RQRPA 2.55(0.13) 2.63(0.13) 3.20(0.17) 3.43(0.25) 3.75(0.21)116Cd 1.25 QRPA 2.99(0.21) 3.11(0.21) 3.74(0.12) 3.86(0.29) 4.35(0.16)

RQRPA 2.64(0.17) 2.75(0.19) 3.21(0.22) 3.34(0.24) 3.72(0.26)1.00 QRPA 2.38(0.17) 2.47(0.17) 2.88(0.17) 2.99(0.23) 3.31(0.21)

RQRPA 2.14(0.14) 2.21(0.14) 2.55(0.17) 2.69(0.19) 2.92(0.21)128Te 1.25 QRPA 3.97(0.14) 4.15(0.15) 5.04(0.15) 5.38(0.17) 5.99(0.17)

RQRPA 3.52(0.13) 3.68(0.14) 4.45(0.15) 4.71(0.17) 5.26(0.16)1.00 QRPA 3.11(0.09) 3.23(0.10) 3.88(0.11) 4.11(0.13) 4.54(0.13)

RQRPA 2.77(0.09) 2.88(0.09) 3.44(0.10) 3.62(0.12) 4.00(0.12)130Te 1.25 QRPA 3.56(0.13) 3.72(0.14) 4.53(0.12) 4.77(0.15) 5.37(0.13)

RQRPA 3.22(0.13) 3.36(0.15) 4.07(0.13) 4.27(0.15) 4.80(0.14)1.00 QRPA 2.55(0.08) 2.93(0.08) 3.52(0.07) 3.69(0.11) 4.11(0.08)

RQRPA 2.15(0.14) 2.66(0.09) 3.17(0.08) 3.29(0.11) 3.69(0.09)136Xe 1.25 QRPA 2.16(0.13) 2.25(0.12) 2.73(0.13) 2.88(0.14) 3.23(0.14)

RQRPA 2.02(0.12) 2.11(0.14) 2.54(0.15) 2.68(0.16) 3.00(0.17)1.00 QRPA 1.70(0.09) 1.77(0.09) 2.12(0.11) 2.21(0.10) 2.47(0.09)

RQRPA 1.59(0.09) 1.66(0.10) 1.97(0.11) 2.06(0.11) 2.30(0.12)

two-nucleons more than the CCM CD-Bonn correlation function. The smallest in magnitudeare matrix elements for the 0 decay obtained with the traditional approach of usingthe MillerSpencer Jastrow SRC and the Fermi hypernetted chain SRCc.

In table 2 we show the calculated ranges of the nuclear matrix element M0 evaluatedwithin the QRPA, RQRPA [25] and SRQRPA [27] in a self-consistent way with the CCM CD-Bonn and Argonne SRC functions by assuming both the standard (gA = 1.254) and quenched(gA = 1.0) axial-vector couplings. These ranges quantify the uncertainty in the calculated

7

J.Phys.G:N

ucl.P

art.Phys.39(2012)124006A

Faessler

etal

Table 2. The calculated ranges of the nuclear matrix element M0 evaluated within the QRPA (column 2), RQRPA (column 4) and SRQRPA (column 6), with standard (geffA = gA = 1.254)and quenched (geffA = 1.0) axial-vector couplings and with the coupled cluster method (CCM) CD-Bonn and Argonne short-range correlation (SRC) functions. Columns 3, 5 and 7 givethe 0-decay half-life ranges corresponding to values of the matrix-elements in columns 2, 4 and 6 for |m | = 50 meV. T 0exp1/2 is the experimental lower bound on the 0-decayhalf-life for a given isotope.

QRPA RQRPA SRQRPANuclei M0 T 01/2 [years] M

0 T 01/2 [years] M0 T 01/2 [years] T 0exp1/2 [y]

76Ge (5.0, 7.2) (3.0, 6.3) 1026 (4.5, 6.2) (4.1, 7.9) 1026 (4.3, 6.2) (4.0, 8.6) 1026 1.9 1025 [5]82Se (4.4, 6.4) (8.5, 18.) 1025 (3.8, 5.6) (1.2, 2.4) 1026 (3.9, 6.1) (9.5, 22.) 1025 3.2 1023 [6]100Mo (3.7, 6.1) (5.9, 15.) 1025 (3.2, 5.0) (8.8, 21.) 1025 (4.0, 5.5) (7.3, 13.) 1025 1.0 1024 [6]130Te (3.6, 5.5) (7.4, 18.) 1025 (3.2, 4.7) (1.0, 2.2) 1026 (3.6, 5.1) (8.5, 17.) 1025 3.0 1024 [7]136Xe (2.1, 3.4) (1.9, 4.8) 1026 (2.0, 3.2) (2.1, 5.5) 1026 (2.4, 3.6) (1.6, 3.7) 1026 5.7 1024 [8]

8


0-decay NMEs of a given QRPA-like approach. By comparing the SRQRPA with theRQRPA results we conclude that the requirement of conserving the particle number have notcaused substantial changes in the value of the 0 matrix elements in that case.

Given the interest in the subject, in table 2 we show also the range of predicted 0-decayhalf-lives of 76Ge, 82Se, 100Mo, 130Te and 136Xe corresponding to the full range of M0 for|m | = 50 meV. This is a rather conservative range within the considered QRPA framework.It represents roughly the required sensitivity of the 0-decay experiment in the case ofinverted hierarchy of neutrino masses, which can be compared with the current bound on the0-decay half-life T 0exp1/2 .

3. decay of deformed nuclei within QRPA

One of the best candidates for searching 0 decay is 150Nd since it has the second highestendpoint, Q =3.37 MeV, and the largest phase space factor for the decay (about 33 timeslarger than that for 76Ge, see e.g. [28]). The SNO+ experiment at the Sudbury NeutrinoObservatory will use a Nd-loaded scintillator to search for neutrinoless double-beta decay bylooking for a distortion in the energy spectrum of decays at the endpoint [29].

However, 150Nd is well known to be a rather strongly deformed nucleus. This stronglyhinders a reliable theoretical evaluation of the corresponding 0-decay NMEs (for instance,it does not seem feasible in the near future to reliably treat this nucleus within the large-scalenuclear shell model (ISM), see, e.g., [14]). Recently, more phenomenological approaches likethe pseudo-SU(3) model [30], the PHFB approach [15], the IBM [16], and the EDF [17]have been employed to calculate M0 for strongly deformed heavy nuclei (a comparativeanalysis of different approximations involved in some of the models can be found in [31]).The results of these models generally reveal a substantial suppression of M0 for 150Nd ascompared with the QRPA result of [11] where 150Nd and 150Sm were treated as sphericalnuclei. However, the calculated NMEs M0 for 150Nd show a rather significant spread.

One of the most up-to-date microscopic ways to describe the effect of nuclear deformationon -decay NMEs M2 and M0 is provided by the QRPA. A QRPA approach for calculating-decay amplitudes in deformed nuclei has been developed in a series of papers [3236].M2 were calculated in [32, 33] with schematic separable forces, and in [34]with realisticresidual interaction. It was demonstrated in [3234] that deformation introduces a mechanismof suppression of the M2 matrix element which gets stronger when deformations of the initialand final nuclei differ from each other. A similar dependence of the suppression of both M2and M0 matrix elements on the difference in deformations has been found in the PHFB [15]and the ISM [14].

In [35, 36], the first QRPA calculations of M0 with an account for nuclear deformationwere done. The calculations showed a suppression of M0 for 150Nd by about 40% as comparedwith our previous spherical QRPA result [11]. In the next section we review the results of[3236].

3.1. Formalism

The NMEs M2 and M0 , as the scalar measures of the decay rates, can be calculated inany coordinate system. For strongly deformed, axially symmetric, nuclei the most convenientchoice is the intrinsic coordinate system associated with the rotating nucleus. This employsthe adiabatic BohrMottelson approximation that is well justified for 150Nd, 160Gd and 160Dy,which indeed reveal strong deformations. As for 150Sm, the enhanced quadrupole moment ofthis nucleus is an indication of its static deformation.

9


Though it is difficult to evaluate the effects beyond the adiabatic approximation employedhere, one might anticipate already without calculations that the smaller the deformation is, thesmaller should be the deviation of the calculated observables from the ones obtained in thespherical limit. In this connection it is worth noting that spherical QRPA results can be exactlyreproduced in the present QRPA calculation by letting the deformation vanish, in spite of theformal inapplicability of the adiabatic ansatz for the wavefunction in this limit.

We give here, for completeness, the formalism of the QRPA calculations of NMEs M2and M0 in deformed nuclei as developed in [3236].

Nuclear excitations in the intrinsic system |K are characterized by the projection ofthe total angular momentum onto the nuclear symmetry axis K (the only projection which isconserved in strongly deformed nuclei) and the parity .

The intrinsic states |K , m are generated within the QRPA by a phonon creation operatoracting on the ground-state wavefunction:

|K , m = Qm,K0+g.s.; Qm,K =

pn

Xmpn,KApn,K Y mpn,KApn,K . (22)

Here, Apn,K = apan and Apn,K = apan are the two-quasiparticle creation and annihilationoperators, respectively, with the bar denoting the time-reversal operation. The quasiparticlepairs pn are defined by the selection rules p n = K and pn = , where is thesingle-particle (s.p.) parity and is the projection of the total s.p. angular momentum onthe nuclear symmetry axis ( = p, n). The s.p. states |p and |n of protons and neutronsare calculated by solving the Schrodinger equation with the deformed axially symmetricWoodsSaxon potential [34]. In the cylindrical coordinates the deformed WoodsSaxon s.p.wavefunctions | with > 0 are decomposed over the deformed harmonic oscillators.p. wavefunctions (with the principal quantum numbers (Nnz)) and the spin wavefunctions| = 12 :

| =Nnz

bNnz|Nnz = |, (23)

where N = n +nz (n = 2n +||), nz and n are the number of nodes of the basis functionsin the z- and -directions, respectively; = and are the projections of the orbitaland spin angular momentum onto the symmetry axis z. For the s.p. states with the negativeprojection = | |, which are degenerate in energy with = | |, the time-reversedversion of equation (23) is used as a definition (see also [34]). The states (, ) comprise thewhole single-particle model space.

The deformed harmonic oscillator wavefunctions |Nnz can be further decomposed overthe spherical harmonic oscillator ones |nrl by calculating the corresponding spatial overlapintegrals AnrlNnz = nrl|Nnz (nr is the radial quantum number, l and are the orbitalangular momentum and its projection onto z-axes, respectively), see appendix of [34] for moredetails. Thereby, the wavefunction (23) can be re-expressed as

| =

B| , (24)

where | =

C jl 12 |nrl = | is the spherical harmonic oscillator

wavefunction in the j-coupled scheme ( = (nrl j)), and B =

C jl 12 AnrlNnz bNnz ,

with C jl 12 being the ClebschGordan coefficient.

The QRPA equations:( A(K) B(K)B(K) A(K)

)(XmKY mK

)= K,m

(XmKY mK

), (25)

10


with realistic residual interaction are solved to get the forward XmiK , backward Y miK amplitudesand the excitation energies miK and

m fK of the mth K state in the intermediate nucleus. The

matrix A and B are defined byApn,pn (K) = pn,pn (Ep + En) + gpp(upunupun + vpvnvpvn )Vpnpn

gph(upvnupvn + vpunvpun )Vpn pnBpn,pn (K) = gpp(upunvpvn + vpvnupun )Vpnpn gph(upvnvpvn + vpunupvn )Vpn pn (26)where Ep + En are the two-quasiparticle excitation energies, Vpn,pn and Vpn,pn are the p hand p p matrix elements of the residual nucleonnucleon interaction V , respectively, u andv are the coefficients of the Bogoliubov transformation.

As a residual two-body interaction we use the nuclear Brueckner G matrix, which is asolution of the BetheGoldstone equation, derived from the charge-depending Bonn (Bonn-CD) one boson exchange potential, as used also in the spherical calculations of [11]. The Gmatrix elements are originally calculated with respect to a spherical harmonic oscillator s.p.basis. By using the decomposition of the deformed s.p. wavefunction in equation (24), thetwo-body deformed wavefunction can be represented as

|pn =pnJ

FJKppnn |pn, JK, (27)

where |pn, JK =

mpmnCJKjpmp jnmn |pmp|nmn, and FJKppnn = B

ppBnn (1) jnnCJKjpp jnn

is defined for the sake of simplicity ((1) jnn is the phase arising from the time-reversedstates |n). The particleparticle Vpn,pn and particle-hole Vpn,pn interaction matrix elementsin the representation (26) for the QRPA matrices A, B (25) in the deformed WoodsSaxonsingle-particle basis can then be given in terms of the spherical G matrix elements as follows:

Vpn,pn = 2

J

pn

pn

FJKppnn FJKpp nn G(pnpn , J), (28)

Vpn,pn = 2

J

pn

pn

FJKpnppnn

FJKpnpp nn

G(pnpn, J), (29)

where Kpn = p + n = p + n.The structure of the intermediate |0+ and |1+ states is only needed within the QRPA to

calculate 2-decay NMEs M2 [34], whereas all possible |K states are needed to constructthe NMEs M0 .

The matrix element M2GT is given within the QRPA in the intrinsic system by the followingexpression:

M2GT =

K=0,1

mim f

0+f |K |K+, m f K+, m f |K+, miK+, mi|K |0+i K,mim f

. (30)

Instead of the usual approximation of the energy denominator in equation (30) as K,mim f =(K,m f + K,mi )/2 (see, e.g., [32, 33]), here another prescription is used in which the wholecalculated QRPA energy spectrum is shifted in such a way as to have the first calculated 1+state exactly at the corresponding experimental energy. In this case the energy denominatorin equation (30) acquires the form K,mim f = (K,m f K,1 f + K,mi K,1i )/2 + 1+1 , with1+1 being the experimental excitation energy of the first 1

+ state measured from the mean g.s.energy (E0i + E0 f )/2.

The two sets of intermediate nuclear states generated from the initial and final g.s. do notcome out identical within the QRPA. Therefore, the overlap factor of these states is introducedin equation (30) [32] as follows:

K+, m f |K+, mi =lil f

[Xm fl f KX

miliK Y

m fl f KY

miliK]Rl f li BCS f |BCSi. (31)

11


The factorRl f li , which includes the overlaps of single particle wavefunctions of the initial andfinal nuclei is given by

Rll = pp|pp (u(i)p u

( f )p + v(i)p v( f )p

)nn|nn (u(i)n u( f )n + v(i)n v( f )n ), (32)and the last term BCS f |BCSi in equation (31) corresponds to the overlap factor of the initialand final BCS vacua in the form given in [32].

The matrix element M0 is given within the QRPA in the intrinsic system by a sum of thepartial amplitudes of transitions via all the intermediate states K :

M0 =K

M0 (K ); M0 (K ) =

s(def) O(K ). (33)

Here, we use the notation of Appendix B in [12], stands for the set of four single-particleindices {p, p, n, n}, and O(K ) is a two-nucleon transition amplitude via the K states inthe intrinsic frame:

O(K ) =

mi,m f

0+fcpcn|Km f Km f |KmiKmi|cpcn 0+i . (34)

The two sets of intermediate nuclear states generated from the initial and final g.s. (labeled bymi and m f , respectively) do not come out identical within the QRPA. A standard way to tacklethis problem is to introduce in equation (34) the overlap factor of these states Km f |Kmi,whose representation is given below, equation (37). Two-body matrix elements s(def) of theneutrino potential in equation (33) in a deformed WoodsSaxon single-particle basis aredecomposed over the spherical harmonic oscillator ones according to equations (27) and (29):

s(def)ppnn =

J

ppnn

FJKppnn FJKpp nn s

(sph)ppnn

(J), (35)

s(sph)ppnn (J) =

J

(1) jn+ jp+J+J J{ jp jn J

jn jp J}

p(1), p(2);J O(1, 2)n(1), n(2);J ,

(36)where J 2J + 1, and O(1, 2) is the neutrino potential as a function of coordinates oftwo particles, with labeling its Fermi (F), GamowTeller (GT) and Tensor (T) parts.

The particle-hole transition amplitudes in equation (34) can be represented in terms of theQRPA forward XmiK and backward Y miK amplitudes along with the coefficients of the Bogoliubovtransformation u and v [34]:

0+f |cpcn|Km f = vpunXm fpn,K + upvnY m fpn,K ,Kmi|cpcn|0+i = upvnXmipn,K + vpunY mipn,K .

The overlap factor in equation (34) can be written as:

Km f |Kmi =lil f

[Xm fl f K X

miliK Y

m fl f KY

miliK

]Rl f liBCS f |BCSi. (37)

Representations for Rl f li and the overlap factor BCS f |BCSi between the initial and finalBCS vacua are given in [32].12


3.2. Calculation results

The NMEs M2 and M0 were calculated according to the above formalism in [3236].These articles contain detailed descriptions of the choice of the model parameters andcomparison between different approximations. Here we only briefly repeat the key pointsof the calculations.

Only quadrupole deformation is taken into account in the calculations [3236]. Thesingle-particle Schrodinger equation with the Hamiltonian of a deformed WoodsSaxon meanfield is solved on the basis of an axially-deformed harmonic oscillator. Decomposition ofthe obtained deformed single-particle wavefunctions is performed over the spherical harmonicoscillator states within the seven lowest major shells. The geometrical quadrupole deformationparameter 2 of the deformed WoodsSaxon mean field is obtained by fitting the experimentaldeformation parameter = 5 QpZr2c , where rc is the charge rms radius and Qp is the empiricalintrinsic quadrupole moment. The experimental values of can be derived from the laboratoryquadrupole moments measured by the Coulomb excitation reorientation technique, or from thecorresponding B(E2) values [39]. Experimental values extracted from the B(E2) have smallerexperimental errors. But deformations extracted from the reorientation effect are in principlethe better values, but have large errors. The fitted values of the parameter 2 of the deformedWoodsSaxon mean field, which allow us to reproduce the experimental , are listed intable 1 of [36]. The spherical limit, i.e. 2 = 0, is considered as well, to compare with theearlier results of [11].

The nuclear Brueckner G matrix, obtained by a solution of the Bethe-Goldstone equationwith the Bonn-CD one boson exchange nucleonnucleon potential, is used as a residual two-body interaction in [3436]. Then the BCS equations are solved to obtain the Bogoliubovcoefficients, gap parameter and chemical potentials. To solve the QRPA equations, one hasto fix the particle-hole gph and particle-particle gpp renormalization factors of the residualinteraction, equations (26). A value of gph = 0.90 was determined by fitting the experimentalposition of the GamowTeller giant resonance (GTR) for 76Ge. The same value of gph wasthen used for all nuclei in question, and led to a good fit to the experimental GTR energy for150Nd, measured very recently [37].

The parameter gpp can be determined by fitting the experimental value of the 2-decay NMEs M2GT MeV1 [38] for each nucleus in question. To account for the quenchingof the axial-vector coupling constant gA, the quenched value geffA = 0.75 gA was used inthe calculation along with the bare value gA = 1.25. The quenching factor of 0.75 comesfrom a recent experimental measurement of GT strength distribution in 150Nd [37]. Thetwo sets of the fitted values of gpp corresponding to the cases without or with quenchingof gA are listed in table 1 of [36]. Note, that this fitting procedure leads to realisticvalues gpp 1.

Having solved the QRPA equations, the two-nucleon transition amplitudes (34) arecalculated and, by combining them with the two-body matrix elements of the neutrino potential,the total 0 NMEs M0 (33) is formed. Such a computation is rather time consuming sincenumerous programming loops are needed to calculate the decompositions of the two-bodymatrix elements in the deformed basis over the spherical ones. Therefore, to speed up thecalculations the mean energy of 7 MeV of the intermediate nuclear excitation energies is usedin the neutrino propagator. The effects of the finite nucleon size and higher-order weak currentswere taken into account. The two-nucleon short-range correlations (SRC) were treated in anextended Brueckner theory (CCM) in a modern self-consistent way, see [25] and section 2,that leads to a change in the NMEs M0 by only a few percent, much less than the traditionalJastrow-type representation of the SRC does.

13


Table 3. Calculated NMEs M0 and corresponding decay half-lives (assuming m = 50 meV)for 0-decays of 76Ge, 150Nd, 160Gd [36], and for 0ECEC of 152Gd, 164Gd, 180W [62] withinthe deformed QRPA. The results obtained for realistic deformations of the nuclei are labeled asdef, whereas those obtained in the spherical limit, i.e. 2 = 0, are labeled as sph.

sph (2=0) defNuclear transition geffA M

0 T 01/2 [years] M0 T 01/2 [years]

0-decay76Ge76Se 0.94 4.10 9.4 1026 4.00 9.8 1026

1.25 5.30 5.6 1026 4.69 7.2 1026150Nd150Sm 0.94 4.52 2.3 1025 2.55 7.1 1025

1.25 6.12 1.2 1025 3.34 4.1 1025160Gd160Dy 0.94 3.76 2.3 1026

0ECEC152Gd152Se (KL1) 1.269 7.50 (8.7 1027, 8.9 1029) 3.23 (4.7 1028, 4.8 1029)164Er164Dy (L1L1) 1.269 7.20 (1.0 1032, 1.1 1032) 2.64 (7.5 1032, 8.4 1032)180W180Hf (KK) 1.269 6.22 (1.4 1030, 2.0 1030) 2.05 (1.3 1031, 1.8 1031)

Table 4. The matrix elements M0 for the 0 decay 150Nd150Sm calculated in differentmodels. The corresponding half-lives T 01/2 (in years) for an assumed effective Majorana neutrinomass m = 50 meV are also shown.

Method Def. QRPA [36] Pseudo-SU(3) [30] PHFB [15] IBM [16] EDF [17]M0 2.95 0.4 1.57 3.24 0.44 2.32 1.71T 01/2 (1025 y) 5.6 1.5 18.7 4.6 1.2 8.54 16.5

An important cross-check of the calculations is provided by a comparison of the presentresults in the spherical limit with the previous ones of [11, 12, 25]. Though formallythe adiabatic Bohr-Mottelson approximation is not applicable in the limit of vanishingdeformation, it is easy to see that the basic equations (33)(37) do have the correct sphericallimit. Details of such a comparison can be found in [36], and an excellent agreement betweenthe NMEs calculated by the genuine spherical code and the deformed code in the sphericallimit was found. Also, the partial contributions M0 (K ) of different intermediate K statesto M0 for the decay 150Nd150Sm were analyzed in [36].

The final results for the NMEs, corresponding to the modern self-consistent treatmentof the SRC [25]), for 0 decays 76Ge76Se, 150Nd150Sm, 160Gd160Dy are listed intable 3. As explained in [36], the difference between the spherical and deformed resultsmainly come from the BSC overlap between the ground states of the initial and final nuclei.The strongest effect of deformation on M0 (the suppression by about 40% as compared toour previous QRPA result obtained with neglect of deformation) is found in the case of 150Nd.This suppression can be traced back to a rather large difference in deformations of the groundstates of 150Nd and 150Sm. As for the gpp dependence of the 0-decay NMEs, it is much lesspronounced than the dependence of the amplitude of 2 decay. A marked reduction of thetotal M0 for the quenched value of gA can be traced back to a smaller prefactor (gA/1.25)2in the definition of M0 (6).

In table 4 the NMEs M0 for 150Nd calculated by other approaches are compared. TheNMEs M0 for 150Nd, obtained within the state-of-the-art QRPA approach that accounts fornuclear deformation [36], compares well with the results of the IBM [16] and PHFB [15]. Thecalculated 0-decay half-life T 01/2 corresponding to the Majorana neutrino mass m =14


50 meV seems to be short enough to hope that the SNO+ experiment will be able to approachthe inverse hierarchy of the neutrino mass spectrum.

4. The resonant neutrinoless double-electron capture

The resonant 0ECEC (neutrinoless double-electron capture) was already considered asa process which might prove the Majorana nature of neutrinos and the violation of thetotal lepton number, by Winter [40] in 1955. The possibility of a resonant enhancementof the 0ECEC in the case of a mass degeneracy between the initial and final atoms waspointed out by Bernabeu, De Rujula and Jarlskog as well as by Vergados about 30 years ago[41, 42]. They estimated the half-life of the process by introducing different simplifications:(i) non-relativistic atomic wavefunctions at the nuclear origin; (ii) qualitative evaluation ofNME of the process; (iii) the degeneracy parameter = MA,Z MA,Z2 was assumed to bewithin the range (0,10) keV representing the accuracy of atomic mass measurement at thattime. MA,Z and MA,Z2 are masses of the initial and final excited atoms, respectively. A listof promising isotopes based on the degeneracy requirement associated with arbitrary nuclearexcitation and on the natural abundance of daughter atom was presented.

In 2004 Sujkowski and Wycech [43] and Lukaszuk et al [44] analyzed the 0ECEC processfor nuclear 0+ 0+ transitions accompanied by a photon emission in the resonance and non-resonance modes. By assuming |m | = 1 eV and 1 error in the atomic mass determinationthe resonant 0ECEC rates of six selected isotopes were calculated by considering theperturbation theory approach.

In 2009 a new theoretical approach to the 0ECEC, a unified description of the oscillationsof stable and quasistationary atoms, was developed by Simkovic and Krivoruchenko[3, 45]. A comprehensive theoretical study of this process for the light Majorana neutrinomass mechanism was performed [3]. It was shown that effects associated with the relativisticstructure of the electron shells reduce the 0ECEC half-lives by almost one order of magnitudeand that the capture of electrons from the np1/2 states is only moderately suppressed incomparison with the capture from the ns1/2 states unlike in the non-relativistic theory. Selectionrules for associated nuclear transitions were presented saying that a change in the nuclear spinJ 2 are strongly suppressed. New transitions due to the violation of parity in the 0ECECprocess were proposed, e.g., nuclear transitions 0+ 0, 1 are compatible with a mixedcapture of s- and p-wave electrons. Based on the most recent data and realistic evaluation ofthe decay half-lives, a complete list of the most promising isotopes for which the 0ECECcapture may have the resonance enhancement was provided. for further experimental study.It includes 96Ru, 106Cd, 124Xe, 136Ce, 152Gd, 156Dy, 164Er, 168Yb, 180W, 184Os and 190Pt [3].By assuming |m | = 50 meV and an appropriate value of NME, half-lives of some of theisotopes were found to be as low as 1025 years in the unitary limit. It is about one order ofmagnitude shorter than the 0 half-life of 76Ge for the same value of the effective mass ofMajorana neutrinos.

The inverse value of the half-life of resonant neutrinoless double-electron capture

ln 2T 0ECEC1/2 (J )

= |Vab(J )|2

2 + 142abab, (38)

where J denotes angular momentum and parity of final nucleus. The degeneracy parametercan be expressed as

= MA,Z MA,Z2 = Q Bab E , (39)

15


where Q stands for a difference between the initial and final atomic masses in ground statesand E is an excitation energy of the daughter nucleus. Bab = Ea + Eb + EC is the energy oftwo electron holes, whose quantum numbers (n, j, l) are denoted by indices a and b and ECis the interaction energy of the two holes. The binding energies of single electron holes Eaare known with an accuracy of a few eV [46]. The width of the excited final atom with theelectron holes is given by

ab = a + b + . (40)Here, a,b is one-hole atomic width and is the de-excitation width of the daughter nucleus,which can be neglected. Numerical values of ab are about up to a few tens of eV. Byfactorizing the electron shell structure and nuclear matrix element for lepton numbers violatingthe amplitude associated with nuclear transitions 0+ J = 01, 11 one gets

Vab(J ) = 14 G2me

(geffA )2

RFab M0ECEC(J ). (41)

Here, Fab is a combination of averaged upper and lower bispinor components of the atomicelectron wavefunctions [3] and M0ECEC(J ) is the nuclear matrix element. We note that byneglecting the lower bispinor components M0ECEC(0+) takes the form of the 0-decayNME for ground state to ground state transition after replacing isospin operators by +. Ris the nuclear radius and gA is the axial-vector coupling constant.

The probability of the 0ECEC is increased by many orders of magnitude provided theresonance condition is satisfied within a few tens of electron-volts. For a long time there wasno way to identify promising isotopes for the experimental search for 0ECEC, because ofthe poor experimental accuracy of measurement of Q-values of the order of 110 keV formedium heavy nuclei. Progress in the precision measurement of atomic masses with Penningtraps [4749] has revived the interest in the old idea of the resonance 0ECEC. The accuracyof Q-values at around 100 eV was achieved. The estimates of the 0ECEC half-lives wererecently improved by more accurate measurements of Q-values for 74Se [50, 51], 96Ru [59],106Cd [53, 52], 102Pd [52], 112Sn [54], 120Te [55], 136Ce [56], 144Sm [52], 152Gd [57], 156Dy[58], 162Er [59], 164Er [60], 168Yb [59] and 180W [61]. It allowed the exclusion of some isotopesfrom the list of the most promising candidates (e.g., 112Sn, 164Er, 180W) for searching for the0ECEC.

Among the promising isotopes, 152Gd has likely resonance transitions to the 0+ groundstates of the final nucleus as it follows from improved measurement of Q-value for thistransition with accuracy of about 100 eV [57]. A detailed calculation of the 0ECEC of 152Gdwas performed in [62] (see table 3). The atomic electron wave functions were treated in therelativistic DiracHartreeFock approximation [63]. The NME for ground state to groundstate transition 152Gd 152Sm was calculated within the proton-neutron deformed QRPAwith a realistic residual interaction [57]. For the favored capture of electrons from K andL shells in the case of 152Gd the 0ECEC half-life is in the range 4.7 10284.8 1029years. This transition is still rather far from the resonant level. Currently, the 0ECEC half-life of 152Gd is 23 orders of magnitude longer than the half-life of 0 decay of 76Gecorresponding to the same value of |m | and is the smallest known half-life among known0ECEC.

The resonant 0ECEC has some important advantages with respect to experimentalsignatures and background conditions. The ground state to ground state resonant 0ECECtransitions can be detected by monitoring the X rays or Auger electrons emitted from theexcited electron shell of the atom. This can be achieved, e.g., by calorimetric measurements.The de-excitation of the final excited nucleus proceeds in most cases through a cascade ofeasy to detect rays. A coincidence setup can cut down any background rate right from the

16


beginning, thereby requiring significantly less active or passive shielding. A clear detectionof these rays would already signal the resonant 0ECEC without any doubt, as there areno background processes feeding those particular nuclear levels. We note that standard modelallowed double-electron capture with emission of two neutrinos,

eb + eb + (A, Z) (A, Z 2) + e + e, (42)is strongly suppressed due to almost vanishing phase space [42].

Until now, the most stringent limits on the resonant 0ECEC were established for 74Se[64, 65], 106Cd [66] and 112Sn [67]. However, following recent theoretical analysis [3] none ofthese resonant 0ECEC transitions is favored in the case of light neutrino mass mechanism.The ground state of 74Se is almost degenerate to the second excited state at 1204 keV inthe daughter nucleus 74Ge, which is a 2+ state [68] and is disfavored by the selection rule[3]. The TGV experiment situated in Modane established the limit on the 0ECEC half-lifeof 1.1 1020 years [66]. The subject of interest was the 0ECEC resonant decay mode of106Cd (KL-capture) to the excited 2741 keV state of 106Pd. For a long time the spin valueof this final state was unknown and it was assumed to be J = (1, 2)+. However, a newvalue for the spin of the 2741 keV level in 106Pd is J = 4+ and this transition is disfavoredagain due to selection rule. A search for the resonant 0ECEC in 106Cd was also carriedout at the Gran Sasso National Laboratories with the help of a 106CdWO4 crystal scintillator(215 g) enriched in 106Cd up to 66% [69]. It was found that the resonant 0ECEC to the2718 keV (J is unknown), 2741 keV (J = 4+) and 2748 (J = (2, 3)) keV excited statesof 106Pd are restricted to T 0ECEC1/2 4.3 1020 yr (KK-capture), T 0ECEC1/2 9.5 1020 yr(KL1-capture) and T 0ECEC1/2 4.31020 yr (KL3-capture), respectively. We note that the 2718excited state decays by 100% into the 3+ state at 1557.68 keV state, which again excludesthe possibility of J = 0, 1 for this state. Further, we already mentioned above that a newmass measurement [54] has excluded a complete mass degeneracy for a 112Sn decay and hastherefore disfavored significant resonant enhancement of the 0ECEC mode for this transition.Recently, a first bound on the resonant 0ECEC half-life of 136Ce of about 1.11015 years wasmeasured [70].

5. Summary

Many new projects for measurements of the 0-decay have been proposed, which hope toprobe effective neutrino mass m down to 1050 meV. An uncontroversial detection of the0-decay will prove the total lepton number to be broken in nature, and neutrinos to beMajorana particles. There is a general consensus that a measurement of the 0-decay inone isotope does not allow us to determine the underlying physics mechanism. It is greatlydesired that experiments involving as many different targets as possible are pursued. Thereis also a revived interest in the theoretical and experimental study of the resonant 0ECEC(neutrinoless double-electron capture), which can probe the Majorana nature of neutrinos andalso the neutrino mass scale. The 0ECEC half-lives might be comparable to the shortesthalf-lives of the 0 decays of nuclei provided the resonance condition is matched with anaccuracy of tens of electron-volts. There is a great deal of theoretical and experiment effortrequired to determine the best 0ECEC candidate.

Nuclear matrix elements of these two lepton number violating processes need to beevaluated with uncertainty of less than 30% to establish the neutrino mass spectrum and CPviolating phases. Recently, there has been significant progress in understanding the source ofthe spread of calculated NMEs. Nevertheless, there is no consensus among nuclear theoristsabout their correct values, and corresponding uncertainty. The improvement of the calculation

17


of the nuclear matrix elements is a very important and challenging problem. We presentedimproved calculation of the 0-decay and 0ECEC NMEs, which includes a consistenttreatment of the two-nucleon short-range correlations and deformation effects. In addition, thepossibility of measuring the 0-decay NME was addressed.

Acknowledgments

This work was supported in part by the Deutsche Forschungsgemeinschaft within the projectNuclear matrix elements of Neutrino Physics and Cosmology FA67/40-1, the VEGA Grantagency of the Slovak Republic under the contract number 1/0639/09 and by the grant of theMinistry of Education and Science of the Russian Federation (contract 12.741.12.0150).

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1. Introduction2. two-nucleon short-range correlations and uncertainties3. decay of deformed nuclei within QRPA3.1. Formalism3.2. Calculation results

4. The resonant neutrinoless double-electron capture5. SummaryAcknowledgmentsReferences

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