Uncertainties in nuclear transition matrix elements for neutrinoless ββ decay within the PHFB model
P. K. Rath, R. Chandra, K. Chaturvedi, P. K. Raina, J. G. Hirsch
aa r X i v : . [ nu c l - t h ] A p r Uncertainties in nuclear transition matrix elements for neutrinoless ββ decay withinthe PHFB model P. K. Rath , R. Chandra , , K. Chaturvedi , P. K. Raina and J. G. Hirsch Department of Physics, University of Lucknow, Lucknow-226007, India Department of Physics and Meteorology, IIT, Kharagpur-721302, India Department of Physics, Bundelkhand University, Jhansi-284128, India Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, A.P. 70-543, M´exico 04510 D.F., M´exico (Dated: August 6, 2018)The nuclear transition matrix elements M (0 ν ) for the neutrinoless double beta decay of , Zr, , Mo,
Ru,
Pd, , Te and
Nd isotopes in the case of 0 + → + transition are calculatedusing the PHFB wave functions, which are eigenvectors of four different parameterizations of aHamiltonian with pairing plus multipolar effective two-body interaction. Employing two (three)different parameterizations of Jastrow-type short range correlations, a set of eight (twelve) differentnuclear transition matrix elements M (0 ν ) is built for each decay, whose averages in conjunction withtheir standard deviations provide an estimate of the model uncertainties. PACS numbers: 21.60.Jz, 23.20.-g, 23.40.Hc
I. INTRODUCTION
Ascertaining the mass and nature of neutrinos requiresthe analysis of observational data obtained from threecomplementary experiments, namely single- β decay, neu-trino oscillation and neutrinoless double beta ( ββ ) ν de-cay. In any gauge theoretical model with spontaneoussymmetry breaking, the observation of ( ββ ) ν decay im-plies non-zero mass of Majorana neutrinos at the weakscale independent of the underlying mechanisms [1, 2].The varied scope and far reaching nature of experimen-tal as well as theoretical studies devoted to ( ββ ) ν decayover the past decades have been excellently reviewed byAvignone et al. [3] and references there in.The observed experimental limits on the half-life T ν / of ( ββ ) ν decay have already provided stringent limits onthe associated gauge theoretical parameters [4]. The reli-ability of extracted gauge theoretical parameters dependson the accuracy of nuclear transition matrix elements(NTMEs). For a given transition, different NTMEs areobtained employing distinct nuclear models, and for agiven model, they also depend on the model space andeffective two-body interaction selected. Other uncertain-ties are related with the inclusion of pseudoscalar andweak magnetism terms in the Fermi, Gamow-Teller andtensorial NTMEs [5, 6], finite size as well as short rangecorrelations [7–10], and the use of two effective values ofthe axial-vector coupling constant g A .The spread between the calculated NTMEs provides ameasure of the theoretical uncertainty [11]. In the caseof the well studied Ge isotope, it was observed thatthe calculated decay rates differ by a factor of 6–7. Theeffective neutrino mass h m ν i is inversely proportional tothe square root of T ν / . Hence, the uncertainty in theeffective neutrino mass is about 2 to 3. For example,from the experimental limit T ν / > . × yr [12],the upper limits on h m ν i range between 0.4 eV and 1.0eV, depending on the NTME [13–15]. If the ( ββ ) ν de- cay were observed in several nuclei, the comparison ofcalculated ratios of the corresponding NTMEs-squaredand the ratios of half-lives could also test the validityof nuclear structure calculations in a model independentway [16].Rodin et al. [17] have estimated the theoretical uncer-tainty employing two models, the QRPA and RQRPA,with three sets of basis states and three realistic two-bodyeffective interactions. Different strategies to remove thesensitivity of QRPA calculations on the model param-eters have been proposed [18, 19]. Further studies onuncertainties in NTMEs due to short range correlationsusing the unitary correlation operator method (UCOM)[7] and self-consistent coupled cluster method (CCM) [8]have been also carried out by Faessler and coworkers.Up to now, the QRPA model and its extensionshave been the most successful models in correlating thesingle- β GT strengths and half-lives of ( β − β − ) ν de-cay and the first in explaining the observed suppres-sion of M ν [20, 21]. Nonetheless, the large scale shellmodel (LSSM) calculations of Strasbourg-Madrid groupare quite promising [22]. Deformation has been includedat various levels of approximation in the QRPA for-malism [23–25]. Recently, the effects of pairing andquadrupolar correlations on the NTMEs of ( β − β − ) ν de-cay have also been studied in the interacting shell model(ISM) [9, 26] and the projected-Hartree-Fock-Bogoliubov(PHFB) model [27, 28].The PHFB model, in conjunction with pairing plusquadrupole-quadrupole ( P QQ ) [29] interaction has beensuccessful in the study of the 0 + → + transition of( β − β − ) ν decay, where it was possible to describe thelowest excited states of the parent and daughter nu-clei along with their electromagnetic transition strengths,as well as to reproduce their measured ββ decay rates[30, 31]. The PHFB model is unique in allowing the de-scription of the ββ decay in medium and heavy massnuclei by projecting a set of states with good angularmomentum, while treating the pairing and deformationdegrees of freedom simultaneously and on equal footing.On the other hand, in the present version of the PHFBmodel, the structure of the intermediate odd Z -odd N nuclei and hence, the single β decay rates and the distri-bution of GT strength can not be studied. Notwithstand-ing this limitation, it is a convenient choice to examinethe explicit role of deformation on the NTMEs. In thestudy of β − β − decay, there are four noteworthy obser-vations in connection with deformation effects [27, 28],namely:(i) There exists an inverse correlation between thequadrupole deformation and the size of NTMEs M ν , M (0 ν ) and M (0 ν ) N .(ii) The NTMEs are usually large in the absence ofquadrupolar correlations; they are almost constantfor small admixture of the QQ interaction and sub-stantially suppressed in deformed nuclei.(iii) In agreement with the observations made byˇSimkovic et al. [32], the NTMEs have a well de-fined maximum when the deformation of parent anddaughter nuclei are similar, and they are quite sup-pressed when the difference in the deformation islarge.(iv) The deformation effects are of equal importance incase of ( β − β − ) ν and ( β − β − ) ν decay.In earlier works, we have calculated NTMEs M ν forthe ( β − β − ) ν [30, 31] and M (0 ν ) for the ( β − β − ) ν de-cay [27] with the P QQ effective interaction [29], and theeffect of hexadecapolar correlations ( HH ) [28] on the cal-culated spectroscopic properties and ( β − β − ) decay rateshas been studied. In the present work, we employ twodifferent parameterizations of the QQ interaction, withand without the HH correlations. Further, the NTMEs M (0 ν ) are calculated with three different parametriza-tions of Jastrow type of SRC employing the four sets ofwave functions. The twelve NTMEs provide a reason-able sample for estimating the associated uncertainties.In Sec II, the PHFB formalism employed to describe the( β − β − ) ν decay with the inclusion of the finite size ofthe nucleons and short range correlations is shortly re-viewed. In Sec. III, the four different parameterizationsof the pairing plus multipole Hamiltonian are introduced,the calculated NTMEs vis-a-vis their radial evolution areanalyzed, and their average values as well as standarddeviations are estimated. Subsequently, the latter areemployed to obtain upper limits on the effective mass oflight Majorana neutrinos. Conclusions are given in Sec.IV. II. THEORETICAL FORMALISM
In the Majorana neutrino mass mechanism, the inversehalf-life of the ( β − β − ) ν decay due to the exchange of light neutrinos for the 0 + → + transition is given by[13, 33, 34] h T ν / (0 + → + ) i − = (cid:18) h m ν i m e (cid:19) G | M (0 ν ) GT − M (0 ν ) F | , (1)where the NTMEs M (0 ν ) k are given by M (0 ν ) k = X n,m (cid:10) + F (cid:13)(cid:13) O k,nm τ + n τ + m (cid:13)(cid:13) + I (cid:11) , (2)with O F = (cid:18) g V g A (cid:19) H ( r ) , O GT = σ · σ H ( r ) (3)and H ( r ) = Rφ ( Ar ) r . (4)The origin of the neutrino potential H ( r ) is due to theexchange of light Majorana neutrinos between nucleonsbeing considered as point particles. To take the finitesize of nucleons into account, neutrino potential H ( r )is folded with a dipole form factor and rewritten as H ( r ) = 4 πR (2 π ) Z d q exp ( i q · r ) q (cid:0) q + A (cid:1) (cid:18) Λ Λ + q (cid:19) , (5)where A = h E N i −
12 ( E I + E F ) . (6)and the cutoff momentum Λ= 850 MeV [27].The short range correlations (SRC) are producedby the repulsive nucleon-nucleon potential generatedthrough the exchange of ρ and ω mesons. They have beenincluded in the calculations of M (0 ν ) for the ( β − β − ) ν de-cay through the phenomenological Jastrow type of corre-lations with Miller-Spenser parametrization [35], effectiveoperators [36], exchange of ω -meson [37], UCOM [7, 38]and self-consistent CCM [8]. It has been observed thatthe effects due to the Jastrow type of correlations withMiller-Spenser parametrization are usually strong [36],where as the UCOM and self-consistent CCM have weakeffects. Further, ˇSimkovic et al. [8] have shown that it ispossible to parametrize the SRC effects of Argonne V18and CD-Bonn two nucleon potentials by the Jastrow typeof correlations within a few percent accuracy. Explicitly,the effects due to the SRC can be incorporated in thecalculation of M (0 ν ) through the prescription O k → f O k f, (7)with f ( r ) = 1 − ce − ar (1 − br ) , (8)where a = 1 .
1, 1 .
59 and 1 . f m − , b = 0 .
68, 1 .
45 and1 . f m − and c = 1 .
0, 0 .
92 and 0 .
46 for Miller-Spencer,Argonne V18 and CD-Bonn NN potentials, respectively.In the next section, the NTMEs M (0 ν ) are calculated inthe PHFB model by employing these three sets of param-eters for the SRC, denoted as SRC1, SRC2 and SRC3,respectively.The three functions f ( r ) are plotted in Fig. 1. They f (r) r(fm)SRC1SRC2SRC3 FIG. 1: Radial dependence of f ( r ) for the three different pa-rameterizations of the SRC. have similar forms, but differ in its value at the origin,and at the position of its maximum, which lies at 1.54,1.15 and 1.09 fm for SRC1, SRC2 and SRC3, respectively.They have clear influence on the radial evolution of the( β − β − ) ν decay matrix elements discussed below.The calculation of M (0 ν ) in the PHFB model has beendiscussed in Ref. [27] and one obtains the following ex-pression for NTMEs M (0 ν ) k of ( β − β − ) ν decay M (0 ν ) k = (cid:2) n Ji =0 n J f =0 (cid:3) − / × π Z n ( Z,N ) , ( Z +2 ,N − ( θ ) X αβγδ ( αβ | O k | γδ ) × X εη (cid:16) f ( π ) ∗ Z +2 ,N − (cid:17) εβ h(cid:16) F ( π ) Z,N ( θ ) f ( π ) ∗ Z +2 ,N − (cid:17)i εα × (cid:16) F ( ν ) ∗ Z,N (cid:17) ηδ h(cid:16) F ( ν ) Z,N ( θ ) f ( ν ) ∗ Z +2 ,N − (cid:17)i γη sinθdθ, (9)where n J = π Z h det (cid:16) F ( π ) f ( π ) † (cid:17)i / × h det (cid:16) F ( ν ) f ( ν ) † (cid:17)i / d J ( θ ) sin ( θ ) dθ, (10) and n ( Z,N ) , ( Z +2 ,N − ( θ ) = h det (cid:16) F ( ν ) Z,N f ( ν ) † Z +2 ,N − (cid:17)i / × h det (cid:16) F ( π ) Z,N f ( π ) † Z +2 ,N − (cid:17)i / . (11)The π ( ν ) represents the proton (neutron) of nuclei in-volved in the ( β − β − ) ν decay process. The matrices f Z,N and F Z,N ( θ ) are given by f Z,N = X i C ij α ,m α C ij β ,m β ( v im α /u im α ) δ m α , − m β , (12) F Z,N ( θ ) = X m ′ α m ′ β d j α m α ,m ′ α ( θ ) d j β m β ,m ′ β ( θ ) f j α m ′ α ,j β m ′ β . (13)The extra factor 1/4 in the Eq. (28) of Ref. [27] shouldnot be there. III. RESULTS AND DISCUSSIONS
The model space, single particle energies (SPE’s), pa-rameters of the
P QQ type of effective two-body inter-action and the method to fix them have been alreadygiven in Refs. [27, 30, 31]. Presently, we use the effectiveHamiltonian written as [28] H = H sp + V ( P ) + V ( QQ ) + V ( HH ) , (14)where H sp , V ( P ), V ( QQ ) and V ( HH ) denote thesingle particle Hamiltonian, the pairing, quadrupole-quadrupole and hexadecapole-hexadecapole parts ofthe effective two-body interaction, respectively. Thequadrupole-quadrupole part of the effective two-body in-teraction V ( QQ ) has three terms, namely the proton-proton, the neutron-neutron and the proton-neutronones, whose coefficients are denoted by χ pp , χ nn and χ pn , respectively. In Refs. [27, 30, 31], the strengthsof the like particle components of the QQ interactionwere taken as χ pp = χ nn = 0 . b − , where b is the oscillator parameter. The strength of proton-neutron component of the QQ interaction χ pn was var-ied so as to fit the experimental excitation energy of the2 + state, E + . In the present work, we also employ analternative isoscalar parametrization of the quadrupole-quadrupole interaction, by taking χ pp = χ nn = χ pn / E + . We will refer to these two parameterizationsof the quadrupole-quadrupole interaction as P QQ
P QQ + state E + [39] can be reproducedwithin about 2% accuracy. The maximum change in E + and E + energies with respect to P QQ B ( E + → + ) transition probabilities, defor-mation parameters β , static quadrupole moments Q (2 + )and gyromagnetic factors g (2 + ) are in an overall agree-ment with the experimental data [40, 41] for both theparametrizations. In the case of P QQ M ν for the 0 + → + transition with respect to P QQ Zr isotope.The HH part of the effective interaction V ( HH ) isgiven as [28] V ( HH ) = − (cid:16) χ (cid:17) X αβγδ X ν ( − ν h α | q ν | γ i×h β | q − ν | δ i a † α a † β a δ a γ , (15)with q ν = r Y ν ( θ, φ ). The relative magnitudes of theparameters of the HH part of the two body interactionare calculated from a relation suggested by Bohr andMottelson [42]. The approximate magnitude of these con-stants for isospin T = 0 is given by χ λ = 4 π λ + 1 mω A h r λ − i f or λ = 1 , , , · ·· (16)and the parameters for the T = 1 case are approximatelyhalf of their T = 0 counterparts. Presently, the value of χ = 0 . χ A − / b − for T = 1, which is exactly halfof the T = 0 case.We refer to the calculations which include the hex-adecapolar term HH as P QQHH . We end up withfour different parameterizations of the effective two-body interaction, namely
P QQ P QQHH P QQ
P QQHH A. SRC and radial evolutions of NTMEs
In Table I, the NTMEs M (0 ν ) evaluated usingthe HFB wave functions in conjunction with P QQ P QQHH P QQ P QQHH , Zr, , Mo,
Ru,
Pd, , Te and
Nd are displayed. The average energy denominator A has been taken as A = 1 . A / MeV following Hax-ton’s prescription [13]. The NTMEs are calculated in thethe approximations of point nucleons (P - 2nd and 3rdcolumns), finite size of nucleons (F - 7th column), pointnucleons with SRC (P+S - 4th to 6th columns), and fi-nite size plus SRC (F+S - last three columns). To obtainadditional information on the stability of the estimationsof NTMEs M (0 ν ) , they are also calculated for A/ M (0 ν ) (in%) due to the different approximations in Table II. Ineach row, i.e. for each set of wave functions, the referenceNTMEs M (0 ν ) are those calculated for point nucleonswithout SRC, given in the second column of Table I.It can be observed that the relative change in NTMEs M (0 ν ) , when the energy denominator is taken as A/ A , is of the order of 10 %. It confirms thatthe dependence of NTMEs on average excitation energy A is small for the ( β − β − ) ν decay and the validity of theclosure approximation is quite satisfactory.The variation in M (0 ν ) due to the different parameter-izations of the Hamiltonians (presented in the differentrows) lies between 20–25%. It is noticed in general butfor Te isotope that the NTMEs evaluated for both pa-rameterizations of the quadrupolar interaction are quiteclose. The inclusion of the hexadecapolar term tends toreduce them by amounts which strongly depend on thespecific nuclei.The inclusion of SRC in the approximation of point nu-cleons (P+S) induces an extra quenching in the NTMEs M (0 ν ) , which can be of the order of 18–23% for SRC1, tonegligible for SRC3. The dipole form factor (F) alwaysreduces the NTMEs by 12–15% in comparision to thepoint particle case. Adding SRC (F+S) can further re-duce the transition matrix elements, for SRC1, or slightlyenhance them, partially compensating the effect of thedipole form factor. It is interesting to note that the ef-fect of F-SRC2 is almost negligible, i.e., nearly the sameas F.The radial evolution of M (0 ν ) has been studied in theQRPA by ˇSimkovic et al. [7] and in the ISM by Men´endez et al. [43] by defining M (0 ν ) = Z C (0 ν ) ( r ) dr. (17)In both QRPA and ISM calculations, it has been estab-lished that the contributions of decaying pairs coupled to J = 0 and J > r ≈ C (0 ν ) for all nuclei undergoing ( β − β − ) ν decay are the maximum about the internucleon distance r ≈ C (0 ν ) due to P QQ Zr,
Mo,
Pd, , Te and
Nd. The radial evolution of M (0 ν ) is studied for eight cases, namely P, P+SRC1, P+SRC2,P+SRC3, F, F+SRC1, F+SRC2 and F+SRC3. In addi-tion, the effects due to the finite size and SRC are mademore transparent in Fig. 3 by plotting them for differentcombinations of P, F and SRC. In case of point nucle-ons, it is noticed that the C (0 ν ) are peaked at r = 1 . C (0 ν ) are increased forSRC2 and SRC3 with unchanging peak position. In thecase of FNS, the C (0 ν ) are peaked at r = 1 .
25 fm, whichremains unchanged with the inclusion of SRC1, SRC2and SRC3. However, the magnitudes of C (0 ν ) change inthe latter three cases. The above observations also re-main valid with the other three parametrizations of theeffective two-body interaction. TABLE I: Calculated NTMEs M (0 ν ) in the PHFB model with four different parameterizations of the effective two-bodyinteraction, three different parameterizations of SRC, with nucleons taken as point particles (P) or with a dipole form factor(F), for the (cid:0) β − β − (cid:1) ν decay of , Zr, , Mo,
Ru,
Pd, , Te and
Nd isotopes.Nuclei M (0 ν ) P P+S F F+S
A A/ Zr P QQ
P QQHH
P QQ
P QQHH Zr P QQ
P QQHH
P QQ
P QQHH Mo P QQ
P QQHH
P QQ
P QQHH Mo P QQ
P QQHH
P QQ
P QQHH Ru P QQ
P QQHH
P QQ
P QQHH Pd P QQ
P QQHH
P QQ
P QQHH Te P QQ
P QQHH
P QQ
P QQHH Te P QQ
P QQHH
P QQ
P QQHH Nd P QQ
P QQHH
P QQ
P QQHH
B. Uncertainties in NTMEs
To estimate the uncertainties associated with theNTMEs M (0 ν ) for ( β − β − ) ν decay calculated using thePHFB model, we evaluate their mean and the standarddeviation, defined as M (0 ν ) = P Ni =1 M (0 ν ) i N (18) and∆ M (0 ν ) = 1 √ N − " N X i =1 (cid:16) M (0 ν ) − M (0 ν ) i (cid:17) / . (19)Recently, it has been shown by ˇSimkovic et al. [8] thatthe phenomenological Jastrow correlations with Miller-Spenser parametrization is a major source of uncertainty.Therefore, it is more appropriate to consider SRC2 or TABLE II: Maximum and minimum relative change in the NTME M (0 ν ) (in %), for all nuclei included in table I, due to theuse of a different average energy denominator ( second column), the inclusion of three different parameterizations of the SRC(SRC1, SRC2 and SRC3) with point nucleons (third to fifth column), the inclusion of finite size effect (F) (sixth column) andfinite size effect plus SRC (F+SRC1, F+SRC2 and F+SRC3 in last three columns). In each row, the results employing one ofthe four different parameterizations of the effective two-body interaction are displayed.Parametrizatios A/ P QQ
P QQHH
P QQ
P QQHH M (0 ν ) and uncertainties ∆ M (0 ν ) for the (cid:0) β − β − (cid:1) ν decay of , Zr, , Mo,
Pd, , Teand
Nd isotopes. Both bare and quenched values of g A areconsidered. β − β − g A Case I Case IIemitters M (0 ν ) ∆ M (0 ν ) M (0 ν ) ∆ M (0 ν )94 Zr 1.254 4.2464 0.3883 4.4542 0.25361.0 4.6382 0.4246 4.8668 0.2759 Zr 1.254 3.1461 0.2778 3.3181 0.12431.0 3.4481 0.3085 3.6376 0.1424 Mo 1.254 7.1294 0.6013 7.4656 0.36351.0 7.8398 0.6826 8.2099 0.4358
Mo 1.254 6.8749 0.6855 7.2163 0.49771.0 7.5660 0.7744 7.9419 0.5769
Pd 1.254 7.8413 0.8124 8.2273 0.61671.0 8.6120 0.9184 9.0370 0.7128
Te 1.254 4.0094 0.4194 4.2175 0.30741.0 4.4281 0.4601 4.6571 0.3355
Te 1.254 4.4458 0.5231 4.6633 0.42691.0 4.9065 0.5837 5.1459 0.4802
Nd 1.254 3.1048 0.4649 3.2431 0.44341.0 3.4334 0.5181 3.5856 0.4952
SRC3 due to the Argonne V18 and CD-Bonn NN po-tentials, respectively. Based on these observations, weperform the statistical analysis of two cases. In case I,we calculate the average and variance of twelve NTMEslisted in the last three columns (F+S) of Table I withthe bare and quenched values of axial vector couplingconstant g A = 1 .
254 and g A = 1 .
0, respectively. Theaverage and standard deviations of eight NTMES M (0 ν ) due to SRC2 and SRC3 are similarly calculated in thecase II. The average NTMEs M (0 ν ) and standard de-viations ∆ M (0 ν ) are presented in Table III. It is no-ticed that the exclusion of Miller-Spenser parametriza-tion reduces the uncertainty by about 55% in Zr to 4% in
Nd isotope. In Table IV, we present the av-erage NTMEs M (0 ν ) of case II along with the recentlyreported results in ISM by Caurier et al. [9], QRPA aswell as RQRPA by ˇSimkovic et al. [8] and IBM by Bareaand Iachello [44]. In spite of the fact that different modelspace, two-body interactions and SRC have been used inthese models, the spread in the NTMEs turns out to beabout a factor of 2.5. Further, we extract upper limitson the effective mass of light neutrinos h m ν i from thelargest observed limits on half-lives T ν / of ( β − β − ) ν de-cay using the phase space factors of Boehm and Vogel[45]. It is observed that the extracted limits on h m ν i for Mo and
Te nuclei are 0 . +0 . − . − . +0 . − . and0 . +0 . − . − . +0 . − . eV, respectively. In the last columnof Table IV, the predicted half-lives of ( β − β − ) ν decayof , Zr, , Mo,
Pd, , Te and
Nd isotopesare given for h m ν i = 50 meV. IV. CONCLUSIONS
We have studied the ( β − β − ) ν decay of , Zr, , Mo,
Ru,
Pd, , Te and
Nd isotopes inthe light Majorana neutrino mass mechanism using aset of PHFB wave functions. The reliability of wavefunctions generated with
P QQ
P QQHH B ( E + → + ) transition proba-bilities, static quadrupole moments Q (2 + ) and g -factors g (2 + ) of participating nuclei in ( β − β − ) ν decay as wellas M ν and comparing them with the available experi-mental data [30, 31]. An overall agreement between thecalculated and observed spectroscopic properties as wellas M ν suggests that the PHFB wave functions generatedby fixing χ pn or χ pp to reproduce the E + are reasonablyreliable.In the present work, NTMEs M (0 ν ) were calculatedemploying the PHFB model with four different parame-terizations of the pairing plus multipolar type of effectivetwo body interaction and two(three) different parameter-izations of the short range correlations. It was found thatthe NTMEs M (0 ν ) change by about 4–14(10–15)%.The mean and standard deviations were evaluatedfor the NTMEs M (0 ν ) calculated with dipole form fac- nnnn ) ( f m - ) A=96A=100A=110A=128A=130 (a) C ( nnnn r ( fm ) A=130A=150 nnnn ) ( f m - ) (b) C ( nnnn r ( fm ) nnnn ) ( f m - ) (c) C ( nnnn r ( fm ) nnnn ) ( f m - ) (d) C ( nnnn r ( fm ) nnnn ) ( f m - ) (e) C ( nnnn r ( fm ) nnnn ) ( f m - ) (f) C ( nnnn r ( fm ) nnnn ) ( f m - ) (g) C ( nnnn r ( fm ) nnnn ) ( f m - ) (h) C ( nnnn r ( fm ) FIG. 2: Radial dependence of C (0 ν ) ( r ) for the (cid:0) β − β − (cid:1) ν decay of Zr,
Mo,
Pd, , Te and
Nd isotopes. In thisFig., (a), (b), (c) and (d) correspond to P, P+SRC1, P+SRC2 and P+SRC3, respectively. Further, (e), (f), (g) and (h) are forF, F+SRC1, F+SRC2 and F+SRC3, respectively.
TABLE IV: Extracted limits on effective light Majorana neutrino mass h m ν i and predicted half lives using average NTMEs M (0 ν ) and uncertainties ∆ M (0 ν ) for the (cid:0) β − β − (cid:1) ν decay of , Zr, , Mo,
Pd, , Te and
Nd isotopes. β − β − g A M (0 ν ) ISM (R)QRPA IBM G T ν / ( yr ) Ref. h m ν i T ν / (y)emitters [9] [8] [44] (10 − y − ) < m ν > = 50 meV Zr 1.254 4.45 ± × [46] 6.41 +0 . − . × +0 . − . × ± +0 . − . × +0 . − . × Zr 1.254 3.32 ± × [46] 20.00 +0 . − . +0 . − . × ± +1 . − . +0 . − . × Mo 1.254 7.47 ± × [47] 1.62 +0 . − . × +0 . − . × ± +0 . − . × +0 . − . × Mo 1.254 7.22 ± × [48] 0.48 +0 . − . +0 . − . × ± +0 . − . +1 . − . × Pd 1.254 8.23 ± × [49] 6.72 +0 . − . × +0 . − . × ± +0 . − . × +0 . − . × Te 1.254 4.22 ± × [50] 8.50 +0 . − . +0 . − . × ± +0 . − . +1 . − . × Te 1.254 4.66 ± × [51] 0.30 +0 . − . +0 . − . × ± +0 . − . +0 . − . × Nd 1.254 3.24 ± × [52] 2.55 +0 . − . +1 . − . × ± +0 . − . +3 . − . × tor and with and without Miller-Spencer parametriza-tion of short range correlations, which were employedto estimate the ( β − β − ) ν decay half-lives T ν / for both g A = 1 .
254 and g A = 1 .
0. The largest standard devia-tion, interpreted as theoretical uncertainty, turns out tobe of the order of 15% in the case of
Nd isotope. Wehave also extracted limits on the effective mass of lightMajorana neutrinos h m ν i from the available limits on ex-perimental half-lives T ν / using average NTMEs M (0 ν ) calculated in the PHFB model. In the case of Te iso-tope, one obtains the best limit on the effective neutrinomass h m ν i < . +0 . − . − . +0 . − . eV from the observedlimit on the half-lives T ν / > . × yr of ( β − β − ) ν decay [51]. Note:
Due to an error in one equation, the NTMEs M F , M GT , M (0 ν ) , M F h , M GT h and M (0 ν ) N given in Ref.[27] must be multiplied by a factor of 2. It implies thatthe limits on the effective light neutrino mass < m ν > must be reduced by a factor of 2 whereas the limits oneffective heavy neutrino mass < M N > must be multipliedby a factor of 2. In both cases the limits are twice morestringent. Acknowledgments
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