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Double beta decay

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1991 Rep. Prog. Phys. 54 53

(http://iopscience.iop.org/0034-4885/54/1/002)

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Rep. Prog. Phys. 54 (1991) 3-126. Printed in the UK

Double beta decay

T TomodaPaul Schemer Institute, CH-5232 Villigen PSI, Switzerland

AbstractRecent developments in the theoretical investigation of nuclear double-beta decayare reviewed. In particular, the neutrinoless mode is discussed in detail, since itis sensitive to lepton number violation as predicted by gauge theories beyond thestandard model and it is expected to give important information on the nature of thenenhinos and the weak interaction. Various approximations made in the theoreticaltreatment of neutrinoless and two-neutrino double beta decay are examined, and thepresent limits on the effective Majorana mass of the electron neutrino as well as thecoupling constants of the right-handed leptonic current are presented.This review was received in July 1990.

0034-4885/91/01W53+'74$14.00 @ 1991 I O P Publihing Ltd 53

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54 T Tomoda

Contents1. Introduction2 . M ajoran a neutrinos2.1. M ajorana neutrinos and their CP properties2.2. Phase conventions2.3 . Types of neutrino m ass matrix3. Th eo reti cal description of pp decay3.1. T he effective Hamiltonian3.2. 2upp decay

3.3. Ovpp decay3.4. p@decay of a single hadron in the nucleus3.5. Ovpp decay involving Higgs bosons4.1. Common features4.2. Nuclear models

5 . pp decay rates and c on stra ints on lepton number violation5.1. O+ -+ O+ Ou and 2u pp decay5.2. O+ - + Ou and 2v pp decay5.3. OuppM decayAcknowledgmentsAppendix 1. Electron wavefunctions and phase space integralsAppendix 2 . Neutrino propagation functionsReferences

4. Nuclear s tru ctu re calculations

6. Summary

Pa g e55595962626565687384878989919696114116

116118118121122

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Double beta decay 55

1. I n t r o d u c t i o nThere has been a growing interest in nuclear double-beta (pp)decay in recent years.This is the process in which an atomic nucleus with Z protons decays to anotherone with two more (or less) protons and th e sam e mass number A , by emitting twoelectrons (or positrons) and, usually, other light particles such as neutrinos:

( A ,Z ) + ( A ,Z + 2) + 2er + anything. (1.1)In order to s tudy pp decay, it is necessary to choose those nuclei in which other decaymodes, especially single P decay and electron capture, are energetically forbidden orstrongly suppressed by selection rules. T he re are som e thirt y even-even nuclei whichsatisfy this condition for P-p- (electron emitting pp) decay, and several candidatenuclei for P+of decay (table 1). A typical case of the 0-P- decay 76Ge-76Se isillustrated in figure 1.Table 1. Double beta transitions for naturally occurring pa rat isotopes. The Qva lue r of he decaye (O+ - +) and thenatural abundances P of the parent isotopesare aken from Wapstraand A u d i (1985)and Ledererand Shirley (1978), respectively.

0-0- ransition Q p s (keV) P (%) 0-0- transition Qss k W p$gCazs - 987 f 4 0.0035 130Te7g ';;Xe76 ,2533 f 4 34.5'tCazs - ;Ti26 4271 f 4 0.187 '"Xes0 - 2iBa7s 847 f 10 10.4$OZn,o + h e 3 8 1001+ 3 0.62 ' $Xes2 -+ ':,6Baso 2479 f 8 8.938G:,44 - Se42 2039.6 0.9 7.8 ':iCesr - t:Ndaz 1417.6 f 2.5 11.1'Sere -+ :Krrr 130 f 9 49.8 't:Ndse - i z S m a r 56 f 17.2"Sers -+ iKrrs 2995 f 6 9.2 ',"Ndss - ,',8Smsa 1928.3 f 1.9 5.7t i K r s o tSrrs 1256 f 5 17.3 'i Ndeo -+ '&?Smas 3367.1 f 2.2 5.6i t Z r s r + u'Mo52 1145.3 2.5 17.4 'i:Smsz - i:Gdso 1251.9 1.5 22.62.8 l eoGdss- fiDya( 1729.5 1.4 21.8YtZr56 - MOWYiMosa - :tRusr 112 f 7 24.1 'Erioz - ~ ~ Y b i o o 653.9 1.6 14.9l o o M o ~ g ::Ru~e 3034 f 9.6 'F Y b lo s - ',';Hfior 1078.8 f 2.7 12.6'ZRuao -+ l:;Pd58 1299 4 18.7 ' ; ~ W I I ~ ~ ~ O s i i o 490.3 f 2.2 28.6l:ZPdsr -+ 'Cds2 2013 f 19 11.8 ';;Oa~ls - '92Pt11, 417 f 4 41.0':tCdes + ':,6Snss 2802 f 7.5 '::Hg~z, - z:;Pblzz 416.5 f 1.9 6.9

F350 f 3

'Cds - ':,'Sner 534 f 4 28.7 ';;Ptizo - i j H g i i s 1048 f 4 7.2$0Sn,, 'z'Te72 2288.1 f 1.6 5.64 zi:Ulre -+ z : f P ~ ~ r r 1145.8f 1.7 99.275

' Te7~ - ' ;Xerr 868 4 31.7' Sn7z -+ ' ?Te70 364 4 4.56 Z2:Thirz - Z2:Uiro 858 f 6 100P + P + tramition Q p e (keV) P (%) P+P+ transition Qss (keV) p ( I" K q 2 ::Serr 833 f 8 0.356 'z:Xero - &?Te7z 821.6 f 2.4 0.096538 f 8 0.106~ R u ~ z 677 f 8 5.5 'g,OBarr- J o X e r s'CdSs - izPd6o 734 f 8 1.25 1 ; i ~ e r e - ~ a s o 366 50 0.190

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56 T Tomoda

c 5 'Y

Figure 1. P P decay of " G e . This is expected to take place as success ive virtual ptransitions via excited states in the intermediate oddcdd nucleus "As to the ground(Ot) or excited (2+) tate in "Se. (Adapted from Ledererand Shirley 1978.)

The principal current interest in pp decay originates in its ability to test thesymmetry properties of the standard model of the electroweak interaction (Glashow1961, Weinberg 1967, Salam 1968) such as lepton number conservation, masslessness ofthe neutrinos and non-existence of right-handed weak currents. In many gauge theoriesbeyond the standard model, however, none of these symmetries are exact and theyare violated to some degree depending on the model. Double beta decay is expectedto yield information on the degree of violation and to set important constraints on themodels.Double beta decay can be classified into various modes according to the light par-ticles besides the electronst associated with the decay. Independently of the possibleviolation of the above symmetries, the 2u (twwneutrino) mode( A ,Z ) + ( A ,Z + 2) + 2e- + 26. (1 .2)

(see figure 2 ( a ) ) is expected to be observed for those nuclei previously mentioned.This decay mode was considered first by Goeppert-Mayer (1935) shortly after Fermi'stheory (1934) of p decay appeared. We note that the neutrino emitted in the process(1 .2) is an (electron-)anli-neulrino which is defined as the neutral lepton accompany-ing the neutron @ decay

n -+ p +e- + 17. (1 .3)According to our standard knowledge, this particle is different (Davis 1955) from an(electron-)neutrino which is defined as the particle that causes inverse @ decay

However, due to the maximal violation of parity in weak interaction (Lee and Yang1956, Wu el al 1957) i t is actually the (almost) opposite helicity of 6. with respectto Y that is responsible for the absence of the reaction (1.4) by a n incident 6.. Thet Since P+p+ decay is unfavourable because of the Coulomb repulsion of the positronsby the nudeus.w e concentrate in the present r e v i e w on the electron emitting case.

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Double bela decay 57possibility still exists that there might be a small admixture of positive (negative)helicity component in U ( V ) which has evaded detection by experiments of the Davistype. Therefore the neutrinos of (1.3) and (1.4) might be different helicity sta tes of anideniical particle. A spin-; fermion which is identical t o its own an ti-particle is calleda Majorana particle (Majorana 1937, Racah 1937), as opposed to a Dirac particlewhich is not. If the neutrino is a M ajo ran a particle, /3/3 decay without neutrinos inthe f inal s tate

( A , Z )- A , Z + 2) + 2e- (1.5)can ta ke place (Furry 1939) in addition to th e ordinary 2u mode. A process in whichthe neutr ino emitted by a neutron is absorbed by another in the nucleus is called aOv mode (neutrinoless mode, figure 2 ( b ) ) and violates the lepton number conservationlaw by two. It is clear from the previous argum ent th at th e helicity m ismatchingbetween the emitted and absorbed neutrinos should be incomplete for this mode tooccur. Thi s is realized if(i) the neutrino has a non-vanishing ma ss; and /or(ii) the neutrino together with the electron can form a right-handed leptoniccharged current and couple weakly to the had ronic current .W ith o ut any of these conditions a M ajoran a neutrino is equivalent (Rya n and Okubo1964) to a two-component Weyl ne ut rin o, which represents half th e degree of freedomof a D irac neutrino (a projection on to a space with left-handed'chirality), and therewill b e no Ovpp decay.

101 I b l

Figure 2. Two-nucleonmechanism for ( a ) tw-neutrino and ( b ) neutrinoless P Pdecay as well as c ) neutrinoless P P decay with Majoron emission.

In the s tan da rd model of the electroweak interactions, th e neu trinos are regardedas massless Dirac particles (or more precisely, Weyl particles) with only left-handedcoupling. T his is, however, an inpu t rather tha n a prediction of the model. In order tounderstand many input assumptions of the standard model which seem to be ratherarbitrary, grand unified theories have been developed (for a review, see Langacker1981). In the simplest theory based on the group SU(5) (Georgi and Glashow 1974)th e neutrinos are predicted to have the same property as in the stand ard model. SinceB - L (baryon number minus lepton num ber) is an exact global sym me try of the SU(5)model, the neutr inos cannot have Majorana masses and Ovpp decay is forbidden.In grand unified theories based on larger groups SO(10) (Georgi 1975, Fritzsch andMinkowski 1975), E(6) (Giirsey el a/ 1976), etc , B - is a local sym me try and can bebroken spontaneously. T he neutrinos are predicted to be Majo rana particles in order toavoid acquiring masses com pa rab le with tho se of quarks or charged leptons (Yanagida1979, Gell-Mann et a1 1979, W itte n 1980). T hese theories predict n eutrino massesroughly in the range of 10-5-1 eV an d also th e existence of right-handed cur ren ts.

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58 T TomodaThere is also a possibil ity t h a t B - L is a global symmetry broken spontaneouslyin the low energy regime (Chikashige e l o l 1980, 1981, Gelm ini and R oncadelli 1981).In such a case, not only the neutrinos acquire Majorana masses but also a masslessNambu-Goldstone boson ap pe ars (called a Majoron). Th is couples to the Majorananeu trino s and gives rise to a neutrinoless pp decay accompanied by M ajoro n emission

(OvPPM)( A , Z )- A , Z + 2) +2e- +MO (1.6)

shown in figure 2(c) (Georgi e t al 1981).Since the OvBP decay am pli tude is proportional to th e M ajorana ne utr in o mass orthe coupling c onstan ts of the r ight-hand ed leptonic cu rrent , experimental informationon Ou decay is expected t o be useful for judgin g which specific gauge mod el is correct.In part icular for the question wh ether th e neutr ino is a Majorana or a Dirac particle,Oupp decay is considered to be th e m os t sensitive way of distinguishing between thesetwo possibilities. For a reliable deduction of the neutrino mass or right-handed cur-rent adm ixtures from exp erime ntal da ta , i t is necessary t o examine crit ical ly variousappro xim ations, both in the derivation of the transit ion operators a nd in their evalu-atio n using specific nuclear models. T hi s is the m ain su bje ct which will b e discussedin detail in the present review.Experimental m ethod s for detecting p.0 decay fall into three categories:(i) direct detection of electron or positron pairs associated with pp decay;(ii) geochemical measurement of the amount of daughter nuclei accumulated in ageologically o ld ore; and

(iii) radiochemical measurement of the amount of daughter nuclei accumulatedunder laboratory conditions.(For a detailed description of these m eth od s, see for instance A vignone and Brodzinski1988, Kirsten 1983, Levine et Q / 1950, respectively. A cosmochemical measurem entusing a meteorite as a sample (M arti and Murty 1985) is a variant of (ii).) Kinematicdata on the electrons or positrons obtained by the first method provide informationon th e mechanisms of pp decay. W ith th e sum energy spectrum of electrons, one candistinguish (see figure 3 in section 3.2.1) among various modes (2u p@ ,Oupp,OuPPM,etc ). With the single-electron energy spectrum and the angular correlation of twoelectrons one can distinguish (see figure 18 in section 5.1) between Oupp decays dueto a finite Majorana mass and the right-handed leptonic current, and in the lattercase, between decays due to its coupling t o the left-handed a nd right-handed hadroniccurrents. Clearly one can only determine th e tota l pp decay rate by the second andth e th ird me thods. We will see in section 5 tha t, in spite of this disadvantage, th etotal decay rates obtained by the geochemical method yield very stringent limits onOupp and OuppM decays.Since the first a tte m pt by Fireman (1948), strenuous efforts have been made bymany experimentalists to observe pp decay. Inghram and Reynolds (1950) found thefirst evidence of pp decay in lsaT e by th e geochemical method and obtain ed th e balf-life 1.4 x loz1 y. Recently Elliott e l a1 (1987a) succeeded in identifying 2upp decayof %je by a direct de tection of two simultaneously em itte d electrons using a timepro jectio n cha mber. T h e ha lf-life of th e decay w as found t o be (1.1+:::) X lozo y,ind ica ting the extrem e difficulty of the exp erime nt. Searches for Oupp decay havealso been undertaken by many groups and lower bounds on the half-lives of orderlo2 - o z 4 y have been obtained.

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Double beta decay 59Reflecting the long history and t he im po rta nc e of the field, many review articleshave been published which partly or extensively deal with pp decay (Primakoff andRosen 1959, 1981, Fiorini 1972, Bryman and Picciotto 1978, Zdesenko 1980, Kirsten1983, Boehm and Vogel 1984, Haxton and Stephenson 1984, Doi el al 1985, V ergados198 6, Avignone a nd Brodzinski 1988, Caldwell 1988, Faessler 1988, Lazarenko 1966,M ut o and K lapdor 1988b). Th e readers ar e referred to these reviews especially fo rt h e topics which are n ot covered in the prese nt article.This review is organized as follows. In section 2 the basic pro perties of M ajor ananeutrinos and their relation to Dirac and pseudo Dirac neutrinos are summarized.Section 3 is devoted to a form al description of pp decay. In section 3.1 an effectiveHam iltonian for th e nuclear weak inte rac tion is introduced and th e orders of ma gnitud eof the leading-order and recoil terms obtained by non-relativistic reduction of thenuclea r currents are discussed. Th e formulae for O + - + and O+ - + 2vpp decayra tes a re given in section 3.2. In section 3.3 w e discuss Oupp decay in detail . T h e decay

rate formulae for Ot - t and O+ + 2+ transitions including the recoil terms of thenuclear cur rents are presented and the ord ers of magnitude of various contributions t othe decay rate a re estimated. In section 3.4 00 decay du e to single hadron transition ssuch as n- A++ (1232 ) or T - + T in the nucleus is discussed, an d in section 3 .5 Ovppdecay with Majoron emission as well as that involving a doubly charged Higgs bosonis considered. Section 4 deals with the forma,l aspects of nuclear structure calculationsfor the transition ma trix elements. Th e closure approximation and the short-rangecorrelations between nucleons are discussed i n section 4.1, an d the m etho ds of pp decaycalc ulat ion s using nuclear models-the shell model, the quasiparticle ran do m phaseapp rox im atio n an d th e projected Hartree-Fock-Bogoliubov method-are presentedin section 4.2. In sec tion 5 calculated 2v and Oupp decay rates are compared withexperimental data. From experimental upp er bounds on O+ - and O 2+ Ovppdecay rates, co nstraints on the q uantities which characterize lepton number violations u c h as the effective masses of light and heavy neutrinos, and the coupling constantsfor the right-handed leptonic curren t are deduced (sections 5.1 and 5.2). Limits on theeffective neutrino-Majoron coupling stren gt h are presented in section 5.3. A summaryof the present review is given in section 6.The natural uni ts ( h = c = 1) and the Bjorken-Drell conventions (1965) for th em etric and Dirac m atrices are used unless othe rwise specified.

2. Majorana neutrinos

I n this section the basic properties of Ma jor an a neutrinos and their relation t o Diracan d pseudo Dirac neutrinos are sum ma rize d. For more comprehensive discussions,reade rs a re referred to t he pape rs by Ca se (1 957), Bilenky and Pontecorvo (1978),Sche chter and Valle (1980 ), Cheng and Li (1980), B e rn a h h and Pascual (1983), Doiet al (1983b, 1985), Kayser (1984) and Bilenky and Petcov (1987).

2.1. Majorana neutr inos and the ir CP p r o p er t i e sLet us assume that there are n generations of charged leptons as well as left- and

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60 T Tomodaright-handed neutrinos:

and that their charged current weak interaction is given by9 -&(Z) = -[/7'(1 - 7s)VLwL; +h"(1+ys)&W&,] 4- HC (2.2)Z J Z

where WL; and W;,, are the gauge boson fields which mediate left- and right-handedinteractions. We assume th at th e n x n mass ma tri x for the charged leptons has alreadybeen diagonalized. Choosing th e phases of the charged lepton fields app rop riate ly, an dassuming CP invariance of the Lagrangian &(z)(cP)cC,d=,) ( c P ) - 1 = Lee(-=, ) (2.3)

where C and P are charge conjugation and space inversion operators, we obtain theuniform CP transform ation property for all the components of VL and uk

where (a) CO?, etc with th e charge conjugation matrix C = C' = -C-' = -CT,CyTC-1 = -Y'. The most general mass term for the neutrinos h a s the form( 2 . 5 )

with the 2n x 2n mass matr ix MO which can be assumed to be symmetric without lossof generality

Here the n x n submatr ix M k gives Dirac mass terms which conserve lepton number,whereas ML and M i re responsible for Majorana mass terms which violate leptonnumber conservation. Assuming CP invariance also for rCm(z),we obtain Mot = MOfrom (2.4). Together with MOT = MO, i t means that MO is a real symmetric matrix.We can diagonalize MO with a real orthogonal matrix 0,

MO = OUTM S0, (2.7)where M j k = Sjkmj,Sjk= 6 j b S j with mj 2 0 and Sj = 2cl. T h e Lagrangian Lm nowtakes the formcm - - ~ T M N (2 .8)

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Double b e t a decay 61with the Majorana neutrino field N given by

where A is a diagonal matrix of arbitrary phases A,k = SjkAj, l A j l = 1. N satisfiesthe Ma jorana condit ionN = A 2 S N C (2.10)

( CP) N( z , t ) ( CP) - ' = iSy 'N( -z , t ) . (2.11)i .e. Nj is identical to its charge-conjugate field N; up to a phase. C P transformationon N is given byT h e f ield N j (2) can he expanded in term s of plane wave solutions of the Dirac equationasN j ( z ) = C J m [p u ,ps e- ip '+ ~ j , , + AjSj(ujp.)ceiP'l.a lP.+l (2.12)and it follows from (2.11) that

(CP)Qj,,(CP)-' = i s j Q j - ps (2.13)for the annihilation operator of the Majorana neutrino with mass mj , 3-momentump and spin projection s. We see tha t S, (or more precisely, iSj) is the intrinsic C Ppar ity of the M ajoran a neutrino N j (Wolfenstein 1981b, Bernabhu an d Pascual 1983,Kayser 1984). It should be noted th at the relationship (2 .13) is free from the arb itraryphase X j . Since the field Nj(z) can create and annihilate the same particle, we havetwo types of neutr ino propag ators

(2.14a)(2,146)

where (I; m) s the usual Feynman propagator for a spin-i particle with mass m.T h e current neutr inos q nd v k appearing in C,, are related to the Ma jorana neu trino" U N L v k = V N , (2.15)

(T[Nj (z)&(y) l ) = i S d z - ; mj)Sjk(T[Nj (z)NF(y) ] ) = iSF(z - ;mj)CTX:Sj6ja

N bywhere NL,R= P L , R N with

and the n x 2 n matrices U and V are given by(2.16)

(2.17)From (2.146)-(2.17) we obtain the propagators of the lepton number violating typefor the current neutrinos

We note these are free from the arbitrary phases X j , as they should be

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62 T Tomoda2.2. Phase conuent ionsAs stated before, A is a n arbitrary phase matrix. There are t w o conventions (Kayser1984) for fixing A which often appear in the literature.Convent ion 1 . A = 1 (Wolfenstein 1981b).(i) U, V are real and given by

(VUS) = O uT (2 .19)(ii) The Majorana condition and the propagator of the second type (2.146) aredependent on the C P parity S :

N = S N C(T[Nj(z)NF(y)]) = iSF( z - y; m ,)CTS,6,,.

(2.20)(2.21)

Convent ion 2.definite A 2 S = 1 (Schechter and Valle 1980 , Doi e t a1 1981a), or to be moresi = + 1A; = .(: si = -1

(i) Uij, Kj are real for S, = +1 and pure imaginary for Sj = -1 , and

( ) = O V T A *(2.22)

(2.23)is a uni tary matr ix .forms independent of the C P par i ty S:(ii) T h e M ajorana condition an d th e propagator of the second type take the simple

(2 .24)(2 .25)

2.3. T yp es o f neutrino mass m a l r i zIn the previous discussion we assumed only C P invariance so Chat the mass matrixwas a general real symmetric matrix. Now let us tu rn to a few special cases.2.9.1. Dirac and pseudo Dirac neutr inos .mass matr ix

If M = M i = 0 in equation (2.6), the

can be diagonalized with(2.26)

(2.27)

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Double beta decay 63where A, B are n x n real orthogonal ma trices which transform th e general real matri xM L to a non-negative diagonal matrix M I :

B M D A = M I . (2.28)TT h e 271 x 2n diagonal matrices M and S ar e given b y

and correspondingly we write(2.29)

(2.30)Equation (2.29) shows that we have obtained pairwise degenerate Majorana neutri-nos with opposite CP parity. Now we introduce $; ( i = 1, . . ,n) which is a linearcombination of the degenerate Majorana neutrinos N I ; and Nil;:

Th en its charge conjugate field @ is given by= & ( S ~ A ; * A ~ N ~s ~ ~ A ; ~ ~ A ~ ~ N ~= &(A;NI - A ~ N I I ) (2.32)

i.e. @ is a Combination of N I and N I I orthogonal to +, The m a s Lagrangian can bewritten asL, = - ~ X M ~ N ~~ % M ~ N ~ ~

=-*MI$ (2.33)and the propagators for $ are given by( T [ $ j ( z ) $ k ( ~ ) l )= iSF(z - ~ ; m j ) & j k(T[$j(~)+z(y)]) = iSF(z - y;mj)CTi[A;(S~A:)A; + A ; ~ ( S I I A : I ) A ; I I ~ ~ ~ ~ *

(2.34a)= 0. (2.34b)

We see that $1 represents a Dirac neutrino with m a s mj and tha t it is formed fromtwo degenerate Majorana neutrinos with opposite C P arity. Th e contributions fromthese two Majorana neutrinos cancel each other exactly in the propagator (2.346) ofthe lepton number violating type. T he curre nt neutrinos and u k are related to$L,R(z) = YS) by

= VL = B ~ $ ~ (2.35)and we obtain from (2.34 ), (2.35) clearly( T [ v ~ ~ ) v & L ( Y ) I )(T [&(Z)V '&(Y) l ) = (T[U~LL(~)V'&L(Y)I) 0i.e. the total lepton number is conserved

(2.36)

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64 T TomodaA Dirac neutrino can also he formed by a combination of Q, and (U$ instead of

Q, and v k (Konopinski and Mahmoud 1953). Let us consider a twc-generation casewith only left-handed neutrinos an d assume t hat the mass m atrix is given byO mo ) . (2.37)

This M i h a s a structure similar t o tha t of (2.26) and it is diagonalized by a Diracneutrino $ = v , ~ ( v , , ~ ) ' . ince Le - L (electron number minus muon number) isconserved in this case, Ovpp decay is forbidden ((T[ve~(z)vTL(y)])0) bu t p- - + e +conversion on nucleip- + ( A ,Z) + + ( A ,Z - ) (2.38)

is allowed ( ( T [ U ( Z ) ~ ~ ~ ( Y ) ] )0).A pair of Majorana neutrinos with opposite C P parity is called a pseudo Diracneutrino when their masses are approxim ately b ut not exactly degenerate (Wolfenstein1981a, Petcov 1982, Valle 1983, Doi e t a l 1983b). In such a case their contribution to(2.18) is non-vanishing hut strongly suppressed.

2.9.2. The seesaw mechanism. For simp licity let us consider the one-generation casein which MO is assumed to be given by

with mR > ImDI. This MO can be diagonalized withmD;;tan 20 =-mRc o s @ - s i n 0

(2.39)

(2 .40)yielding

T he current n eutrinos are expressed in term s of mass eigenstate neutrinos asV L = X ; C O S @ N I L + X ; ~ ~ ~ @ N ~ LU;= X ; s i n 0 N l ~ f X ~ c o s 0 N ~ ~ (2.42)

In left-right sym me tric gr and unified theories such as those based on the group rep-resentations SO(10) or E(G), the neutrinos are treated on the same footing as otherfermions. Consequently under min imal assum ptions, a neutrino would acq uire a Diracmass of the sam e order as those of othe r fermions, which clearly contradicts th e experi-men tal limits on the neutrino m asses. A ma ss matrix of the typ e (2.39) was introducedas a remedy for such a situation. The left-handed neutrino V L of equation (2 .42) con-sists mainly of the M ajorana neutrino N1 with mass m l % mL/mR. A s mR becomeslarger, m l becomes correspondingly smaller (called a seesaw mechanism, Yanagida1979, G ell-Mahn et a/ 1979). One would get ml - 1 eV for mo - 1-103 MeV andmR - 103-109 GeV.

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Double b e t a decay 653. T h e o r e t i c a l d e s c r i p t i o n of f i f i decay9. I . The effective HamiltonianIn the previous section we assumed the charged current interaction for leptons of theform (2 .2) . T he relevant par t for decay can he written as

(3.1)9&c(z) = -[J[wc,, + jkw;,,] + HC2 J zwhere the left- and right-handed leptonic currents are given by

J[ = eY(l - 5)VeL = w(1 + 7 5 ) & ( 3 4with the electron field e and th e curr ent electron-neutrino fields (see (2 .15) )

(3 .3)Here Ni is a Majorana neutrino field with mass mi. One should remember t ha t aDirac neutrino can also be expressed as a superposition of Majorana neutrinos as in(3.3) . In th e following we adopt convention 2 for the choice of the arbitrary phases X i(see equations (2.22)-(2.25)). Th e gauge bosons W L an d W R are related t o the masseigenstates WI and W, (with masses M I and M2 by

cos< sinC2)( - s i n C cost) ( 1 (3.4)and in general # 0. Adding left- and right-handed nuclear currents J [ , R ~ ~ ~ 8 ~oth e leptonic coun terparts jL,R n ( 3 . l ) , where Bc is the Cabibbo-Kobayashi-Maskawaangle, we can write the effective weak interaction Hamiltonian for decay due to Wboson exchange in the form (Bhg el al 1977, Doi e l a l 1983a)Hw = ( G C O S ~ C / ~ ) ( J L ~ J [ ~K ~ L , , J ; ~r l j ~ , , J [ + X~RJ;~)+ HC (3.5)where

We use i n t h e following G = 1.16637 x lo- GeV-2, cosBc = 0.9737 (Particle DataGroup 1984) and regard the coupling constants K , 7 and X as small parameters (< 1).Assuming that the nucleons in a nucleus behave in the same way as free nucleons(impulse approxim ation), we write the nuclear currents in te rms of a nucleon fieldli, = Z ) as

J[~(z)= I l ( z ) r + ( g v r - gwuypv - AY75 + g P y d ) + ( ~ )J;(z) = 4 ( z ) ~ + ( g v y y w ~ q y YAY% - gpy5qP)+(+) (3.7)

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66 T Tomodawhere T+ = + (T I +in) conver t s a neutron into a proton, q, = p, - p ; , = ia, - -ia,)is the 4-momentum transfer, and gv, A , gw a n d g p are the vector, axial vector, weakmagnetism and pseudoscalar form factors (see e.g. Commins and Bucksbaum 1983).T h e first two of these a t q 2 = 0 a r e

-.t

a n d gw(0) s given by the (well tested) conserved vector current (CVC) hypothesis as(3.9)0gw(0)=-2 M K O = 3.70

where M and KO are the mass and the isovector anomalous magnetic moment of thenucleon. Their q 2 dependence can he well approximated by

with A - 1 GeV. The pseudoscalar form factor gp is given by (3.10)(3.11)

assuming the partially conserved axial vector current (PCAC) hypothesis. By th eFoldy-Wouthuysen transformation (1950), the nuclear currents (3.7) are reduced tothe nou-relativistic form (Rose an d Osborn 1954, R i a r 1966).JLt(z) = 726(z - )A [gVV(')p +gwW(')' - gAA(k)P- p P ( k ) " ] ,

= l k = O . l , - . . (3.12)AJ $ ( z ) = r26(2 - ) [gvV(k)p+ gw ')'' + g ,A(k )Pf gpP(')p]n= l k = O , l , ...

where k indicates the order in 1 / M . The terms up to order l / M Z are listed in table2 . T h e m o m e n t a p , and p; should be interpreted as -iB/aT, standing to the rightand left, respectively, of 6(z - rn) n equation (3 .12) . It follows in the case of smallenergy transfer

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Double beta decay 67Table 2. Non-relativistic reduction of the nuclear currents. The ime and spatialcomponents of the t e m defined n (3 .12) are listed according to the order kin 1/M.Here qo = po - a , q = p - p', 4 = p + p', where p" and p'p are the initid and final4-momentaof a nucleon. w is the Pauli spin matrix.

the neutr ino emitted by one nucleon is absorbed by another . I ts typical momentump is estimated to be of order - 1/F" - 100 MeV, where PNN - 2 fm is t he meaninternucleon distance, so that the recoil terms in (3.12) become much larger than inthe case of single p o r 2vPP decay. Substi tut ing the est imates(3 .14)

in to th e various term s of equation (3.12) we obtain the est imates l isted in table 3. Inthe theoret ical descr ipt ion of Ovpp decay in section 3.3 th e recoil terms of order q / Mor Q / MC" = A $'

= (P" + PL).U " / 2 MDn= V;)+ (gw/gv)W;')

= [p,+ p; - p p u n x ( p , - ; ) ] / 2 M(3.150)

p1p = KO + 1 = 4.70 (3.156)wil l be retained. Compared with the leading terms V(')' = 1 and A(') = U , heseterms have a different property under par i ty transformation, and consequently theyenter in to Ovpp decay ampli tudes in a different manner. T he recoil te rm D, for iu-stanc e, does not give simply an ap proxim ately 25% correct ion t o the decay am pli tudes.As will be shown later, it gives the dominant contribution to Ot - t Ov@odecaydue t o the in te rac t ion propor tional to 7 in (3.5), which represent,s the coupling of theright-handed leptonic curre nt to th e left-handed nuclear curre nt. The reco i l te rmsA(')' a n d V(')were taken into account in the formalism for O+ -+ O OvPP decay byDoi e t a1 (1983a) , and then Tomoda e t a l (1985) included also ' the weak magnetismte rm W(') . One sees f rom tab le 3 that the spatial component of the pseudo-scalart e rm gpPfl) may be even larger than the term gvD. Its effect is, however, to givea correction of - x (-30%) - 10% to the leading term gAA('), where the factor5 comes from the angle average. If the recoil term C should become im po rtan t , we

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68 T Tomodawould have to include also the time component of the pseudoscalar term gPP(')O. Wewill neglect the recoil terms associated with gp in the following. The finite extensionof the nucleon represented by the q2 dependence of the form factors will also be ne-glected unless the effective twonucleon transition operators for Ovpp decay becomeshort-ranged (see (3.69),3.70)).

Table 3. Order of magnitude of the ter m of the nuclearcurrents (3.12) for the c ~ s eof virtu$ emission of neutrino with lpvl p Y 100 MeV.

k = O k = l k = 2

9.2. 2uPp decayT h e Zupp decay (figure Z ( Q ) ) is described as a second-order process in the effectiveweak interactio n (3.5).Since those processes which involve only left-handed currentsclearly give the dominant contribution, we neglect right-handed currents in the caseof Zupp decay. T h e differential decay r ate is given by

(3.16)where

x P - P ( N i , N j ) lX ~Pzb:(Y)YV(1 - YS)N~~l,,(ll)~~,s;(z)Yr(l 5 ) r r ( z ) , (3,17)

1 +WI + E N- EIHere eps(z) and Njrs(z) re the wavefunctions of the emitted electron and the Ma-jorana neutrino Ni f mass m i , with energy, momen tum and spin projection ( z , P ,s)

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Double beta decay 69an d (w, k , s ) , respectively, and normalized as

(3.18)

Th e operator P(I1, / z ) interchan ges th e particles Il and /2. Th e ini t ial , intermediatea n d final nuclear state s a nd their energies are labelled by I , N and F. Th e summat ion&,in is taken over the spin projections of the electrons, the neutrinos and the finalnuc lear sta te , an d taken over thos e light neu trino species the emission of whichis kine ma ticall y allowed. We assume in t he following th a t the masses of these lightneutrinos are all much smaller than the Q-value of the pp decay

Q p p = Er - Ep - 771, (3.19)and tha t ~ i , j l U ei U ej lz 1 . ( In the oue-generat ion example given in section 2.3.2,~ ; , j lUe ;Uej l z= IUe1I4= COS*Bwith IS1

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(3.21)

Th e phase-space factors j :) and j i ; ) are th e pr od uc ts of electron radial wavefunctionsgiven by (A1.9) and (A1.lO) in appe ndix 1. Since the typical nuclear excita tion energyEN- E1 (- 10 MeV) due to the Gamow-Teller ope rato r r+u is usually much largerthan the lepton energies (- 1 MeV), ICN and L N can be approximated as

(3.24)and similarly for L N ,where WO s t h e tota l released energy

WO= Q p p + 2m, = Er - E F (3.25)and ( I C N ) , (LN) are defined as ICN. L N of equatio n (3.22) with EN eplaced by someaverage energy (EN) f the intermed iate nuclear s tate s . Using this approx imation , wecan write the half-life in the factorized form (Do i e t a / 1985)

[r:/z(o+ o+)]- = WZ F Z l M Z ~ I Z (3.26)where

is the nuclear t rans i t ion am pli tude and(3.27)

(3.28)

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Double beta decay 71the lepton phasespace integral . Fz, is not very sensitive to the choice of the actualvalue of (EN) ince the dependence of the integrand is largely compensated by that ofth e factor outside th e integral. If EN n equat ion (3 .27) is also replaced by a n averagevalue (EN),he summ at ion over the intermediate nuclear states can be completed togive (Primakoff and Rosen 1959)

with

where

(3 .29)

(3 .30)

(3 .31)

This closure approximation has been used frequently in the li terature. Formally theapproximat ion is always valid if ( E N ) s de$ned by equat ions (3.27) and (3.29). Inpractice, however, i t is rather difficult to estimate (EN)correctly without preciseknowledge ab ou t th e value of Mz, which is ju s t th e quant i ty to be ca lcu la ted . This isthe case especially when cancellation occurs am ong the term s under the sum m ationin equat ion (3 .27) (see section 4.1) .Figure 3 show s the differ ential decay ra tes d W zu /d fl an d dWz,/dT.,,, whereT,,, = 1 + c2 - 2m,, obtained by condit ional integrat ions in (3.28), compared tothe corresponding spectra of Ovpp and OvpPM decay. The different characteristics ofthese spectra are useful for distinguishing among the different decay modes.

3.2.2. The half-life for O f - + 2 v P p decay is givenby equat ions analogous to (3.26)-(3.28) (Molina and Pascual 1977, Doi e t a l 1981b,Haxton a n d Stephenson 1984):O+ -+ Z+ Zvgo decay.[r$(o+ -+ 2 t ) l - I x F ; ~ ~ J A & ; + ~ ~ (3 .32)

with the nuclear t ransi t ion ampli tude(3 .33)

and the lepton phase-space integralf l ( y ) ( ( K ~ ) ( L N ) ) ~ W Z ,u i dwz dr l d rz . (3 .34)J(4Wo + (EN)-Fl; = In 2In th e closure ap proxim ation, equat ion (3 .33) is reduced to

(3 .35)

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72 T TomodeE 101 I I

B ' S e d ' K r l O ~ l

0 1 2 3T (MeV1

Figure 3. ( a ) Single-electron SpectNm for pp decay of "Se. The full c u r v e w a scalculated by integration except t i in (3.28) nd normalized to the central value of theexperimental 2 v half-life (Elliott el I 1987a) 1.1 x 1 y. The brokencurve shows theresult of an analogous calculation using (3.94) for Ovpp decay with Majoron emissionwith a normalization to the experimental bound (Moe t ai 1988) OuM > 1 . 6 X 10'' yand magnified by a factor 10. (See igure 18 for the case of OvOP decay.) ( b ) Sumenergy spectra of the two emitted electrons for ZvPp, OvDBM and OYPP decay of8 z S e . The differential rates dW /dTaumwith T,um= (1 + LZ - 2m.c' for the first t w omodes were calculated analogously to ( G ) . The vertical line at Q o p = 2.995 MeVindicates the position of an expected peak for 0u.PP decay.

1/2

where

with c-3 denoting a spherical tensot product (Edmonds 1957). In contrast to equation(3.21) or (3.28) for Ot + Ot decay, the Ot -+ 2+ phase space integral (3.34) containsonly the digerenee - L N , which becomes much smaller t h a n the s u m K N + L Ndue to cancellat ion. T h e rat io

(3.37)

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Double beta decay 73is estimated to be of order - $ x 0.52/102 ,.-1/800 for the typical case of the de-cay 76Ge+76Se(2:). T h is me ans a further suppression of Oc 2+ 2vpp decaywhich is already unfavoured because of its smaller &-value compared with Ot -decay. Since the leadin g S-wave contr ibutio n is suppressed, P-wave leptons maygive a comparable contribution (Doi el 01 1985). In fact the decay amplitudefor the process in which electrons and neutrinos are emitted in the combinationS;/,P; 2S; zP;12,S;12P;/,S;12Pg,2or P;,2Pj12S;,2S~ is associated with the sumI C , + L N . k h e suppression factor due to a P-wave to d w av e ra tio i s es timated to be4aZ x ?gR- 1/8) x (1/120) - 1/96 0 for th e 76Ge decay, which should be com paredwith the factor .- 1/800 estimated previously.3.3. OvBP decaySince the first formulation by Fur ry (1939), the theory of Ouop decay has been devel-oped (Primakoff and Rosen 1959, Molina and Pas cual 1977, Doi el a l 1981c, 1983a,Hax ton and Stephenson 1984, Tom oda e t 01 1986) in accordance with the progressin our unde rstan ding of the weak interaction. Here we follow mainly the notationadopted by Tom oda e l a l (1986). Firs t we rewrite the Ha milto nian density (3.5) intoth e following form

2HW= (G COSSc/JZ) C [ j ~ i p j [ , + R i @ j : i ] + HC (3.38)

where j & a re the left- a nd right-hand ed leptonic curre nts formed out of the electronand maSs eigenstate neutrino fields e a n d N;:i = l

jLi = ey @ ( 1 - Y S ) N L & = e y @( l+ Y5)NiR (3.39)and jLki are the nuclear currents coupled to these leptonic currents:

.fL = Liei(JLt + KJ$) JRi - V r (U[ + q J [ ' ) . (3.40)The t e rm nJ[' will be neglected in the following since K enters into pp decay ampli-tudes always in the combination 1& K and we expect ~ < 1. T he differential 0vp.Odecay rate is given by

with(3.41)

(3.42)where P, (a L , R) is the projection operator (2.16), and the summation zapinstaken over the spin projections of the electrons (si, s;) and the f inal nuclear state.

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74 T TomodaOther notat ions are the same as i n th e case of 2v@@ ecay (see the paragraph belowequation (3.17)) . Note that e p s ( z ) a n d Nik,(Z) ar e th e wavefunctions which are usedto expa nd th e field ope rators e and Ni.The summat ion E, f the numerator of the l a s t factor in equation (3.42) over theneutrino spin yields (cf also (2.18))

14Pp,,;(y)y,Pp-(wyow - k.7 + m;)e"'("-")P aypez,s;(z). (3 .43)Using th e a nticom muta tion relations of y-matr ices and P,Pp = 6,pP,, we obtain

This equat ion shows tha t the Ov@@ ecay amplitude contains a factor proportion al to(i) the neutrino mass for the processes involving only the left- or right-handedleptonic current ((e@) (LL) or (RR) ) ; and(ii) the neutrino energy or momentum for those involving both the left- and right-handed leptonic currents ((a@) (LR) or (RL)) .Th e contribution from the processes exclusively d ue t o th e right-handed leptonic cur-rent ((.@) = (R R )) will be neglected in the following since it is of second order in thesmall coupling constan ts X and 0. See , however, th e discussion ab out th e contributionof heavy neutrinos in equations (3.76) and (3.77). )In contrast to the case of 2upp decay, the neutrinos in Ovp@ decay are virtualparticles exchanged between nucleons and their typical energy is much larger than th etypical excitation energy of the intermediate nuclear states (see (3.14)):

w - 00 MeV >> EN- E, - 2 / M - 0 MeV. (3.45)Therefore the variatioii of EN in the energy denominator can be safely neglected(Primakoff and Rosen 1959). Replacing EN by an 'average' (E N ), and using th eclosure relation ENN ) ( N I = 1, we can complete the sum ma tionover the intermediatenuclear state s. In th is closure app rox ima tion , equa tion (3.42) now readsx Paype; , , ; (~ ) (3 .46)

where A, = E , + ( E N ) - E,. The second term in the square bracket of this equationresults from t he antisymmetrizat ion of t h e ou tgo ing electrons. We see th a t the effectof the a ntisym metriza tion is construc tive for the terms proportional to mi or E . 7 ,nddestructive for that proport ional to w y o . Integrating these terms over the neutrinomomentum k we obtain the neutr ino p ropagation functions H ( r ) , H ' ( r ) = (d/dr)H(r)and 2 ( r ) given in appendix 2 .For the electron wavefunctions (see appendix 1) we take into account S I / Z(g-1,

f1) and PI/* ( g l , -1) waves for O+ 4 O+, and Sl I2 and P3/2 (g-2, f 2 ) waves forOt - + Ov@@decay, respectively, As for the nuclear currents (3.40) , we includethe recoil terms A(')O = C and V(')+ (gW/gv)W(')= D (see equations (3.15)) inaddit ion to the lowest order terms V(')O = 1 and A(') = b .

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Double beta decay 7 53.9.1. O+ O+ O v p p decay. The differential rate for O+ -+ O+ Ovpp decay withthe energy of the 'first' electron 1 and the angle between the two emitted electronseI2 s given by (Doi e l a / 1983a, Tomo da et n l 1986)

(3 .47)where

f

(3.49)

Th e phase-space factors (= fill)are th e produ cts of electron radial wavefunctionsgiven in appendix 1, and Xi are th e following combinations of nuclear m at rix elements

where(m,) = C"iu:i( A ) = A X ' U e i V i

i

i

(7)= VZ'UeiVei

(3.50)

(3 .51)

Th e ampl i tude X2 which represents the contribution of the processes involving solelyright-handed leptonic current ap) ( R R ) in (3 .42) ) has been neglected asmentionedbefore (see, however, (3 .76) and (3 .77) ) . T he summa t ion in equat ion (3.51) shouldbe taken over those light neutrinos (mi

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76 T Tomodanoted, however, that even i f mi # 0 for any i, th e effective electron-neutrino m ass(mu)an be much smaller than any mi when the contributions of neutrinos withopposite CP parity cancel each other . On the oth er h an d, if (A) # 0 and/or ( q ) # 0,ther e will be a finite probability for Ovpp decay, and this in tu rn induces a finite mass(m,) even if (m,) = 0 a t a tree level (Schechter and Valle 1982, Nieves 1984, Takasugi1984). However, since the Ov decay rate can still be expressed in terms of (my),A )and (q), we consider these as independent parameters in the following.

T h e nuclear m atrix elements M,$;) and x in equations (3.50) are given by:(3.520)GT0 4 - H(r1z)m b z )(3.526)

and their combinationsf + = r . k G Tx i = - X F Z X G T, - 2xk) .

Here(3.53)

(3.54)

with 6 = a/lal for any vector Q, (012)as been defined by equation (3 .31) , and H(r)etc ar e given in appendix 2. One obtains from (A2.8) the following useful relations(Tomoda et a / 1986)(3.55)

(3.560)(3.56b)

Insofar as we restrict ourselves t o t h e case where a t least one of the electrons is emittedin an S1/2 wave, there is no con tribu tion which is first-order in the other recoil term C(3.150) t o Ot -+ O+ decay, under th e present assumption of th e closure appr oxim ation .

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Double beia decay 77Integrating equation (3.47) over f 1 , we obtain

(3.57)where the numerical coefficient wa s introduced for convenience (see (3.59)) and

(3.58)

(3.62)defined analogously to (3.58). The angular correlation between t h e outgoing electronsis represented by the ratio

(3.63)whereas the single electron spectrum is obtained by integrating (3.47) over cosBI2:

(3.64)The quant i t ies d i ) efined by (3.49) ar e expressed analogously t o (3.60) with thereplacement of Fj l ) by fji) n (3.61) .

--wo, - Q ( o ) ( f l ) W o v ( f l ) .dci

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78 T Tomoda3.3.2. Recoil malriz elements in O -t O Ou,f?P decay. The nuclear matrix eleme ntMk(0) (3.52f) which originates from the recoil term of the nuclear vector current,can be decomposed into centra l, tensor, mom entum -depend ent and spin-orbit pa rts(Tomoda el a1 1986):

Here HRC(r) and HRT(T) re given by

(3.65)

(3.66)

( 3 . 6 7 ~ )(3.676)

where A = (a /& ) , the closure energy A is defined by (A2.2) , and equation (A2.7)was used. The operator S in (3.66) has been defined by (3.54), and P,, and l , ,are the CM momen tum and relat ive orbital angular mom entum of the nucleons n andIn:

Prim = P +P , k m = rnm x 5(pn- P~). (3.68)In the above derivation we neglected the q 2 dependence of the form factors (i.e. th efinite extension of the nucleon ). Since this ap proxim ation is not justified when th eresulting two-body ope rators are short-ranged, we take into account (3.10) at bothends of the n eutrino pr opa gato r and replace the 6-function and l/r2 in equation (3 .67a)by

- - e - A r [ i + A?-+ ~(A?-)21 (3.69)641rand

u l ( r , A ) = (3.70)respectively. For all the o the r transition op erato rs of equation (3.52) we neglect thenucleon finite size eff ec ts sin ce they are relatively long-ranged or do not contr ibutevery much t o the total Ovpp decay rate.

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Double beta decay 793.3.3. U + -+ 0 OuPP decay mediated by heavy neutrinos. Let us now considerthe contribution of heavy neutrinos with mi >> A 1 GeV. For these neutrinos, A(

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80 T Tomoda9.9.4. Rough esl imation of O c -+ O t OuPP decay amplitude (Tomoda et a l 1985).We can regard the OvBP decay ampl i tude as consisting of the contr ibutions from aneutrino propagator, electron wavefunctions and nuclear currents. Table 4 schemat-ically shows the factor decomposition of the partial decay amplitudes @Xj (seeequation (3 .49)) into these contr ibutions. S i and Pi in this table denote the radialand P I / ? wavefunctions of electrons at the nuclear surface (see appendix 1):

S i = &.;)(E;, R) or f : - ) ( ~ i , R) i = 1,2P ; = g i -) ( 6 i . R) or -f ;)(Ei,R) i = 1,2 . (3.78)

(a) T h e numerator of th e neutr ino pro pagator consists of three term s mi,w y Q ndE . 1 (see equation (3 .46) ) . As mentioned earlier, the contribution of the wyo termgets suppressed due t o the antisymmetrizat ion of the out goin g electrons:

(W+ Az)-' - W+ A i ) - ' ~ i :A i - Az)/w2 = (1- FZ)/W'- l/lOO)w-'for the typical values of w sz k - 100 MeV and 1 - 2 - 1 MeV. In the case of thek. 7 term, the negative pari ty of k has to be compensated with eithe r an electronP-wave or the nucleon recoil current D.

Table 4. Factor decomposition of the O t - + OvPP decay amplitudes into variouscontributions. Only the relative magnitudes are 8iv.n for each column. Here w andk are the energy and the momentum of the virtual light neutrinos, k = JkI z w ,and L, the merge s of the emitted electrons. The unit vectors i l z and P + n , whichoriginate from the momen tm k of he neutrino ( f i 2 ) and the electmn P - w a v e ( i i zand F t 1 2 ) , are included in the contributions of the nudear currents in order to showtheir way of coupling to U and D . The common factor ~ t . 2 as been omitted inthe last column. See text for the definition of Sj and P i .

Type of (a)Neutrino (b) Electron (c) Nuclear currentsmatrix element propagator wavefunctions

(b ) An electron P-wave has a n ampli tude $a2 + $ ( E f m,)R re la t ive to an S-wave at the nuclear surface (see (A1.7)) . In the typical case of %e decay we have+a2- and j c f n,)R - m.R - &, This means tha t t he P-wave to S-wave ratiois enhanced by an order of magnitude due to the Coulomb attraction. If one usesplane waves multiplied with the Fermi function, this enhancement cannot be takeninto account and the Contribution of the matr ix element Xs x p ) is underest imated.

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Double beta decay 81Table 5 . Order of magnitwits of the various contributions to the O+ + O+ OuP Pdecay amplitudes. Only the relative magnitudes are given for each column.

Type of (a)Neutrino (b ) Electron ( c ) Nuclear Product ofmatrix element propagator wadunctions currents (4 (b )x l (c ) l

2

a I f one u s e s plane wave8 multiplied by thc Fermi function

On the othe r hand the domina nt aZ erm s cancel each othe r in th e phase-space factorassociated with X4 x ) G ~ , ~ , ~ ) ,nd we have

Thus the radial wavefunctions of P-wave electrons are smaller at the nuclear surfaceth an those of S-wave electrons by a t least a n order of magnitude.(c) We expect th at the nucleons in relative S-states give the most imp ortan t cou-tributions to the nuclear matrix elements in the case of O - + Ovpp decay. Thismeans the dominance of spin-singlet (S = 0) states since the two nucleons whichexchange a virtual neutrino are both initially and finally in isospin-triplet (T = 1)states. Therefore we expect u 1 u 2- 3 and IS121< 1. The nuclear current con-tribution becomes small also in the case of the X , (2;) term because the opera-tor u l - u2 has a non-vanishing matrix element only between S = 0 and S = 1states. We obtain I(u1 - u z ) . ( i l z x i + l z ) l - 0.3 by comparison of the calcu-lated nuclear matrix elements x'p and xbT of table 12 assuming r12 - 2 fm andr+12 2R - 10 fm. Th e l ast en t ry i n the table is the contr ibution of the nuclearrecoil current D . Neglecting the non-central parts, its magnitude is est imated t o be

Table 5 summ arizes the est imated order of magnitudes for the quantities of table4. ALthough th e estimation has been m ad e for the case of "Ge decay, these numbe rsare not expected to change very much for other parent nuclei of practical interest. Wesee from table 5 that the processes involving P-wave electrons (Xq nd X5 terms) arenot important in the case of O+ f Oupp decay and that the X s and X s terms aredominant (11(X) and SOO(0)) for th e decay caused by the right-handed leptonic cu rren tcoupled t o right- and left-handed nuclear cu rre nt, respectively. From these estima teswe expect that the following four parameters correspond roughly to the same Ov,B,B

1612. ( ~ iD2 - 2 x Di)l - p p k l ~ 1u 2 ( / 3 M - 1.

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82 T Tomodadecay probability:

(m,) - 1 eV((m;')) - GeV-'( A ) - .5 X( q )- x 10-8, (3 .79)

9.9.5. O+ + 2+ Ovpp decay. Since the combination of an SI/?a n d a P3lz electronyields the total angular momentum of 2 with the lowest single-electron orbital angu-lar m omenta, i t gives the dom inant contribution to O+ + 2+ Ovpp decay. For thecontrib ution of neutrin o pro paga tor a nd the nuclear curr ent s, the following two caseshave to be considered separately. Firs t, in the processes caused by th e right-handedleptonic curren t , the k 7 erm in the neutrino propagator combined with the lowestorder nuclear current terms V(O)' = 1 and A(') = U compensates the negative parityof a P-wave electron, and gives the dom inant contribution (Molina and Pascual 1977,Doi et QI 1981c, Haxton and Stephenson 1984) . Second, in the processes du e to a finitemass of the ne utrin o, the negative p arity of th e electron system has to be comp ensatedwith the nuclear recoil-current terms C or D . T h e half-life for 0' -+ 2+ Ovpp decayis given by

(3.80)where F,* are the phase-space integrals obtained by replacing fji) in equation (3 .62)with the combination fj* ( A l . l l ) of electron wavefunctions. T he nuclear m at rixelements M a (U = A , q , m) are defined by

(3.81)

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Double beta decay 83with

(3.83)

Here h,, = -rnmH'(rnm). Possible contributions of heavy neutrinos have beenneglected. Similarly to the case of O+ -+ 0' decay, we can estimate the relativeorder of magnitude of various contributions to the decay amplitude. Table 6 givesthe summary of the estimation, where only the contributions of M I , M4 and thesecond term of M, in equations (3.82) were considered for simplicity. As we see bycomparing the last columns of tables 5 and 6, O+ - + OuPP decay has differentrelative sensitivities to the parameters (m ), ( A ) and (7) from those of O c +decay. We expect that the following three parameters correspond roughly to the sameOt + Ovpp decay probability:(m,)/m, - 13eV A ) Y 1.5 x (7)- x (3.84)i.e. O+ + 2+ decay is relatively more sensitive to ( A ) and less sensitive to (m,)compared with O+ -+ Ot decay. Therefore experimental dataon Ot - + decay wouldbe useful for determining which of the parameters is responsible for 0vPp decay. Ofcourse the absolute Ot - + decay rate for a given parameter (e.g. ( A ) ) becomesmuch smaller than that of Ot f decay since(i) the &-value is smaller;(ii) O+ - + decay necessarily involves a P3/2 wave electron whose amplitude at(iii) the nuclear matrix elements M I and (g v/g~ )' M4 are smaller by - $ thanthe nuclear surface is - R / 3 -m,R - 1/80 of a n S-wave amplitude; andtheir O + + counterparts M$ )& and M$ ;)& as we will see in section 5.2 .

Table 6. Order of magnitudes of the various contributions to lhe O+ -, + OuPPdecay amplitudes. Only the relati- magnitudes are given for each column.Type of (a)Neutrino (b) Electron ( c ) Nuclear Product ofmat. el. propagator wavefunctions currents (4 (b)x ( c )MW (m)lm 1 0.5RAgV 0.6(mu)/m.MA 200(X) 1 g 2 - 2 IOO(X)

5Wrl)gvMn 200(rl) 1 9* + R 1 .

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84 T Tomoda3.4. pp decay of a single hadron in the nucleusIn the preceding subsections we discussed pfl decay in which two successive virtualdecays of d- to u-quarks ta ke place in different nucleons. Clearly th is is the onlypossibility if we restrict the degree of freedom of a nucleus t o tha t of nucleons, Inactual nuclei, however, virtual mesons are exchanged and nucleons can be excited toisobars. Th en it is also possible for successive d-quark decays to occur in a singlebary on or meson in the nucleus. Let us consider such cases in t he following.3.4 .1 . N - tmnsation. T he first exa mp le is the pp tran sition s between the nucleonand the A isobar (1232 MeV) shown in figure 4. These processes can cause nuclearOvpp decay v i a virtual excitation of the A isobar as depicted in figure 5 (Primakoffand Rosen 1969). Neutrinoless pp decay due to this mechanism (referred to as theA mechanism) is on the one h and suppressed by th e large m m difference betweenthe nucleon and the A isobar, but on the other hand enhanced because of the shorterdistance over which the virtual neutrino has to travel. In the case of 2 4 0 decay,however, the contribution of an analogous processes is expected to be negligible sincethere is no advan tage in th e small distance between t w o decaying d-quarks. Assumingtha t the baryons can be well described by the non-relativistic constituent q u a rk model(Kokkedee IQGQ),we can reinterpret the transition operators derived in the previoussubsections as acting on quarks in a baryon. Suppose such an operator has the form

(3.85)

where C~,?~:a:jh') and A(A2), ith rank A I and A z coupled t o a total rank A , act. onthe internal and center-of-mass coordinates of a baryon, i.e. they consist of pi - ~ j ,p;-p, , LT,, uj, nd T ~ + T ? + T Q ,p, + p 2 + p , , espectively. In or de r for the operato r(3.85) to contribute t o a O -+ J" nuclear transition in th e A mechanism , th e followingtwo selection rules must be satisfied:( i) the op erator Q$' ) has a rank XI = 1 or 2 in spin space, a rank zero in orbitalspace, and a positive parity ?rl = +; and(ii) A = J , and the parity i ~ 2f A(Az) is equal to r.In earlier treatments of the A mechanism (Primakoff a nd Rosen 1969, Sm ith et 11973, Pic ciotto 1978, Halprin et a l 1976) these selection rules were not taken intoaccount until Doi et a l (1981b) paid attention to the first rule. Haxton e l al (1981)then argued that a O+ -+ O+ transition is strictly forbidden regardless of the choiceof the weak interaction Hamiltonian i f the second rule is also taken into account.Certainly th e opera tors (3.82) canno t contribute to a O+ - + transition even if theyare interpreted as acting on quark degrees of freedom, and none of the operators in(3.52) con tain a component of t h e form (3.85) with A 1 = A2 = 1or 2 , and TI = az = +.However, including the Contribution of electrons emitted i n a n S l l Z and a P 1 / 2 waveas well as the recoil curre nts of quarks, we obtain an operator of the form

(3.86)where L is the orbital angular mom entum of the baryon. After taking th e ma trixelement between t he i ntern al wavefunctions of the nucleon a nd the A isobar, we ob tai n

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Double beta decay 85f rom equat ion (3.86) an opera tor proport ional to (X)[T=2S+ 7 t ) = 2S t ] ., hereS is the t ransi t ion spin (Sug awa ra and von Hippel 1968) a n d 7 the transition isospinof r ank 2 (Tomoda 1988). This clearly contributes to O+ -+ O+ Oup@decay in theA mechanism. We obtain also similar contribut ions proport iona l to (m,) and (v)if we avoid th e closure app roxim ation. However, these con tribut ions du e to the Amechanism a re no t expected to be impo rtan t s ince the O+ -+ O+ Ou@@ decay in thetwo-nucleon m ech anis m is dom inate d by processes involving only S , l Z electrons.

Figure 4. Ov,30 transitions between nucleon and A isobar

Figure 5 . Nuclear 0 4 3 ecay via virtual excitation of A isobar (A mechanism).

Th e s i t ua t ion is quite different in the case of O - f t rans it ions . T he opera tor(3.87)

(see equations (3.83)) , when act ing on quarks, satisfies both of the selection rulesabove, and yields an operator proport ional to 7 =2S+ (Tt)p=*S,where S is thetransi t ion spin of rank 2 (Tomoda 1988), after ta king the m atrix element between theinternal wavefunctions of the nucleon and the A isobar. The processes of figure 5 canbe described by the effective ope rator actin g on the nucleons n and m

where VAN is the st rong interact ion potent ial for the NN- AN t ransi t ion and6m the m ass difference between t he A isobar an d nucleon. T h e explicit form of theeffective operator was derived by Tomoda (1988) using a one-boson (T+ p ) exchangepotent ial for VAN.

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86 T Tomoda3.4 .2 . A - A tmnsition. Since the isospin of the A isobar is larger than $, af lp transition like those of figure 4 but within the isospin multiplet A- - A + o rA' - A t t is also possible (Haxton and Stephenson 1984). As shown in figure 6,a transi t ion of this type necessitates virtual excitation of A in both the initial andfinal nuclei so tha t i t s contr ibut ion is expected to be much less than that of figure 5.A potentially favourable case is the short-ranged O t - t OvPP process due to theeffective ma ss ((m;')) or ( ( m ; ' ) ) ~ f heavy neutrinos. Wh ile th e operators app earin gin equations (3.73), interpreted as acting on quarks (with gA , A - co), cannotcontr ibute to th e N H A transition because of th e first selection.rule above, they cancause A ++ A t ransi t ion an d con tribute to th e process of figure 6.

P Pl I

.....

Figure 6. OvPO decay involving A - transition.3.4.3. r- - + Imnsit ion.. Figure 7 ( a ) shows another example of pP transitionwithin a n isospin multiplet an d c on tribute to nuclear PP decay through th e diagram7 ( b ) . T h e processes of figures 6 and 7 ( b ) were considered previously by Pontecorvo(1968) in the context of a hypothetical superweak interaction. Vergados (1982a) de-rived effective two-nucleon op er at or s which describe th e process offigure 7 ( b )mediatedby virtua l heavy neutrinos an d calculated the nuclear m atrix elements for the O + - tOupp decay of 4aCa . Assuming the non-relativistic qu ark model for the pion, he foundthat the Contribution of the diagram 7 ( b ) was comparable to th at of the conventional2N mechanism (figure 2 ( b ) ) .

" *^

wTI-

Figure 7 . a ) O v4 O transitions between T - and st. a ) Nuclear O@P decay dueto Ov transition of pion exchanged between nucleons.

In relation to the discussion in sections 3.4.1 and 3.4.2, it should be mentionedthat there are processes involving the A isobar which can contribute to O+ - O t0 ) .Oupp decay (figures 8 ( d ) a n d ( e ) ) . Fazely and Liu (1986, 1987) est im ate d ANN .A$A : R ~ [ A A A ] 1 : -2 : -3, where A, AgA and AA& denote respectively the

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88 T Tomoda(m i

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Double beta decay 89Based on the same assumption Haxton e l a1 (1982b), however, obtained an effectiveop era tor which is different from th at of the previous author s mainly by a n overall factor

of 1 / 4 r . Their shell model calculation gave a rate three orders of magnitude smallerthan tha t obtained by M ohapatra and Vergados.Furthermore, since the singly charged Higgs boson Q- in figure 9 should actuallybe a physical Higgs boson, which is predominantly the member H- of the tripletand couples with quarks only through a small admixture of the charged member ofthe doublet, the contribution of figure 9 will be highly suppressed (Wolfenstein 1982,Schechter and Valle 1982).In view of both of these suppression factors, the Ovpp decay involving a doublycharged Higgs boson seems to be unimportant although such a suppression might beavoided by i ntrod ucin g othe r typ es of coupling or more complicated Higgs sectors(Schechter and Valle 1982, Picciotto and Zahir 1982, Escobar and Pleitez 1983).

4. N u c l e a r structure c a l c u l a t i o n sThis section deals with the formal aspects of nuclear structure calculations for thepp transition matrix elements. The results of numerical calcnlations will be given insection 5.4.1. Common features4.1.1. Closum opprozimotion. The closure approximation was used frequently inthe literature. It facilitates numerical calculcations considerably since in this approx-ima tion one needs only the wavefunctions of th e initial and final nuclear sta tes of ppdecay. Wh ile it is expected to be good in the case of Oupp decay (see (3.45)), itsvalidity is not trivial in th e case of 2 v p p decay. Formally the approximation is alwaysvalid if (EN)s defined by equ atin g th e right-han sides of equations (3.27) and (3.29),I.e.

(4.1)If it is possible to calculate the right-hand side of this equation, which is jus t thedecay amplitude M,,, there is actua lly no need t o invoke the closure approximation.In practice, however, it may be technically difficult to perform an explicit summ a-tion especially in the case of a large-scale calculation. Then it becomes necessary toestimate the energy (EN)which satisfies (4.1) as well as possible. Vergados (1976)estimated it by

(4.2)This (EN) s the energy expectation value for th e Gamow-Teller collective st at eNr+alOf), where N is a norm alization cons tan t. It can be calculated relativelyeasily but tends to overestimate the correct value of the closure energy because of thelarger weight for higher-lying states.

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90 T TomodaTo improve upon this method Haxton and Stephenson (1984).estimated (EN) romthe E--weighted sum :

and instead of performing a shell-model calculation, which is just as difficult as thecalculation of the right-hand sid e of (4. 1), they used t he param etrization of Gamow-Teller (GT) strength distribution made in statistical studies of 13 decay by th e W asedagroup (Takahashi and Y am ada 1969, Koyama e l a l 1970, Takah ashi 1971).One can also think of using the GT l+ s tr eng th I(1;11r-ul10$)12 with respect tothe f inal nuclear state in equations (4.2) or (4.3) instead of the 0- strength from theinitial st at e. I t would give a sm aller value for (EN) ince the G T pt stre ng th is usuallydistributed a t lower energies th an the p- countrrpart .Any of these methods will fail badly if there occurs a considerable cancellationamong the terms under the summation in the right- and/or left-hand side of (4.1).It is therefore desirable to avoid the closure approximation in the case of 2u decaywhenever possible (Huffman 1970, Skouras and Vergados 1983, Grotz e t a/ 1983,Tsuboi el a/ 1984).

4.1.2. Shor t - range correlalions. Nuclear models employed in @@ decay calcnla-tions are usually based on the independent particle picture. While long-range cor-relations are taken into account by mixing of hasis states within their model spaces,the short-range repulsive correlations due to the nucleon hard care are absent in themodel wavefunctions. The ir effec t is especially i m po rt an t for the calculation of ma trixelements such as X G T ~ , F ~n d ,& since the range of the relevant operators is of theorder of th e hard core radiu s. An approximate way to correct for this is to multiplytwo-nucleon wavefunctions by a correlation function f(l.1 - 2I) when we calculatetransition ma trix elements. T hi s amoun ts to the replacement

(ppJpll~(J)(lnnJn) ppJpllfUcJ)fllnnJ) (4.4)in equation (4.24) etc below. As a form of f ( r ) ,a simple step function with the hardcore radius (Halpriu e l a l 19 76 , Vergados 1981) or a more realistic one (Miller andSpencer 1976)

f(r) = 1 - - n r D ( l- r2) (4.5)with a = 1.1 fm- an d b = 0.68 fm-2 has been used frequently in the literature (e.g.Haxton e l a l 1982a, Vergados 1983, To m od a el al 1986, Engel et a/ 1988, Muto el a l1989b).A more fundamental approach wa s undertaken by Wu e l a l (1985), who derivedthe effective transition ope rator f U f for the case of 4aC a decay. Th eir results usingthe Reid and Paris potentials show weaker correlations than the phenomenologicallydetermined f ( r ) of (4.5).

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Double beta decay 914.2. Nuclear models4.2.1. Shell model. Th e nuclear shell model has a nice feature tha t i t can provide uswith microscopic wavefunctions which have a given set of quantum numbers such aspar i ty , angular momentum and isospin corresponding to the symmetry properties ofth e H amilton ian. T hese wavefunctions autom atically include any correlations causedb y the interaction part of the Hamiltonian provided that the model space is t akent o he l a rge enough . Th is last co nditi on, however, is often difficult to satisfy. Int h e case of PP decay, 4aC a is probably th e only nucleus which can be t r ea ted in theconventional shell model reasonably well. As on e goes to heavier nuclei the numberof basis stat es increases explosively. Although it is not conceivable that all of theseare equally i mp ort an t, i t is diff icult to select im porta nt ones w ithout the knowledgeab ou t ' true' wavefunctions.An assum ption of weak coupling between the proton and neu tron system s gives aposs ib le way to t runc ate the huge model space to a tracta ble one. In order t o t reatt h e lza'lsaTedecays, Vergados (1976) assumed a weak coupling limit, i.e. completedecoupling of the proto n and n eutron syste ms. T h u s each nucleus w a s described by as ingle product of proton and neutron wavefunetions.Pro ton- neu tron correlations c an , in principle, be taken into account if linear com-bina tion s of pro du ct wavefunctions are use d. For the description of 'Qe, "Se an d1za/130Tedecays , Haxtou e l a l (1981, 1982a, 1984) first solved the eigenvalue equa-t ion to ob ta in t h e lowest fifty states for each of the pure proton and neutron systems.Th en the Ham iltonian including the proton-neutron interaction was diagonalized inth e space spanned by the pro duc ts of these pro ton and neutron wavefunctions. As wewill see in section 5, th e nuclear m at rix elements for 0" decay are not very sensitive toproton-neutron correlations an d expected to be calculated reasonably well by thesewavefunctions. In th e case of 2 u decays, however, th e proton-neutron correlations areessential and the weak coupling scheme does not give reliable results.4.2.2. Quasiparticle random ph as e n ppr or t m a t aon . The quasiparticle random phaseapproximat ion (QRPA) (Baranger 1960) has been widely used for various excitationmodes in even-even nuclei. Halbleib an d Sorensen (1967) applied this formalismt o a proton-neutron mode (ex citatio n fro m even-even t o odd-odd nuclei) for thecalculation of Gamow-Teller decay s.

T h e QRPA pho non s for a proton-neutron mode with angu lar mom entum and parityJ are defined by

where the subscript p or n denotes the quantum numbers, except the z-component ofangular mo me ntum , of a proton or neutron state. T h e tildes indicate the t ime reversalrij, = (-l)j+"'aj-,,, and the qua sipa rticle crea tion ope rator ~ j , t is related to thenucleon ope rator s cj,t, ?jm by the Bogoliubov-Valatin transformation (Bogoliubov1958, Valatin 1958)

(4.7)j m + = u jc jm -vjcj,.Th e coefficients u j and u j ( u j , v j 2 0 , + U = 1) can be interpreted as vacancy andoccupation a mplitud es, respectively, an d are determ ined in the sam e way as in the BcS

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92 T Tomodntheory of superconductivity (Bardeen e t a/ 1957), i.e. by minimizing the expectationvalue of th e nuclear Ha miltonian in th e quasiparticle vacuum with fixed average pro tonand neu tron numbers. T h e forward- and backward-going amplitudes zpn,i and ypn,ias well as the energy eigenvalues wi are obtained f rom the QRPA equation

[- -BA] [;] = [61 (4.8)with the subm atrices A and El defined byA p , , p w = 6pn,p,n~(

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Double beta decay 93with

( 1 f b + d l O , + ) = C ( P ( ( r + a ( ( n ) ( u p u , l p n , j+ "pUnYp",j) (4.14a)p n

(4.14b)0,+ +Ir W I I Q ) = C(PIIr+aIIn)(vpu,Zp, ,k + G ~ G v ~ ~ . ~ )p nwhere we omitted the superscript lt n the QRPA amplitudes zpn,j nd yp,,j, and thebarred quantities indicate that quasiparticles and phonons are defined with respect toth e final ground state 0;. T h e overlaps (J; 1 JT ) between two J' state s belonging t otwo different sets { IJ ; ) } and { I J ; ) ) are given by

(4.15)

In the case of Ov@ decay, virtual excitation of intermediate states with J" # 1+has t o be also taken into account. We define ( 0 1 2 ) by(012) = (o;ll[cP,' @ ~"~l(J)llJ;)(J;IJ~)(J;ll[~P' @ EJJ)II0:)

pp"'J r j k J '

x ( - l ) tP ' t J t J ' (2J '+ 1) [ , ;,} (pp'J'(r;r$Olzlnn'J')(4.16)

where the las t factor on th e right-hand side is a non-antisymmetrized two-body matrixelement. We obtain this form from (3.42) after integration over z, and k,with theopera tor O l z dependent on th e energy E ( J T ) of the individual state IJ,"). It reducest o (012) = ( O f i f E,,,, r$r~O,,lOf) (equation (3.31)) if w e replace the energy E ( J ; )by some average ( E N ) nd if both sets of the intermediate states { IJ ; ) ) and {IJ;)}are complete (closure approxim ation). T h e mat rix elements of the one-body transitiondensities in QRPA ar e given analogously to (4.14) by

( J J [ C P + @ E"I'J'll0:) = m ( u p u n z ; ; , j + upunyp,,j) (4.170)(o;ll[cp' @ E ] ' q J ; ) = m(ijptl"i;i,hcpv"V;;,k). (4.176)Equation (4.16) with (4.17) will be used (T om od a and Faessler 1987, see also Engelet a l 1988) for the calculation of Ov matrix elements such as (3.52) and (3.73). Itis possible to deal with 0 1 2 dependent on the individual energy E ( J ; ) and test thevalidity of the closure approximation in the case of OvPP decay. Since this will tu rnou t t o be reliable enough as expected, we usually replace E ( J ; ) by some average ( E N )t o simplify numerical calculations.T he Gamow-Teller ope rator in (4.14) connects only single-particle sta tes with thesa m e orb ital qu an tu m numbers. Under the usual situ atio n of open-shell nuclei withlarge neutron excess, the product upun which contribute s to (4.14) can be large (- 1)for a pair of orbitals with high occupation by neutrons and low occupation by protons(see figure 10). In contrast, v p u n is always small (< 1) since the orbitals occupiedby protons are occupied by neutrons as well. Combining this with the property that

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94 T Tomodathe backward-going amplitude y is in general smaller than the forward-going one z,we see that the P-t transition matrix elements (4.14a) are dominated by the firstterm in the parentheses. In contrast to this, the first and th e second term s in the P + tmatrix elements (4.146) are generally both small and can be of the same order. Thefirst and second terms vanish in the limit of no pairing correlations ( f i P U , = 0) andno RPA-type ground-state correlations ( t j = O), respectively, indicating the sensitivityof the P+ transitions to these correlations.

Figure 10. Schematic representation of the occupation amplitudes for proton (up)and neutron (U ) as well as the products uv with ZL = (1 - ' ) ] I 2 , he broken linescorrespond to Lhe case of smaller pairing correlations,and the horizontal lines showthe proton and neutron Fermi levels.

For the important single-particle orbitals between the proton and neutron Fermilevels, up a n d U are of order 1 while u p , U, < 1. Therefore the ph (particle-hole)interaction (O(1)) dominates over the pp (particle-particle) interaction (O(uZ,) orO ( u ; ) ) in the subm atr ix A (4.90). O n the other hand the products of the uu actors inthe submatr ix B are all of the same order (O(vpun)), and inclusion of pp interactionaffects B substantially. It enhances B in the case of 1+ mode since the ph interactionis repulsive while the pp interaction i s attractive. (It can be shown that the relation(pn- ' I V1p'd-l J ) = - pn JI V Ip'n' J ) (4.18)holds exac tly for the spec ial case of a local potential without spin-isospin dependence.)The lower half of the QRPA equat ion (4.8) can be cast into the formy = - (A +w)-'Bz. (4.19)

The backward-going amplitude y becomes larger for larger E , and since A is positivedefinite an d the m atri x elements of B are largely positive, the elements of y tend tohave the s ign opposite to that of the dominant elements of I (Vogel and Zirnbauer1986). Therefore the effect of the t er m s involving y in (4.140) a nd (4.146) is to h i n d e rthe G T transitions. Since the con tributio ns of the two terms in (4.14b) are of thesame order as previously mentioned, the degree of hindrance in the P+ ampli tudesis sensit ive t o the magn itude of the subma trix B , and this in turn depends cruciallyt 0 refers to e* emitting (virtual) transition of an even-even nudeus.

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Double beta decay 95on th e inclusion and t he s t reng th of PP interaction (Vogel and Zirnbauer 1986). Aswe will see in section 5, almost complete cancellation between these two contributionsseem to occur in actu al nuclei .

4.2.3. Pmjected Har2ree-Foc~-Bogoliubou method. The two impor tant aspects ofopen-shell nuclei , pair ing correlation a n d qua dru pole deforma tion, can be taken intoaccount by a quas ipar t ic le vacuum s ta te I@) atisfying a u l a ) = 0 for any a, here thequasiparticle creation operator aut is defined by(4 .20)

which is a generalization of (4.7). For simplicity we assume that this transformationdoes not mix: ( i) proto ns and neu trons ; ( ii) s tates of different parities; and ( i ii)s ta tes of different ang ular mom entum projection on z-axis (axial symm etry). Th equasiparticle vacuum I@) s a superposit ion of s tates not only with different protonand neutron numbers , as in the spherical BCS vacuum, but also with different totalangular momentum. We project out a s ta te wi th g iven proton (Z) and neutron (N)numbers , angular momentum J , nd its z-component M=O (Schmid et a l 1984):

I Z N J , M = 0) = NP:&l@) (4.21)where N = (@lPf&l@)-1/2s a normal izat ion factor , and the project ion operator isgiven by

PF$K = / d f i D f i K ( f i ) k ( f i ) (4.22)with

2 1 + 1e(6) exp[-i(4,fip + +,fin)lR(n)d 6 = d+p d& dQ.

D ? i K ( f i )= (2?r)28?r2 exp[-i(+pZ + + n N ) l D L K ( f l )(4 .23)

Here Np(fin)s a pro ton (neu t ron) number opera to r , k(n) and D L K ( Q ) re the I-ta t ion operator and th e D-funct ion (E dmo nds 1957) in the ordinary three-dimensionalspace. T he coefficients of the quasiparticle transformation in (4 .20) are determinedby minimizing the expec tation value of th e nuclear Hamiltonian( Z N J M I H I Z N J M )

(variation a fter projection).wr i t ten as ( T o m o d a e l a l 1986)Th e m at r ix e lement of any two-body operator U() for O f

-+ J f pp decay can be

(4.24)

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96 T Tomodawhere (pp'JpllU(J)llnn'Jn) is a n antisymmetrized two-body matrix element. T h e ma-trix elements of two-body transition density are calculated as(JF+II[[cp'@ c ~ , ~ ] ( ~ ~ )[En @ E n , ] ( J ' ) ] ( J ) l l O ~ )

= NFNJ 1 f i D $ i (fi) (@FI@,(i) Or)x (pmpp'mp, J p Mp )(nmnn'mn,J, -M.)(-l)Jn+M=(JpMp J,M. I O)

(4.25)where Z and N are the proton and neutron numbers of the initial state and

(4.26)

(4.27)obtained from (4.20) was used in deriving (4.25),where U and V are t h e quasiparticletransformation matrices for I@p).

T h e present m ethod is especially suited for trea ting heavier deformed nuclei usingrealistic interactions and a large num ber of single particle orbitals. We can also applyit without any difficulty to the calculation of O+ - + transitions, which is not trivialin the QRPA approach. On the other hand it will overestimate 2u decay rates for thesame reason as in the weak coupling shell model.5. pp d e c a y rates and constraints on lepton number violation5.1. U+ - + Ou and 2u p@decayT h e first calculation of the nuclear m atr ix elements for pp decay was reported morethan thirty years ago by Beliaev and Zakhar'ev (1958), who used shell-model wave-functions to calculate the 2 v p p dec ay ra te of 4aCa. T his nucleus, which is the lightest? ecay can didate except for 46Ca nd h as a large Q value ( & p ~= 4.272 MeV), hasappear ed frequently in the l i terature on p/3 decay. Kh ode l (197Oa, b , 1974, see alsoFaya ns an d Khodel 1977) calculated the Ou as well as 2v decay matrix elements ofthis nucleus treating the vertex renormalization in the theory of finite Fermi system(Migdal 1967). While counter experiments only set limits on p p decay of 48Ca (seee.g. Bardin et a1 197O), occurrence of 130 decay in l3'Te becam e more an d more evi-

de nt by geochemical m easurem ents (Inghr am a nd Reynolds 1950, Takaoka an d O ga ta1966, Gerling et al 1967, Kirsten e t a1 1968). The 2v matrix element for this nucleuswas calculated by Huffman (1970) in the QRPA. The Gamow-Teller ( G T j interaction,i.e. th e isospin flip pa rt of

XU1: U z T i . 2 2 (5.1)

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Double beta decay 97with a constant x , was used as the effective N N interaction and only the particle-hole matrix elements were retained in ( 4 . 9 ~ ) nd (4.9b).For the calculation of Mz,,he approximated (O$llr+ulll:) by ( O ~ ~ ~ r + u ~ ~ l ~ )n (4.13) so that a l l the barredquantities in (4.14) and (4.15) were replaced by those without bars ((1:llf) =6 k j ) . His matrix element IMz,l = 0.204 MeV- and the pha ses pac e integralFz. = 1.27 x lo- y- MeV2 of tab le 25 (app end ix 1) give .I/Z = 1.9 x 10 Y,which is two orders of magnitude shorter than the limit from the geochemical data1.5 x 10 y < r,p < 2.75 x 10 y (Kirsten e l Q 1986). (It should he noted tha t thephase-space inte gra l of Prim ako ff an d Ro sen (1959) used by Huffmau was underesti-mated by a factor of about five for the 130Te decay (Haxton et a/ 1982a) due t o thenon-relativistic appro xim ation for th e Co ulom b correction.)T h e p/3 decay of 12a/13aTeas well as 46Ca w a s investigated by Vergados (1976)using th e shell model. He obt aine d a much smaller value JMz,~ 0.0388 MeV- forth e I3Te decay fro m IM,$?$)I= 0.248 and A = 12.8 MeV (see (3.29), (A2 .2 ) ) in theclosure appro xim ation. Th e model space used in his calculation , however, seems tohave been too small. There the weak coupling limit, i.e. a complete decoupling ofproton a nd neutron systems, w a s assum ed, an d th e seniority-zero wavefunction in t heOhll/z shell w a s used for th e neutron sys tem .

Table 7. Nuclcar matrix elements calcukted in the shell modcl and closurc energiesA for O+ + O t 2 v and Ov@P decay (Haxton and Stephenson 1984).

Mg;) 0.222 1.278 0.938 1.474 1.4830.005 0.016 0.011 0.013 0.014(fm-) 0.115 0.411 0.331 0.412 0.413

XF -0,149 -0,200 -0.177 -0.225 -0.2251.087 1.141 1.142 1.175 1.179

-0,155 -0.231 -0.204 -0.272 -0.271xF -0.110 -0.013 -0.021 -0.030 -0.025X? -0.191 0.269 0.441 0.337 0.2%XPX+

~~A (MeV) 7.72 9.41 10.08 12.54 13.28

A series of more systematical shell-model calculations of Ov and 2v matrix ele-me nts were performed by Haxton et a / (1981, 1982a, h, 1984). They took (Of7/2. Of5/2,par ticle orbita ls for the calculation of 48Ca, 76Ge/Se and 12d/130Te ecays, respec-tively, and used the effective interaction of Kuo and Brown (1968) modified to repro-duce the spectra of the neighbouring nuclei in the respective region better (McCroryet a/ 1970 , Baldridge an d Vary 1976). T h e Cou lom b interaction between proton s w a salso included. Except for the case of 4d Ca , h e model spaces were trun cate d under theassumption of weak coupling of the proton and neutron systems (see section 4.2.1).Th e calculated 2v and Ov matrix elements (both in the closure approximation) as wellas the closure energies A are listed in table 7. The 2v matrix element Mg$) for the4Ca decay is much smaller than those for other nuclei (see also Vergados (1976), Sk-ouras and Vergados (1983)), and can be interpreted (Zamick and Auerbach 1982) asdu e t o th e Lawson-Nilsson I< selection rule (Lawson 1961). T he double Fermi m atrix

1 ~ 3 j 2 , P I / z ) . Of5/z, 1 ~ 3 / ~ ,P I / Z ,g9p) and (Og7/2, & /z, & /Z , 2 s 1 p Ohii/z) single

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98 T Tomodoelements M p ) ar e defined by replacing U I 2 with 1 in th e definition (3.30) of M&?;),an d can have non-vanishing values du e t o isospin mixing. T h e second line of the tab leshows th at they are in fact small enough to be safely neglected. T he dependence onnuclei is weaker for the Ov m atr ix element M, $,) than for M&?;). (For instanceof *Ca is not so strongly suppressed because the Ov transition operator has a radialdependence.) Concerning ot he r Ov matrix elements (relative to M , $ , ) )we see thatxLTs 1, an d e xcept for those of4Ca, X F = xk w -f(gv/gA) = -0.21 and I & 3 b 1 . 3 0 ~ 0 . 0 5 e t>5t 1.5 < T , , ~ < 2.75t>5.OC 1 .2*O. l t 1 . 8 i 0 . 7 i t 0 . 7 5 i 0 . 03 f t

1 , 0 3 t 0 . 3 3 d9 f l d -0.421,1t0.8h-0 .3

I(mv)l (ev)

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Double beta d e c a y 99(less string ent) because of the interference te rm s in (3.60). In view of the discrepancybetween th e calculated an d experime ntal 2v decay rates , Haxton a nd Stephenson con-sidered a possible suppression of Ov matr ix e lements by the same amount as for the2v ones. If this is the case, the limits deduced from 76Ge an d '%e decays should bemultiplied by a factor -2. Those deduced from the i2a/i30Te ratio remain the sam e ifall the Ov and 2v ma trix elements for these tw o nuclei are reduced by th e sam e factor(i.e. by a factor -IO), which is not obvious when there occurs a large suppression.The discrepancy in the 2v decay rates remained a serious problem. I t w a s evenlarger when the amplitudes M z , were calcu lated w itho ut invoking th e closure approx-im ation . Th e shell model calculations for 4aCa (Skouras and V ergados 1983, Tsu bo i e la1 1984) and the calculation for heavier nuclei in the BCS model with particle-numberprojection ( P B C S ) (Grotz and Klapdor 1986) showed that the closure energies A esti-mated by Haxton el a1 using (4.3) were a factor -2 too large.

Table 9. Half-lives for Z w O O decay calculated in the PBCS method (Klapdor andGr otz 1984). Collective Ah excilation and the ground state correlation due toquadrupole-quadrupole nteraction were also taken into accwnt.

Pairing + VGT 1.3 0.051 0.062 0.015+Ah 1.9 0.072 0.087 0.020+VQQ 2.2 0.15 0.57 0.12

In investigating the mechanism of suppressio n of Zvpp decay, Klapdor and Grotz(1984) applied the pairing-plus-quadrupole model with residual proton-neutron forcesby Halbleib and Sorensen (1967) to the calculation of 2v decay rates , taking intoacco unt also particle-number projection an d isobar-hole (A h) excitation. Th e GTin teract ion (5.1) was first diagonalized in the 2qp (two quasiparticle) space for thein termediate 1+ states , and the Oqpf4qp space for the init ial and final O+ s ta tes .Apar t f rom the number project ion , these are par t of th e c ontribution s included inthe QRPA treatment of Huffman (1970), in which (6qp, lOqp, . . .) and (8qp, 12qp,. .) components are also included in 1+ a n d O+ sta tes , respectively. T he closureappro xima tion was avoided by th e explicit trea tm en t of th e 1+ s ta tes . T h e calculatedhalf-lives are given in the first line of table 9. The half-life for 130Te was similar toHuffman 's value. It increased -30% by tak ing in to acc oun t the exc itatio n of collectiveA h state s . In the next s tep they considered the ground sta te correlation d ue toquadrupole ( 2 + ) mode. This induces 4 q p compon ents of the typ e

and consequently, by recoupling the angular momenta

in the ground states of the init ial and final nuclei, which add constructively to thecomponen ts of the same type induced b y th e GT interaction . T h e half-l ife for 13'Teincreased further by a factor 6 so that i t became shorter than the geochemical l imitby one order of magnitude instead of two.

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100 T TomodaTable 10. Nuclear matrix elements for Ot - t OuOO decay calculatedin the P B C Smethod (Grotz and Klapdor 1985a).

A&?.) (fm-) 1.02 0.79 0.83 0.78XF -0.24 -0.23 -0.24 -0.24

Whether there is a similar suppression for the Ou mode is an interest ing andim po rta nt issue. In subseq uent work, Gr otz and Klapdor (1985a) calculated th e Oumatrix elements M: ) and XF (ta ble 10) employing essentially the sam e wavefunctions(wi thout the A h components) as for the 2u matrix elements above. They found th atthe co rrelations (both in the O t and 1+ states) affected the Ou matrix elements onlyweakly. Th e tota l effect of t h e correlations was to reduce M:;) by - 0% from thepure BCS result.

Table 11. Recoil matrix elements for the Ot - t OYOP decay of G e (Tomodae t al 1986). Here m and mg; denote the contributions of the t e a proportionalto A in (3.67), nd xk = (gv/s~)M$~)/M&).he effects of he short-rangecorrelations(incolumns (2) nd (3))and the finite extension of the nudeon for mg4and mg (incolumn (3)) were also taken into account.

(1) Point (2)Point (3)ExtendednoSRC SRC SRC

-71.603.10

-0.20-68.70-0.6470.003

-0.004-0.6490.6530.116

-68.5878.82