Z. Phys. A 347, 151-160 (1994) ZEITSCHRIFT FURPHYSIKA �9 Springer-Verlag 1994

Nuclear structure calculation of/]+//+,//+/EC and EC/EC decay matrix elements M. Hirsch 1, K. Muto 1, T. Oda 1, H.V. Klapdor-Kleingrothaus 2

Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo, 152 Japan 2 Max-Planck-Institut ffir Kernphysik, Postfach 103980, D-69029 Heidelberg, Germany

Received: 6 September 1993

Abstract. Nuclear matrix elements for double positron emisson ~+/?+), positron emission/electron capture (/?+/ EC) and double electron capture (EC/EC) in the 2v/?/? decay mode and for fl+/~+ and/~+/EC decay in the 0 v/?/~ mode are calculated for the experimentally most promis- ing isotopes 58Ni, 78Kr, 96Ru, l~ 124Xe, 13~ and 136Ce within pn-QRPA. We point out that the matrix element for the 2vfl+/EC decay differs from the 2vfl+fl + matrix element, an effect not considered previously. For the neu- trino accompanied decays our calculation predicts for the fl+/EC and the EC/EC mode half lives which are shorter typically by 4-7 orders of magnitude than those for the double positron emission. However, even for the best candidates typical values for 2vfi+/EC (2vEC/EC) are still in the range of ~ 1022 ((some) 1021) years. For 0 v/~/~ decay we have calculated all matrix elements rele- vant for both, the mass mechanism and the right-handed currents for the first time complete. A detailed discussion of the differences between the 0v/~+/q +, the Ovfl+/EC and Ovfi-fl decay is given.

PACS: 21.60; 23.40.

1. Introduction

Double beta decay is known as a sensitive tool to explore neutrino properties ever since Furry [1] suggested the neutrinoless double beta decay mode (0 vBB) as a pos- sibility to decide whether the neutrino is a Dirac or a Majorana particle. Though only experiments can finally prove which description of the neutrino is correct, double beta decay has also been subject of much theoretical work since quantitative information about, for example, the effective neutrino mass requires a reliable calculation of nuclear matrix elements.

Lower limits on half lives for 0 vBB decay have been improved considerably [2-7], but up to now no conclu- sive evidence for 0 vBB decay has been reported. On the

other hand, experimental progress has finally led to the observation of 2 v,aB decay for the isotopes 76Ge [2, 3], 82Se [41, l~176 [5], 128'13~ [6, 71 and 238U [8] - all of which a r e / ? - / ? - decays. However, there exists no posi- tive evidence for any/~ +/~ + decay.

The main reason why/1 - p - decay has attracted much more interest is simple: Even the largest Q - v a l u e for double positron emission, Qp + p + (t24Xe)--- 1.02 MeV, is low compared to those of the best B - / ~ - decay candi- dates. This results - due to the strong dependence of the phase space on the available energy - in half life esti- mations far out of reach of present day experiments [9].

In any case, if double positron emission is possible energetically, electron capture always acts as a competi- tive process and in double beta plus decay we therefore have to deal with three different possible decay modes,

2v/?+/?+: (Z,A)--~(Z-2, A)+2e+ +2Ve, (1.1a)

2v/?+/EC: e - +(Z,A)-- , (Z-2, A)+e + +2re, (1.1b)

2vEC/EC: 2 e - + ( Z , A ) ~ ( Z - 2 , A )+2v e. (1.1c)

In the following we will refer to the fl + /EC decay more briefly as the mixed mode. The most important difference between processes (1.1 a)-(1.1 c) is perhaps their different Q-values,

Q~+p+ = M ( A , Z ) - M ( A , Z - 2 ) - 4 m e c2, (1.2a)

Q~+/Ec-=M(A,Z) -M(A,Z-Z) -2mee 2, (1.2b)

Q~cmc = M (A, Z) - M (A, Z - 2). (1.2c)

Therefore, even for I24Xe the Q-value for EC/EC decay is larger than the one for double positron emission by a factor of ~3. Shorter half lives for processes (1.1b) and (1.1 c) are thus expected.

2 vflB decay half lives for double positron emission have been calculated earlier by phase space estimations [9]. Later on, Staudt et al. [ 10] calculated 2 vB + B + half lives within the pn-QRPA model of [11, 12], essentially confirming the pessimistic estimations of Haxton and Ste-

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phenson [9]. Kim and Kubodera [13] estimated half lives also for the electron capture modes within a relatively simple model, assuming a constant nuclear matrix ele- ment. Recently more advanced phase space estimations for the mixed mode and the EC/EC mode have been published by Doi and Kotani [14]. Though in the work of [14] the important nuclear matrix elements have not been calculated, their results indicated that the half lives for 2v/~+/EC and 2vEC/EC decay might be much shorter than those for double positron emission and prob- ably be within reach of advanced/~B decay experiments. However, in order to draw firm conclusions we have to calculate the nuclear structure matrix elements. This is the first purpose of this paper.

In this context we would like to mention that recently we received a preprint by Suhonen [15], dealing with the calculation of matrix elements for 2vEC/EC decay. However, our present work differs from this one in two respects. The first one is that, while [15] deals only with the neutrino emitting mode of EC/EC decay, we consider also the mixed mode as well as the neutrinoless modes. More important is that our results for the 2vEC/EC decay differ considerably from those given by Suhonen [ 15], so that we arrive at a somewhat different conclusion.

As mentioned above, one of the most interesting decay modes of double beta decay is the 0v~/? decay. From 0 vBB decay not only information on the effective neu- trino mass can be obtained, but it could also be possible if the weak interaction has a right-handed component (RHC) [9, 16, 17]. Matrix elements and phase spaces for the RHCs differ from those for the mass mechanism. To obtain complete information it is necessary to calculate various nuclear matrix elements.

Also 0 v ~ + ~ + decay matrix elements for the mass mechanism have been calculated in [10], but no calcu- lation for the matrix elements for the RHCs exists up to now in the literature. Moreover, for the more interesting 0 vB +/EC decay mode to the best of our knowledge there exists no calculation. The complete calculation of the decay rate coefficients for these both modes is the second purpose of the present paper.

Though experimental limits on 0 vB+fl + and 0 vfl+/EC decay are much weaker than those of the best / ? - / ? - decay candidates (and hence are derived limits on the effective neutrino mass) one important result of the pre- sent work is an interesting enhancement effect in the (,~)-terms of the RHCs in 0 vfl +/EC decay. A detailed discussion will be given in Sect. 3.3. However, the situ- ation can be summarized by stating that the 0 vfl +/EC decay is - similarly to the 0 vfl - fl - (0 + ~ 2 +) transitions - much more sensitive to the RHC mode than to the mass mechanism, but having much shorter half lives than those expected for 0 + ~ 2 + transitions. On the other hand, as discussed in Sect. 3.3, the same enhancement effect will make it markedly difficult to determine exact limits on the neutrino mass and right-handed parameters from the nonobservation of 0 vfl +/EC decay alone; additional in- formation from 0 vfl +fl + or 0 v f l - f l - decay will be re- quired.

This paper is organized as follows: in Sect. 2 we con- sider the 2 v decay mode. Matrix elements are calculated

within the pn-QRPA formalism [11, 12]. Predictions of 2vflf l decay half lives are given for all three possible neutrino emitting modes for the isotopes 5*Ni, 78Kr, 96Ru, l~ 124Xe, 13~ and 136Ce. By the special example of l~ however, we would also like to discuss the limi- tations of present 2 vflfl decay calculations. Section 3 is then concerned with the calculation of the matrix ele- ments for the neutrinoless modes, followed by a discus- sion of the implications of our results on the determi- nation of the neutrino mass and the right-handed param- eters. We then close with a short summary and outlook.

2. 2 v,fl/? decay

2.1. General considerations

For 2 v/~/~ decay the inverse half life can be expressed as [16, 17],

[T~ 2 v 1 - 1 - a2v ~' ~' (2.1) 1/2J in(2) f do2 Y, Ao, o,, a,a"

where a2~ =(GgA)4/(32rcTh) and the relevant nuclear structure information has been confined into the coeffi- cients A,,~,, see below, dt2~v stands for the leptonic phase space and e distinguishes between the different decay modes e = fl + B +, 13 +/EC or EC/EC. Phase spaces can, in principle, be calculated unambigously and also very accurately [14, 16]. We follow essentially the descrip- tion of Doi and coworkers and refer for brevity to their original papers [ 14, 16].

Here instead we would like to discuss one aspect of the nuclear structure calculation in some detail; this is the matrix elements of the 2 vfl +/EC and the 2 vB + B + decay are different contrary to previous belief. The co- efficients A~,,, in (2.1) can be expressed as

A2, ., =(0ff lit+ ~11 1,+ > <la + II t+ II0+ >

[I t+ II 1+ ) ( 1+ II t+ II 0+ )

• (KoKo, + G G, + 1KoL., + �89 (2.2)

where l0 + ) and (0}1 are the wave functions of the initial and final nuclear states and one has to sum up over all intermediate states I1+). The factors K a and L, are the typical energy denominators of the second-order perturbation and for double positron emission given by

1 1 Ka + (2.3a)

E , - - E i + e l + V l Ea- -E i+e2+v 2

1 1 L. - }- (2.3b)

E a - - E i + e l + v 2 Ea- -E i+ea+v 1

For a captured electron, however, we have to replace one (continuum) positron energy by the energy of the cap- tured electron, ex--* - e c. This leads to a smaller energy denominator for the mixed mode than for the other two modes and consequently to an enhancement of the matrix element for fl +/EC decay, as can be best seen with the help of the following consideration.

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Assume that the lepton energies in the denominators can be replaced by e + v "~�89 Q + me, irrespectively of the indices. Then one can separate the phase space integra- tion and the calculation of the nuclear matrix elements, leading to the well-known expression

2v - 1 _ F2V ~M ~2 (2.4) [T1/21 - - ~, G T Y '

where

MG~-=2 <~176 llt+~176 (2.5) a E , - E i + � 8 9

and F 2v as the leptonic phase space integral. Note that by this assumption the four possible combinations of de- nominators have become equal. For ( E , - E~) >> e~ + v~ we expect that the half life calculated by (2.4) approaches that of the exact (2.1). On the other hand, especially if there is a large cancellation among the different terms in (2.5) deviations between the results of (2.1) and (2.5) have to be expected. Moreover, for the mixed mode this ap- proximation will not be well satisfied since the energy denominators are asymmetric with respect to the number of leptons in continuum states. In order to give an equiv- alent expression also for the mixed mode it might be possible to assume that the available energy is shared equally between the three emitted leptons. (Then the de- nominator containing the captured electron will be smaller (�89 Q) than the one with the positron in the continuum state (2 Q).) Numerically, however, we found that though this approximation better reproduces the result of the exact calculation (2.1) there still remain some differences. In the calculation of the half lives we therefore carried out the integration over leptonic energies in (2.1) for each calculated intermediate state exactly.

2.2. Matr ix elements calculated in QRPA

Nuclear matrix elements and the resulting half lives have been calculated within pn-QRPA (proton-neutron quasi- particle Random phase approximation), pn-QRPA has extensively been applied to the calculation of 2 v/~- /?- decay half lives throughout the past few years [11, 12, 17-25]. We will therefore not repeat the details of the model here but briefly summarize some main results.

The RPA of charge-changing transitions has been de- veloped by Halbleib and Sorensen [26]. Early calcula- tions, however, mainly based on the shell model (for a review of theoretical fib decay calculations predating 1984 see [9]), overestimated the 2 v decay rates by quite large factors. Vogel and Zirnbauer [18] then could show that the inclusion of the particle-particle force, first considered by Cha [27], leads to a strong reduction of the 2 vBB decay matrix element in better agreement with experi- mental data. In this work [18] a zero-range interaction was taken, however, Civitarese et al. [9], using the Bonn potential [28], proofed that the suppression mechanism persisted also for more realistic nucleon-nucleon inter- actions. Muto et al. [11] then, by a careful fit of the strength of the particle-particle interaction to single/~ + decay data, were able to calculate 2 v f l - B - decay half

lives for the experimentally most interesting isotopes consistent with existing measurements.

The strong dependence of calculated 2 v BB decay ma- trix elements on the particle-particle interaction initiated several subsequent studies employing various refine- ments. For example, in [23] the effects of a particle-num- ber projection on the QRPA calculation has been studied, whereas Staudt et al. [25] calculated 2 vB - B - decay ma- trix elements including effects of core polarization and the so-called folded diagrams in the effective interactions. This latter work [25] showed that the use of different effective interactions does not drastically affect calculated 2 v matrix elements. However, the principle features of the earlier 2 v calculations remained basically unchanged in these studies [23, 25].

In our present work we follow essentially the descrip- tion of Muto et al. [11, 17]. Single particle energies are obtained from a Coulomb-corrected Woods-Saxon po- tential, with the parameters taken from Bohr and Mot- telson [29]. Typically we account for two major oscillator shells in the numerical calculation. A realistic interaction, the Paris potential [30, 31], is used consistently in both the BCS and the RPA calculations. We adjust the strengths of the pairing interactions g~air and gPair such that the experimental pairing gaps [32] are reproduced. For the strength of the particle-hole interaction gph we take the empirical formula gph = 1 + 0.002 A [ 11 ], which leads to a better reproduction of the excitation energy of the Gamow-Teller giant resonance. The fit of gph, how- ever, influences the final 2 v matrix element only very weakly [11, 17]. For the choice of the more important particle-particle strength parameter gpp see the discussion below.

Since the earlier B - B - decay calculations [11, 12, 17-25] have shown that the particle-particle interaction plays a crucial role for the calculation of 2 v B B decay half lives, the influence ofgpp on the fl + B + decay matrix elements has been investigated in detail. A typical result is displayed in Fig. 1 for the example of 13~ For the plot gpe has been varied between 0 and 1. Note however, that such a free variation has no physical meaning; it should be purely understood as a study of the model parameter dependence of the final result. For the sepa-

I

�9

0.8

0 . 0

--0.4

~-.-- _ _ _ _ ~ 13OBa

--.~-...

\,'x

0 ' . 5 1 . 0

gpp

Fig. 1. Calculated 2 v B + B "- (full line) and 2 v B +/EC (dashed line) decay matrix elements for the representative example of ~3~ as a function of the particle-particle strength parameter gpp. The results obtained are similar to those of previous 2 vB fl- decay calcula- tions

154

ration of the nuclear matrix elements and the leptonic phase space integration we assumed for the B + B + matrix elements the common (�89 Qpp + me)-approximation for the average leptonic energies (full line), whereas for the mixed mode the available kinetic energy was distributed equally among the 3 emitted leptons (dashed line).

In agreement with the expectation, matrix elements are decreasing functions of gpp. At a certain value of gpp the matrix elements vanishes completely. In the most probable range of gpp calculated matrix elements are strongly suppressed, in agreement with the results of ear- lier 2 vfl - B - decay calculations.

Moreover, Fig. 1 shows the relative enhancement of the mixed mode matrix element compared to the fl + B + decay matrix element. From the discussion in Sect. 2.1 it is clear that both matrix elements would become equal in the limit (E a - E i ) ~ o o . On the other hand, the lower the average excitation energy in the intermediate nucleus is the more enhanced will be the mixed mode matrix element. The enhancement is therefore different for dif- ferent isotopes and dependent on gpp, typical values are in the range of (10-30)%.

While for the isotopes 58Ni, 7aKr, 124Xe, 13~ and 136Ce we consider the p n-QRPA calculation to be reliable, the nuclei 96Ru and l~ revealed also some limitations of the present approach. The case of 1~ shall be dis- cussed in some detail now (quite similar arguments can be applied to 96RH).

Recall that within QRPA a B-transition is described as the creation of a QRPA phonon out off the ground state. The double beta decay matrix element can be de- composed into B - and fl + transitions from the ground states of the mother and daughter isotopes to the excited states in the intermediate nucleus. In QRPA the relevant Gamow-Teller transition matrix elements are then given by

/~ - -transitions:

41,+ It_

- - Z @l l " l l n> - r#" (2.6a) p~n

fl +-transitions :

(la+ It+ 01 o+ )

@11' Iln> - rW upv ), (2.6b) p,n

where @ II ~ [] n) are the reduced matrix elements of o-, X f" and Yf" are the forward and backward going am- plitudes of the RPA phonons, whereas u and v are the occupation amplitudes determined by the BCS calcula- tion.

The RPA assumptions then ensures us that if there are many individual terms contributing with a compa- rable magnitude to the sums in (2.6) the calculation should be reliable and moreover relatively stable against small variations of model parameters. The importance of the ground state correlations (i.e. the Y-terms) for the fiB decay matrix element can also be understood easily.

Though in RPA we assume X>> Y, there will be some contributions from terms where up v, >> Vp un, so that prod- ucts Xvp un and Yup v, can be of comparable magnitude. The enhancement of the ground state correlations by the particle-particle force can therefore lead to a strong sup- pression of the 2 vBB decay matrix element.

On the other hand, the application of RPA might sensitively depend on the choice of model parameters if the sum in (2.6) is dominated by only a few (or even only one) terms. An example for such a case is the isotope l~ For l~ the 2 vBfl matrix element is dominated by a few strong single-particle transitions in gg/2-gT/2 orbitals. We have therefore calculated 2 v matrix elements for ]~ for two sets of input parameters for the Woods- Saxon potential. The first calculation has been carried out with the standard choice of parameters described above, while for the second one only the strength of the spin-orbit force of the Woods-Saxon potential has been changed by ~9% from its standard value [29]. While such a change generally only leads to a small shift in the calculated single-particle levels, the main effect of this variation for 1~ is to decrease the energy difference between the vds/2 and the vg7/2 by about ,,~ 0.3 MeV. The results of these calculations are shown in Fig. 2. Taking the standard Woods-Saxon parameters the 2 vBB decay matrix element of t~ stays nearly constant over a wide range of gpp, but then starts to increase just before the QRPA calculation collapses. On the other hand, by the small change of the strength of the spin-orbit force one gets a sharp decrease of the 2 v matrix element, leading to a strong suppression of MGr around gp~---0.85. The explanation for this unusually strong model parameter dependence is found along the lines discussed above. Un- fortunately, no experimental information on the energies of single-particle levels in this mass region are available, which at least in the case of 1~ would be very helpful. However, as can be seen from Fig. 2, by lowering the strength of the spin-orbit force the calculation yields a suppressed 2 vflfl decay matrix element in better agree- ment with the expectation. The numerical instability of the calculation makes the prediction of a 2vBB decay half life for t~ very difficult; the half life of 1~ (and 96Ru) given in Table 1 should therefore be understood only as a lower limit.

~ " . ,1oo C d J \

2.0 \

1.0

0 . 0 - - -

015 1.0 goo

Fig. 2. Calculated 2 v f l + fl + and 2 v f l + / E C decay matrix elements for 1~ as a function of gp~ for two different sets of input param- eters. For a discussion see text

155

TaMe la, b. Predicted 2 vfl/? decay half lives for the three possible decay modes. Half lives are given in years. Note that the half lives for 96Ru and m6Cd should be taken as lower limits, as is discussed in the text. For completness also existing experimental limits at 68% c.l. are given. With the exception of ~24Xe the experimental numbers refer to the sum of all decay modes of double beta decay

a 2vfl + fl + decay half lives

Isotope T~i~ T~/~2 p > Ref. (exp)

78Kr 2.3 • 1026 96RH 5.8 • 1026 3.1 • 1016 [33] 1~ 4.2 • 1026 2.6 X 1017 [34] 124Xe 1.4 • 1027 2.0 • 1014 [35] t3~ 1.7 x 1029 136Ce 5.2• 1031

b 2 v,B +/EC and 2 vEC/EC decay half lives

Isotope TI~/~(/?+/EC) TI~/~2~(EC/EC) T~7~(fl+/EC)> Ref. (exp)

58Ni 5.5 • 1025 3.9 X 1024 6,2 X 1019 [34] 7SKr 5.3 X 1022 3.7 X 10 22 96Rl1 1.2• 1022 2.1 • 1021 6,7• 1016 [33] l~ 4.1X 10 21 8.7• 1020 5.7X 1017 [34] 124Xe 3.0 M 1022 2.9 • 1021 4.8 • 1016 [35] 13~ 1.0• 1023 4 .2 • 1021 136Ce 9.2• 1023 1.7• 1022

The dependence of the calculated matrix elements on gpp makes predictions of 2 vflfl decay half lives somewhat uncertain. The similarity between the/~ +/~ + decay ma- trix elements of our present work and those of [11 ], on the other hand, gives us some confidence that the follow- ing procedure might be applicable] We take the most probable values of gpp from the fit to single B + decay data [ 17] and calculate average half lives in the spirit of [11 ] by summing over calculated decay rates within the (1 a)-interval of the most probable value of gpp.

Calculated half lives for the 3 different neutrino-ac- companied decay modes are given in Table 1. Though, as discussed above, we have to expect a considerable un- certainty for the 2 vflfl decay calculation there are some interesting facts which can be learned from the quoted values. In the case of double positron emission expected half lives are very large, confirming the earlier calcula- tions.[9, 10]. Note, that our values do not coincide with those given in [10], which can be traced back mainly to the different phase space treatment.

More interesting, however, is that the other two decay modes have half lives which are shorter by many orders of magnitude, but even for the best candidates we expect the mixed mode half life to be still in the range of 1022 years. These numbers should be compared to existing experimental limits [33-35], for completeness also given in Table 1. On the other hand, advanced Bfl decay ex- eriments might improve these limits by many orders of magnitude. Just for example, Ohsumi and Ejiri [36] es- timated the sensitivity of the ELEGANTS V detector [37]

1 The application of this procedure to the calculation of the half life of 76Ge led to the prediction of T~/2"-3• 1021y [11] in good agreement with the most recent measurement T~/2---1.4• 1021 y [3]

with which a limit of ~ 3 . 5 • 1020 years for the mixed mode of l~ might be reached. Nevertheless, it seems that much experimental work remains to be done before the 2 vfl +/EC decay will be detected.

As given in Table 1, half lives for double electron cap- ture are even shorter than those of the mixed mode; with the difference increasing with increasing proton number Z. The latter effect can be explained simply by the higher probability to find an electron inside the nucleus for heav- ier isotopes. Experimentally, however, the detection of 2 v E C / E C decay has also some disadvantages. While there are two 511 keV y-rays as a signal in 2v f l+/EC decay, there are only two X-rays from the deexcitation of the orbital electrons in 2 v EC/EC decay as observable. On the other hand, [37] quotes a sensitivity of ~ 7 • 102o years for the double electron capture decay of 1~ which is not too far away from the expectation. There might therefore be some hope that the detection of 2 v E C / E C decay becomes possible in the not too far future.

Finally, we would like to compare our results with those of Suhonen [15]. In this work [15] 2 vEC/EC decay half lives of the following 4 isotopes have been calculated. SSNi(Tl /2~-3.9• 1023y), 96Ru(T1/2~2 .8 x 1021y), l~215 102~ and ~36Ce(T~/z~-6.4• 1019y). Suhonen [15] also made use of the pn-QRPA approach (in the particle-number projected version [23]) but took the Bonn potential [28] for the effective interaction. Com- pared to our calculation, however, these half lives appear to be surprisingly short. For the isotopes 96Ru and i~ this might be explained with the unusual behaviour of the 2vflf l decay matrix elements which has been dis- cussed for the case of 1~ above. A similar feature can be found in the calculation of Suhonen (see Fig. 1 of[15]), the matrix elements for these isotopes are - using the standard Woods-Saxon parameters - nearly unsupressed by the inclusion of the particle-particle force, in contrast to the results of previous 2 v f l - f l - decay calculations [11, 17-25].

To summarize the results on 2vflfl decay we state that in agreement with the expectation half lives for 2 vfl + / E C and 2 v EC/EC decay are shorter than those for double positron emission. Half lives for the experi- mentally most Promising isotopes are given in Table 1, which we hope may be useful as a guideline for future B + fl + decay experiments.

3. Neutrinoless decay modes

3.1. Decay rate formulas

In the case of 0v f l+ f l + and 0v/~+/EC decay the theo- retical description is quite similar to that of the more common 0 v f l - f l - decay [38]. As in the neutrino-accom- panied decays, in principle also (neutrinoless) double electron capture could occur. However, as pointed out by Vergados [39] and discussed in some detail by Doi and Kotani [38], the 0 v EC/EC decay mode appears to be hopelessly slow; the relative order of magnitude of its decay rate compared to the neutrinoless mixed mode is

156

expected to be at least < 10 -4 [38]. Therefore, we will consider only the 0 vB +/7 + and the 0 v/7 +/EC decay here.

We adopt the following effective weak hamiltonian,

G Hw= ~ (jLuJ~t 4- Kjr~J~ ~-

+ r/JR~ J~* + ~-JR~ J~*) + h.c., (3.1)

where small (capital) letters denote leptonic (hadronic) currents and the subscripts L and R indicate left-handed and right-handed couplings, respectively. Note that in this convention the right-handed parameters K, ~/and 2 are chosen such that the ordinary left-left handed inter- action has a relative strength of unity.

The derivation of the decay rate(s) follows closely that outlined for the/7-/7 decay [12, 16, 17, 38]. Neglecting terms proportional to ~c, which should be a good ap- proximation since they always appear as 1 • ~c and K ~ 1 is expected, and considering only light neutrino eigen- states, the inverse half life for 0 v/7/7 decay can be ex- pressed in a factorized form as:

[ TI~.~ fl'8'~ (07 "'~ O; )] --1

2

=Cram \ m e - / 4-C;n<gl)2+C~ a<z)2

+ <u > me

(my) + c ; , - - <,t > + G, <n > 7, (3.2)

me

where ~ is a discriminator (cr = / 7 - ,6 , ,6 +/7 + and ,8 +/EC), indicating that the coefficients C~y are different for the different decay modes.

In writing down (3.2) we made use of the following definitions for the effective neutrino mass and right- handed parameters

( m , , ) = ~ , ' U2. mj, ( 3 . 3 a )

J

@} = 1 / Z ' Uej V~j-, (3.3b) J

( 2 ) =2 E" Uej geJ ' (3.3c) J

where the prime indicates that the sums are restricted to light neutrino mass eigenstates (mj < 10 MeV).

The coefficients C~y contain products of nuclear ma- trix elements and leptonic phase spaces. Within the non- relativistic impulse approximation there appear 9 dif- ferent matrix elements Ma (/7 = GT, F, GTco, Fog, GTq, Fq, T, P and R) and also 9 phase space integrals,

Cm~m = ( M a r - Me) 2 G~, (3.4a)

C ~ , = ( M a r - M F ) [ - S ~ M 2 _ G ; + M , + G 2 ] , (3.4b)

C~, =(Mar--MF)[SeM2+ G ~ - M a _ G~

- Sp (Mp G~ - S e M R Gg )], (3.4c)

c;, = M2L a;

- } s SeM,+ M z_ G~ - M ( + G2] , (3.4d)

G 2 - ~ f e [ 2 S ~ M , _ M2+ G~-M?_ G2]

- S ~ M e M R G r (3.4e)

C,~, = - 2{M~+ M 2_ G~ - } f e [Se (M,+ 3/2+

+ M , _ M 2 _ ) G ; - M , + M , _ G2]}. (3.40

For shorthand in (3.4) we made use of the following relations,

MI+ - =Mcrq-6Mr+__ 3 Mrq, (3.5a)

M2• : M a ro~ 1 • MFo) -- g M1 T- " (3.5b)

The 9 nuclear matrix elements in (3.4) are (within the closure approximation) the same for the different decay modes but appear in different combinations. They involve 5 different kinds of transition operators and 4 types of "neutrino potentials". A complete definition of them is given in [ 12]. The factor Sp takes Sp = + 1 for the/7 -/7 - and S~ = - 1 for the/7 +/7 + and the/7 +/EC mode, be- cause of the different character of the Coulomb potential for emitted electrons and positrons. The factor S e is

I + 1, f o r / 7 - / 7 - and/7+/7+ St = - 1, for/7 +/EC, (3.6)

which appears due to the different number of fermions in the final states [38]. Finally,

I1 1, for f l - / 7 - and/7+/7+ -- (3.7)

f~ 3e for /7+/EC, 2meR '

which comes from the fact that a captured electron and an emitted positron feel different nuclear charges [38]. c~ in (3.7) is the fine structure constant and R is the nuclear radius.

3.2. Numerical calculations

We have calculated the various nuclear matrix elements appearing in the decay rate formula (3.2) within pn- QRPA [12]. For the evaluation we performed the closure approximation [9]. Though closure is known to fail badly in the case of 2 vflB decay, the comparatively large mo- mentum transfer by the virtual neutrino in 0 v/7/7 decays ensures that in the neutrinoless modes closure should be a good approximation. We can further strengthen this statement with the results of two recent numerical studies [40, 41 ], which showed that the use of closure introduces only minor corrections to the matrix elements for neu- trinoless decays.

For the determination of the coefficients Cxy matrix elements have been calculated with the most probable values of the particle-particle interaction strength gpp [ 11 ]. Also the other model parameters have been chosen in the

same way as for the 2 vflfl decay calculation discussed in Sect. 2. This procedure is mainly motivated by the attempt to calculate 0 vBB decay matrix elements in an as similar model as possible to that for the 2 vBB decay calculation.

It should be noted, however, that there are some dif- ferences between the 2 v and the 0 v calculations due to the virtual nature of the neutrino in the latter. One im- portant difference is that while in 2 v decay, due to the selection rules for Gamow-Teller transitions, the inter- mediate states are I1 +)-states exclusively, in 0 v decay also higher multipolarities contribute to the total matrix element. It is known [12] that the particle-particle inter- action is attractive mainly in the 1 + channel, while the higher multipolarities are only weakly affected. The sup- pression of the 0 vflfl decay matrix elements by the in- clusion of the particle-particle interaction is therefore less strong than that of the 2vBfl decay matrix elements. 0 vflB decay matrix elements are consequently less af- fected by uncertainties in the best values of gp~, and might more reliably be calculable than 2 vBB decay half lives.

0 vfl + B + decay matrix elements show a very similar model parameter dependence than that reported in earlier 0 v f l - B - decay calculations [12]. We will therefore not repeat the discussion here. However, we would like to mention that a reasonable change of gpp changes the cal- culated matrix elements usually by less than --~ 50%.

157

Consistent with the expectation and the discussion presented above even for the special case of l ~ 0 v f l f l decay matrix elements are only weakly affected by a varia- tion of gpp and other model parameters. A change of the strength of the spin-orbit force, as discussed in Sect. 2.2, would change the results of the 0 vflfl decay calculation by at most a few percent. Also a variation of gpp within 2 a of the most probable value changes the different ma- trix elements by only (10-40)% from the mean.

The various matrix elements resulting from the pn- QRPA calculation are given in Table 2. With the help of the phase space integrals of [33] we are then able to determine the coefficients of the decay rate formula. They are summarized in Table 3. A detailed discussion of these results will be given in the next section.

3.3. Discussion of the results for the neutrinoless modes

The coefficients C~y summarized in Table 3 are the main result of the present work for the neutrinoless modes; their implications for the effective neutrino mass and right-handed parameters shall be discussed briefly.

In agreement with simple phase space considerations the coefficients for the neutrinoless mixed mode are larger than those for the Ovfl+fl + decays. The relative en- hancement of the different coefficients for a given isotope is, however, not a common factor as could naively be

Table 2. Ca lcu la ted 0 vBfl decay ma t r i x elements for the 7 mos t interest ing isotopes

I so tope 58Ni 78Kr 96Ru l~ 124Xe 13~ 136Ce

M a r 1.36 3.28 2.62 3.34 3.92 4.02 2.44 M F --0.30 - 1.42 - 0 . 9 8 - 1.22 - 1.35 - 1.50 - 1.02 Mar,o 1.20 3.19 2.47 3.14 3.72 3.86 2.44 Me,o - 0 . 2 7 - 1.23 - 0 . 8 8 - 1.09 - 1.19 - 1.32 - 0 . 9 1 MGr q 1.19 2.13 1.81 2.35 2.82 2.70 1.40 MFq - 0 . 2 6 - 1.31 - 0 . 8 4 - t.05 - 1.21 - 1.34 - 0 . 8 9

M r - 0.026 - 0.67 - 0.31 - 0.38 - 0.67 - 0.87 - 0.66 M e 0.99 - 0.89 0.82 1.43 - 0.61 - 0.63 - 0.51 M R 0.92 3.83 3.24 4.10 4.73 5.44 3.90

Table 3a. Coeff ients in the decay rate fo rmula for the double pos i t ron emission mode

I so tope 78Kr 96Ru l~ 124Xe 13~ 136C e

Cram 1.6 • 10-16 3.0 x 10-17 5.4 • 10-17 8.7 • 10-17 1.6 • 10-17 1.1 • 10-18 C,,, - 4 . 3 • 10 -14 - 9 . 2 • 10 -15 - 1 .6x 10 14 - 2 . 1 X 10 -14 - 4 . 8 x 10 -15 - 4 . 0 • 10 -16 Cruz - 9 . 3 • 10 18 - 1.3 • 10-18 - 2 . 6 • 10-18 - 5 . 8 X 10-19 7.3 X 10 -19 7.2• 10 -20 C,, z 6 .0• 10 -12 1.4• 10 -12 2 . 4 x 10 -12 2 . 8 x 10 -12 6 .5 • 10 -13 5 .9• 10 -14 Caa 1.9• 10 -17 2 .0 • 10 -18 4 .4 • 10 -18 8 .5• 10 -18 5 . 5 x 10 -19 1.8• 10 20 C,a - 3.3 • 10-18 - 6.5 • 10-19 - - 1.6 • 10-18 - 4.9 • 10-18 - 6.0 • 10-19 - 4.2 • i0 2o

Table 3b. Coefficients in the decay rate fo rmula for the electron capture pos i t ron emiss ion mode

Iso tope 58Ni 78Kr 96Ru l~ 124X e 130Ba 136Ce

Cmm 9 . 1 • -I8 4.0X 10 -16 3 . 5 • -16 7 . 7 • -16 1 . 6 • -15 1.5• 10 - Is 5 .6• t0 -16 Cm, 1 . 2 • -15 7 .4 • 10 -14 7.1X 10 -14 1 .5 • -13 2 . 9 • -13 2 .8• 10 -13 1 .1 • -13 C,,,a 3 . 9 • -17 2 .4 • 10 -15 2 .0 • 10 -15 4 . 5 • 10 -15 9 . 2 • -15 8 .0• 10 - i s 2 . 9 • -15 C , , 1 .2• 10-13 8 .9• 10 -12 9 .4 • 10 -12 2 . 0 • -11 3 .3• 10 -11 3 .5• 10 -11 1.67<10 11 C;~z 6 .2 • 10-17 5.9 X 10-15 4 . 8 x 10 - i s 1 . 1 • -14 2 .2 • 10 -14 1.7• 10 -14 5 . 8 • 15 C,z - 6 .2 • 10-I7 - 2 . 5 • - 2 . 8 • 10 -15 - 6 . 6 • -15 - 1 .4• 10 -14 - 9 . 8 • -15 - 2 . 9 • 10 15

158

expected. Just to give an illustrate example, for 124Xe o n e �9 B + / E C / ? + p + ,~ finds for the rano C~m /Cr~ m -18, whereas

C~x+/(zc/C~+~+~2600! Generally, all coefficients in- volving ( 2 ) have much larger enhancement factors than those for ( q ) or (m v).

The large enhancement factors of the coefficients Cxx, Cma and C,~ in the mixed mode can be explained as follows. The (possible) contributions of a right-handed interaction are proport ional to the neutrino four-mo- mentum q = (co, q). It is convenient to divide it into two parts, the co-terms and the q-terms (with their associated matrix elements given in Table 2). Usually one would expect the m-terms to be the dominant ones, since q acts as a pari ty-odd operator between two parity-even nuclear states (remember that in f i r decay mother and daughter isotopes always have I 0 + ) ground states), which requires an additional operator vanishing in the usual approxi- mation for allowed transitions 2. However, it is not so for double lepton emission since for the co-terms there ap- pears a large cancellation among different combinations of energy denominators, as is explained in detail in [ 16]. The associated kinematical factors G 2 and G 3 therefore carry factors of (e 12/m~)2 and (e 12/m~) respectively, where e ~ 2 = e ~ - e 2 is the energy difference between the two emitted leptons. These factors lead to a strong damping of G 2 and G 3 in double lepton emission, since e ~2 is a very small quantity on average. On the other hand, in the mixed mode e~2=e +e c, where e c is the energy of the captured electron, is nearly equal to Qa +/Ec which cannot be called "small". Therefore, while G 1 and G 4 - G 9 are larger in the mixed mode by moderate factors, mainly due to the larger available energy, both G z and G3 are enhanced strongly. In combination with the nuclear ma- trix elements this enhancement acts then constructively for C ~ , C ~ and C,~ 3

Combining the coefficients C~y with available exper- imental limits on 0 vfl +/EC and 0 vfl + fl + decay half lives [33-35] the upper bounds on (m~) , ( 2 ) and ( i / ) summarized in Table 4 are obtained. Presently existing experimental limits on 0 vfl + fi + and 0 vfl +/EC decays are much weaker than those for the best f l - f l - decay candidates [2-7]. Consequently also the limits on (my) , ( ~ ) and ( I / ) are far less stringent. Just for compari- son we mention the currently best limits deduced f rom the non-observation of 76Ge Ovflfl decay [3]: (my) < 1.4(1.2) eV, @ ) < 2.2(2.2)• 10 -6 and (11) < 1.4(1.2) • 10 - s for maximum ("on axis", which means

that the other two parameters are assumed to be zero), with coefficients taken f rom [12]. Note, however, that the use of the coefficients calculated by the Tiibingen group [42] would only lead to minor changes for the limits f rom 76Ge"

While f rom the discussion above it is obvious that the 0 vfl +/EC decay is relatively more sensitive to the ( 2 ) - mechanism, Table 4 on the other hand shows that f rom the non-observation of 0 vfl + / E C decay alone more strin- 2 This is the origin of the P-wave and recoil terms which appear in the coefficients C,~ and Cmn, compare (3.4) 3 The enhancement of G2 and G3 is not important for C,, and C~, since these are dominated by the contribution from the nuclear recoil terms, which are multiplied by the large G 9 and G 6 respectively

T a b l e 4a, b. Limits on (m~), (2) and (r/) deduced from the non- observation of 0 v fl + fl + and 0 v fl +/EC decay. Experimental data taken from [33-35]. Upper values given are those obtained letting the other two parameters vary arbitrarily, whereas the lower ones are "on axis" assuming the other two parameters to be equal to zero

a B + B + decays

Isotope (rn~) < [eV] (2) < (t/) <

96Ru 7.8 • 10 s 4.1 • 10 ~ 7.1 • 10 -3 5.4• 105 4.0• 10 ~ 4.9• 10 -3

l~ 2.0• 105 9.5• 10 l 1.8• 10 -3 1.4• 105 9.4• 10 -I 1.3• 10 -3

124Xe 1.2 • l0 s 5.3 • 10- * 1.2 • 10- 3 8.5x 104 5.3• 10 -I 9.2• 10 .4

b fl +/EC decays

Isotope (m~) < leVI ()~) < (t/) <

58Ni 1.2• 10 6 7.4• 10 -x 1.2x 10 .2 2.2• 10 4 1.6• 10 -~ 3.7• 10 .4

96Ru 5.6)< 106 2.4• 100 4.2)< 10 -~ 1.1 • 105 5.6• 10 .2 1.3• 10 -3

106 Cd 3.7x 106 1.5• I0 ~ 2.8• 10 .2 2.5 • 104 1.3 x 10 -2 3.0 X 10 -4

124Xe 6.6 • 105 2.7 • 10-1 5.7 • 10- 3 1.2• 10 4 6.2)< 10 -3 1.6• 10 -4

gent limits on the Bfl decay parameters are difficult to get; while values "on axis" from 0 v p + / E C decays are smaller by large factors compared to those of 0 vfl + r + decays it is not so for an arbitrary variation of the other two parameters ("maximum") . The reason is that the ellipsoids of the mixed mode are highly deformed, since not only the coefficients Cxx but also the Cm2'S and C~x's are enhanced - by comparable factors. (Only if Caa is scaled by ~2 and Cruz and Cn~ by ~ also, the maximum allowed value of ( 2 ) would be more restrictive by (exactly) a factor of ~.) The situation is therefore com- parable to that in 0 v f i r (0 + ~ 2 + ) decays, which are more sensitive to ( ; t ) and ( r / ) than to the mass mechanism, such that the existence of right-handed weak currents would be the most likely explanation if 0 vflfl (0 + ~ 2 +) decay ever would be observed 4.

Finally we would like to add some speculations about the usefulness of 0 v p + / E C decay data. While it is dif- ficult to get valuable information on the f i r decay pa- rameters f rom the experimental data at present, the situation would change completely if neutrinoless double beta decay ever would be observed. (For the following discussion we have to assume some definite numbers, however, the qualitative features of the arguments pre- sented would not be changed if 0 vflfl decay would be observed with another value of the half life. Simply scale all quoted numbers by a common factor.) Assume, for example, ( m v ) = 1.0 eV, then we would expect that the

4 Even the observation of 0 v fl,8 (0 + ~2 +) decay would not be a proof of the existence of right-handed weak currents, as has some- times been stated [43], since the contribution from the mass mech- anism vanishes only in first order. Contributions from the mass mechanism become possible, though suppressed, for example, al- lowing the outgoing two leptons to be in S~/2-D~/2 or P~/2-P3/2 states [16]

159

0 v/3# decay half life of 76Ge would be equal to 2.3 • 10 24 years. On the other hand, this value could also be ob- tained for a nearly arbitrarily small value of (my) , but <~) = 1.8x 10 -6 or QT> = 1.0• 10 -8. For a complete analysis further information would be required.

There are, in principle, different possibilities to decide whether the decay is dominated by the mass mechanism or by right-handed current interaction. One is to measure the angular distribution of the outgoing leptons, in ad- dition to the half life [16] (the angular distribution for the (~) - te rms differ from the ones for the mass mecha- nism). However, such an experiment would require quite large a statistics and moreover can not be done within an experiment using semiconductors, such as 76Ge. Also the observation of 0 v #13 (0 + ~ 2 +) decays might be help- ful, as mentioned above. However, in that case very long half lives need to be measured. For example, taking the coefficients of [43] for 76Ge 0 v/~# (0 + --*2 +) decay the following half lives can be estimated: ( 2 ) = - 1 . 8 • 10 .6 ((~/> = 1.0 x 10- s) leads to T~/2 (0 + --+ 2 +) ~- 2 • 10 29 years (T1/2 (0 + --*2+) ~ - 1.2• 1032 years).

In principle, a third possibility would be to measure the 0 v#13 decay half lives of different isotopes, since the values of, for example, (m v > and (,~) which lead to the same half life for 76Ge give slightly different half lives for 136Xe (or any other isotope). However, in such an analysis one would have to account for both, the experi- mental error and the theoretical uncertainties in the cal- culation of the matrix elements. Accounting for these uncertainties unfortunately limits the usefulness of this idea as long as one considers only 0 v # - f l - decays, as is shown in Fig. 3a. For the plot we assumed a OvBI3 decay half life of (1.5 _ 0.5) • 10 24 y for both 76Ge and 136Xe (larger or smaller values for the half lives again lead only to a scaling factor). The region between the full lines (dashed lines) are allowed by 76Ge (~36Xe), and the shaded area is allowed by both experiments.

On the other hand, if we insert the values for (my> , (2 > and (I/> quoted above we get estimations of mixed mode and/~+ B + decay half lives ("on axis") presented in Table 5. For 124Xe, for example, we expect the mixed mode half life to be shorter by more than an order of magnitude, if the 76Ge decay is dominated by ( ; . ) than if it is dominated by the mass mechanism. While a full 3-dimensional analysis need to be done in principle, even the non-observation of the mixed mode decay could be helpful (still assuming that at least one 0 vBB decay has been observed), see below. The expected "on axis" half lives are quite large compared to existing limits, however shorter than those expected for the 76Ge (0 + --+ 2 +) decay by 4 (7) orders of magnitude for the (2 ) - t e rms (the ( , / ) - terms).

A graphical representation of the large differences between the neutrino mass induced neutrinoless mixed mode decay and that caused by the (k )-contributions is given in Fig. 3b, c. Assuming again T1}~ p p = (1.5 +0.5) X 10 24 y for 76Ge and T]}~ pa = (1.5 +__ 0.5) x 10 25 (10 26) y for 124Xe in Fig. 3b (Fig. 3c), one finds that while the shorter value used in Fig. 3b would still be marginally consistent with an arbitrarily small effective neutrino mass large regions of parameter space could be excluded if the

<A> X l O 6

<my>

<A> XIO 6

( ~ - ' - - - - - - . . . 3.0 L

p//

_ s . 0 f - ___ /

<my>

<A>• 6

_ ~_O~0 <mz,>

Fig. 3. a Allowed region of parameter space if 0 vflfl decay would be observed in two 0 vp p - decay experiments. Shown is only the <Z)-(my> plane for simplicity. For the plot a OvBB decay half life of (1.5 4- 0.5) • 1024 y has been assumed for 76Ge (full lines) and 136Xe (dashed lines). The shaded area is consistent with both ex- periments. For discussion see text. b As a, but for one 0 v B - # - decay experiment (76Ge) and one experiment measuring the neu- trinoless mixed mode decay of |24Xe. For 124Xe a 0 v/~/7 decay half life of T1}~ ~e= (1.5 +_ 0.5)• 10 25 has been assumed, e As b, but for a 0 vt3B decay half life of TI}~ ea --- (1.5 4- 0.5) • 1026 for 124Xe

Table 5. Expected "on axis" half lives in years for 0 vfl+fl + and Ovfl+/EC decays for (m~>=l.0eV, (2>=1.8x10 -6 and (r/> = 1.0 • 10-s. The table shows the different sensitivity of these modes to the mass mechanism and the right-handed current parameter <~). For discussion see text

Isotope Mode (m v ) (2) (r/)

78Kr 0 v/? + ,8+ 1.6X 1027 1.6X 1028 1.7X 1027 0vfl+/EC 6.5• 10 26 5.2x 10 25 1.1 • 10 27

96Ru 07/~+fl + 8.7• 1027 1.5N 1029 7.1 • 1027 0v/?+/EC 7.5• 1026 6.4• 1.1•

1~ Ovfl+fl + 4.8X 1027 7.0• 4.2X 1027 0 vp+/EC 3.4• 10 26 2.8X 10 25 5.0X 10 26

~24Xe Ofl+fl + 3.0• 10 27 3.6• 10 28 3.6X 10 27 Ov,O+/EC 1.6x 1026 1.4x 10 25 3.0• 10 26

13~ Ovfl+fl + 1.6• 10 28 5.6x 1029 1.5• 10 28 Ovfl+/EC 1.7x 10 26 1.gx 10 25 2.8M 10 26

136Ce Ovf l+~ + 2.4X 1029 1.7• 103I 1.7• 0v/~+/EC 4.7x 1026 5.3 X 1025 6.3X 1026

160

larger half life would be correct. Moreover, if 0 v13B de- cay would ever be observed in a 0 v / ~ - f l - decay exper- inaent even the non-observation of the neutrinoless mixed mode at some given level would allow similar conclusions (limits allow consistent solutions only inside the ellip- soids, compare.Fig. 3, therefore also larger limits can be used to exclude regions in parameter space).

To summarize the discussion on the neutrinoless fl +fl + and fl + / E C decays we state that while presently existing half life limits lead to only rather loose bounds on (m v), ( 2 ) and ( r / ) , an interesting enhancement effect for the (2 ) - t e rms has been found for the neutrinoless mixed mode. I f 0 vBB decay ever would be observed, an experiment on the neutrinoless mixed mode decay there- fore might offer an additional possibility to decide whether the decay is dominated by contributions from a finite neutrino mass or right-handed weak currents.

4. Summary

We have calculated 2 vflfl decay half lives for the exper- imentally most promising isotopes for all three possible decay modes (/~+B +, ~ + / E C and EC/EC) . While the calculated 2 vB + ,8 + decay half lives are very large, con- firming earlier studies [9, 10], the predicted half-lives for the mixed mode and the double electron capture are shorter by many orders of magnitude.

For O vflB decay we have calculated all matrix ele- ments for both the mass mechanism and the right-handed weak current terms for the first time complete. For fl + fl + and fl + / E C decay coefficients of the decay rate for the isotopes 58Ni, 78Kr, 96Ru, l~ 124Xe, 13~ and 136Ce

are tabulated. While limits on the effective neutrino mass and right-

handed parameters deduced from experimental limits are at present not very restrictive when compared to those of the best / ~ - ~ - decay candidates, an interesting en- hancement effect for the (2 ) - t e rms in the neutrinoless mixed mode has been pointed out. We think that if neu- trinoless double beta decay ever would be observed the consideration of neutrinoless mixed mode decays offers a possibility to decide whether the observed decay is dom- inated by the mass mechanism or by right-handed weak currents.

Let us finally mention that while the pessimistic esti- mates for the 2 v double positron emission half lives might have discouraged experimentalists, we hope that the con- siderably shorter half lives for the other two decay modes leads to renewed interest in double beta plus decay ex- periments.

This work is supported in part by a Grant-in-Aid for Scientific Research (05243204) from the Ministry of Education, Science and Culture. One of us (M.H.) would like to thank the Japanese Min- istry of Education, Science and Culture (Monbusho) for financial support. He also acknowledges valuable discussions with S.S.

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