Nuclear moments for the neutrinoless double beta decay

NUCLEAR PHYSICS A Nuclear Physics A 628 (1998) 170-186

EL-SEWER

Nuclear moments for the neutrinoless double beta decay C. Barber0 a,**1,F. KrmpotiC a,‘, D. TadiC b a Departamento de Fisica, Facultad de Ciencias, Universidad National de La Plata, C.C. 67, 1900 Lu Plats, Argentina h Physics Department, University of Zagreb, Bijenic’ka c. 32-P.O.B. 162, 41000 Zagreb, Croatia

Received 14 August 1997; revised 15 October 1997; accepted 28 October 1997

Abstract A derivation of the neutrinoless double beta decay rate, specially adapted for nuclear structure calculations, is presented. It is shown that the Fourier-Bessel expansion of the hadronic currents, jointly with angular momentum recoupling, leads to very simple final expressions for the nuclear form factors. This greatly facilitates the theoretical estimate of the half-life. Our approach does not require the closure approximation, which, however, can be implemented if desired. The method is exemplified for ,f3/? decay 48Ca ----t48Ti, both within the QRPA and a shell-model like model. @ 1998 Elsevier Science B.V.

1. Introduction The standard model (SM) of electroweak interactions has been brilliantly confirmed by a host of experiments. However, it says very little about the properties of neutrinos. Actually, in the SM it is postulated that: (i) the neutrinos are the only fermions without

right-handed partners; and (ii) their masslessness is dictated by the global lepton-number symmetry, and not by a fundamental underlying principle, such as gauge invariance for the photon. Further, whether neutrinos behave so “trivially” as required by the SM is one of the most fundamental open questions of present-day physics. It has been known for a long time [ l-5) that neutrinoless double beta decay (/3&,) is a very sensitive probe of lepton number violating terms in the Lagrangian, such as * Corresponding author. ’ Fellow of the CONICET of Argentina. 037%9474/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PII SO375-9474(97)00614-3

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the Majorana mass of light neutrinos, right-handed weak couplings as well as the Higgs exchange [6], right-handed weak coupling involving heavy Majorana neutrinos [711, massless Majoron emission [2,8-11], and R-parity breaking in the supersymmetric model [ 12,13]. Thus, if flflo~ decay is someday observed experimentally it would hint at new physics beyond the SM. But, even if it is not observed, the measured limits on its transition probability, which are steadily improving [ 14], could be translated into more stringent constraints on the parameters of the new theoretical developments mentioned. The extraction of these constraints from the data is only possible when we know how to deal with the nuclear structure involved in the flflo~ decay. This is not at all an easy task, because of: (i) the nuclear Hamiltonian is only roughly known to the extent that the choice of the appropriate parametrization is an art; (ii) there is, in general, a very large number of nuclear states involved in the calculation; and (iii) the formulas for the flflo~ decay rate are rather complex and difficult to implement in a nuclear structure calculation. In the present work we derive simple expressions for the nuclear matrix elements, especially tailored for the nuclear structure calculations. The simplification mainly comes from the Fourier-Bessel expansion of the term exp[ik - (rl - r2)] in the transition amplitude, and in performing the integrations in the following order: first on d o t , then on drl and dr2, and finally on k2dk [15]. Thus far, the same procedure has been applied for the evaluation of the matrix elements MF, M~r [ 16,17] and MR [ 1 1] that arise from the electron s-wave. Here we also deal with the p-wave matrix elements that are relevant when the admixture of the right-hand lepton current is considered. Other studies on the subject are those of Vergados et al. [18], who derived the formulas directly in momentum space, and those of Suhonen et al. [19], who worked in the framework of a relativistic quark confinement model. This paper is organized as follows. In Section 2 we discuss the basic mechanism for flflo~ decay, presenting the effective Hamiltonian and the transition amplitude in a form convenient for the multipole expansion, which is carried out in Section 3. In Section 4 we give detailed formulas for the nuclear matrix elements and discuss the nuclear structure calculations involved in the problem. Summarizing conclusions are drawn in Section 5.

2. Effective Hamiltonian and the half-life The Ovflfl half-life [T0y(0 + ~ 0 + ) ] _ 1 = Fo~ ln2'

(1)

for the decay from state II) in the ( N , Z ) nucleus to state IF) in the ( N - 2, Z + 2) nucleus (with energies El and EF and spins and parities J'~ = 0+), is evaluated via the

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second-order Fermi golden rule. Thus the decay rate (in h = c = me units) is [20]

dp____Z_l d p ~ [Roy(el, e2)[2t~(EI ÷ ~2 + EF -- El) (277.) 3 (277.) 3 ,

For = 2~r ~

(2)

SeI Se2

with

dk Rop(el,e2) = ~ - - ~ - ' ~ N

(2¢r)3


sv

E~ -

EN

-- el

--

e , , vlnwII>

to

(3)

where e - (e,p, se) (v -- (to, k,s~)) stands for the energy, momentum and spin projection o f the electron (neutrino), and N runs over all levels in the ( N - 1, Z + 1) nucleus. The effective weak Hamiltonian reads [2,4,5]

Hw = --~

d x [ j L e ~ ( x ) J ~ ( x ) + j R e , ( X ) J ~ ( X ) + h.c.],

(4)

e=l where the summation is over the number of lepton generations,

j~,Rg(X) = 2~(X)TI~PL,RNL,Rg(X);

PL,R= l ( 1 q: T5),

(5)

are the leptonic currents, formed from the electron field ~ ( x ) and the Majorana neutrino field N e ( x ) o f mass me, and J~(X)

=Uegd~t(x),

.7~g~(X)= Vee,(aJ~t(x) + r / J ~ t ( x ) )

(6)

contain the hadronic (V :F A) currents J~.R" 2 Uee and Ve, are the neutrino mixing matrices for the left- and right-handed sectors, and A and r / a r e the strengths o f admixtures of the (V + A) current. Within the non-relativistic impulse approximation the hadronic currents read

JI~,R(X) = [ pV(X) q: p A ( x ) , j v ( X ) T jA(X) ],

(7)

where 3

p v ( x ) : gv ~

r+~(x -- rn), R

pA(X) = gA ~-~7+n[tr n . p n 6 ( x _ r , ) 2MN B gv ~ j r ( X ) = 2MN jA(X) =gA ~

+ 6 ( x - - r n ) t r n .pn],

r+ [pn6(X _ rn) + 6 ( x -- rn)Pn + fw~7×trnt~(x - rn) ], n

r+Orn6(X -- r~),

(8)

17

2 We do not consider the admixture of the hadronic (V + A) current into J~, since its contribution to tiff decay amplitudes is negligible [4]. 3 See Eq. (3D-18) in Ref. [21]. The correspondence between the non-relativistic approximations used here and that prevailingly employed in the studies of/3fl0v decay [ 1-5] can be found on p. 516 of Walecka's book 122].

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1"73

are the one-body vector (V) and axial-vector (A) densities and currents, MN is the nucleon mass and f w = 4.7 is the effective weak-magnetism coupling constant. Merging (4) into (3) and performing the s~-summation one obtains [4]:

G2 2n

f

Ro,.=~ ~. . . .~ Z J dxdy/~(F J~:(y)eik'YIN)(N[J:~ (x)e-ikXll) ~=L,~ ( 2 . j× [ 1 -- P(e~, e2)] ~ ( e 2 , y ) 7 . P ~ (oJy ° - k .~, + m~) P.yu~O c (e~, x) w(el + w + EN -- Et) ,

(9)

where ~b(el,x) and ~/,(e2,x) are the wave functions of the emitted electrons, and the operator P(el,e2) interchanges the particles el and e2. The structure of Eq. (9) suggests that it might be convenient to introduce the Fourier transforms of the quantities defined in (8), i.e.,

p ( k ) = . f dx p ( x ) e-ik.x, j ( k ) = / dx j ( x ) e -ik'x.

(10)

Next, following the usual procedure [2,4,5], we evaluate the sl/2 and PV2 contributions of the electron wave functions to the amplitude R0~. The first gives rise to the following k and N dependent nuclear moments

MF ( k, N) = (FlPv( - k ) l N) (U[pv ( k ) I1), Mc,v( k, N) = (FIJA(--k )IN) " (NIJA( k )II), MR ( k, N) = - i R k . ,

( 11 )

where R is the nuclear radius, and the second gives rise to

M'F ( k, N) = 2x/'3i(V[P(v°)(-k ) lU ) (Xlpv( k ) II), M'GT( k, N) =6i(FIj(a°l) ( - k ) lX) • (Nlja ( k ) ll), M T ( k , N ) = 2i(FIj(a21)(-k )[U) • (UlJA( k )tl), M p ( k, N) = - v ~ i [ (FIj(A'°) ( - k ) lU >(Ulpv (k) l/)

- • (Ulp(vl ) ( k ) ll) ] ,

(12)

where we have introduced the tensor operators

p ( J ) ( k ) = f dx p ( x ) (k ® x)(J)e -ik'x,

j(LJ)(k) = L J -1

f axt,i(x) ®(!, (x) x)(L)](J)e -ik'x,

(13)

with L = v/2L 4- 1. The explicit form of the matrix elements defined in (1 1) and (12) are shown in Appendix A.

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We now define the nuclear matrix elements

Mx-_

g. ~

d k v ( k , CON)Mx(k,N)

for X = F , GT, F ' , G T ' , P , R , T ,

(14)

and

R

Mx~, - 4~.g2 ~N

/

dkv~o(k, t o N ) M x ( k , N )

for X = E GT,

(15)

with

v(k, COu) -

2

1

~" k ( k + CON)'

Vo~( k, t O N ) -

2

1

7"r (k + WN) 2'

(16)

and CON = EN -- 71 (El + EF).

(17)

In deriving expression (17) we have approximated the electron energies as et,2 ~ (EI - E F ) / 2 . We have also neglected the neutrino mass in comparison with k, i.e. we have taken oJ ~ k. For the transition amplitude we obtain

g2 G 2

5

R0y(el, e2) - 4~-Rx/~ Z

ZkLk(el, e2),

(18)

k=l

where ZI =

( m y ) ( M F -- M a r ) ,

z2= (rl)(MaT~o

+ MFw) +

(A)(MF~o

--

MaT, o),

z3= 4(~7)MR, z4= ~i[ (,1)(M'cr -- 6Mr + 3M'F) -- (~)( M'6r - 6Mr - 3M'F) ], Z~= 4i(~7)me

(19)

encompass the hadronic matrix elements, as well as the parameters t

(my) = ~

2

mgUeg,

g

t

(rl) = rlZ'U.eV~e,

(20)

g

where the summation ~ t t goes only on the light neutrinos [4,5]. The leptonic matrix elements Ll (el, E2) are displayed in Appendix B.

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Finally, by performing the integrations (summations) on the electron states indicated in (2), we obtain the familiar expression for the 01,tiff half-life [2,4] [Toy(0 + --~ 0+)1 - l = (mu)2C, + (A)2C2 + (r/)2C3 + (my)(/~>C4

+(m~)(rl}C5 + (a)('r})C6,

(21)

where C1 = ( M r - M G T ) 2 ~ I , 1 2 2 C2 = M~_~2 4- ~MI+~4 - ~M2-MI+~3,

C3 ----M2+~2 4- ~1 M 2I - ~ 4

_

~M2+M,-G34- M~G9+

M R M p G 7 4- M 2 G s ,

C4 = (MF - MaT) [M2-G3 - MI+G4] , C5 = --( MF -- MGT) [M2+G3 - M1-G4 -4- MRG6 4- MeGs] , C6 = -2M2-M2+G2 4- ~ [M2-M1- 4- M2+M,+] G3 - g2M I - M I + G 4

(22)

contain the usual combinations of the matrix elements

M~Gr - 6Mr -t- 3M~F,

MI+

=

M2+

= M a T t o 4-

MFo, -- ~Ml:r,

(23)

and the kinematical factors

32R27r 5 In 2

1 - - e---Z~m~z

5rk(T°)'

(24)

The electron phase-space factors .Tk(T0), as a function of the maximum kinetic energy To = El - EF -- 2, are listed in Appendix B. It is important to stress that, within the procedure followed here to derive the result (18), we do not need to refer at all to the so-called closure approximation (CA). (Remember that the CA implies: (i) supplantation of the energies EN by average values (EN), and (ii) use of the closure relation ~-~.NIN)(NI = 1 for the intermediate states.) When reworked in the CA, the moments (14) and (15) are directly comparable to those that appear in the literature [ 1-5].

3. Multipole expansion and angular momentum recoupling The starting point for the multipole expansion of the hadronic current is to use the Fourier-Bessel relation e ik'r =

47r Z

iLjL(kr) (YL(k) • YL(~') )

L

=--4rr Z iL(--1)LLjL(kr) [YL(k) ® Yt.(i')]0 L

(25)

176

C. Barbero et aL/Nuclear Physics A 628 (1998) 170-186

in the equations exhibited in Appendix A. Then we perform the angular momentum recoupling, and rewrite the nuclear moments ( 11 ) and (12) in terms of the one-body spherical tensor operators

+ ~. ^ Tn rn ga (krn) YJM( rn ),

Y~jM(k) n

r n+ rnja(krn) [o'n ® YL(rn) ] JM, ~.

SALJM(k ) = ~ n

PLJM(k) = ~ r+jL(kr,)[P, ® YL(P,)]Jg.

(26)

n

Finally, the angular integration on dot, is performed. We illustrate the procedure by sketching in Appendix C the derivation of a part of the final formula for the nuclear matrix element MR. Proceeding in a similar way with the remaining matrix elements we obtain:

MF=4rrR\-~A /

~/v(k, wN)Ic2dk(flY°j(k)lN).(Nl~j(k)lI),

(27)

MGT 47"rR~--~(- 1),+L+J I v(k, wu)k 2 dk(FIS°Lj(k)IN). (NIS°/j (k) II), (28) ,I

LJN

(gv)ZZiC-J+l MtF =-sTrR\-~afl LJN

(JIIL)(JIIL)

× f v(k,,oN)k 3 dk(FIY~j(k)fU). (Ul~j(k)II), M Gr t = 8"rrR ~

(29)

iC-L'+J ( -1)L'+J(L'llL)(L'llL)

LL~JN X /

v ( k , CON)k 3

2~R 2 gv

Z

MR - M ~ gA LL'JN

dk(FIS1Luj(k)IN). (NISO,L,j(k)IF),

iL+L'( -- l ) J

(30)

f v(k, CON)k3 dk(FlS°Lg(k)IN)

•{ f w k [ 8LL, -- ( J l IL) (Jl IL') ] (NISO,L,j(k)II) 1 L' MT = 4OrrR X f

J

(LllL')(NIPL'j(k)]I)

'

(31)

iL+L'+I L2(1LI f ) ( ILIL t) -LLtJ~JN

v ( k , CON)k3

J' L L'

dk(FlS[j,j(k)IN). (NISO,L,j(k) I1),

J' J L'

(32)

C. Barbero et

al./Nuclear PhysicsA 628 (1998) 170-186

Mp=87rV~RgAZiL+J+lj(Jl[L)(JllL){~ gv LJN X f

177

L 1} 1J

o ( k , O)N)k 3 dk(FlS[jj(k)IN) • (UlV°j(k)II),

(33)

where (L11 J) is a short notation for the Clebsch-Gordon coefficient (L0101J0) The formulas for the matrix elements MF~o and MGTo~are obtained from those for MF and Mcr with the replacement v(k, ~oN) ~ v,~(k, WN) (see Eqs. (14) and (15)). The evaluation of the Ovflfl matrix elements encompasses: (i) appraisal of the scalar product

(FIT j( k ) IN). (NIT j( k )1I),

(34)

where T j (k) represents any of the one-body operators displayed in (26), and (ii) integration on the neutrino momentum k. More details of these two steps are given in the next section.

4. Nuclear structure calculations To evaluate the matrix elements (34) it is convenient to rewrite the operators (26) from the Hilbert space to the Fock space [21], i.e.

-rjM(k) = J-' ~-~£(plLmJ(a)lln)(@a~)j

g

•

(35)

pn

In this way we obtain

(FITj(k)IN). (NlTg(k)ll) =- ~ M

(OylTj(k)lJ~M) ' (J~M[Tg(k)[O~-)

azrM

= (-)J

Z

(PlITj(k)IIn)pph(pnp'n';J~)(P'IITj(k)IIn'I'

(36)

aqrpnp'n I

where

p ,,,, (pnp , ,n. , ,J~) = )-2(0~11

(apa,)J=llJ~)(J~ll(a~,,a,,)J~llo+,) t ~ ,~

(37)

is a two-body state-dependent particle-hole (ph) density matrix, and the index a labels different intermediate states with the same spin J and parity 7r. Within the CA we can sum over a, and deal with the state-independent ph density matrix ph ~P g nnt Pcl e nt ; jTr.,I = ~

pph(pnptnt; j~r) ot

- Y-' (o~ I [ (a],a.)

j.(a],,a,, )j. 1o[0+),

(38)

which is related to the particle-particle (pp) density matrix

pPP(pp'nn'; J~') = ] - ' (0}[ rt~,tara t p p,)j-(aha~,)j~]o[O?),

(39)

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by a Pandya-like relation ph (pnp t n t ; J~') = Z ( - )j,,+jt,t+J+l~2 ( jp jn J } pp Pcl tLJn' Jp' I Pt-(PP rnn~;l'r) •

(40)

The reduced single-particle pn form factors for the one-body operators defined in (26) are [23,24]

(pl lY~aj( k ) ll n) = ( 47r)-]/2Wjoj(pn) R~a(pn; k ), (Pl IS~LJ (k) [in) = (47r) -]/2WLij(pn)R~(pn;k), (p[IPLj(k) IIn) = (47r)-V2[W(Lf)(pn)R(L-)(pn; k) + W ~ ) (pn) R~+) (pn; k)], (41) with angular parts 4

WLSJ(pn) =v/-2sJLIjn)p(InL[lp)

L S J In½J.

,

W~ij ) (pn) = qzi(-1) t"+J"+J+l/2JLlpL). (l. + ½ q: ½)]/2(lpLlln 9= 1) In J

In In :t= 1 lp

'

(42)

and radial parts K . R ~ ( p n ; k ) - RL(lp,np,ln,nn, k) =

/

• Unp,l,,(r)un,,,t,,(r)jL(kr)r

2+K

dr,

J o OQ

R(c+)(pn;k) =

f

Un"le(r)

(d 21.+I±I) -~r ±

2r

un"'l"(r)jl"(qr)r2 dr"

(43)

o To carry out the numerical calculation of Eqs. ( 2 7 ) - ( 3 3 ) it is convenient to group separately the angular and the radial parts. For instance, Mar can be cast in the form

MGr = -- Z (-- ) L Z LJ~

PPh(pnp' n' ; J~) WLu(pn) WLlj(p' n')

pp'nn'

x 7-~°L(pnplnl; (oj~),

(44)

where the two-body radial integrals are defined as

7-~L, (pnp'nt ; (ojz )

=

R[dkk2+Kv(k;oJjg)RO(pn;k)

R Io, ( p , n , ;k).

(45)

. 1

One way to include the effects of the finite nucleon size (FNS) and the two-nucleon short-range correlations (SRC) on the/3/30~ moments has been reported in Ref. [ 16], with the result: 4 We use here the angular momentum coupling[( ½, I)j).

C. Barbero et al./Nuclear Physics A 628 (1998) 170-186 ( u( k' tON ) ---+v( k' t°N)

179

A2 ) 4 1 in k + k c Af'~ k2 - 7rkk--'-~c k - kc

+ ~

n

~

,

(46)

I;f=]

where A = 850 MeV is the cutoff for the dipole form factor in the FNS correlations, A2 x~: = A2 + (k -4- kc) 2'

(47)

and kc = 3.93 fm-1 is roughly the Compton wavelength of the w-meson in the SRC correlations. Integration on the neutrino momentum k is simplified when the harmonic oscillator radial wave functions are employed. The following relations among the one-body radial integrals are then valid:

Rl (pn; k)

(2t,) --1/2 { (2/. + 2n. + 3)l/ZR°(k; It,, np, l. -4- 1, n.)

- (2nn)U2R°(k;Ip,nt,,l. + 1,n. - 1 ) } , R~+)(pn;k) =-4-

(2/. + 2 n . + 2 q : 1)U2R°(k; L~ lp,np , l n T l,nn)

+ (2nn + 1 ± l)l/2R°(k;lp, np,l. qz 1,n. -4- 1 ) } ,

(48)

where t, = M w / h is the oscillator parameter, and the k-integration in the matrix elements ( 2 7 ) - ( 3 3 ) only involves the radial integrals (45). Their explicit forms in this case are shown in Appendix D. _ p h [ ~ _ t nt" The densities pph ( pnp~n~; j~7r ) and Pcl tent, , J'~) are supplied by nuclear structure calculations. As an example, we discuss below the/3/3 decay 48Ca ---+48Ti. We first consider the case when the intermediate nucleus 4SSc and the final nucleus 48Ti are described, respectively, as one-particle one-hole and two-particle two-hole excitations on 48Ca, i.e.

IJ~aM) = Z

(pn[J~) (a~an) j=M I0+),

pn

10}> =

Z N ( p p ' ) N ( n n ' ) (pp'nn'; I~'10}) p >~p'n>~n'l ~ x[~'ata ' (a~a~,)t,]olO+), p pt ~)1~

(49)

with N ( p p ~) = (1 + 8pp,)-l/2. One obtains from (37)

Pph(pln]p2n2;

J~a)= (J~lpzn2) ~

~

l(p3n3lJ2) ^ 7r

p ~ p ~ n > / n t p3n3l ~

x (0~ [pp'nn'; 1 ~) ( - ) "' +p~+J+t

C. Barbero et al./Nuclear Physics A 628 (1998) 170-186

180

×

{Pln3p3nlJ1}

[51(ppt)~l(nnt)t~PP~t~nnlt~P'p3~n'n3'

(50)

and ph Pcl

(Pl HIP2n 2," J~) =

E p >/p'n~n' Pr

't(O~fIpp'nn';U)(-)"'+'2+J+'

n2 P2 I

Pl(PP')Pl(nnt)tSPP~t~nn'6p'P2t~n'n2'

(51)

where

P j ( p p ' ) = N ( p p ' ) [ l - (-)J+P+P'(p +-* p ' ) ] .

(52)

Within the CA one can use the closure relation

Z

(p3naJJ~r) ( J~Jp2n2) = 6p2pat~n2n3,

(53)

which leads from (50) to (51). On the other hand, within the QRPA formulation, and after solving the BCS equations for the intermediate nucleus 4SSc [25], the two-body density matrix becomes

pph(pnp'n'; 3~) = [UnUpXj; (pn) + UpVnYc(pn)] I

!

I

I

× [ut,,v,,Xc( p n ) + un,vp,Yj~,(p n )] ,

(54)

where all notations have the standard meanings [ 17,25]. One should bear in mind that when the QRPA is used, the energies toj~ that appear in the matrix radial integrals are the solutions of the RPA problem and not the excitation energies of the intermediate nucleus relative to the initial nucleus. In particular, in the single mode model [ 16], where there is only one intermediate state for each j r (and which seems to be a reasonable first-order approximation for the tiff decays of 48Ca and l°°Mo nuclei [26,27] ),

pph (pnpn; j~r ) = UpVnUnVp where G ( J ~) • .!

= G(pnpn;J~),

1+

~oo

,

too = - [ G ( p p p p ; O +) + G(nnnn;O+)]/4,

(55) and

. .!.

= G ( J J JJ , j~r) are the particle-particle matrix elements. The intermediate states for

48Sc are: [Of7/z(p)Of7/2(n)]j~, and the values of the ratios G(J+)/too for the 8 force can be found in Table 1 of Ref. [26].

5. Summarizing discussion A straightforward derivation of the flflo~ decay rate, based on the Fourier-Bessel expansion of the transition amplitude, and the posterior application of Racah algebra,

C. Barbero et al./Nuclear Physics A 628 (1998) 170-186

181

has been performed without invoking the closure approximation. If necessary, this approximation can be implemented at any step of the calculation. It has been used for deriving the/3/30,, formulas in Refs. [ 1,2,4], but not in Refs. [ 18,19]. To evaluate the nuclear matrix elements exhibited in Eqs. ( 2 7 ) - ( 3 3 ) we only have to perform summations on the angular momenta and the intermediate virtual states. The successive terms rapidly decrease, because the radial integrals (45) steadily diminish in magnitude when the multipolarities L and U are augmented [16,17]. The formulas become particularly simple when the harmonic oscillator basis is used. Then the Horie and Sasaki method [ 15] can be exploited for the evaluation of the radial form factor~ (45), and the equations displayed in Appendix D can be used. The present formalism is especially suitable for the nuclear structure in which the summation on the intermediate states is unavoidable, such as the QRPA. The closure approximation then just connotes that the variation of the energy denominators with nuclear excitation is not considered. Evidently this does not lead to a major simplification in the numerical calculation. For example, because of pph(pnpln~; J~) given by (54), the summation in (44) on different states o~ with the same J~ persists, although we do the replacement oJg,7 --+ (wg~} = ~Og,. In contrast, the use of the closure approximation is mandatory, and can be implemented easily as described in the last section, when the study is performed in the shell-model ph tYamework, i.e. when one possesses information only on Pc~, or equivalently on the 0 nuclear wave functions for the initial and final states. In this case the matrix element (44) reads ,,,,.

,,

,,

0

,,

Pcl t p n p n ; J ~ r ) W L i j ( p n ) W c i j ( p n ) ~ L L ( p n p n ;COg,).

M ~;T = LJ r

pptnn~

The main difference between the formalism presented here and those published so far [ 1,2,4,18,19] is its simplicity. As such it is more suitable for numerical calculations. Let us underscore a few points in this regard: (1) While in the neutrino potential formalisms [1,2,4] one deals with two-body matrix elements, which lead to rather complicated analytic expressions for the /7/3o,, moments, we only have to handle the well known one-body operators (26). It would be illustrative to compare our result (31) for the matrix element MR with Eqs. ('3.65) to (3.68) in Tomoda's report [4]. (2) At variance with the formalism developed by Vergados et al. [ 18], the results reported here are not limited to the employment of harmonic oscillator one-particle wave functions. Besides, we totally avoid usage of the Moshinsky-Brody transformation brackets, which can be cumbersome. (3) There are also several substantial differences to the works of Suhonen et al. [ 19], where the Fourier-Bessel expansion was also used. First, they obtain different and more complex results for ME and MGT. Second, they do not exhibit the explicit structure of the remaining matrix elements, given here by Eqs. ( 2 9 ) - ( 3 3 ) , but only show their general layout. Yet, this layout cannot be used lor any practical purpose. Third, instead of dealing with the plain nuclear shell model, they operate in a relativistic quark confinement

C. Barbero et al./Nuclear PhysicsA 628 (1998) 170-186

182

model. Fourth, their formulation is limited to the QRPA approximation as well as to the harmonic oscillator basis. In summary, we believe that the present formalism simplifies the nuclear structure evaluation of the flflo~ matrix elements to a large extent. The formulation is also applicable to matrix elements that appear in some supersymmetric contributions [ 13].

Appendix A: Matrix elements Mx(k, N) After integrating on dx and dy, as indicated in Eq. (9), the matrix elements (11 ) and (12) read

MF(k,N) =gZ(F I ~,-"~'+eik'r"lN)(N [ Z Tree + -ik.r., I1), ~ n n

M c r ( k , g ) =g2a(fl Z r n o+' n e

IN). (NI ~-~ -+-T", t~me--ik'r'lI ) ,

ik •

17

(A.2)

m

M~(k, N) = -2ig2v(fl ~ ' + k .

r, eik'r"]N)(NI Z T , , ,+e

17

I ( ,kN ) MGT

(A.I)

m

= -2ig~(F[ S

-ik'rn,

1~ t),

(A.3)

Ill

T+k " rnCrneik'r"lN) " (NI Z rmO'me + -ik.r.,li), I1

(A.4)

m

.Rgmgv K. ,_ { (F[ Z 7",+o',e ik.r,, IN) MR(k,N)=--,-~-~-N n

× (Ul~-~'.,+ [Pme-ik'r., +e-ik'r.,pm

+ fwV×O'm e-ik'r']

m

I/)}, (A.5)

2 v ~ 2 '[ Z ' r + [ o'n ® ( k Q rn)(2)l(1) eik'r"[N) • (N[ Mr(k, N) = ---~-iga(F t1 V

× Z_~ rmCrme- i k . r . , II)' TM

+

(A.61

m

MR(k, N) = --v/2igAgv { X/3(F[ Z , r+[~n ® (k ® rn) (l) ] (Oleik'r"[U)(Ul × ~-~Tn+e--ik'r'[l) -- (F I ~-'~Tn+ O'nei k . r "IN) m

•(NI

n

~-~T~m(km

® rm)(l)e-ik'r'lI)} "

(A.7)

C. Barbero et al./Nuclear Physics A 628 (1998) 170-186

183

Appendix B: Electron matrix elements and phase-space factors

The leptonic factors in Eq. (18) are: L~ (el,e2) = (-1)U2-Yx~, ' [ g _ j ( e l ) - f l ( e l ) o ' " Pl]

(B.1)

x [fl (e2)o" • P2 + g-I (e2)]X-,;, L2(el,e2) = (el

- -

e2) (--1)l/2-s;x~,L

(B.2)

X [g-I ( e l ) f l (e2)tr •/~2 + fl ( e l ) g - j (e2)or./~l ] X-.~;. I

L3(el,e2) = ~ ( - 1 )

U2+s'~ ,~f

(B.3)

-As,I

x[g-l(el)g-t(e2)

+ fj (ei)fl(e2)o'.

P l o r •/~2]X-s;

i l)l/2+s2x~,I { [ f l ( e t ) f - l ( e 2 ) + g l ( e l ) g - i ( e z ) ] t r - p l L4(EI, e2) = ~-R(-

- [f-l(el)fl(e2)

(B.4)

+g-l(el)gJ(e2)]o"P2}X-4,

i 1 ) V2+s'. L5(ej, e2) = ~-~(- X sf , { [ g - l ( e l ) f - j ( e 2 ) + f - t ( e l ) g - l ( e 2 ) ]

- [gl(el)fl(e2) +fl(el)gl(e2)]o''~lo"~z}X-s;,

(B.5)

where all notations have the usual meanings [2]. The electron phase-space factors .Tk(To) that appear in Eq. (24) are: f'l (To) = To(30 + 60To + 40Tg + 10T3 + T4)/30, .f'2(T0)

=

T~o(70 + 77ro + 14T2 + To3)/420,

.Y3(To) =To3(10 + 10To + To2)/30, f'4(To) =To2(30 + 35To+ 10T2 + T 3) 135, 5r-5(To) = To[60To + 80T2 + 30To3 + 3T~o+ s¢

(B.6)

x (60 + 90To + 40Tg + 5To3)]/45, .Y6(To) =2To( 12 + 18To + 8To2 + T3)/(3R), f7(To) = 4To [60To + 100To2 + 55To3 + 12Tdo+ T~ + ~:

(B.7)

x (60 + 90To + 45Tg + 10To3 + T~)]/(45R), Us(To) =To[ 100Tg + 150T~ + 73T4 + 14T5 + T6 + 2~: x (60To + 100To2 + 55To3 + 12T~ + To5)

(B.8) (B.9)

+(2 (60 + 90To + 45Tg + 10T3 + T~) ] / 135, f'9(To) = 4To(60 + 90To + 45Toz + 10T3 + T~o)/(15R2),

(B.10p

3o~Z ~:= R

(B.l I

with

184

C. Barbero et al./Nuclear Physics A 628 (1998) 170-186

Appendix C: Derivation of the final formulas for the nuclear moments

Below we give details of the derivation of the last term in (31)• First we rewrite (A.5) as MR(k, N) = - - t•-RgAgv ~ - NN- k . (El

Z q'+O'neik'r'lN) n

×
(c•1)

Ill

and then we express the vector product, involving the nucleon momentum term in (C. 1), in spherical coordinates [vl~p)(k, N) = Rgagv Mn x/2~ v2 -",~, t - 1 ) U ( l v l v ' l l t z ) k " vv~ tz + --v i k r ×(fl~-~z,o', e IN)(NI~-~'~ rme + - ' ~' " ° 'Pm -~ 11

'

tl) •

(C.2)

nl

After performing the multipole expansion (25) and handling some straightforward Racah algebra, we obtain M(RP)(k' N) -

Rgagv__

MN

v/-6(47r)2k

,

Z

iL-C (--1)M'L

{LJ

LUJJ'MrMM~ Kpp

I

1 K

-vL'M'IJM) ×Y,¢p(ic)Y~'M'(~¢)(FISOLj'M'(k) IN)(NIPL'JM(k) II), × (LO]OI KO) ( 1, --m~pl J', -M 1) ( l,

(C.3)

where the tensor operators S°Lj,(k) and PL,j(k) are defined in (26), Finally, angular integration allows us to perform the summations on the angular momentum projections and obtain

f da'MT("N) = RgAg'V(4")'

L {LIJI L' I}

LL'J

× (FIS°Lj(k)IN). (NIPr,j(k)I1)-

(C.4)

This result, together with Eq. (14), yields the last term in Eq. (31).

Appendix D: Radial form factors for the harmonic oscillator wave functions

Following the Horie and Sasaki method [ 15] the radial integral (45) can be expressed as: T¢~L, (pnpl nl; o~j~) = [ M(p, n ) M ( f , n r) ] -I/2 x Z mm t

am (p, n)a.,, (p', n')f~L' (m, m'; wj~),

(D. 1)

C. Barbero et al./Nuclear Physics A 628 (1998) 170-186

185

where

M(nplp, n , l , ) = 2""+""np!nnI(21t, + 2np + I ) !!(21, + 2n, + 1 ) !!,

(D.2)

(..k') k+U=s

> (2lp + 2np + 1)!!(21n + 2n. + 1)!! (2lp+2k+l)[!(2ln+2U+l)!!

f~L,(m,m';wc)

=Za2u

~ L , [ -m L

m ' -2 L ' L ' ) J ~ ( w j : ) ,

'

(D.3)

(D.4)

,u

and O<3

J ~ ( oJj,7 ) = (2u) -/~ R / dk k2U+2+~e -~/2u/., ( k; ~oj~ ).

(D.5

0

References I I W.C. Haxton and G. Stephenson, Prog. Part, Nucl. Phys. 12 (1984) 409. 12 M. Doi, T. Kotani and E. Takasugi, Prog. Theor. Phys. Suppl. 83 (1985) 1; Phys. Rev. D 37 (1988) 2575. 13 D. Vergados, Phys. Rep. 133 (1986) 1. 14 T. Tomoda, A. Faessler, K.W. Schmid and E Griimmer, Nucl. Phys. A 452 (1986) 591: T. Tomoda, Rep. Prog. Phys. 54 (1991) 53. 151 M. Doi and T. Kotani, Prog, Theor. Phys. 89 (1993) 139. 16 ] R.N. Mohapatra and J. Vergados, Phys. Rev. Lett. 47 ( 1981 ) 1713. 171 A. Halprin, P. Minkowski, H. Primakoff and S.P. Rosen, Phys. Rev. D 13 (1976) 2567: R.N. Mohapatra, Phys. Rev. D 34 (1986) 3457; M. Hirsch, H.V. Klapdor-Kleingrothaus and O. Panella, Phys. Lett. B 374 (1995) 7. I 81 G.B. Gelmini and M. Roncadelli, Phys. Lett. B 99 ( 1981 ) 411. 191 C.P. Burgess and J.M. Cline, Phys. Lett. B (1993) 141; Phys. Rev. D 49 (1994) 5925: C.D. Carone, Phys. Lett. B 308 (1993) 85. 101 R Bamert, C.P. Burgess and R.N, Mohapatra, Nucl. Phys. B 449 (1995) 25. I I ] C. Barbero, J. Cline, E Krmpotid and D. Tadid, Phys. Lett. B 371 (1996) 78; Phys. Lett. B 392 (1997) 419. 12[ J.D. Vergados, Phys. Lett. B 184 (1987) 55. 131 M. Hirsch, H.V. Klapdor-Kleingrothaus and S.G. Kovalenko, Phys. Lett. B 352 (1995) 1; Phys. Rev. Lett. 75 (1995) 17; Phys. Rev. D 53 (1996); A. Faessler, S. Kovalenko, E Simkovic and J. Schwieger, Phys. Rev. Lett. 78 (1997) 183. 141 H.V. Klapdor-Kleingrothaus, Prog. Part. Nucl. Phys. 32 (1994) 261. 151 H. Horie and K. Sasaki, Prog. Theor. Phys. 25 (1961) 475. 161 E Krmpotid, J. Hirsch and H. Dias, Nucl. Phys. A 542 (1992) 85. 171 E Krmpoti6 and S. Shelly Sharma, Nucl. Phys. A 572 (1994) 329. 181 J.D. Vergados, Nucl. Phys. A 506 (1990) 482; G. Pantis and J.D. Vergados, Phys. Lett. B 242 (1990) 1; A. Faessler, W.A. Kaminski, G. Pantis and J.D. Vergados, Phys. Rev. C 43 ( 1991 ) 21; G Pantis and J.D. Vergados, Phys. Rep. 242 (1994) 284; G Pantis, E Simkovic, J.D. Vergados and A. Faessler, Phys. Rev. C 53 (1996) 695.

186

C. Barbero et aL/Nuclear Physics A 628 (1998) 170-186

]191 J. Suhonen, S.B. Khadkikar and A. Faessler, Phys. Lett. B 237 (1990) 8; Nucl. Phys. A 529 (1991) 727; Nucl. Phys. A 535 (1991) 509. [201 H. Primakoff and S.P. Rosen, Rep. Prog. Phys. 22 (1959) 121. 121 I A. Bohr and B.R. Mottelson, Nuclear Structure, Vol. 1 (Benjamin, New York, 1969). [221 J.D. Walecka, Theoretical Nuclear and Subnuclear Physics (Oxford University Press, New York, 1995). [23] M.E. Rose and R.K. Osbom, Phys. Rev. 93 (1954) 1315; Phys. Rev. 93 (1954) 1326. 1241 T. deForest and J.D. Walecka, Adv. Phys. 15 (1966) 1. [25] E Krmpoti6, T.T.S. Kuo, A. Mariano, E.J.V. de Passos and A.ER. de Toledo Piza, Nucl. Phys. A 612 (1997) 223. /26] E Krmpoti6, Rev. Mex. Ffs. 40 (1994) 285. [271 H. Ejiri, K. Fushimi, K. Hayashi, T. Kishimoto, N. Kudomi, K. Kume, K. Nagata, H. Ohsumi, K. Okada, T. Shima and J. Tanaka, Nucl. Phys. A 611 (1996) 85.