NMEs for 0νββ(0^+\rightarrow2^+) of two-nucleon mechanism for ^{76}Ge
aa r X i v : . [ nu c l - t h ] J a n NMEs for νββ (0 + → + ) of two-nucleon mechanism for Ge Dong-Liang Fang a,b and Amand Faessler c a Institute of Modern Physics, Chinese Academy of Science, Lanzhou, 730000, China b University of Chinese Academy of Sciences, Beijing, 100049,China and c Institute for theoretical physics, Tuebingen University, D-72076, Germany
In this work we present the first beyond closure calculation for the neutrinoless double betadecay (0 νββ ) of Ge to the first 2 + states of Se. The isospin symmetry restored Quasi-particlerandom phase approximation (QRPA) method with the CD-Bonn realistic force is adopted for thenuclear structure calculations. We analyze the structure of the two nucleon mechanism nuclearmatrix elements, and estimate the uncertainties from the nuclear many-body calculations. We find g pp plays an important role for the calculations and if quenching is included, suppression for thetransition matrix element M λ is found. Our results for the transition matrix elements are about oneorder of magnitude larger than previous projected Hatree-Fock-Boglyubov results with the closureapproximation. PACS numbers: 14.60.Lm,21.60.-n, 23.40.Bw
I. INTRODUCTION
In the standard model, the nuclear weak decay is inter-preted as the low-energy effective theory for weak inter-action. This decay is mediated by the left-handed gaugeboson W ± . The mass of W ± are acquired through thespontaneous symmetry breaking by the so-called Higgsmechanism. However, the Yukawa coupling of Higgs par-ticle to neutrinos is absent in the standard model due tothe absence of the right-handed neutrinos. The discov-ery of neutrino masses from oscillation experiments thenasks for new physics beyond the standard model. As anextension to Standard model, the L-R symmetric model[1–3] introduces the right-handed SU (2) R gauge symme-try and a hence heavy right-handed gauge boson fromsymmetry breaking with extra Higgs bosons at a higherenergy scale beyond electroweak scale. In such a the-ory, the introduction of lepton number violating neutrinoMajorana mass terms together with normal Dirac massterms gives naturally the tiny neutrino mass through theso-called See-Saw mechanism [4]. Such extensions to theStandard Model could also affect the rare nuclear processcalled neutrinoless double beta decay (0 νββ ). The par-ticipation of right-handed weak gauge bosons will alsoinduce the emission of right handed leptons. The si-multaneous presence of weak currents with both chiralitywill introduce a momentum term into the neutrino prop-agator. These terms are not suppressed like the massterms due to the smallness of neutrino mass. The right-handed weak currents will contribute to the decay to theground states with extra terms and change the electronspectra [5]. Nevertheless, these terms are suppressed bythe new physics parameters as well as the electron wavefunctions for p partial waves. Therefore, they are hin-dered in normal neutrinoless double beta decay comparedto the neutrino mass mechanisms. On the other hand,the decay to the 2 + states, are dominated by the helic-ity changing mechanisms (V+A terms, see [6]). In thissense, the branching ratio of neutrinoless double beta de-cay to the 2 + state (hereafter 0 νββ (2 + ), the spin-parity of the final states of the decay are included inside theparenthesis ) could help to reveal the underlying mecha-nisms of this very rare decay. Nevertheless, experimen-tally such a process is extremely difficult to observe dueto the large 2 νββ (0 + ) background around the position of0 νββ (2 + ) Q value. Despite the difficulties, the observa-tion of 0 νββ (2 + ) together with that for the decay to theground states will determine the underlying mechanismsof the neutrinoless double beta decay and pave our wayto new physics beyond the standard model. For exam-ple, the observation of 0 νββ (2 + ) could possibly rule outa category of mechanisms where no right-handed gaugebosons or fermions are present.There are numerous publications dedicated to thenuclear many-body calculations for neutrinoless doublebeta decay with various approaches, e. g. the ShellModel calculations [7, 8], the QRPA calculations [9, 10],the IBM calculations [11] and the nuclear meanfield cal-culations [12], especially the recently developed ab ini-tio methods [13]. However all these works focus on0 νββ (0 + ), there aren’t to many theoretical investiga-tions available for 0 νββ (2 + ) in the literature for the ma-trix elements (NME). An earlier calculation [14] withthe Projected Hartree Fock Bogoliubov method (PHFB)suggests, that the nuclear matrix elements (NME) forthis decay mode are much smaller than for the decayto ground states. The QRPA method is widely used indouble beta decay calculations [10, 15, 16]. The QRPAcan also be adopted to describe the vibrational 2 + states.Therefore, there are attempts to use QRPA to calculate ββ -decays to the first 2 + state [17, 18]. And most of themfocus on the 2 νββ case. In this study, we go step forwardby carrying out the QRPA calculations for 0 νββ (2 + ) for Ge. Taking advantage of the QRPA method, we takeinto account the contributions from all the intermediatestates. Also we can include the isoscalar particle-particleresidual interaction which is missing in PHFB calcula-tions. At this first attempt, we do not include too manyexamples. We focus on one nucleus, Ge which is alsothe candidate treated in ref. [14]. It has been shown in[14], that besides the nucleon mechanism, the N* couldalso play an important role, we will not discuss this in thecurrent study. Also, as in [7], the induced weak currentwill further reduce the NME, this will be neglected in thiswork. As well as the heavily suppressed neutrino massmechanism for 0 νββ (2 + ) through nuclear recoil [19].The current article is arranged as follows: first we givethe general formalism of our many-body calculations, andthen we show the results and discuss possible uncertain-ties, and finally we present the conclusions and outlook. II. FROMALISMS
The half-lives of 0 νββ (2 + ) can be expressed in a simpleform, while we consider the light neutrino only [6, 14]: τ − = F ( h λ i M λ − h η i M η ) + F ( h η i M ′ η ) (1)Here F are the phase space factors expressed in [6,14]. h λ i and h η i are the new physics parameters which are model dependent. In the L-R symmetric model [2]: SU (2) L × SU (2) R × U (1) B − L , we have [6]: h λ i = λ X j U ej V ej h η i = η X j U ej V ej (2)Here U ej and V ej are matrix elements for the generalizedPMNS matrix [20]. λ ≈ ( M W L /M W R ) are the squareof the ratio of the masses between the mass eigenvaluesof the light and heavy gauge bosons, η ≈ tan ξ is themixing angle between the left-handed gauge boson andthe heavy gauge boson mass eigenvalues. M λ , M η and M ′ η are the nuclear matrix elments(NMEs) as combinations of different components [6]: M λ = X i =1 C λi M i , M η = X i =1 C ηi M i , M ′ η = X i =6 C ′ ηi M i (3)The different coefficients C Ii are given in table I of [14].The general form of above NMEs can be expressed as: M I = X J π m i m f X J ′ J J ′ j p j p ′ J j n j n ′ J ′ J J ′ h j p j p ′ J ||O I || j n j n ′ J ′ i× ( − J ′ + J √ J ′ + 1 h +1 f || g [ c † p ˜ c n ] J ′ || J π m f ih J π m f || J π m i ih J π m i || [ c † p ˜ c n ] J || + i i (4)A general derivation of the single particle matrix ele-ments are given in the appendix. For the related op-erators, we have the form [14]: O = σ · σ [ˆ r ⊗ ˆ r ] (2) h ( r ) (5) O = [ σ ⊗ σ ] (2) h ( r ) (6) O = [[ σ ⊗ σ ] (2) ⊗ [ˆ r ⊗ ˆ r ] (2) ] (2) h ( r ) (7) O = [ˆ r ⊗ ˆ r ] (2) h ( r ) (8) O = [( σ + σ ) ⊗ [ˆ r ⊗ ˆ r ] (2) ] (2) h ( r ) (9)And: O = [( σ − σ ) ⊗ [ˆ r ⊗ ˆ r ] (1) ] (2) r + r h ( r ) (10) O = [( σ − σ ) ⊗ [ˆ r ⊗ ˆ r ] (2) ] (2) r + r h ( r ) (11)Here r is the relative distance between the two decayingnucleons ~r = ~r − ~r and ~r + = ( ~r + ~r ) / M , M and M are the space-space componentsof the current-current interaction, while M is the time-time components and M , M and M are the time-spacecomponents. These time-space components of the NME’sappear only in the L-R symmetric case and are missing in the neutrino mass mechanisms. In all these NME’s,we find a similarity between M and M νGT for 0 νββ (0 + )as well as M and M νT . The GT operator σ or the tensoroperator [ σ ⊗ ˆ r ] replace the scalar products in 0 νββ (0 + ).We also find an analog similarity between M and M νF ,where ˆ r ’s come from the p-wave electron and the mo-mentum term form a tensor product instead of a scalarproduct in 0 νββ (0 + ).The neutrino potential differs from that of the massmechanism due to the momentum terms in the neutrinopropagator [5]: h ( r ) = 2 Rπ r Z F ( q ) j ( qr ) q dqq + E N (12)By deriving this, we assume that the two electrons sharethe decay energy, therefore E N = E m + M m − ( M i + M f ) / E m is the excitation energy of the m th excitedstates of the intermediate nucleus. The nuclear radius R = 1 . A / [¯ f m ] introduced here makes the final NMEdimensionless. For the form factor F ( q ) we use a dipoleform with the parameters are as in [9].Usually, an extra radial function f ( r ) should be multi-plied to the above expression to take into considerationthe strong repulsive nature of nucleon-nucleon interactionat short range. This is usually called the short range cor-relation (src) function, and in our calculation we choosethe CD-Bonn or Argonne src extracted from the corre-sponding nuclear force with the form given in [21]. For the reduced one-body density for transitions fromintermediate states to final 2 + , the expression is compli-cated [22]: h + f || [˜ c † p c n ] J ′ || J π m i√ J ′ + 1 = p J + 1)[ X p ′ ≤ p ( − j p ′ + j n p δ pp ′ (cid:26) j p ′ j p j n J ′ J (cid:27) ( u p u n X + f p ′ p X J π mp ′ n − v p v n Y + f p ′ p Y J π mp ′ n )+ X p ′ ≥ p ( − j p + j n p δ pp ′ (cid:26) j p ′ j p j n (cid:27) ( u p u n X + f pp ′ X mp ′ n − v p v n Y + f pp ′ Y mp ′ n ) − X n ′ ≤ n ( − j n + j p √ δ nn ′ (cid:26) j n ′ j n j p (cid:27) ( v p v n X + f n ′ n X mpn ′ − u p u n Y + f n ′ n Y mpn ′ ) − X n ′ ≥ n ( − j n ′ + j p √ δ nn ′ (cid:26) j n ′ j n j p (cid:27) ( v p v n X + f nn ′ X mpn ′ − u p u n Y + f nn ′ Y mpn ′ )] (13)Here X ’s and Y ’s are the amplitudes for pn-QRPA(proton-neutron Quasi-particle Random Phase Approxi-mation) describing the intermediate states and X ’s and Y ’s are the amplitudes for CC-QRPA (Charge Conserv-ing QRPA) describing the final 2 + state [22]. And u ’sand v ’s are the BCS coefficients.The reduced one body density for transitions from theinitial states to the intermediate states can be expressedas [9]: h J π m i || [ c † p ˜ c n ] J || + i i√ J + 1 = X pn ( u p v n X J π ,m i pn + u n v p Y J π ,m i pn )(14)We also introduce the overlap between the initial andfinal intermediate states with the form [9]: h J π m f || J π m i i = X pn ( X J π ,m i pn X J π ,m f pn − Y J π ,m i pn Y J π ,m f pn ) × ( u ip u fp + v ip v fp )( u in u fn + v in v fn ) f h BCS | BCS i i (15)For simplicity, we set f h BCS | BCS i i ≈ III. RESULTS AND DISCUSSION
For our QRPA calculations, the single particle ener-gies are taken from the solutions of Schr¨odinger equa-tions with a Coulomb corrected Woods-Saxon potential.For the single particle wave functions, we use the Har-monic Oscillator wave functions. For the pairing part, we use the realistic CD-Bonn force derived from Br¨ucknerG-matrix. This is also used for the pn-QRPA and CC-QRPA residual interactions. A fine tuning of the inter-actions is needed to reproduce the experimental values.For the pairing part, we fit the two parameters g ppair and g npair to reproduce the odd-even mass staggering. For pn-QRPA, we multiply the G-matrix by overall renormaliza-tion factors g ph and g pp ’s for particle-hole and particle-particle parts, respectively. We set g ph = 1. And for g pp , we fit the iso-scalar channel ( g T =0 pp ) and iso-vectorchannel ( g T =1 pp ) separately. g T =0 pp is fixed by reproducingthe experimental 2 νββ and g T =1 pp are fixed to put to zerothe 2 νββ Fermi matrix elements due to isospin symme-try restoration [23]. For more details, one can refer tothe our previous work [22]. For a baseline calculation,we consider the CD-Bonn src bare axial vector couplingconstant g A = 1 .
27. And we use a model space with N max = 6 which consists of 28 single particle levels forboth neutrons and protons.In table.I, we present the NMEs for each operator. Asa comparison, we also present the results from PHFB cal-culations. The current results (the baseline results) differfrom the PHFB results by factors from five to more thanone order of magnitude case by case. The largest devia-tion we see in M for the M ′ η part. We obtain M muchlarger than M in magnitude. Our results have have alsodifferent phases for these two NMEs, this then combinedwith the C ′ η coefficients leads to the enhancement insteadof the cancellations of M ′ η . Therefore, our M ′ η is muchlarger than a previous PHFB calculation. PHFB getsan approximate cancellation between M and M , whichleads to an negligible M ′ η .For the M λ ( η ) part, we find the NME’s have basicallysimilar phases as the PHFB results except for M . Onthe other hand, our results are about one order of magni- M M M M M M Λ M η M M M ′ η PHFB[14] 0.151 0.027 -0.002 -0.049 -0.004 0.002 0.061 0.074 0.042 0.001Baseline 0.705 -0.253 -0.046 -0.153 -0.048 0.150 0.469 0.527 -1.270 1.519 N max = 5 0.629 -0.208 -0.014 -0.124 -0.069 0.151 0.438 0.661 -1.369 1.688 N max = 7 0.640 -0.256 -0.048 -0.145 -0.063 0.121 0.439 0.643 -1.251 1.564w/o src 0.701 -0.234 -0.049 -0.154 -0.051 0.128 0.451 0.485 -1.182 1.410Argonne src 0.705 -0.250 -0.046 -0.153 -0.048 0.149 0.467 0.519 -1.261 1.505L.O. 0.749 -0.347 -0.051 -0.154 -0.041 0.228 0.540 0.823 -1.756 2.152w/o F ( q ) 0.695 -0.241 -0.047 -0.154 -0.050 0.136 0.457 0.529 -1.272 1.521Closure Energy 0.696 -0.267 -0.043 -0.144 -0.041 0.177 0.472 0.522 -1.247 1.493 g T =0 pp = 0 0.611 -0.169 -0.054 -0.161 -0.065 0.029 0.376 0.540 -1.240 1.496 g T =1 pp = 0 0.795 -0.246 -0.034 -0.156 -0.034 0.206 0.516 0.501 -1.437 1.665 g A = 0 .
75 0.695 -0.241 -0.047 -0.154 -0.050 0.008 0.317 0.529 -1.272 1.249TABLE I: The NME values for 0 νββ (2+). Here the baseline calculation is explained in text. And also various approximationsand parameters will be discussed in text. M N max = 5N max = 6N max = 7 −0.050.00 M −0.0250.0000.025 M −0.050.00 M −0.0250.0000.025 M −0.250.00 M + − + − + − + − + − + − + − + − + − + − + − J π −0.50.0 M FIG. 1: (Color online) The dependence of the NMEs on the model space for different multipoles. Here N max refers to thelargest principle quantum number for the outermost shell. tude larger, although the relative ratios among differentNMEs ( M − M ) are similar. Of these NMEs, M is thelargest. The second largest is M and third is M . M and M are relatively small and hence less important.For M λ , if we multiply the NMEs with the correspond-ing C λ ’s, we find that M and M contributes coherently, they are then cancelled by M , while the rest two NME’scontributes less than 10%. For M η , all these three NMEsgives additive contributions, this makes M η about threetimes larger than M λ . This is the reason, why M η islarger than M λ as also observed in [14]. In our case,this ratio is about three. This is similar to PHFB cal- M L.O.w/o F(q ) FullClos. Energy −0.050.00 M −0.0250.0000.025 M −0.050.00 M −0.0250.0000.025 M −0.250.000.25 M + − + − + − + − + − + − + − + − + − + − + − J π −0.50.0 M FIG. 2: (Color online) The NMEs for a Coulomb type neutrino potential (blue bars). The orange bars are NMEs withoutform factors and red bars are with excitation energies replaced by a closure energy. The green bar are our baseline calculationsexplained in the text. culations, however, in their calculation, the strong can-cellation gives a negligible M λ , the ratio is about 30. Inshort, in their calculation, only M η is important while M λ and M ′ η can be neglected due to the cancellationsbetween different parts of the NME. We get quite differ-ent results, and M ′ η is the most important contribution,with a value about three times larger than M η . This willaffect the constraints on new physics models and needsfurther investigation.As we show above, no obvious suppression of 0 νββ (2 + )NME’s as claimed by [14] is found from current calcula-tions compared to 0 νββ (0 + ), this also agrees with the q terms in 0 νββ (0 + ) calculations [24]. This is the majordifference of the current work and [14]. This is also dif-ferent from 2 νββ (2 + ), where the NME is suppressed bythe cubic dependence of the energy denominator [6]. Inthis sense, suppression of 0 νββ (2 + ), if it exists, must berelated to other issues. This may come from the uncer-tainties of the many-body approaches, such as the size ofthe model space or other structure ingredients for the 2 + states which may lead to different transition rates fromthe intermediate states for various transitions. A morethorough comparative study could give us mor detailed hints.Compared to previous calculations with PHFB [14],the QRPA calculation goes beyond the closure approx-imation. We calculate explicitly the contribution fromeach intermediate state. In fig. (1-4), we present theindividual contributions from different multipoles of theintermediate states and we will also show how the dif-ferent approximations may affect these results. In eachgraph, the results are compared with our present baselinecalculations with standard conditions described above.For details of the structure of the nuclear MME’s, westart with our baseline calculation ( e.g. orange bars infig.1). For M , the multipoles give positive contributionswith several exceptions. Unlike 0 νββ (0 + ) ( M νGT ), wherethe largest contributions come from low-spin intermedi-ate states and the NME values decrease as spins increase, M has its largest contribution from 4 − . We find a roughtrend that the NME’s first increase and then decrease asspin increases. And as one would expect, the NME’sfrom states with very high spin can be safely neglected. M has basically the same characters as M except themuch smaller magnitude. A large contribution from 1 − is observed for M but not for M . For most multipoles, M w/o srcCD-BonnAV-18 −0.050.00 M −0.0250.0000.025 M −0.050.00 M −0.0250.000 M −0.250.00 M + − + − + − + − + − + − + − + − + − + − + − J π −0.50.0 M FIG. 3: (Color online) The NME dependence on the short range correlations. M has different signs as M , this contradicts conclu-sions in [14]. Not all multipoles contribute equally withsimilar spins, we find that the states with negative par-ity generally contribute more. In some sense, these twoNME’s behave like M νGT for 0 νββ (0 + ) as we mentionedabove. The smallness of final M comes partially fromits magnitude and partially from the cancellations be-tween low-spin and high-spin multipoles. This is analogto M νT , although these two NME’s depend differently onˆ r . M , the time-time component of the NME is on theother hand, very close to M νF . The two NME’s have onething in common: Only intermediate states with naturalparity ( π = ( − J ) have non-zero contributions. In thecurrent calculation, all multipoles contribute negativelyexcept 1 − . All these multipoles contribute basically withthe same magnitude as M and M is the major cancel-lation to M λ .For the space-time components, M , M and M , wefind a quite different behavior. M from each multipole isabout one order of magnitude smaller than M and M .Due to the strong cancellations among different multi-poles, M is one of the smallest of all the NME’s. Al-most all multipoles contribute positively except a strong cancellation from 3 − . For M , the important contribu-tions comes from three multipoles (3 − , 5 − and 7 − ), allthe other multipoles contribute much less. The lack ofcancellations from other multipoles thus makes M thelargest from all the NME’s.A possible cause of the smallness of NMEs in ref. [14]may come from the small model space used whith onlytwo major shells. To test this, we plot in fig.1 the resultswith N max = 5 (blue bars) and N max = 7 (green bars).The sensitivity of the NME’s to diffeent model spaces aredifferent. Also different is the sensitivity of the individ-ual multipoles for each NME. For M , the influence frommodel space is generally small, there is no unique trendfor different multipoles under the change of the modelspace. For some multipoles, the NME decreases with en-larged model space, but most cases we find that the extraorbitals will first enhance but then reduce the NME. Theorbitals of different parity contribute to the NME differ-ently. The addition of N=6 shell brings in the positiveparity orbitals which enhance the NMEs for multipolessuch as 3 − or 5 + , while the negative parity orbitals fromN=7 shells reduce the NME’s. Unlike the case of M , theincrease of the model space enlarge M , especially for 1 + ,2 − etc . For 1 + , a strong suppression is observed when M [0.0, 0.93][0.58, 0.93] [0.62, 0.93][0.62, 0.00] −0.050.00 M −0.0250.0000.025 M −0.050.00 M −0.0250.0000.025 M −0.250.00 M + − + − + − + − + − + − + − + − + − + − + − J π −0.50.0 M FIG. 4: (Color online) The NME dependence on g pp ’s. The values in the bracket are g T =0 pp and g T =1 pp respectively. The bluebars are results with g T =0 pp = 0, and the orange bars are g T =0 pp values which reproduce the 2 νββ NME with g A = 0 . g A . Thered bars are results with g T =1 pp = 0. And the green bars here again are the baseline calculations. the N=6 shell is added. But such addition gives strongenhancement to 2 − or 3 + . The addition of the N=7 shellto the model space causes milder changes. We see en-hancement from 1 − and 2 − but reductions from 3 − and4 − intermediate states. A similar behavior shows the M ,where we find a strong enhancement from 1 + too. Forall other multipoles, the change due to model space en-largement is relatively small. For M , low spin multipolesare much more sensitive to the model space changes anddifferent multipoles behave differently, though we cannotfind any specific patterns. This is also true for the space-time components of the NME. In general, they are lessaffected by the change of model space in our calculations.If we look at the total changes of each NME from N max = 5 to N max = 6, M increases about 10%. thisis the smallest change among all the NMEs. Meanwhile,all the other NMEs changes about 20% or more. By per-centage M changes by more than 200%. By the abso-lute values of NMEs, M , M , M and M get enhancedwhile the rest get reduced. When the N = 7 shell isadded, the change is relatively milder, especially for M , M and M , this implies a general trend of convergence of the results with a larger model space. But for M , M , M and M , we find a slower trend of convergence thanfor the above NMEs. In either cases, the changes fromadding the N = 7 shell is smaller than adding the N = 6shell. Current results suggest that the errors of adoptingthe current model space are generally smaller than 20%.Also, these results suggest that the deviation of our cal-culations and those in [14] is not caused by the smallmodel space, which they adopted. It is most probablethat the smallness of their results are caused by differentnuclear structures. Additional work is needed to clarifythis.The form factor with a dipole form is widely used in0 νββ (0 + ) calculations [9]. In our calculations, we use thesame form for g V ( q ) and g A ( q ). We find that the formfactors are not very important except for the 1 + statesof M . This may suggest that with the actual neutrinopotential the low momentum parts where the q satisfies g A ( q ) ∼ g A (0) dominate, and the high momentum partsare either small or cancels each other. A careful checksuggests the latter should apply. These behaviors alsohelps to explain the large reduction for some NME’s whenthe realistic neutrino potential is considered. With lowmomenta, the intermediate energies become important.In this sense, the choice of the intermediate energies be-comes important. For calculations with closure approxi-mation, a closure energy is needed. So we also check howthe use of closure energy would affect the NME’s. Thisis illustrated with the red bars in fig.2. A closure energyof 7 MeV is used. We find for most multipoles of mostNME’s, the proper choice of closure energy brings smallerrors to the calculation and we can draw the conclusion,that using the closure energy in 0 νββ (2 + ) barely changesthe final results, the errors are within several percents,this agrees with 0 νββ (0 + ) QRPA calculations. Anotherimportant correction comes from the induced weak cur-rent [9], it is not included in the current calculations andwill be implemented in our future study. If that is in-cluded, the NME will most probably be further reducedand new potentials needs to be introduced [5].In 0 νββ (0 + ), src only plays an important role for theheavy neutrino mass mechanism [12, 26], while for thelight neutrino mass mechanism, the correction is rela-tively small, up to only several percents [12, 26]. In cur-rent calculations, we adopt two src’s [21]: the CD-Bonntype and Argonne type, they are obtained by fitting therespective nuclear potentials. From fig.3, we find similartrends as 0 νββ (0 + ), the general correction from src isabout several percent and for most cases, the two src’sgive similar amount of corrections. For almost all mul-tipoles, we find that the src enhances the NMEs moreor less. However, the net effects to the NMEs are slightreductions for M and M due to cancellations amongdifferent multipoles. And enhancements of NME valuesfor other operators are observed. The largest correctioncomes from M , then M and M . For M we can alsofind slight difference between the two src’s.The most important parameters in QRPA calculationsare the particle-particle interaction strength g pp ’s. For0 νββ (0 + ), one finds that M GT for both 2 νββ and 0 νββ depends sensitively on the isoscalar strength g T =0 pp while M νF are sensitive to the isovector strength g T =1 pp . So wealso test such dependence in current calculations. As wehave shown in [22] for the case of 2 νββ to 2 + states,the GT type decays are also sensitive to g T =1 pp since theisovector particle-particle residue force affects the struc-ture of 2 + states in QRPA calculations. In fig.4, we firstswitch off the isoscalar interaction (the blue bar). Thenthe effects of this interaction to NMEs can be estimatedby comparing the blue and green bars. The results sug-gest that, while M and M are not closely related to g T =0 pp , M comes out to be the most sensitive one like its0 νββ (0 + ) counterpart M νGT . Similar as M νGT , 1 + comesout to be the most sensitive multipole, the increase ofthe g T =0 pp drastically changes the NME most probablydue to the SU(4) symmetry restoration[23]. And the ef-fects of isoscalar residue interactions to the specific mul-tipoles of specific NMEs are quite different, some NMEsare enhanced and some are reduced. In total, the intro-duction of isoscalar interactions enhances M , M and M , but reduces other NMEs. And the magnitudes ofthese enhancements and reductions are really case de-pendent. The similar thing happens to isovector interac-tions (this interaction has been switched off for the redbars in fig.4), for some cases they are more importantthan the isoscalar interactions. In general, the particle-particle interaction is one of the most important sourceof the errors for QRPA calculations. M is at least sen-sitive to g pp ’s while M and M are really sensitive to g T =0 pp and g T =1 pp respectively. Especially M , the absenceof isoscalar interactions will reduce the NME by morethan 30%.The special attention should be paid to the case of thequenched g A . In all our above analysis, we assume that g A is not quenched, however, in nuclear medium quench-ing of g A is observed. In current calculations, we use thesimplest treatment for quenching, that is simply changethe value of g A , while there exists more fundamental ap-proaches for the quenching of g A using the chiral two-body currents[27]. In our case, the difference betweenquenched g A calculations and our baseline calculationsare the different fit of g T =0 pp . Therefore, the differencefor the individual NMEs are small as in fig.4. But thequenched g A will also change the coefficients C ’s. As aresult, the NMEs M λ , M η and M ′ η have been changedtoo. In general, these NME’s are reduced due to the con-vention used in[14]. For M ′ η a reduction of a rough factorof g A /g A is expected, since the two components M and M has the same dependence on g A . For M λ and M η , dif-ferent components have a different g A dependence. Andif we take g A = 0 . g A we find a rough cancellation for M λ as predicted in [14], this comes from the interplayamong different components. Meanwhile the reductionfor M η is basically 25%. The severe reductions of M λ emphasize the importance of where the quenching of g A originates and how to treat it properly. IV. CONCLUSION AND OUTLOOK
In this study, we calculate the NME of 0 νββ to 2 +1 for Ge. We got quite large results compared to previouscalculations. We estimate the errors of current calcula-tion by changing several parameters we use. We find that g A may be a very important issue for the final NME’s.Further investigations, such as the role of the inducedweak current, the anharmonicity beyond QRPA, and thedecay mechanism mediated by N*, are needed for muchmore detailed conclusions. acknowledgement This work is supported by the ”Light of West China”program and the ”From 0 to 1 innovative research” pro-gram both from CAS.
Appendix A: Derivation of single particle matrixelements in particle-particle channel
The seven decay operators are taken from [14] andpresented above. They can be written in the formsof combination of three parts: the relative coordinate( O r ( ~r ) ≡ O J (ˆ r ) h ( r )), the center of mass coordinate( O R ( ~R ) ≡ O J ( ˆ R ) f ( R )) and spin part ( O S ( ~σ , ~σ )).These operators can then be expressed in a general form O (2) I = [[ O J (ˆ r ) ⊗ O J ( ˆ R )] ( J ′ ) ⊗ O J ( ~σ , ~σ )] (2) h ( r ) f ( R ).We assume the quantum numbers for the single orbitalare ( n , l , j ) and ( n , l , j ) for protons and ( n ′ , l ′ , j ′ )and ( n ′ , l , j ′ ) for neutrons. The single matrix elementsof these operators under harmonic oscillator basis can beexpressed as: h pp ′ J || [[ O J (ˆ r ) ⊗ O J ( ˆ R )] ( J ′ ) ⊗ O J ( ~σ , ~σ )] (2) || nn ′ J ′ i = X nl N L LS A pp ′ J ,LS h n l , n l , L | nl N L , L i× X n ′ l ′ N ′ L ′ L ′ S ′ A nn ′ J ′ ,L ′ S ′ h n ′ l ′ , n ′ l ′ , L ′ | n ′ l ′ N ′ L ′ , L ′ i× h nl N L L ; s , s , S ; J ||O (2) I || n ′ l ′ N ′ L ′ L ′ ; s ′ , s ′ , S ′ ; J ′ i (A1)Here A pp ′ ( nn ′ ) J,LS is the 9j-symbol for JJ to LS couplingtransformation: A ττ ′ J,LS = (2 S + 1)(2 L + 1) p (2 j τ + 1)(2 j τ ′ + 1) × l τ j τ l τ ′ j τ ′ S L J (A2) And h n l , n l , L | nl N L , L i is the Brody-Moshinskitransformation coefficients [28].Using techniques from e.g. [29], we could further getthe expressions of each operator: h nl N L L ; s , s , S ; J ||O (2) I || n ′ l ′ N ′ L ′ L ′ ; s ′ , s ′ , S ′ ; J ′ i = p J + 1)(2 J ′ + 1) L L ′ J ′ S S ′ J J J ′ × p (2 J + 1)(2 J ′ + 1)(2 J ′ + 1) l l ′ J L L ′ J L L ′ J ′ × h nl || O J (ˆ r ) f ( r ) || n ′ l ′ ihN L|| O J ( ˆ R ) f ( R ) ||N ′ L ′ i× h s , s , S || O J ( ~σ , ~σ ) || s ′ , s ′ , S ′ i (A3)All these reduced matrix elements with different O J (ˆ r ) etc. can be calculated analytically in harmonic oscillatorbasis and we omit their derivations in current article, onecould refer to references such as [29]. [1] J. C. Pati and A. Salam, Phys. Rev. D , 275(1974);[2] R. Mohapatra and J. C. Pati, Phys. Rev. D ,2558(1975);[3] G. Senjanovic and R. N. Mohapatra, Phys. Rev. D ,1502 (1975); R. N. Mohapatra and G. Senjanovic, Phys.Rev. Lett. , 912 (1980); Phys. Rev. D , 165 (1981).[4] S. F. King, Rept. Prog. Phys. , 107-158 (2004)[5] D. Stefanik, R. Dvornicky, F. Simkovic and P. Vogel,Phys. Rev. C , 055502 (2015)[6] M. Doi, T. Kotani and E. Takasugi, Prog. Theor. Phys.Suppl. , 1 (1985)[7] E. Caurier, F. Nowacki, A. Poves and J. Retamosa,Nucl. Phys. A , 973c (1999); J. Menendez, A. Poves,E. Caurier and F. Nowacki, Nucl. Phys. A , 139(2009); E. Caurier, F. Nowacki and A. Poves, Phys. Lett.B , 62 (2012).[8] J. Men´endez, J. Phys. G , 014003 (2018)[9] F. Simkovic, G. Pantis, J. D. Vergados and A. Faessler,Phys. Rev. C , 055502 (1999)[10] F. ˇSimkovic, V. Rodin, A. Faessler and P. Vogel, Phys.Rev. C , 045501 (2013)[11] J. Barea, J. Kotila and F. Iachello, Phys. Rev. C ,014315 (2013); ibid , 034304 (2015).[12] L. S. Song, J. M. Yao, P. Ring and J. Meng, Phys. Rev.C , 054309 (2014); J. M. Yao, L. S. Song, K. Hagino,P. Ring and J. Meng, Phys. Rev. C , 024316 (2015); L. S. Song, J. M. Yao, P. Ring and J. Meng, Phys. Rev.C , 024305 (2017).[13] J. M. Yao, B. Bally, J. Engel, R. Wirth, T. R. Rodr´ıguezand H. Hergert, Phys. Rev. Lett. , no.23, 232501(2020)[14] T. Tomoda, Nucl. Phys. A , 635-646 (1988)[15] M. T. Mustonen and J. Engel, Phys. Rev. C , no.6,064302 (2013)[16] J. Hyv¨arinen and J. Suhonen, Phys. Rev. C , 024613(2015)[17] J. Suhonen and O. Civitarese, Phys. Lett. B , 212-215(1993)[18] J. Schwieger, F. Simkovic, A. Faessler and W. A. Kamin-ski, J. Phys. G , 1647-1653 (1997)[19] T. Tomoda, Phys. Lett. B , 245-250 (2000)[20] Z. Z. Xing, Phys. Rev. D , 013008 (2012)[21] F. Simkovic, A. Faessler, H. Muther, V. Rodin andM. Stauf, Phys. Rev. C , 055501 (2009)[22] D. L. Fang and A. Faessler, Chin. Phys. C , 084104(2020)[23] V. Rodin and A. Faessler, Phys. Rev. C , 014322 (2011)[24] K. Muto, E. Bender and H. V. Klapdor, Z. Phys. A ,187-194 (1989)[25] D. L. Fang, A. Faessler, V. Rodin and F. Simkovic, Phys.Rev. C , 034320 (2011)[26] D. L. Fang, A. Faessler and F. Simkovic, Phys. Rev. C , 045503 (2018)[27] J. Engel, F. Simkovic and P. Vogel, Phys. Rev. C ,064308 (2014)[28] M. Moshinsky, Nucl. Phys.13