Ali, Borisov, Zhuridov

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ISSN 1063-7788, Physics of Atomic Nuclei, 2010, Vol. 73, No. 12, pp. 20832096. c Pleiades Publishing, Ltd., 2010.

Original Russian Text c A. Ali, A.V. Borisov, D.V. Zhuridov, 2010, published in Yadernaya Fizika, 2010, Vol. 73, No. 12, pp. 21392152.

NUCLEITheory

Mechanisms of Neutrinoless Double-Beta Decay:

A Comparative Analysis of Several Nuclei

. Ali1), . V. Borisov2)*, and D. V. Zhuridov3)

Received February 10, 2010

AbstractThe neutrinoless double beta decay of several nuclei that are of interest from the experimentalpoint of view (76Ge, 82Se, 100Mo, 130Te, and 136Xe) is investigated on the basis of a general Lorentz-invariant effective Lagrangian describing physics effects beyond the Standard Model. The half-lives andangular-correlation coefficients for electrons are calculated for various decay mechanisms associated, inparticular, with the exchange of Majorana neutrinos, supersymmetric particles (withR-parity violation),leptoquarks, and right-handedWRbosons. The effect of theoretical uncertainties in the values of relevantnuclear matrix elements on decay features is considered.

DOI:10.1134/S1063778810120136

1. INTRODUCTION

The majority of Standard Model extensions aimedat explaining the observed smallness of neutrinomasses predict the Majorana nature of the neutrinos.Such theories include those employing the seesawmechanism [1] and models of neutrino-mass gener-ation by radiative corrections [2, 3]. An experimentalobservation of neutrinoless double-beta decay (02)would be a decisive argument in favor of the Majorananature of the neutrino, as contrasted against the Diracnature of all of the remaining known fermions (for anoverview of experiments performed now and plannedfor the future to seek such decays, see [4]).

We recall that 02decay is forbidden in the Stan-dard Model by the law of lepton-number conserva-tion. However, the Lagrangians of various StandardModel extensions may include terms that lead tolepton-number nonconservation and, hence, to thedecay in question. Possible mechanisms of lepton-number violation may involve the exchange of Majo-rana neutrinos [57], supersymmetric Majorana par-ticles [813], scalar bilinears [14] (for example, dou-bly charged Higgs particles [3]), leptoquarks [15],right-handed bosons WR [6, 7, 16], and so on. It

is necessary to separate the aforementioned differentcontributions to the decay process in order to extractinformation about the properties of lepton-number-violation sourcesin particular, about the masses

1)Deutsches Elektronen-Synchrotron, DESY, 22607 Ham-burg, Germany.

2)Moscow State University, Moscow, 119991 Russia.3)Scuola Normale Superiore, piazza dei Cavalieri 7, I-56126

Pisa, Italy.*E-mail: [email protected]

of (Majorana) neutrinos and the parameters of theirmixing. On this path, measurements of the propertiesof the02decay of various nuclei and their compar-ative analysis will be helpful in pinpointing the maindecay mechanism [1719]. It should be noted that thepossibility of identifying different decay mechanismswas first demonstrated in [6, 7]. It was shown in [20]that the calculated ratios of nuclear half-lives arehighly sensitive to the choice of model for calculatingrelevant nuclear matrix elements, and it was proposedthere on this basis to test nuclear models by com-

paring these ratios with (future) experimental data on02decay of several nuclei.

In our previous studies [21, 22], we investigatedthe effect of various mechanisms of the neutrino-less double-beta (02) decay of the 76Ge nucleuson the angular correlation of final-state electrons.In the present article, we generalize the results ofthose studies by performing a comparative analysis ofthe half-lives and electron angular-correlation coef-ficients for the 76Ge, 82Se, 100Mo, 130Te, and 136Xenuclei.

d

d

u

e

e

u

M

Fig. 1.Long-range mechanism of02decay.

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2084 ALI et al.

Table 1.Nonzero coefficients for various models

Model Nonzero

RC [7, 16] VAVA

RPV SUSY [12] SPS+P, VAVA,

TRTR

LQ [15] S+PSP, V+AVA

2. COMPARATIVE ANALYSIS OF DECAYMECHANISMS FOR SEVERAL NUCLEIFollowing [22], we restrict ourselves to the class

of long-range mechanisms of 02 decay, whichinvolve the exchange of light Majorana neutrinosM.A general structure of the corresponding Feynmandiagrams is shown in Fig. 1, where four-fermion ver-tices are described by a general effective low-energyLagrangian including a complete set of Lorentz-invariant terms:

L = GFVud2

(Uei+ VAVA,i)jiVAJ+VA, (1)

+,

ijiJ

+ +H.c.

.

Here, J+ = uOd are quark currents; ji= eOi

are lepton currents, where the fieldsicorrespond tolight-mass eigenstates of Majorana neutrinos; sum-mation is performed over the dummy indices , =V A, S P, TL,R(OT = 2P, = i2 [, ],P = (1 5)/2is the chiral projection operator, and= L, R); the prime means that the term for which= =V Ais excluded from the sum; Ueistandfor lepton-mixing-matrix elements [23]; and Vud isa quark-mixing-matrix element [24]. The coefficientsi parametrize new-physics effects [that is, the de-

viation of the Lagrangian from the Standard Modelform of the product of currents of the(V A) (VA) type and mixing with non-SM neutrinos]. Thenonzero coefficients

= Ueii (2)

are presented in Table 1 for a number of versions of theStandard Model extension:Mand RPV SUSY (su-

persymmetry with R-parity violation) [12] and Mandright-handed currents (RC) associated with right-handed WRbosons [6, 7, 16] or leptoquarks (LQ) [15].

The differential decay width for the 0+(A, Z) 0+(A, Z+ 2)ee transitions has the form [22]

d

d cos =

ln 2

2 |MGT|2A(1 Kcos ), (3)

where is the angle between the electron momentain the rest frame of the primary nucleus; MGT is the

GamowTeller nuclear matrix element; andKis theangular-correlation coefficient,

K= B/A, 1< K


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MECHANISMS OF NEUTRINOLESS DOUBLE-BETA DECAY 2085

Table 2.Quantities factors A and Bin (3) and (4) for specifically chosen

A BVAVA A0+ 4C1|||VAVA|c02+ 4C1|VAVA|2 B0+ 4D1|||VAVA|c02+ 4D1|VAVA|2

VAV+A

A0+ 4C0|||VAV+A|c01+ 4C1+|VAV+A|2 B0+ 4D0|||VAV+A|c01+ 4D1+|VAV+A|2

V+A

VA A0+ C3|||V+A

VA|c2+ C5|V+A

VA|2

B0+ |||V+A

VA|(D3c2+ D3s2) + D5|V+A

VA|2

V+AV+A A0+ C2|||V+AV+A|c1+ C4|V+AV+A|2 B0+ |||V+AV+A|(D2c1+ D2s1) + D4|V+AV+A|2

SPSP A0+ 4CSP0 |||SPSP|c04+ 4CSP1 |SPSP|2 B0+ 4DSP0 |||SPSP|s04+ 4DSP1 |SPSP|2

SPS+P A0+ 4CSP0 |||SPS+P|c03+ 4CSP1 |SPS+P|2 B0+ 4DSP0 |||SPS+P|s03+ 4DSP1 |SPS+P|2

S+PSP A0+ CSP2 |||S+PSP|c4+ CSP3 |S+PSP|2 B0+ |||S+PSP|(DSP2 c4+ DSP2 s4) + DSP3 |S+PSP|2

S+PS+P A0+ CSP2 |||S+PS+P|c3+ CSP3 |S+PS+P|2 B0+ |||S+PS+P|(DSP2 c3+ DSP2 s3) + DSP3 |S+PS+P|2

TLTL A0+ 4CT0 |||TLTL |c06+ 4CT1|TLTL|2 B0+ 4DT0|||TLTL|s06+ 4DT1 |TLTL |2

TLTR

, TRTL

A0 B0TRTR

A0+ CT2

|

||TRTR

|c5+ CT3

|TRTR

|2

B0+ DT2

|

||TRTR

|c5+ DT3

|TRTR

|2

Table 3. Numerical values of Ci and Di multiplied by 1015 for various nuclei within the QRPA model without pnpairing [25]

Nucleus C1 C0 C1+ C2 C3 C4 C576Ge 7.208 6.680 6.191 4.601 1520 11.54 3.152 10582Se 29.97 29.50 29.03 15.45 2297 81.56 2.627 105100Mo 151.3 20.04 2.656 46.82 2.446 104 245.1 6.426 106130Te 46.67 46.30 45.93 35.89 9670 115.7 2.480 106136

Xe 45.54 48.16 50.94 37.88 9896 112.0 2.581 106

Nucleus D1 D0 D1+ D2 D3 D4 D576Ge 5.872 5.442 5.044 0.6179 17.31 9.073 2.309 10582Se 26.42 26.00 25.59 3.292 3.428 72.72 2.154 105100Mo 133.5 17.69 2.344 26.62 773.7 219.1 5.390 106130Te 39.62 39.30 38.99 5.405 152.7 92.26 1.951 106136Xe 38.45 40.67 43.01 7.952 171.7 86.66 2.014 106

(A) all = 0, |m| = 0(SM+Majorana neutri-nos);

(B) = 0, |m| = 0 (zero effective Majoranamass);

(C) = 0, |m| = 0,ci cos i= 0,but this also permits investigating a more generalsituation,

(D) = 0, |m| = 0,ci= 0.We restrict ourselves to analyzing the contribu-

tions of the parameters VAVAbecause the necessary

extended [with respect to case (A)] sets of nuclearmatrix elements were calculated in [25] only for these

parameters. As far as we know, there are no numericalvalues of non-SM nuclear matrix elements (for theirdefinition, see [22]) in more recent studies (we aregrateful to F. Simkovic for a discussion on this issue).

(A) All = 0= 0= 0, |||mmm| = 0| = 0| = 0.Using formulas (3)

(5) and the results from [22], we obtain the half-lifeand the angular-correlation coefficient in the form

T1/2=|MGT|2C1||21 , (6)

PHYSICS OF ATOMIC NUCLEI Vol. 73 No. 12 2010


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Table 4.Numerical values ofCiandDimultiplied by 1015 for various nuclei within the QRPA model with allowance forpn pairing [25]

Nucleus C1 C0 C1+ C2 C3 C4 C5

76Ge 3.526 6.188 10.86 0.745 1065 3.037 3.153 10582Se 59.15

24.39 10.06

54.58 3206 226.1 2.627

105

100Mo 0.1805 5.737 182.3 1.530 860.9 37.98 6.428 106130Te 30.83 44.73 64.91 25.58 7878 80.66 2.480 106136Xe 42.05 47.99 54.77 30.60 9509 88.69 2.581 106

Nucleus D1 D0 D1+ D2 D3 D4 D5

76Ge 2.873 5.041 8.846 0.7080 12.21 2.634 2.309 10582Se 52.13 21.50 8.867 11.42 4.815 186.5 2.154 105100Mo 0.1593 5.062 160.9 0.5702 26.72 27.92 5.389 106130

Te 26.17 37.97 55.09 4.971 124.1 63.12 1.951 106

136Xe 35.50 40.52 46.25 4.889 165.0 70.31 2.014 106

Table 5. Ratios of the half-lives,R andR, for variousnuclei according to calculations on the basis of the QRPAmodel with (without)pn pairing [25]

Nucleus R = RVAVA RVAV+A RV+AVA RV+AV+A82Se 0.4 (0.2) 0.4 (2.8) 2.1 (3.1) 0.3 (0.03)100Mo 1.1 (195.2) 52.9 (0.6) 1.1 (0.5) 1.1 (0.8)130

Te 0.2 (0.1) 0.2 (0.2) 0.2 (0.1) 0.2 (0.04)136Xe 0.5 (0.2) 0.4 (0.4) 0.4 (0.2) 0.4 (0.1)

K=D1/C1= B01/A01.

Here, C1 = A01(1 F)2, where F= (gV/gA)2 MF/MGT, with MF being the Fermi nuclear matrixelement andgV andgAbeing, respectively, the vectorand axial-vector nucleon form factors, and A01 andB01 are kinematical factors (integrals over the elec-tron phase space) [22]. Taking into account the dipolemomentum-transfer dependence of the form factors

gV and gA at various values of the cutoffparameter(A >V ), we obtain the tensor contribution MTto the total nuclear matrix elements. As a result,M0 =MGT(1 F) is replaced by a more reliableform,M0 =M0+ MT(according to [27, 28]).

For the chosen set of nuclei and the mass inter-val of interest from the experimental point of view,the half-life dependences of the square of the ef-fective Majorana mass [see Eq. (6)] in Fig. 2 forthe nuclear matrix element M0 (M0) calculated

on the basis of the QRPA model without and withallowance for pn pairing [25] (RQRPA and QRPAmodels in [28]) are represented by, respectively, thesolid and long-dashed curves (short-dashed and dot-ted curves). The two vertical straight lines in Fig. 2indicate the planned sensitivity limits at the first andsubsequent stages of future experiments aimed atsearches for02decay:

T1/2(GERDA, Majorana,76

Ge)= 2 10264 1027 yr [29, 30],

T1/2(SuperNEMO,82Se) = (12) 1026yr [31, 32],

T1/2(MOON,100Mo) = 2.1 10261 1027 yr [33],

T1/2(COURE,130Te) = (2.16.5) 1026 yr [31, 34],

T1/2(EXO,136Xe) = 6.4 10252 1027 yr [31, 35].

From Fig. 2, one can see that, for the chosen setof decaying nuclei and at a fixed value of the half-life T1/2, the calculated effective Majorana mass |m|depends substantially on the choice of nuclear model.In particular, a limit on |m|2 for the 76Ge nucleus isapproximately five times more stringent in the case ofthe QRPA model withoutpnpairing than in the caseof the QRPA model with allowance forpn pairing. Onthe other hand, Fig. 2 demonstrates the possibilityof discriminating between the predictions of differ-ent nuclear models by using the results for severaldecaying nuclei. For all four of the nuclear modelsconsidered here, the 130Te nucleus shows the highest



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10

2

10

1

|

m

|

2

, eV

2

10

25

T

1/2

, yr10

2

10

3

10

0

10

1

10

0

10

3

Xe

10

2

10

1

|

m

|

2

, eV

2

10

25

T

1/2

, yr10

2

10

3

10

0

Mo

10

1

10

0

10

3

10

1

10

25

T

1/2

, yr10

2

10

3

10

0

Te

10

2

10

3

10

1

10

0

10

2

10

1

10

0

Ge

10

0

Se

10

3

10

1

10

2

10

3

10

1

10

0

10

3

|

m

|

2

, eV

2

10

1

10

2

10

2

10

0

Fig. 2.Square of the effective Majorana mass,|m|2, as a function of the half-life T1/2for various nuclear models (see mainbody of the text).

sensitivity to the effective Majorana mass|m|itpermits testing the smallest values of this mass (ata fixed value ofT1/2).

In order to compare the half-lives and angular-correlation coefficients for different nuclei, we con-sider the ratios

R(AX) = T1/2(AX)

T1/2(76Ge), K(AX) = K(

AX)

K(76Ge), (7)

where, as a reference nucleus, we use the isotope76Ge, which has received the most detailed study.Concurrently, we notice that relative quantities usu-ally are less sensitive to the theoretical uncertainty

in the nuclear matrix elements than their absolutecounterparts (see [36] and references therein). Thenumerical values of the ratios in (7) for the standarddecay mechanism (= 0,

|m

| = 0) are presented

in the second columns of Tables 5 and 6.(B)

= 0= 0= 0,|||mmm| = 0.| = 0.| = 0. By analogy with (7), we

define the ratios

R(AX) =T1/2(

,AX)

T1/2(, 76Ge)

, (8)

K(AX) = K(, AX)

K(, 76Ge),



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Table 6.Ratios of the angular correlation coefficients Kand Kfor various nuclei in the QRPA model without (with)allowance forpn pairing [25]

Nucleus K = KVAVA KV+AVA B09A09AX

A09B09

76Ge

KV+AV+A B02A02AX

A02B02

76Ge

82Se 1.08 1.12 (1.12) 1.13 1.13 (0.95) 1.06

100Mo 1.08 1.15 (1.15) 1.13 1.14 (0.85) 1.05

130Te 1.04 1.07 (1.07) 1.07 1.01 (0.90) 1.01

136Xe 1.04 1.07 (1.07) 1.06 0.98 (0.91) 1.01

Table 7. Numerical values ofi [in units of1012 yr] forV+AV+A= 0

Nucleus 1 2 3 476Ge 7.942 10.10 5.140 6.31082Se 3.429 3.846 1.246 1.413100Mo 8.796 9.840 5.359 6.072130Te 1.917 2.403 0.8231 0.9698136Xe 4.112 5.315 1.824 2.161

which characterize the contributions of non-SM de-cay mechanisms individually (only one of the co-

efficients is nonzero). These ratios do not de-pend on only for|m| = 0, but they depend on and through various sets of coefficients Ci andDi appearing in the expressions for these ratios (see

Table 2). The values ofR andK

for the chosennuclei are presented in Tables 5 and 6. The ratios of

the half-lives are markedly different, with the excep-tion of the ratiosRVAVA coinciding with the ratioRfor the standard decay mechanism. However, thesevariations are largely due to theoretical uncertaintiesin the nuclear matrix elements. This is illustratedin Table 5 by comparing the results of the calcula-tions within the QRPA model with and without al-lowance forpnpairing. In particular, the ratio RVAV+A(R = RVAVA) for the 100Mo nucleus is two ordersof magnitude larger than the analogous ratios for

the other nuclei in the QRPA model without (with)allowance for pn pairing. This is because the ratioRVAV+A(AX)(R = RVAVA(AX)) is proportional to thenuclear-matrix-element combination|MGT|2|F+1|2 (|MGT|2|F 1|2), with|MGT|and|F+ 1|(|MGT| and |F 1|) being smaller for the 100Mo nu-cleus than for the other nuclei by factors of about 3 (2)and 5 (10), respectively: only for the 100Mo nucleusis the value ofF accidentally close to1(+1). We

Table 8.Numerical values of1,2[in units of1012 yr] and3,4[in units of108yr] for

V+AVA= 0

Nucleus 1 2 3 476Ge 7.657 10.45 1.947 2.39082Se 3.286 4.008 4.030 4.573100Mo 8.518 10.16 2.109 2.390130Te 1.903 2.419 0.3864 0.4553136Xe 4.130 5.293 0.7885 0.9338

note that, forRV+AVA, there is no similar accidentalenhancement because of a more intricate structureof the corresponding expressions {see expressions (6)and (7) and Table 2 above; see also Eq. (28) in [22]}.

On the other hand, the ratios of the angular-correlation coefficients K and KVAVAdo not depend on

the nuclear matrix elements and assume the formK = KVAVA = KVAV+A =

B01A01

AX

A01B01

76Ge

. (9)

Also, the coefficients KV+AVAare virtually independentof the uncertainties in the nuclear matrix elements,because the following approximate relations hold:

KV+AVA B09A09

AX

A09B09

76Ge

, (10)

KV+AV+A B02A02

AX

A02B02

76Ge

.

We recall that, here, Aiand Biare purely kinemat-ical quantities (phase-space integrals). The values ofthe right-hand sides of Eqs. (10) are given in thefourth and sixth columns of Table 6 for the sake ofcomparison. We note, however, that these ratios ofthe angular-correlation coefficients do not provide apossibility for discriminating between the predictionsof the different theories since, within the assumedexperimental error of about 20%, their values arecompatible with unity.



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m

|

2

, eV

2

10

25

T

1/2

, yr10

1

10

1

(

c

)

10

0

10

2

10

2

10

18

|

V

AV

+

A

|

2

10

25

T

1/2

, yr10

1

10

1

(

d

)

10

0

10

2

10

2

|

m

|

2

, eV

2

10

1

(

a

)

10

2

10

14

|

V

+

AV

+

A

|

2

10

1

(

b

)

10

2

10

0

Fig. 3.Correlations (a,c) between|m|andT1/2and (b,d) between||andT1/2for afixed value ofK = 0.8and a nonzero

value of the parameter (a, b) V+AV+A or (c, d) V+AVA. The solid, long-dashed, short-dashed, dotted, and dash-dotted curves

represent the results for the decays of the 76Ge, 82Se, 100Mo, 130Te, and 136Xe nuclei, respectively.

The 100Mo nucleus shows the highest sensitivityto non-SM decay mechanisms in terms of the prop-erties considered here; this is not so only in the case

of the ratio RV+AVA, for which the 82Se nucleus is themost sensitive.

According to half-life measurements, the 100Mo130Te and 82Se130Te pairs of nuclei are the mostsensitive to the effects of the non-SM parametersVAV+Aand

V+AVA, respectively.

Measurements of angular-correlation coefficientsshow that the 76Ge100Mo pair of nuclei is the mostsensitive to the effects ofV+AVA.

(C) = 0,= 0,= 0,|||mmm| = 0,| = 0,| = 0, ci= 0.ci= 0.ci= 0. With the aid of

Eqs. (4) and (6) and the data in Table 2, the re-

duced Majorana mass = m/me and the param-eterof the non-SM Majorana mass

(according

to the choice of indices and in Table 2) canbe expressed in terms of the half-life T1/2 and theangular-correlation coefficient K(which are measur-able quantities) as

||2 = (1 2K)/T1/2, (11)||2(or4||2) = (3+ 4K)/T1/2,

where the coefficients are

1= Di/i, 2= Ci/i, (12)

3 = D1/i, 4= C1/iwith i= |MGT|2(C1Di D1Ci). In particular, wehavei= 4(5) forV+AV+A(

V+AVA).

The values of i for the nuclei considered hereare given in Table 7 (8) for a nonzero value of theparameterV+AV+A(

V+AVA).

Figures 3, 4, 5, 6, and 7 show correlations between|m| andT1/2and between || andT1/2for the cho-sen nuclei in the QRPA model without allowance forpn pairing [25] at a nonzero value of the parameter (,b) V+AV+Aor (c, d)

V+AVAfor the angular-correlation co-

efficient fixed at K= 0.8, 0.4, 0, 0.4, and0.8, re-spectively. Figure 8 shows correlations between |m|and Kand between|| and Kfor the same nucleiand at a fixed half-life value of T1/2 = 10

25 yr for

nonzeroV+AV+AorV+AVA. The case ofK= 0.8(Fig. 3)

approximately reproduces the standard decay mech-anism (SM + Majorana neutrino) considered in Sec-tion (A)see Fig. 2. From Figs. 37, one can seethat, smaller values ofKcorrespond to larger values



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, eV

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T

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, yr

10

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10

2

(

c

)

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0

10

2

10

18

|

V

AV

+

A

|

2

10

25

T

1/2

, yr

10

1

10

1

(

d

)

10

0

|

m

|

2

, eV

2

10

1

(

a

)

10

14

|

V

+

AV

+

A

|

2

(

b

)

10

0

10

1

10

2

10

2

10

1

10

3

10

0

10

2

10

3

Fig. 6.As in Fig. 3, but forK =0.4.

|

m

|

2

, eV

2

10

25

T

1/2

, yr

10

1

10

3

(

c

)

10

0

10

4

10

18

|

V

AV

+

A

|

2

10

25

T

1/2

, yr

10

1

10

1

(

d

)

10

0

|

m

|

2

, eV

2

10

2

(

a

)

10

14

|

V

+

AV

+

A

|

2

10

1

(

b

)

10

0

10

2

10

2

10

2

10

0

10

2

10

3

Fig. 7.As in Fig. 3, but forK =0.8.



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2092 ALI et al.

|

m

|

2

, eV

2

0.2

(

c

)

1.0

0.1

0.5 0 0.5 1.0

K

0

0.3

0.4

0.510

16

|

V

AV

+

A

|

2

0.2

(

d

)

1.0 0.5 0 0.5 1.0

K

0

0.4

0.6

0.8

|

m

|

2

, eV

2

0.2

(

a

)

0.1

0

0.3

0.4

0.510

12

|

V

+

AV

+

A

|

2

0.2

(

b

)

0

0.4

0.6

1.0

0.8

Fig. 8. Correlations (a,c) between |m| andKand (b, d) between ||andKfor afixed value ofT1/2 = 1025yr and a nonzero

value of the parameter (a,b)V+AV+Aor (c,d)V+AVA. The notation for the curves is identical to that in Fig. 3.

ofV+AVA, the 76Ge, 82Se, and 100Mo nuclei being the

most sensitive to angular correlations. Figures 3a and3care almost identical (the same is true for Figs. 46 and 8), because the values of1,2at V+AV+A= 0arevery close tothose at V+AVA= 0 fora fixed nucleus; thedifference is significant only at small values of|m|(Fig. 7). In Figs. 7and 7c, there are curves only forthe 82Se and 100Mo nuclei since, for the remainingnuclei, 76Ge, 130Te, and 136Xe, the values of|m| areoverly small. We note that, in Figs. 6 and 7 (bandd), the curves corresponding to the 76Ge and 100Monuclei coincide with each other.

Figure 9 illustrates correlations between (a,c, e)the mass MWR of the right-handed WR boson, (b,

d, f) the mixing angle , (d) the half-life T1/2,and (e,f) the angular-correlation coefficientKwith-in theS U(2)L SU(2)R U(1)leftright symme-try model [22, 37] for the case of equal form fac-tors for left- and right-handed hadron currents (gV =gV) and the mixing-parameter value of= |UeiVei| =106 [22].

From Fig. 9, one can see that, the largerT1/2 atfixed Kor the closerKto unity atfixed T1/2, the more

stringent the lower limit on MWR (upper limit on),the maximum being reached for the 130Te nucleus.However, a significant effect of the WR boson on02decay forT1/2 < 1026 yr is impossible becauseof a stringent limit on its mass from an electroweakfit [24]: MWR >715 GeV. We note that, in Figs. 9cand 9d, the curves corresponding to the 76Ge and100Mo nuclei merge together.

(D) = 0,= 0,= 0, |||mmm| = 0,| = 0,| = 0, ci= 0.ci= 0.ci= 0. Using relations (4)and (6) and the data from Table 2, we obtain

A =C1||2 + 4Ci||||c0k+ 4Cj ||2, (13)AK=D1||2 + 4Di ||||c0k+ 4Dj ||2

for= S

P, V

A,TL(with the exception of =

TRand= TL) andA =C1||2 + Ci||||ck+ Cj ||2, (14)AK=D1||2 + Di ||||ck+ Dj ||2

for=S+ P,V + A,TR(with the exception of=TLand= TR). Here, we have A = (|MGT|2T1/2)1[see Eq. (6)]. In the particular case of a nonzero valueof the parameter V+AV+A (

V+AVA), we have i= i

= 2,j =j = 4, andk = 1(i= i = 3,j =j = 5,k = 2).



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2

1.0 0.5 0 0.5

1

10

M

W

R

, GeV

200

(

e

)

1.0

(

f

)

0.5 0 0.5

150

300

500

70010

3

||

5

10

25

T

1/2

, yr

2

1

M

W

R

, GeV

10

25

T

1/2

, yr

150

(

c

)

0

(

d

)

10 30100

200

250

300

10

3

||

5

KK

20 10

1

10

0

10

2

0.2

0.1

300

(

a

) (

b

)

100

500

700

0.5

1.0

Fig. 9.Correlations between (,c,e)MWR , (b,d,f)and (d)T1/2, (e,f)Kat (,b)K = 0.8, (c, d)K =0.8, and (e, f)T1/2 = 10

26yr. The notation for the curves is identical to that in Fig. 3.

The correlation between |V+AV+A| and |m| at the fixedvalues ofc1= 1, 0, and 1 is shown in Fig. 10. Inorder to illustrate the fact that the results dependsignificantly on the theoretical uncertainty in the nu-clear matrix elements [28, 38], we use, as before,two versions of the QRPA model: without and withallowance forpn pairing [25]. One can readily see thata transition from one model version to the other leadspredominantly to a parallel shift of the curves, whichis maximal (minimal) for the 100Mo (136Xe) nucleus.

It is obvious that, in the case of an experiment witha single isotope, the set of two Eqs. (13) or (14) gives

no way to determine three quantities: {||, c0i,||}

or {||, ci, }. For this, it is necessary to measurethe half-lives of three isotopes or the half-lives of twoisotopes and the angular-correlation coefficientKforat least one of them. We consider in detail the second

possibility, supplementing the set of Eqs. (13) or (14)for one nucleus whose angular-correlation coefficientKwas measured with the equation

A = C1||2 + 4Ci||||c0k+ 4Cj ||2 (15)or with the equation

A = C1||2 + Ci||||ck+ Cj ||2 (16)for another nucleus whose coefficientKis unknown.



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2094 ALI et al.

0.04

0.5

130

Te

00

|

m

|

, eV

10

7|

V

+

AV

+A|

0.08 0.12

1.0

1.5

2.0

0.04

0.5

136

Xe

00

|

m

|

, eV0.08 0.12

1.0

1.5

2.0

0.04

0.5

100

Mo

00

0.08 0.10

1.0

1.5

2.0

1

1

100

Mo

00

2 3

2

3

4

0.1

2

76

Ge

00

0.3 0.4

6

8

10

0.05

0.5

82

Se

00

0.10 0.15

1.0

1.5

2.5

0.02 0.06

0.2 0.5

4

10

7

|

V

+

AV

+

A

|

0.20

2.0

Fig. 10.Correlation between|V+AV+A|and|m|atc1 =1, 0, 1(dotted, solid, dashed curves, respectively) for various nuclei;three curves below on the left (above on the right) correspond to the QRPA model without (with) allowance forpnpairing.

From (14) and (16), we obtain

|| =

, || =

, (17)

ck =A C1 Cj

Ci

,

where

=1122 1221, (18) = 122 212, = 211 121

with the coefficients being

11= C1 CjC1/Cj, (19)12= Ci CjCi/Cj, 1= A ACj/Cj ,

21= D1 Dj C1/Cj, 22= Di Dj Ci/Cj ,2= AK ADj/Cj .

Similar expressions can be obtained from Eqs. (13)and (15).

It follows that, having measured the half-lives oftwo isotopes and the angular-correlation coefficientfor one of them, one can calculate, on the basis ofEqs. (17)(19), the effective Majorana mass |m| ||me and the parameters||and ci characterizingnon-SM decay mechanisms.

3. CONCLUSIONSOn the basis of an effective Lagrangian that de-

scribes long-range mechanisms of 02 decay via



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the exchange of a light Majorana neutrino and heavynon-SM particles, we have analyzed half-lives andelectron angular-correlation coefficients for the de-cays of several nuclei that will be used in future ex-periments. A comparison of the results of measure-ments for the above properties of decay of severalnuclei will make it possible to minimize the theoretical

uncertainties in the relevant nuclear matrix elementsand to extract information about a dominant decaymechanism. Measurement of the angular-correlationcoefficient for electrons will permit refining this in-formation [21, 22]. It should be noted that the an-gular distributions of electrons in the two-neutrinobeta decay of the 100Mo and 82Se nuclei have alreadybeen measured in the NEMO3 experiment, which iscurrently under way. In the future, it is planned toperform similar measurements for02decay uponaccumulating sufficiently large statistics of respectiveevents [39]. Other planned experiments that will beable to measure the angular distributions of elec-

trons include SuperNEMO [40], MOON [33, 41], andEXO [42].

ACKNOWLEDGMENTS

We are grateful to the reviewer of this article forenlightening comments.

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