Uncertainties in nuclear transition matrix elements for neutrinoless $ββ$ decay II: the heavy Majorana neutrino mass mechanism

0.38 MB·English

by  P. K. Rath

Uncertainties in nuclear transition matrix elements for neutrinoless ββ decay II: the
heavy Majorana neutrino mass mechanism
P. K. Rath1, R. Chandra2,3, P. K. Raina3,4, K. Chaturvedi5, and J. G. Hirsch6
1Department of Physics, University of Lucknow, Lucknow-226007, India
2Department of Applied Physics, Babasaheb Bhimrao Ambedkar University, Lucknow-226025, India
3Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur-721302, India
4Department of Physics, Indian Institute of Technology, Ropar, Rupnagar - 140001, Punjab, India
5Department of Physics, Bundelkhand University, Jhansi-284128, India
6Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de M´exico, 04510 M´exico, D.F., M´exico
(Dated: January 28, 2013)
2
Employing four different parametrizations of the pairing plus multipolar type of effective two-
1
body interaction and three different parametrizations of Jastrow-type of short range correlations,
0
the uncertainties in the nuclear transition matrix elements M(0ν) due to the exchange of heavy
2 N
Majorana neutrino for the 0+ → 0+ transition of neutrinoless double beta decay of 94Zr, 96Zr,

n 98Mo, 100Mo, 104Ru, 110Pd, 128,130Te and 150Nd isotopes in the PHFB model are estimated to
a be around 35%. Excluding the nuclear transition matrix elements calculated with Miller-Spenser
J
parametrization of Jastrow short range correlations, theuncertainties are found tobe smaller than

6 20%.
1
PACSnumbers: 21.60.-n,23.40.-s,23.40.Hc

]
h
t
- I. INTRODUCTION ments(NTMEs)inthe QRPAandlimits onthe effective
l
c light neutrino mass mν , heavy neutrino mass MN ,
h i h i
u right handed heavy neutrino M , λ , η and mix-
n In addition to establishing the Dirac or Majorana na- ing angle tanξ have been obtaihnedR.iThheihehaviy neutrino
ture of neutrinos, the observation of (ββ) decay is a
[ 0ν mechanism has also been studied in the QRPA without
convenient tool to test the lepton number conservation,
2 [9]andwithpn-pairing[10]. IntheheavyMajorananeu-
possible hierarchies in the neutrino mass spectrum, the
v trino mass mechanism, Sˇimkovic et al. [11] have studied
origin of neutrino mass and CP violation in the leptonic
0 the role of induced weak magnetism and pseudoscalar
sector. Further, it can also ascertain the role of various
6 terms and it was found that they are quite important
5 gauge models associated with all possible mechanisms, in 48Ca nucleus. The importance of the same induced
1 namely the exchange of light neutrinos, heavy neutri-
currents in both light and heavy Majorana neutrino ex-
. nos, the right handed currents in the left-right symmet-
6 change mechanism has also been studied using the pn-
ric model (LRSM), the exchange of sleptons, neutrali-
0 RQRPA [12] as well as SRQRPA [3].
1 nos, squarks and gluinos in the Rp-violating minimal
1 super symmetric standard model, the exchange of lep- In spite of the remarkable success of the large scale
: toquarks, existence of heavy sterile neutrinos, compos- shell model (LSSM) calculations of Strassbourg-Madrid
v
i iteness,extradimensionalscenariosandMajoronmodels, group[13],thereisanecessityoflargeconfigurationmix-
X allowingtheoccurrenceof(ββ) decay. Stringentlimits ing to reproduce the structural complexity of medium
0ν
r ontheassociatedparametershavealreadybeenextracted and heavy mass nuclei. On the other hand, the QRPA
a
from the observed experimental limits on the half-life of and its extensions have emerged as successful models by
(β−β−) decay [1] and presently, all the experimental including a large number of basis states and in correlat-
0ν
attempts are directed for its observation. The experi- ingthe single-β GT strengths andhalf-livesof(β−β−)2ν
mental and theoretical studies devoted to (ββ) decay decayin additionto explaining the observedsuppression
0ν
over the past decades have been recently reviewed by of M2ν [14, 15]. In the mass region 90 A 150, there
≤ ≤
Avignone et al. [2] and references there in. is a subtle interplay of pairing and quadrupolar correla-
tionsandtheir effects onthe NTMEs of(β−β−) decay
Presently,thereisanincreasedinteresttocalculatere- 0ν
liable NTMEs for (β−β−) decay due to the exchange have been studied in the interacting shell model (ISM)
0ν [16, 17], deformed QRPA model [18–21], and projected-
of heavy Majorana neutrinos, in order to ascertain the
Hartree-Fock-Bogoliubov(PHFB) model [22, 23].
dominant mechanism contributing to it [3, 4]. The lep-
ton number violating (β−β−) decay has been studied The possibility to constrain the values of the gauge
0ν
by Vergados by taking a Lagrangian consisting of left- parameters using the measured lower limits on the
handed as well as right-handed leptonic currents [5]. In (β−β−) decayhalf-livesreliesheavilyonthemodelde-
0ν
the QRPA, the (β−β−) decay due to the exchange of pendent NTMEs. Different predictions are obtained by
0ν
heavy Majorana neutrinos has been studied by Tomoda employing different nuclear models, and within a given
[6]. The decay rate of (β−β−) mode in the LRSM has model, varying the model space, single particle energies
0ν
beenderivedbyDoiandKotani[7]. Hirschetal.[8]have (SPEs) and effective two-body interaction. In addition,
calculated all the required nuclear transition matrix ele- a number of issues regarding the structure of NTMEs,
2
namely the effect of pseudoscalar and weak magnetism neutrino, calculated in the PHFB model with four dif-
termsontheFermi,Gamow-TellerandtensorialNTMEs ferent parameterizations of pairing plus multipolar type
[24,25],theroleoffinitesizeofnucleons(FNS)aswellas ofeffectivetwo-bodyinteraction[23]andtwo(three)dif-
short range correlations (SRC) vis-a-vis the radial evo- ferent parametrization of Jastrow type of SRC [27]. In
lution of NTMEs [16, 26–28] and the value of the axial- confirmation with the observation made by Sˇimkovic et
vector coupling constant g are also the sources of un- al. [27], it was noticed that the Miller-Spenser type of
A
certainties and remain to be investigated. parametrization is a major source of uncertainty and
Itwasobservedby Vogel[29]thatincaseofwellstud- its exclusion reduces the uncertainties from 10%–15%
ied76Ge,thecalculateddecayratesT0ν differbyafactor to 4%–14%. Presently, the same procedure has been
1/2
of 6-7 and consequently, the uncertainty in the effective adopted to estimate the theoretical uncertainties asso-
neutrinomass m isabout2to3. Thus,thespreadbe- ciated with the NTMEs M(0ν) for (β−β−) decay due
h νi N 0ν
tweenthecalculatedNTMEscanbeusedasthemeasure to the exchange of heavy Majorana neutrino. In Sec.
ofthetheoreticaluncertainty. Incasethe(ββ) decayof II, a brief discussion of the theoretical formalism is pre-
0ν
differentnucleiwillbe observed,BilenkyandGrifols[30] sented. The resultsfordifferentparameterizationsofthe
havesuggestedthattheresultsofcalculationsofNTMEs two-body interaction and SRC vis-a-vis radial evolution
of the (β−β−) decay canbe checkedby comparing the of NTMEs are discussed in Sec III. In the same section,
0ν
calculated ratios of the corresponding NTMEs-squared theaveragesaswellasstandarddeviationsarecalculated
with the experimentally observed values. for estimating the theoretical uncertainties. Finally, the
Bahcall et al. [31] and Avignone et al. [32] have cal- conclusions are given in Sec. IV.
culated averages of all the available NTMEs, and their
standarddeviationis takenasthe measureoftheoretical
uncertainty. On the other hand, Rodin et al. [33] have II. THEORETICAL FORMALISM
calculatednineNTMEswiththreesetsofbasisstatesand
three realistic two-body effective interactions of charge In the charged current weak processes, the current-
dependent Bonn, Argonne and Nijmen potentials in the current interaction under the assumption of zero mass
QRPA as well as RQRPA and estimated the theoretical neutrinos leads to terms which, except for vector and
uncertainties by making a statisticalanalysis. It wasno- axial vector parts, are proportional to the lepton mass
ticedthatthevariancesaresubstantiallysmallerthanthe squared, and hence negligible. However, it has been re-
average values and the results of QRPA, albeit slightly portedbySˇimkovicetal.[24,25]thatthecontributionof
larger,arequiteclosetotheRQRPAvalues. Faesslerand the pseudoscalar term is equivalent to a modification of
coworkers have further studied uncertainties in NTMEs the axial vector current due to PCAC and greater than
due to short range correlations using unitary correlation the vector current. The contributions of pseudoscalar
operator method (UCOM) [26] and self-consistent cou- and weak magnetism terms in the mass mechanism can
pled cluster method (CCM) [27]. change M(0ν) upto 30% and the change in M(0ν) is con-
The PHFB model has the advantage of treating the N
siderably larger. In the shell-model [16, 38], IBM [39]
pairinganddeformationdegreesoffreedomonequalfoot-
andGCM+PNAMP[40],thecontributionsofthesepseu-
ing andprojectingout states withgoodangularmomen- doscalar and weak magnetism terms to M(0ν) have been
tum. However, the single β decay rates and the distri-
alsoinvestigated. However,ithasbeenreportedbySuho-
bution of GT strength, which require the structure of
nen and Civitarese [41] that these contributions are rel-
the intermediate odd Z-odd N nuclei, can not be stud-
atively smallandcan be safely neglected. Therefore,the
ied in the present version of the PHFB model. In spite
investigation of this issue is of definite interest and is
of this limitation, the PHFB model in conjunction with
reported in the present work.
pairingplusquadrupole-quadrupole(PQQ)[34]hasbeen In the two nucleon mechanism, the half-life T0ν for
successfully applied to reproduce the lowestyraststates, 1/2
the 0+ 0+ transition of (β−β−) decay due to the
electromagnetic properties of the parent and daughter → 0ν
exchange of heavy Majorana neutrino between nucleons
nuclei, and the measured (β−β−) decay rates [35, 36].
2ν having finite size is given by [6, 7]
InthePHFBformalism,theexistenceofaninversecorre-
lation between the quadrupole deformation and the size
[o2f2,N2T3M].EFsuMrth2eνr,,Mit(h0νa)s abneednMnoN(t0iνc)edhatshabteethneoNbsTeMrvEeds T10/ν2 0+ →0+ −1 = Mmp 2G01 MN(0ν) 2, (1)
astraenutsfuoarllsymlaalrlgdeeffoorrmapataiiorno,fssupphperreicssaeldnudcelpeei,nadlimngosotncothne- h (cid:0) (cid:1)i (cid:18)h Ni(cid:19) (cid:12)(cid:12) (cid:12)(cid:12)
where m is the proton mass and (cid:12) (cid:12)
differenceinthedeformation∆β ofparentanddaughter p
2
nucleiandhavingawelldefinedmaximumwhen∆β =0
2 M −1 = U2m−1, m >1 GeV, (2)
[22, 23]. h Ni i ei i i
In Ref. [37], a statistical analysis was performed for X
(0ν)
extracting uncertainties in eight (twelve) NTMEs for and in the closure approximation, the NTMEs M is
N
(β−β−) decay due to the exchange of light Majorana of the form [12, 26, 27]
0ν
3
The exchange of heavy Majorana neutrinos gives rise
to short ranged neutrino potentials, which with the con-
MN(0ν) =−MFh+MGTh+MTh, (3) sideration of FNS are given by
where
Mα = 0+F Oα,nmτn+τm+ 0+I (4) H (r ) = 2R f (qr )h (q)q2dq (8)
Xn,m(cid:10) (cid:13)(cid:13) (cid:13)(cid:13) (cid:11) αh nm (mpme)π Z αh nm α
with
O = H (r ) (5)
Fh Fh nm
where f (qr ) = j (qr ) for α = F as well as GT
O = σ σ H (r ) (6) αh nm 0 nm
OGTh = [3n(·σ mr GT)h(σnmr ) σ σ ]H (r ) and fTh(qrnm)=j2(qrnm).
Th n nm m nm n m GTh nm
· · − ·
(7) Further, the h (q), h (q) and h (q) are written as
F GT T
b b
g 2 Λ2 4
h (q) = V V (9)
F g q2+Λ2
(cid:18) A(cid:19) (cid:18) V (cid:19)
g2(q2) 2 g (q2)q2 1 g2(q2)q4 2g2 (q2)q2
h (q) = A 1 P + P + M
GT g2 − 3g (q2)2m 3g2(q2)4m2 3 g24m2
A (cid:20) A p A p(cid:21) A p
Λ2 4 2 q2 1 q4 g 2 κ2q2 Λ2 4
A 1 + + V V (10)
≈ (cid:18)q2+Λ2A(cid:19) " − 3(q2+m2π) 3(q2+m2π)2# (cid:18)gA(cid:19) 6m2p (cid:18)q2+Λ2V (cid:19)
g2(q2) 2 g (q2)q2 1 g2(q2)q4 1g2 (q2)q2
h (q) = A P P + M
T g2 3g (q2)2m − 3g2(q2)4m2 3 g24m2
A (cid:20) A p A p(cid:21) A p
Λ2 4 2 q2 1 q4 g 2 κ2q2 Λ2 4
A + V V (11)
≈ (cid:18)q2+Λ2A(cid:19) "3(q2+m2π) − 3(q2+m2π)2# (cid:18)gA(cid:19) 12m2p (cid:18)q2+Λ2V (cid:19)
where the form factors are given by usingeffectivetransitionoperator[42],theexchangeofω-
meson [43], UCOM [26, 44] and the self-consistent CCM
Λ2 2
g (q2) = g A [27]. The SRC can also be incorporated phenomenologi-
A A q2+Λ2 callyby Jastrowtype ofcorrelationswithMiller-Spenser
(cid:18) A(cid:19)
Λ2 2 parametrization [45]. Further, it has been shown in the
g (q2) = κg V self-consistentCMM[27]thattheSRCeffectsofArgonne
M V q2+Λ2
(cid:18) V (cid:19) and CD-Bonn two nucleon potentials are weak and it is
g (q2) = 2mpgA(q2) Λ2A−m2π (12) possible to parametrizethem by Jastrowtype of correla-
P (q2+m2) Λ2 tions within a few percent accuracy. Explicitly,
π (cid:18) A (cid:19)
with g = 1.0, g = 1.254, κ = µ µ = 3.70, Λ =
V A p n V
0.850 GeV, ΛA =1.086 GeV and mπ−is the pion mass. f(r)=1 ce−ar2(1 br2) (13)
Substituting Eq. (5)–Eq. (11) in Eq. (3), there is − −
one term, associated with h , Eq. (9), contributing to where a = 1.1, 1.59 and 1.52 fm−2, b = 0.68, 1.45 and
F
M , while M has four terms, denoted by M , 1.88 fm−2 and c=1.0, 0.92 and 0.46 for Miller-Spencer
Fh GTh GT−AA
M ,M andM ,whichcorrespondto parametrization,CD-Bonnand Argonne V18 NN poten-
GT−AP GT−PP GT−MM
thefourtermsinh ,Eq. (10). Thetensorcontribution, tials, respectively. In this work the NTMEs M(0ν) are
GT N
M , has three terms, denoted by M , M and calculated in the PHFB model for the above mentioned
Th T−AP T−PP
M , which correspond to the three terms in h , three sets of parameters for the SRC, denoted as SRC1,
T−MM T
Eq. (11). Theircontributionstothetotalnuclearmatrix SRC2 and SRC3, respectively.
element are discussed in Sec. III. In Fig.1, we plot the neutrino potential H (r,Λ)=
N
The short range correlations (SRC) arise mainly from H (r,Λ)f(r) with the three different parametrizations
Fh
the repulsive nucleon-nucleon potential due to the ex- ofSRC.Itisnoticed,thatthepotentialsduetoFNSand
changeofρandω mesonsandhavebeenincorporatedby FNS+SRC3 are peaked at the origin where as the peaks
4
duetoFNS+SRC1andFNS+SRC3areatr 0.6fmand
450 ≈
r 0.5 fm, respectively. The shapes of these functions
400 F ha≈vedefinite influence ontheradialevolutionofNTMEs
350 F+SRC1 M(0ν) for (β−β−) decay due to the exchange of heavy
N 0ν
Majorana neutrino as discussed in Sec. III.
300 F+SRC2
Λ) 250 F+SRC3
r,
(
N 200
H
150
100
50
0
0 1 2 3 4
r(fm)
ThecalculationofM(0ν) inthePHFBmodelhasbeen
N
FIG. 1: Radial dependence of H (r,Λ)= H (r,Λ)f(r) for discussed in our earlier work [22, 37] and one obtains
N Fh
the threedifferent parameterizations of the SRC. In thecase the following expression for NTMEs M(0ν) of (β−β−)
of FNS, f(r)=1. α 0ν
decay [37]
π
M(0ν) = nJi=0nJf=0 −1/2 n (θ) (αβ O γδ)
α (Z,N),(Z+2,N−2) | α|
(cid:2) (cid:3) Z0 αXβγδ
f(π)∗ F(ν)∗
Z+2,N−2 Z,N
εβ ηδ
sinθdθ (14)
(cid:16) (cid:17) (cid:16) (cid:17)
× 1+F(π)(θ)f(π)∗ 1+F(ν) (θ)f(ν)∗
Xεη Z,N Z+2,N−2 εα Z,N Z+2,N−2 γη
h(cid:16) (cid:17)i h(cid:16) (cid:17)i
and the expressions for calculating nJ, Hamiltonian given by [23]
n (θ), f and F (θ) are given in
(Z,N),(Z+2,N−2) Z,N Z,N
Refs. [22, 37]. H =H +V(P)+V(QQ)+V(HH) (15)
sp
The calculation of matrices f and F (θ) re-
Z,N Z,N
quires the amplitudes (u ,v ) and expansion coeffi- where H , V(P), V(QQ) and V(HH) denote the
im im sp
cients C , which specify the axially symmetric HFB single particle Hamiltonian, the pairing, quadrupole-
ij,m
intrinsic state Φ with K = 0. Presently, they are quadrupole and hexadecapole-hexadecapole part of the
0
| i
obtained by carrying out the HFB calculations through effective two-body interaction, respectively. The HH
theminimizationoftheexpectationvalueoftheeffective partofthe effectiveinteractionV(HH)iswrittenas[23]
χ
V(HH)= 4 ( 1)ν αr4Y (θ,φ)γ β r4Y (θ,φ)δ a†a† a a (16)
− 2 − h | 4,ν | ih | 4,−ν | i α β δ γ
(cid:16) (cid:17)αXβγδXν
with χ = 0.2442 χ A−2/3b−4 for T = 1, and twice of citation energy of the 2+ state E for the consid-
4 2 2+
this value for T = 0 case, following Bohr and Mottelson ered nuclei, namely 94,96Zr, 94,96,98,100Mo, 98,100,104Ru,
[46]. 104,110Pd, 110Cd, 128,130Te, 128,130Xe, 150Nd and 150Sm
ascloselypossibletotheexperimentalvalues. Thisisde-
In Refs. [22, 35, 36], the strengths of the like par-
noted as PQQ1 parametrization. Alternatively, one can
ticle components χ and χ of the QQ interaction
pp nn employ a different parametrization of the χ , namely
were kept fixed. The strength of proton-neutron (pn) 2pn
PQQ2 by taking χ = χ = χ /2 and the ex-
2pp 2nn 2pn
component χ was varied so as to reproduce the ex-
pn
5
citation energy E can be reproduced by varying the
2+
χ . AddingtheHH partofthetwo-bodyinteractionto TABLEI:CalculatedNTMEsM(0ν)inthePHFBmodelwith
2pp N
PQQ1andPQQ2andbyrepeatingthecalculations,two four different parameterization of effective two-body interac-
tion and three different parameterizations of Jastrow typeof
moreparameterizationsoftheeffectivetwo-bodyinterac-
SRC for the β−β− decay of 94,96Zr, 98,100Mo, 104Ru,
tions, namely PQQHH1 and PQQHH2 were obtained (cid:0) (cid:1)0ν
110Pd, 128,130Te and 150Nd isotopes due to the exchange of
[37].
heavy Majorana neutrino exchange. (a), (b), (c) and (d) de-
The four different parameterizations of the effective
note PQQ1, PQQHH1, PQQ2 and PQQHH2 parameteri-
pairing plus multipolar correlations provide us four dif-
zations, respectively. See the footnote in page 3 of Ref. [37]
ferentsetsofwavefunctions. Withthreedifferentparam-
for further details.
eterizationsofJastrowtypeofSRCandfoursetsofwave
Nuclei F F+S
functions, sets of twelve NTMEs MN(0ν) are calculated SRC1 SRC2 SRC3
forestimatingthe associateduncertaintiesinthe present 94Zr (a) 236.9498 77.5817 138.2606 191.3897
work. The uncertainties associated with the NTMEs (b) 220.3794 72.4285 128.7496 178.0783
M(0ν) for (β−β−) decay are estimated statistically by (c) 205.8370 72.9303 124.3248 168.5705
N 0ν
calculating the meanand the standarddeviationdefined (d) 211.0437 68.9323 122.9710 170.3572
by 96Zr (a) 177.7479 56.4909 102.4434 142.8831
(b) 185.5251 59.5338 107.2877 149.3117
M(0ν) = ki=1MN(0ν)(i) (17) ((dc)) 117750..48713909 5564..02734862 10918..24906531 114317..12284700
N P N 98Mo (a) 355.1915 117.0804 208.2494 287.5615
(b) 346.1118 116.4967 204.5667 281.0515
and
(c) 358.5109 118.0563 210.1150 290.2080
1/2 (d) 343.4160 115.2077 202.6977 278.7158
N
∆M(0ν) = 1 M(0ν) M(0ν)(i) 2 100Mo (a) 365.8004 122.2000 215.8882 296.9869
N √N 1" N − N # (b) 361.9877 122.6611 214.7455 294.4297
− Xi=1(cid:16) (cid:17) (c) 368.4056 123.2364 217.5391 299.1598
(18)
(d) 328.9795 111.4464 195.1601 267.5869
104Ru (a) 274.0700 89.7666 160.7925 222.1151
(b) 264.9015 88.1515 156.2893 215.1076
III. RESULTS AND DISCUSSIONS (c) 258.2796 84.6746 151.6002 209.3600
(d) 247.0603 82.3208 145.8435 200.6645
The model space, SPE’s, parameters of PQQ type of 110Pd (a) 424.6601 140.3359 249.6835 344.3187
(b) 379.9404 127.4915 224.6563 308.6907
effective two-body interactions and the method to fix
(c) 407.2163 134.6824 239.4733 330.1888
them have already been given in Refs. [22, 35, 36].
(d) 390.3539 130.6314 230.5392 316.9996
It turns out that with PQQ1 and PQQ2 parameteri-
128Te (a) 190.5325 62.4373 111.5143 154.1796
zations, the experimental excitation energies of the 2+
(b) 231.8024 77.4559 136.7936 188.1893
state E [47] can be reproduced within about 2% ac-
2+ (c) 220.7156 73.5158 130.0810 179.0960
curacy. The electromagnetic properties, namely reduced (d) 235.4814 78.6367 138.9052 191.1366
B(E2:0+ 2+)transitionprobabilities,deformationpa- 130Te (a) 236.0701 81.5493 141.3447 192.7610
→
rameters β , static quadrupole moments Q(2+) and gy- (b) 231.5921 79.3844 138.1901 188.8492
2
romagnetic factors g(2+) are in overall agreement with (c) 233.0024 80.4020 139.4400 190.2194
the experimental data [48, 49]. (d) 230.5282 78.9888 137.5307 187.9675
150Nd (a) 163.8037 55.8968 97.8169 133.6912
(b) 130.1364 43.8840 77.3178 105.9993
(c) 160.2720 54.6713 95.6942 130.8005
A. Short range correlations and radial evolutions
(d) 131.9781 44.6741 78.5433 107.5715
of NTMEs
In the approximation of finite size of nucleons with
100Mo are presented in Table II for PQQ1 parametriza-
dipole form factor (F) and finite size plus SRC (F+S),
tion. From the inspection of Table II, the following ob-
the theoretically calculated twelve NTMEs M(0ν) us-
N servations emerge.
ing the four sets of HFB wave functions generated with
PQQ1, PQQHH1, PQQ2 and PQQHH2 parameteri-
(i) The contribution of conventional Fermi matrix el-
zations of the effective two-body interaction and three
ements M = M is about 20% to the total
Fh F−VV
different parameterizations of Jastrow type of SRC for
matrix element.
94,96Zr, 98,100Mo, 104Ru, 110Pd, 128,130Te and 150Nd iso-
topes are presented in Table I. (ii) The Gamow-Teller matrix element is noticeably
To analyze the role of different components of NTME modified by the inclusion of the pseudoscalar and
(0ν)
M ,the decompositionofthe latter intoFermi, differ- weak magnetism terms in the hadronic currents.
N
enttermsofGamow-Tellerandtensormatrixelementsof While M increases the absolute value of
GT−PP
6
600.0
− −
TABLEII:DecompositionofNTMEsforthe β β decay
(cid:0) (cid:1)0ν F
of 100Mo including finite size effect (F) and SRC (F+S) for 500.0
thePQQ1parametrization. F+SRC1
NTMEs F F+S -1m)400.0 F+SRC2
SRC1 SRC2 SRC3
f
MF 68.6223 35.8191 54.0101 64.2516 νννννννν)(300.0 F+SRC3
00
MGT−AA -370.5960 -144.5650 -242.3340 -316.3250 (CN200.0
MGT−AP 174.4640 43.3631 93.9737 137.5100
MGT−PP -66.3082 -8.3767 -28.5936 -48.0727 100.0
MGT−MM -41.7693 16.3949 7.6213 -13.3421
MGT -304.2095 -93.1837 -169.3326 -240.2298 0.0
0 1 2 3 4
MT−AP 9.4369 9.2393 10.0332 10.0610
MT−PP -3.6622 -3.5528 -3.9226 -3.9386 r (fm)
MT−MM 1.2567 1.1163 1.3438 1.3722
MT 7.0314 6.8028 7.4544 7.4945
FIG. 2: Radial dependenceof C(0ν)(r) for the β−β− de-
N (cid:0) (cid:1)0ν
M(0ν) 365.8004 122.2000 215.8882 296.9869 cay of 100Mo isotope.
(cid:12) N (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
J >0 almost cancel beyond r 3 fm and the magnitude
MGT−AA, MGT−AP has a significant contribution of C(0ν) for all nuclei undergo≈ing (β−β−) decay have
withoppositesigninallcases. ThetermM 0ν
GT−MM their maximum at about the internucleon distance r 1
issmallerthanothers,andtheintroductionofshort ≈
fm. These observations were also made in the PHFB
range correlations changes its sign. model [28, 37]. Similarly, the radial evolution of M(0ν)
N
can be studied by defining
(iii) The tensor matrix elements have a very small con-
tribution, smaller than 2%, to the total transition
matrix elements. M(0ν) = C(0ν)(r)dr (19)
N N
Z
(iv) The inclusion of short range correlations changes
the nuclear matrix elements significantly, whose ef- TheradialevolutionofM(0ν) hasbeenstudiedforfour
N
fects arelargefor the Gamow-TellerandFermima- cases, namely F, F+SRC1, F+SRC2 and F+SRC3. To
trix elements but small in the case of tensor ones. maketheeffectsoffinitesizeandSRCmoretransparent,
we plot them for 100Mo in Fig. 2. In case of finite sized
(v) The Miller-Spencer parameterization of the short
nucleons,theC(0ν) arepeakedatr 0.5fmandwiththe
range correlations, SRC1, cancels out a large part N ≈
additionofSRC1andSRC2,the peakshiftstoabout0.8
of the radial function H , as shown in Fig. 1. The
N
fm. However, the position of peak is shifted to 0.7 fm
same cancellation reduces the calculated matrix el-
for SRC3. In Fig. 3, we plot the radial dependence of
ementsto aboutonethirdofitsoriginalvalue. The
other two parameterizationsof the short range cor- CN(0ν) forsixnuclei,namely96Zr,100Mo, 110Pd, 128,130Te
relations, namely SRC2 and SRC3, have a sizable and150Ndandthesameobservationsremainvalid. Also,
effect,whichisinallcasesmuchsmallerthanSRC1. the same features in the radial distribution of C(0ν) are
N
noticedinthecasesofPQQ2,PQQHH1andPQQHH2
With respect to the point nucleon case. the change parametrizations.
in M(0ν) is about 30%–34% due to the FNS. With the
N
inclusion of effects due to FNS and SRC, the NTMEs
change by about 75%–79%, 58%–62% and 43%–47% for B. Uncertainties in NTMEs
F+SRC1, F+SRC2 and F+SRC3, respectively. It is
noteworthy that the SRC3 has practically negligible ef- The uncertainties associated with the NTMEs M(0ν)
fect onthe finite size case. Further, the maximum varia- for (β−β−) decay are estimated by preforming a Nsta-
tion in MN(0ν) due to PQQHH1, PQQ2 and PQQHH2 tistical anal0yνsis by using Eqs.(17) and (18). In Table I,
parametrization with respect to PQQ1 interaction are sets of twelve NTMEs M(0ν) of 94,96Zr, 98,100Mo, 110Pd,
about 24%, 18% and 26% respectively. N
128,130Teand150Ndisotopesaredisplayed,whichareem-
IntheQRPA[26,27],ISM[16]andPHFB[28,37],the
(0ν)
radial evolution of M(0ν) due to the exchange of light ployed to calculate the average values MN as well as
(0ν)
Majorana neutrino has already been studied. In both uncertainties ∆M tabulated in Table III for the bare
N
QRPAandISMcalculations,ithasbeenestablishedthat axial vector coupling constant g =1.254 and quenched
A
the contributionsofdecayingpairscoupledto J =0 and value of g =1.0.
A
7
700.0 250.0
(a) (b)
A=96
600.0
A=100
200.0
500.0 A=110
) )
-1m A=128 -1m150.0
400.0
f A=130 f
( (
νννννννν) 330000..00 AA==115500 νννννννν)
00 00 110000..00
(N (N
C200.0 C
50.0
100.0
0.0 0.0
0 1 2 3 4 0 1 2 3 4
r (fm) r (fm)
400.0 500.0
(c) (d)
400.0
300.0
) )
1 1
-m -m300.0
f f
(200.0 (
νννννννν) νννννννν)
00 00 220000..00
(N (N
C C
100.0
100.0
0.0 0.0
0 1 2 3 4 0 1 2 3 4
r (fm) r (fm)
FIG. 3: Radial dependence of C(0ν)(r) for the β−β− decay of 96Zr, 100Mo, 110Pd, 128,130Te and 150Nd isotopes. In this
N (cid:0) (cid:1)0ν
Fig., (a), (b),(c) and (d) correspond to F, F+SRC1, F+SRC2and F+SRC3,respectively.
Itturnsoutthatinallcases,theuncertainties∆M(0ν) IV. CONCLUSIONS
are about 35%for g =1.254and g =1.0. Further, we
A A
(0ν)
estimatetheuncertaintiesforeightNTMEsM calcu- We have employed the PHFB model, with four differ-
N
lated using the SRC2, and SRC3 parameterizations and ent parameterizations of pairing plus multipole effective
theuncertaintiesinNTMEsreducetoabout16%to20% two body interaction, to generate sets of four HFB in-
withtheexclusionofMiller-Spensertypeofparametriza- trinsic wave functions, which reasonably reproduced the
tion. InTable IV,averageNTMEs for caseII alongwith observedspectroscopicproperties,namelytheyrastspec-
NTMEs calculated in other models have been presented. tra, reduced B(E2:0+ 2+) transition probabilities,
It is noteworthy that in the models employed in Refs. static quadrupole mome→nts Q(2+) and g-factors g(2+)
[6, 8, 9], effects due to higher order currents have not of participating nuclei in (β−β−) decay, as well as
2ν
been included. We also extract lower limits on the ef- their M [35, 36]. Considering three different parame-
2ν
fective mass of heavy Majorananeutrino MN from the terizationsofJastrowtypeofSRC,setsoftwelveNTMEs
largestobservedlimitsonhalf-livesT10/ν2 ohf(β−iβ−)0ν de- MN(0ν) forthestudy(β−β−)0ν decayof94,96Zr,98,100Mo,
cay. The extracted limits are M > 5.67+0.94 107 104Ru, 110Pd, 128,130Te and 150Nd isotopes in the heavy
h Ni −0.94 ×
GeV and > 4.06+0.64 107 GeV, from the limit on half- Majorana neutrino mass mechanism have been calcu-
life T10/ν2 >3.0×10−204.6y4r×of 130Te [56] for gA = 1.254 and lated.
g =1.0, respectively. The study of effects due to finite size of nucleons and
A
SRC reveal that in the case of heavy Majorana neutrino
exchange,the NTMEschangebyabout30%–34%dueto
finite size of nucleons and the SRC1, SRC2 and SRC3
8
SRC play a more crucial role in the heavy than in the
(0ν) (0ν)
TABLEIII:AverageNTMEsM anduncertainties∆M light Majorana neutrino exchange mechanism.
N N
for the β−β− decay of 94,96Zr, 98,100Mo, 104Ru, 110Pd,
(cid:0) (cid:1)0ν Finally, a statistical analysis has been performed by
128,130Teand150Ndisotopes. Bothbareandquenchedvalues employing the sets of twelve NTMEs M(0ν) to estimate
of g are considered. Case I and Case II denote calculations N
A the uncertainties for g = 1.254 and g = 1.0. It turns
with and without SRC1, respectively. A A
out that the uncertainties are about 35% for all the con-
− −
β β gA Case I Case II sidered nuclei. Exclusion of Miller-Spenser parametriza-
(0ν) (0ν) (0ν) (0ν)
emitters MN ∆MN MN ∆MN tion of Jastrow type of SRC, reduces the maximum un-
94Zr 1.254 126.2146 44.9489 152.8378 27.1912 certainties to a value smaller than 20%. The best ex-
1.0 142.9381 49.1752 172.1620 29.3965 tracted limit on the effective heavy Majorana neutrino
96Zr 1.254 100.5313 36.8858 122.5048 21.9209 mass M from the available limits on experimental
N
1.0 114.4851 40.3246 138.6328 23.5263 h i
98Mo 1.254 202.5006 71.6345 245.3957 41.8882 half-lives T10/ν2 using average NTMEs M(N0ν) calculated
1.0 230.1520 78.3244 277.2795 44.9878 in the PHFB model is > 5.67+0.94 107 GeV and
100Mo 1.254 206.7533 73.0792 250.1870 43.7119 >4.06+0.64 107 GeV for 130Te iso−t0o.9p4e.×
1.0 235.0606 79.9885 282.7964 47.1334 −0.64×
104Ru 1.254 150.5572 53.9389 182.7216 31.9382
1.0 171.8075 59.0467 207.1750 34.3939
110Pd 1.254 231.4743 82.4924 280.5688 49.1588 Acknowledgments
1.0 263.4339 90.3033 317.3947 53.0150
128Te 1.254 126.8285 46.3381 153.7370 29.4676
1.0 143.9772 50.6942 173.5263 31.8554 This work is partially supported by the Department
130Te 1.254 136.3856 46.9164 164.5378 27.2226 of Science and Technology (DST), India vide sanction
1.0 154.3797 51.2511 185.2849 29.1907 No. SR/S2/HEP-13/2006, DST-RFBR Collaboration
150Nd 1.254 85.5467 31.4473 103.4294 20.9802 via grant no. RUSP-935, Consejo Nacional de Cien-
1.0 97.3640 34.5024 117.0160 22.8729 cia y Tecnolog´ıa (Conacyt)-M´exico, European Union-
Mexico Science and Technology International Coopera-
tion Fund (FONCICYT) Project 94142, and Direcci´on
change them by 75%–79%, 58%–62% and 43%–47%, re- General de Asuntos del Personal Acad´emico, Universi-
spectively. Further, it has been noticed through the dad Nacional Aut´onoma de M´exico (DGAPA-UNAM)
study of radial evolution of NTMEs that the FNS and Project IN102109-3.
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′
TABLE IV: Average NTMEs M (0ν) =(g /1.254)2M(0ν) for the β−β− decay of 94,96Zr, 98,100Mo, 110Pd, 128,130Te
N (cid:16) A N (cid:17) (cid:0) (cid:1)0ν
and 150Nd isotopes. Both bare and quenched values of g are considered. The superscripts a and b denote the Argonne and
A
CD-Bonn potentials.
′
β−β− g M (0ν) QRPA QRPA QRPA QRPA SRQRPAa SRQRPAb T0ν( yr) Ref. hm i(GeV)
A N 1/2 N
emitters [6] [8] [9] [10] [3] [3]
94Zr 1.254 152.84±27.19 1.9×1019 [50] 2.57+0.46×104
−0.46
1.0 109.48±18.69 1.84+0.31×104
−0.31
96Zr 1.254 122.50±21.92 99.062 9.2×1021 [51] 2.68+0.48×106
−0.48
1.0 88.16±14.96 1.93+0.33×106
−0.33
98Mo 1.254 245.40±41.89 1.0×1014 [52] 9.70+1.70
−1.70
1.0 176.33±28.61 6.97+1.13
−1.13
100Mo 1.254 250.19±43.71 155.960 333.0 56.914 76.752 259.8 404.3 4.6×1023 [53] 3.43+0.60×107
−0.60
1.0 179.84±29.97 191.8 310.5 2.47+0.41×107
−0.41
110Pd 1.254 280.57±49.16 6.0×1017 [54] 2.43+0.43×104
−0.43
1.0 201.84±33.71 1.75+0.29×104
−0.29
128Te 1.254 153.74±29.47 122.669 303.0 101.233 1.1×1023 [55] 2.06+0.39×106
−0.39
1.0 110.35±20.26 1.48+0.27×106
−0.27
130Te 1.254 164.54±27.22 108.158 267.0 92.661 239.7 384.5 3.0×1024 [56] 5.67+0.94×107
−0.94
1.0 117.83±18.56 176.5 293.8 4.06+0.64×107
−0.64
150Nd 1.254 103.43±20.98 153.085 422.0 1.8×1022 [57] 5.99+1.21×106
−1.21
1.0 74.41±14.55 4.31+0.84×106
−0.84
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