(1)

Revista Mexicana de Física 40, Suplemento 1 (1994) 285-296

Double beta transition mechanismF. KRMPOTlé

Depttf'tamento de Física, Facultad de Ciencias ExactasUniversidad Nacional de La Plata, C.C. 67, HOO La Plata, Argentina

Received 7 January 1994; accepted 17 March 1994

ABSTRACT. After briefly reviewing /3/3decay as a test of the neutrino mass, 1 examine the nuclearstructure involved in this process. Simple formulas (a la Padé) are designed for the transitionamplitudes and the general behavior of /3/3decay amplitudes in the quasiparticle random phaseapproximation are discussed. Results of a calculation for 48Ca, 76Ge, 82Se, looMo, 128Te and 130Tenuclei are presented, in which the partic1e-partic1e interaction strengths have been fixed by invokingthe partial restoration of the isospin and Wigner SU(4) symmetries. An upper limit of (m.) '" 1 eVis obtained for the effective neutrino mass.

RESUMEN.Luego de una breve revisión del decaimiento /3/3,como sonda de prueba para la masadel neutrino, examino la estructura nuclear involucrada en el proceso. Son diseñadas formulassencillas (a la Padé) para las amplitudes de transición en la aproximación de fases al azar. Sepresentan resultados de un cálculo para los núcleos 48Ca, 76Ge, 82Se, 1ooMo, 128Te y 130Te, en elcual las intensidades de interación partícula-partícula son fijadas invocando la restauración parcialde simetrías de isospin y SU(4) de Wigner. Un límite superior de (m.) '" 1 eV es obtenido para lamasa efectiva del neutrino.

PAes: 23.40.Hc; 21.10.Re; 21.60.Jz

1. INTRODUCTION

The double beta ({J{J) decay is a nice example of the interrelation between the ParticlePhysics and the Nuclear Physics: we can get information on the properties of the neutrinoand the weak interaction from the {J{J decay only if we know who to deal we the nuclearstructure in volved in the process. There are already several well-known reviews on theneutrino physics [1, 21 and therefore, after a brief historical overview, 1 will limit this talkto the nuclear facet of the problem. More precisely, sorne recent developments performedby our group will be summarized.

Because of the pairing force there are approximately 50 nuclear systems in which theodd-odd isobar, within the isobaric triplet (A, Z), (A, Z + 1), (A, Z +2), has a higher massthan its neighbors. Within such a scenario the single {J decay is energetically forbiddenand the initial nucleus can disintegrate only via the {J{J decay. This is a second-orderweak interaction process, similar to electromagnetic processes such as the atomic Ramanscattering and the nuclear 11 decay [31.The modes by which {J{J decay can take place are connected with the neutrino (v)-

antineutrino (Ji) distinction. If v and Ji are defined by the transitions:

n ~ p + e- + Ji

v+n-p+e 1

285

286 F. KRMPOTlé

the decay (A, Z) - (A, Z + 2) can occur by successive {3decays:

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

passing through the intermediate virtual states of the (A, Z + 1) nudeus.Yet thc neutrino is the only fermion in lacking a additionaJly conserved quantum number

that di£ferences between v and v. Thus it is possible that the neutrino is a Majoranapartide, Le., equal to its own antipartide (i> la ,,0).1 When v = v the process

(A, Z) - (A, Z + 1) + e- + v == (A, Z + 1) + e- + v

- (A, Z + 2) + 2e- (3)

is also aJlowed. In absence of the helicity suppression (as would be natural before theobservation of parity violation) this neutrinoless ({3{3ov) mode is favoured by phase spaceover thc two-neutrino ({3{3zv) mode by a factor of 107_109: Tov ~ (1013 - 1015) Y whileTzv ~ (10zo_lOZ4) y. Several searches for the {3{3decay has been made by the early 1950swith the result that T 2: 1017 y. This seemed to point that v # v and prompted theintroduction of the lepton number L to distinguish v from v: L = +1 was attributed toe- and v alHl L = -1 to e+ and v. The assumption that the additive lepton number isconserved then aJlows the {3{3zv decay but prohibits the {3{3ov one, for which C1L = 2.

But with the discovery in 1957 that the parity is not conserved for the weak interactionit was realizcd that the MajoranajDirac character of the ncutrino was still in question. If

n - p + e- + VIII!

VLI! + n - p + e-

(4)

then the second process in (3) is forbidden because the right handed neutrino has thewrong helicity to be reabsorbed. Therefore the !'5-invariancc of the weak interaction couldaccount for no {3{3ov decay, regardless of the Dirac or Majorana nature of the neutrino.Otherwise, this decay can be observed only when the lepton number is not conservedand the neutrino is a massive Majorana particle.2 This cvent discouraged experimentalsearchcs for a long time, but with the development of modern gauge thcories the situationbegan to change. In fact, there are many reasons for the renaissance of interest in (3{3-decayover the past decade. The most important one is that, if thcre is any new physics beyondthe standard SU(2JL x U(I) gauge model of electroweak interactions, the {3{3ov decay willplaya crucial role in shaping the ultimate theory. Moreover, no solid theoretical principieprccludcs ncutrinos from having rnass and the 1Il0st attractive cxtcnsions of the standard

1 A Dirac particle can be viewed as a combination of two Majorana particles with equal rnassand oppositc CP propertics and thcir contribution to the {3{3o",uceay cancel.

2 \Vc assurne fo! simplicity that weak interactions with right.hamlcd currcnts do not play anessential role in the neutrinoless mode.

DOUBLE BETA TRANSITION MECHANISM 287

model require neutrinos to be massive. The theory is neither capable of predicting the scaleof neutrino masses any better than it can fix the masses of quarks and charged leptons.5ince the half-lives can be cast in the simple form

(where g' s and M' s are, respectively, the phase space factors and nuclear matrix elementsand (mv) is the effective neutrino mass) it is clear then that we shall not understand the{3{3ov decay unless we understand the {3{32v decay. The last is the slowest process observedso far in nature and offers a unique opportunity for testing nuclear physics techniques forhalf-lives ~ 1020y. Thus, the comprehension of the {3{3transition mechanism cannot buthelp advance knowledge of physics in general.lt is worth noting that more than 30 {3{3decay experiments are underway or in stages of

planning and construction. Until now positive evidence of the {3{32v decay mode has beenfound for 76Ge, 825e, 100Mo,128Te,130Teand 238U.Yet, despite the colossal experimentalprogress the neutrinoless, lepton violating decay, if it. exists, has escaped detection untilnow,3The {3{3decays occur in medium-mass nuclei that are rather far from closed shell5, and

we all know that shell-model calculations are practical only when the number of valencenucleons is relatively small. Therefore, at the present time, the nuclear structure methodmost widely used is the quasiparticle random phase approximation (QRPA). Within thismodel the {3{3-decayamplitudes are very sensitive to the interaction para meter in theparticle-particle (PP) channe!, usually denoted by gPP. It is still more interesting that,close to the expected value for gPP, the {31}matrix elements go to zero. But when aphysical quantity has a zero (or near zero) a conservative law should, very likely, be atits origino Thus, resorting to a toy model, 1 will first discuss the general behavior of thenuclear matrix elements [5, 6]. Later, 1 will show that they have zeros because of therestoration of the isospin and Wigner 5U(4) symmetry [4, 7]. This is not surprising sincethe Fermi (F) and Gamow-Teller (GT) operators Tolo and aTx, relevant in the (3{3decay,are infinitesimal generators of 5U(2) and 5U(4), respectively. Finally, we use the conceptof restoration of these symmetries to fix the PP interaction strengths and to estimate the(3{3matrix elements. lt can be argued that, the 5U(4) symmetry is badly broken in heavynuclei and that therefore our recipe is quite arbitrary. 1 will show, yet, that the residualinteraction is capable to overcome most of the SU(4) breaking caused by the spin-orbitsplitting.

2. GENERAl. BEIIAVIOR OF M2v AND Mov IN TIIE QRPA

Independently of the nucleus that decays, of the residual interaction that is used, and ofthe configuration space that is employed, the {3{3-momentsas a function of gPP alwaysexhibit the following features:(i) The 2v moments have first a zero and latter 00 a pole at which the QRPA collapses.

3 The {3{3ov sensitivities have chaoged from - 5 x 1015 in 1948 to - 5 X 1023 in 1987.

288 F. KRMPOTIé

0.04 -- ••••• ::~::: •••••••••••••••••••••••• __ •••.'."_ .._ .. _ .. _ .. _ ..,

N 0.0::<1

-o."

-0.8

48eH82Se

7liCe

128Te

100Mo1:10Te

, 5 ... _.. ......

" 10o::<I 5

O0.0 0.5

-"-1.0 1.5 2.0

FIGURE 1. Calculated matrix elements M2v (in units of [MeVI-1), the Ov moments for J' = 1+(Mov(J' = 1+)) and total moments Mov as a function of the partide-partide S = 1, T = Ocoupling constant t.

(ii) The zeros and poles of Mov for the virtual states with spin and parity J. = 1+ arestrongly correlated with the zeros and poles of M2v.(iii) The total {3{3ov moments also possess zeros but at significantly larger values of gPP.The behaviour of the {3{3moments for several nuclei is illustrated in Fig. 1. These

results have been obtained with a {j force, using standard parametrization [71. Instead ofthe parameter gPP, I use here the ratio between the triplet and singlet coupling strengths inthe PP channel, Le., t = vfP Iv~p. Calculations with finite range interactions yield similarresults [21.I will resort now to the single mode model (SMM) description [5, 6] of the {3{3-decays

in the 48Ca -; 48Ti and lOoMo -; lOoRu systems. This is the simplest version of theQRPA with only one intermediate state for each r. It allows to express the moments ala Alaga [8]' ¡.e., as the unperturbed matrix elements Mgv and M8v(J+) multiplied bythe effective charges:

(5)

(6)

DOUBLE BETA TRANSITION MECHANISM 289

-0.70 -100 - -Mo

0.35 48Ca"N

::E 0.00I

-0.35

-0.700.5

- -- - - ...•• ...••

1.5

FIGURE 2. Exacl (solid lines) and SMM (dashed lines) malrix elemenls M2v (in unils of [Mevt1 J.as a funclion of lhe coupling conslanl tito. lo is lhe value of t for which M2v is null.

Here G(J+) == G(pn,pn; J+) are lhe PP matrix elements (proportional to t (or to gPP)),wO is the unperturbed energy, and WJ+ are the perturbed energies. It will be assumed herethat the isospin symmetry is strictly conserved, in which case (as it will be seen latter on)M2v(O+) = Mov(O+) == o. When the pairing factors are estimated in the usual manner,onc gcts

(7)

for the single pair conligurations [Oh¡2(n)Oh¡2(p)IJ+ in 48Ca and

W = woJI + 4F(45 + F/wO)/225wo + G(270 + 172F/wo + 49G/wO)/225wo, (8)

for [Og7¡2(n)Og9¡2(p)lJ+ in 10°1'.10. Therefore, while the numerators in Eqs. (5) and (6)depend only on the PP matrix elements, their denominators depend on thc particle-hole(PH) matrix elements F(J+) == F(pn,pn; J+), as wcl!. The numbers in the last twocquations arise from the pairing factors. As illustrated in Fig. 2, the SMM is a fair lirst-order approximation for thc /3/32v decays in 48Ca and 10°1'.10 nuclei.

The role playcd by thc ground state correlations (GSC) in building up Eqs. (5) and (6)can be summarized as fol!ows:

(a) The numerator, i.c., thc factor (1 +G /wO), comes from the interferencc between theforward and backward going contributions. Thcsc contribute coherently in thc PP channeland total!y out of phasc in thc PH channcl.

(b) The G2 and F2 tcrms in thc denominator arc vcry strongly quenchcd by the GSC,while the GF tcrm is cnhanced by the samc effect. In particular, for 48Ca the ter mquadratic in G does not contri bu te at al!.

It can be stated thereforc that, within thc SMM and because of the GSC, the M2vmatrix element is mainly a bilinear function of G(I +). Besides, it passes through zero alG(I+) = -wo and has a pole when WI+ = o. Similarly, al! MOv(J+) moments turn out to

290 F. KRMPOTlé

TABLE J. The "'18"(J+) momenls and lhe factors G(J+)/wo wilhin lhe single mode lllodel for'"Ca and 100Mo.

"Ca 100Mo

J -.A18v -G(J+)/wo -.iVf8l.' -G(J+)/wo

O 1.0159 s

13439 Uf 2.8316 lQt21 27

2 0.1573 5 0.2741 ~(t + f,.')21$

3 0.2143 1J!t 0.2086 13077 69;1 t

4 0.0446 9 0.1263 162 ( 20)nS 1001 t + 8Ts

5 0.1081 235 0.0585 12.1 tTIiOT t 1287

6 0.0122 25 0.0842 20( 14)4295 ili t + 2"75

7 0.0988 lit 0.0177 190 t429 3861

8 0.0925 490 (t R)2431 + 95

he quotients of a linear function of C(J+) and lhe square root of another linear funclionof C(J+). Both the zero and the pole of MOv{1 +) matrix element coincide with thoseof the 2v momento Besides, as the magnitudes of C( J) and F( J) decrease fairly rapidlywith J (see Table 1), the quenching e!fect, induced by the PP interaction, mainly concernsthe allowed Ov momento For higher order multipoles it could be reasonable to expand thedenominator in Eq. (6) in powers of C(J+)/wo and to keep only the linear termo Thisterm strongly cancels with a similar tenn in the numerator and the net result is a weaklinear dependence of the Mov(J+ O; 1+) moments on the PP strength. Obviously, for thelast approximation to be valid, the parameter t (or gPP) has to be small enough to keepWI+ real. Briefly, the SMM can account for all four points raised aboye, and leads to thefollowing approximations

and

~ 1 - titoM2v = M2v(t = O) 1 _ t/II '

~ ,+ 1 - l/loMov = Mov(J = 1 ; t = O) /VI - I I1

+ Mov(J' O; 1+; I = 0)(1 - 1/12),

(9)

(10)

where I1 2: lo and 12» 11, and the condition I ~ II is fulfilled. It is self evident that theseformulae do not depend on lhe type of residual interaction, and that analogous expressionsare obtained when the parameter gPP is used (wilh gPP's for I's).The common behavior of the /3/3 moments for all nuelei, together with the similarity

between the SMM and the full calculations for 48Ca and 100Mo (shown in Figs. 1 and 2,respectively), suggests to go a step further and try to express the exact calculations within

DOUBLEBETATRANSITIONMECIIANISM 291

TABLE 2. The coelficienls to, t" and t2 and lhe malrix elemenls M2", Mo"(J' = 1+), andMo"(J' f- 1+) Cor t = O, in lhe paramelrizalion oC lhe 2v and Ov /3/3momenls. The malrixelemenls M2" are given in nnils oC [MeV¡-'. The valnes oC lhe PP coupling slrenglhs, which leadlO maximal resloralion oC lhe SU(4) symmelry (t = I.ym), are shown in lhe lasl row.

"Ca 76Ge 82Se ,00Mo 128Te IJOTe-M2t1 0.173 0.308 0.321 0.451 0.381 0.331to 1.394 1.161 1.206 1.469 1.265 1.2611, 1.754 1.680 1.691 1.649 2.131 2.268

-Mo"(J' = 1+) 1.506 4.242 4.179 5.015 4.599 4.182-Mo"(J' f- 1+) 1.501 6.924 7.495 9.762 7.997 7.486

lo 1.227 1.155 1.141 1.372 1.377 1.4071, 1.768 1.741 1.764 1.711 2.236 2.345t2 12.82 13.23 12.14 6.527 13.39 11.08

t!lym '" 1.50 ~ 1.25 '" 1.30 ~ 1.50 ~ 1.40 '" 1.40

the framework of Eqs. (9) and (10). At a lirst glance this seems a diflicult task, beca use:(i) lhe SMM do es nol include the effecl of lhe spin-orbit splitting, which plays a veryimportant role in the f3f3-decay through lhe dynamical breaking of lhe SU(4) symmetry,and (ii) the full calculations involve a rather large conliguration space (of the order of 50basis vectors).The paramelers lo, 1" and 12 that lit lhe f3fJ moments displayed in Fig. 1 are lisled

in Table 2, logether wilh momenls M2", Mo"(J' = 1+), and Mo"(J+ i 1+) for I = O.The reliability of formulae (9) and (10) is surprising, lo lhe extent that it is not possibleto distinguish visually the exact curves from the litted ones. As illustrated in Fig. 3, thissituation persists even within the number projected QRPA [9]. Why the exact calculationscan be accounted for by Eqs. (9) and (lO)? 1 do not know a fully convincing answer. Yet,let me note that for a n dimensional conliguration space, M2" can always be expressedby the ratio of the polynomials of degrees 2n - 1 and 2n in C(1 +) [4), ¡.e.,

(U)

Thus the above results seem to indicate that cancellations of the type (a) and (b) are likelyto be operative to all orders, and that linear terms in C(J +) are again the dominant ones.General expressions for Mo", analogous to (U), are not known, but some cancellationmust be taking place in these as well.

3. RESTORATION OF TIIE ISOSPIN AND SU(4) SYMMETRIES

An important question in the QRPA calculations is, how to lix gPP or I? Several attemptshave been made to calibrate gPP using the experimental data for individual GT positron

( 12)

292 F. KRMPOTlé

decays. The weak point of this procedure is that the distribution of the /3+ strength amonglow-Iying states in odd-odd nuclei is certainly affected by the charge-conserving vibrations,which are not included in the QRPA. For exalllple, the single beta transitions lOoTc -lO°/vlo and lOoTc _ lOoRu have been discussed recently in the standard QRPA [10]' wherethe looTc states are described as pure pn-quasiparticle excitations, while the suggestedwave function for the ground state in 1OI/vIois (cL ReL [11i) is only "" 35% of quasiparticlenature:

11/2+) = 0.59181/2,00) - 0.57181/2,20) + 0.32181/2,40)

-0.26Id5/2,22) - 0.26Idl/2, 42) - 0.21Ig7/2, 24).

(In the basis state Ij, NI) the quasiparticle j aJl(1 the N bosons of angular 1Il0mentum 1are coupled to the total spin 1/2.)\Ve gauge t by resorting to the resto ratio n of the \Vigner SU( 4) symmetry [41.Unlike the

rnethod mentioned aboye, this rnethod involves the total GT strength, which dependent ofthe charge-conserving vibrations only very weakly. \Ve are aware, however, that the SU(4)sYlllmetry is badly broken in medium and heav)' nuclei, and therefore before proceeding,it is necessary to spedfy what we mean by reconstruction of this synunetry.For a system with N # Z, the isospin and spin-isospin symrnetries are violated in

the mean field approximation, even if the nuclear hamiltonian commutes with the corre-sponding excitation operators /3" (T'f and aT'f)' l3ut, we know that when a non-dynamicalviolation occurs in the I3CS-Hartree Fock (I3CS-IlF) solution, the QRPA induced GSCcan be invoked to restore the symmetry. There are subtleties involved in the restorationmechanism: the GSC are not put in evidence explicitly, but only implicitly via their effectson the one-body moments /3" between the groUllll state and the excited states. l3esides, forthe F excitations and when the isospin non-conserving forces are absent, a self-consistentinclusion of the GSC leads to the following:1) all the /3- strength is concentrated in the collective state, and2) the /3+ spectrull1, which in QRPA can be viewed as an extension of the /3- spectrull1

to negative energies, is totally quenched.Tlle self-consitency is only attained when the same S = O, T = 1 interaction coupling

strcngths are used in the pairing and PP channels, i.e., when Vféur = vfP 1 and the extent lowhich the aboye conditions are fulfilled may be taken as a measure of the isospin sYlllmetryrestoration. In Fig. 4 is shown the behavior F strength /3+ as a function of the para meters = vfP /vrair.l3esides being spontaneously broken by the IlF-I3CS approximation, the SU(4) sYll1me-

try is also dynamically broken by the spin-orbit field and the superlllultiplet destroyingresidual interactions. l3ut, the last two effects have a tendency to cancel cach other. Infact, within the TDA the energy differences betwecn the GT and F resonances can beexpressed as [12]

EGT - El' = [2.1, - (vrh - v~h) N2~ Z] /vIeV,

where 2.1, "" 20A-1/3 is the mean spin-orbit splitting and v~h and vrh are, respectively,the singlet and the triplet coupling constants in the PH channe!. As vrh > vrh thc residual

DOUBLEBETATRANSITIONMECHANISM 293

0.4 76- - -- --- Ge0.2

,, ,N

E 0.0 ,I ,,

-0.2 ,,\

\\-0.4 \\

0.0 0.5 1.0 1.5

FIGURE3. Calculated double beta decay matrix elements M2" (in units of [MeVj-l) for 76Ge, asa function of t. Solid and dotted curves correspond to the projected (PQRPA) and unprojected(QRPA) results, respectively.

interaction displaces the GT resonance towards the lAS with increasing N - Z. What ismore, the energetics of the GT resonances are nicely reproduced by (see Fig. 5)

(N - Z)EGT - EF = 26A-1/3 - 18.5 A MeV, (13)

which has the same mass and neutron excess dependence as (12). Briefly, the experimentaldata show that the SU(4) symmetry destroyed by the mean field is partiaUy rcstored bythe residual interaction.4 The GSC are likely to alter Eq. (12) very little. But, withinthe QRPA the <J"T+ transition strength is strongly quenched and the GT resonance issomewhat narrowed, as compared with the TDA results. As such the global effect ofthe pn residual interaction on the GT strengths {3I. (<J"T",) is qualitatively similar to thecorresponding effect on the F strengths {3I. (T",), in the sense that the conditions 1) and 2)are approximately fulfiUed, and we say that the SU(4) symmetry is partiaUy restored. ltseems reasonable then to assume that the maximal restoration is achieved for the value oft where the GT strength {3+ is minimum, and this is the way how we fix the parameter t.

4. RESULTS FOR M2" AND (m")

From the results displayed in Table 3, it can be said that with t = tsyrn the calculated M2"moment for 48Ca does not contradict the experimentallimit and that the 2v measurementin 82Se is weU accounted for by the theory. On the other hand, the calculated 2v matrix

4 Neither in 208pb, where the GT strength ls located at the energy of the lAS, the SU(4)symmetry is totally restored, as indicated by the resonance width of '" 4 MeV.

294 F. KRMPOTlé

0.3

2.0

1.5

,,,,

1.5

126Tt'

130T

t" •. _._._._'.

1.0

<:'\.\101>-TELLEH

FEH\lI

1.0

. .~.:. ..;:."' ..•.. .:. ...

0.5

---IOO~h)

0.5

":'.0.2

0.1 -:::. ••..••..... :.:.:.:••. o",

:.;:-' :.".- :.:':':':-:'::':::":,:, ..c 0.0

~t 0.0•-•I 0.8•

""-0.6

0.4

0.2

0.00.0

FIGURE4. Fermi and Gamow- Teller transition strcngths {3+ for the nuclei 48Ca, 76Ge, 82Se, lDoMo,128Teand !JoTe, as a funetion of particle-particle eouplings s (S = O, T = 1) and t (S = 1, T = O)respeetively.

TABLE3. Experimental and ealculated 2v moments for t = t,ym (in units of [MeVj-I) .

4.Ca 76Ge •2Se 1DOMo 128Te ¡3OTe

IM2.["'P < 0.081 O 280+0.006 0141 +0.00' O 294+0.029 0.038~gg: 0027~gg:. -0.010 .. -0.014 .. -0,033

Meal 0.091 0.100 0.121 0.102 0.118 0.0962.

elements tum out to be too small for 76Ge and 100Mo and too large for 128Te and ¡30Te(in both the cases by a factor of '" 3). Yet, one should bear in mind that: i) the ealeulatedvalues of M2" vary rather abruptly near t = t,ym and therefore it is possible to accountfor the M2" in all the cases with a comparatively small variation « 10%) of t, and ii) theminimum value of the GT (3+ strength critically depends on the spin-orbit splitting overwhieh we still do not have a complete control.

Besides the issue of the procedure adopted for fixing the partide-partide strength pa-rameter within the QRPA, there are some additional problems in calculations of the matrixclement M2", as yet not fully understood. They are related with the type of force, choiceof the single partide speetra, treatment of the differenee between the initial and final nu-clei, etc. AH these things are to some extent uncertain and therefore it is open to questionwhether it is possible, at present, to obtain a more reliable theoretical estimate for the 2vhalf lives that the one reported here.The upper limits for the effective neutrino mass (m"), obtained from the measured

Ov half.lives and the calculated matrix elements are shown in Table 4, where also are

DOUBLEBETATRANSITIONMECHANISM 295

4e z,

í~ L! • Nb• Me

• 2 • ~: n ; , So

:> 3

LX Tmo Pb~

::i

1 f~• Q~ 2w,

! a~~

Iw

~o0.10 0.12 0.14 0.16 0.18 0.20 0.22

(N-Z) lA

FIGURE5. Plot of EGT - EF versus (N - Z)/A. When the experimental results overlap (for 90.92Zrand 20SPh) we displace them slightly with reseet to the eorreet value of (N - Z)/A for the sake ofclarity. The values ealculated by Eq. (12) are indicated by full eireles.

TABLE 4. Upper bounds on the effective neutrino mass (mv) (in eV) ohtained from the QRPAealculations oí the nuclear matrix elements. Far the sake oí comparison, in aH the cases the sameexperimental data, as well the same effective axial vector coupling constant (gA = -gv) have beenused.

'SCa 76Ge "Se IooMo 12sTe 1JOTe

Pasadena (reL [13]) 4.4 20 20 1.8 22Heidelberg (reL [14]) 22 2.0 7.4 26 1.5 21Tübingen (reL [15]) 3.1 12 3.8 31our resuIts (reL [7]) 71 1.5 5.3 8.8 1.0 12

presented the results obtained by other groups. The difference in a factor of about 2-3between both: i) the results obtained by the Pasadena group and the groups of Tübingenand Heidelberg for T6Ce and 82Se nuclei, and ii) lhe previous and present calculationsfor lOoMo, ¡28Te and ¡30Te nuclei, is just a refleetion of the unavoidable uneertainty oflhe QRPA calculations, and it is difficult to assess which one is "better" and which is"worse" .

ACKNOWLEDGMENTS

1 am grateful to the organizers of the workshop for their kind invitation and hospitality,and in particular lo Professor Alfonso Mondragon.

296 F. KRMPOTlé

REFERENCES

1. See W.C. Haxlon ano G.J. Stephenson, Prog. Parto Nud. Phys. 12 (1984) 409; M. Doi, T.KOlani and E. Tagasuki, Prog. Theor. Phys. Suppl. 83 (1985) 1; F.T. Avignone III and R.L.Brodzinski, in Neutrinos, ediled by H.V. Klapdor (Springer-Verlag, New York, 1988), p. 147;K. Mulo and H.V. Klapdor, ibid. p. 183; F. Krmpolié, in Lectures on Hadron Physics, Ed. byE. Ferreira (World Scienlific, Singapore, 1990) p. 205.

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