Nuclear Physics A452 (1986) 591-620 @ North-Holland Publishing Company

NEUTRINOLESS J?/? DECAY

AND A NEW LIMIT ON THE RIGHT-HANDED CURRENT

T. TOMODA, AMAND FAESSLER and K.W. SCHMID

Institut fiir Theoretische Physik, Uniuersitiit Tiibingen, Tiibingen, West Germany

F. GRtiMMER

Institut fiir Kerphysik, Kernforschungsanlage Jiilich, Jidich, West Germany

Received 9 August 1985 (Revised 11 October 1985)

Abstract: We have calculated the OV@ and 2vfip decay rates of the transition 76Ge(0:) + %e(O:). We took into account a relativistic correction to the nuclear current including weak magnetism. The nuclear two-body transition operator for the Ovpp decay originating from this correction acquires a finite range due to the finite extension of the nucleon and the short-range NN correlations reduce its matrix element only moderately. The contribution from this second-forbidden transition plays a dramatic role in the Ovpp decay caused by a specific admixture of a right-handed leptonic current because of the high momentum of the virtual neutrino exchanged between nucleons and systematic cancellations in allowed and other second-forbidden Ou# transitions. A new limit on the right-handed current coupling strength /(?)I < 6 x lo-* was obtained, which is more stringent by an order of magnitude than that obtained recently by the Osaka group.

1. Introduction

A growing interest has been focused on the nature of the neutrino in the course

of the recent development of the grand unified theories [for a review, see ref. )I.

In many of these theories (SO,,, E6, etc) the neutrino is regarded as a Majorana

particle (i.e. identical to its own antiparticle) in order not to acquire a mass

comparable to those of quarks or charged leptons. Such theories predict ) the

neutrino mass roughly in the range of 10--l eV and also the existence of right-

handed currents. Since the question of whether the neutrino is a Dirac or Majorana

particle can be answered practically only by studying neutrinoless pp decay (Ovp/?

decay), much effort - both experimental and theoretical - has been devoted to the

problem of nuclear pp decay*.

The Ou/3p decay, which violates lepton-number conservation, takes place if the

neutrino is a Majorana particle under the conditions (i) that it has a non-vanishing

mass and/or (ii) that there is an admixture of a right-handed leptonic current

coupled to nuclear currents. By comparing experimental data with theoretical

calculations one can deduce important information about the character of the

neutrino and the magnitudes of its mass and right-handed current admixtures.

l For reviews see refs. 2-8). See also refs. 9-6) and refs. 7-9) for recent experimental and theoretical works, respectively.

591

592 T Tomoda et al. / Neutrinoless pp decay

In most of the theoretical investigations of the Oz@ decay [see e.g. ref. )I the

non-relativistic limit of the nuclear weak current was used. This is justified in a type

of Or@ decay caused by a non-vanishing mass of the neutrino since this is an

allowed decay. In another type of Or@/3 decay caused by a specific admixture of

a right-handed leptonic currrent, however, the contributions of second-forbidden

processes become important because those of allowed processes cancel 7*8,20*)

each other to a great extent. One kind of these second-forbidden Oz@ transitions

consists of processes in which one of the two electrons is emitted in a P-wave. The

transitions of this kind were taken into account by the authors of refs. h*7.22) who

deduced an upper limit on the admixture of the right-handed current. There is,

however, another kind of second-forbidden Oupp transition in which both of the

electrons are emitted in S-waves but a relativistic correction to the nuclear current

plays a role. Since the contributions of the transitions of this second kind are

expected to be of the same order as those of the first kind, they should be taken

into account in order to be consistent, as was pointed out by Doi et al. 23). They

included the relativistic correction term in their formalism 23*24) and calculated the

electron phase-space integrals 24320 ) also for this correction term. The magnitude of

its contribution to the Ov/3p decay rate, however, was unknown, for there was not

even a rough estimate of the relevant nuclear matrix element.

In our previous paper 2) we pointed out that weak magnetism should also be

taken into account in order to ensure the consistency of the approximation. We

then demonstrated that a simple estimate yields a very large contribution from the

relativistic correction term including the weak magnetism and that we should obtain

an upper limit on the right-handed current admixture which is smaller (i.e. more

stringent) by two orders of magnitude than that obtained by Haxton and

Stephenson 6). The contribution from the relativistic correction term becomes

dominant because the neutrino exchanged between nucleons is virtual and the

momentum transfer to it is limited only by the amount which a nucleon in a nucleus

can provide (= twice the Fermi momentum ~540 MeV/c). This is in sharp contrast

to the single-/? decay, where both the electron and the neutrino are real particles

and the momentum transfer from a nucleon to these leptons is restricted by the

Q-value (a few MeV) of the decay. Motivated by the above estimate we calculated 2)

for the first time the Or@/3 decay rate including the relativistic correction and found a limit on the right-handed current admixture which is -A as large as that of ref. )

if we neglect the short-range nucleon-nucleon (NN) correlations. But since an

exchange of a neutrino between two nucleons yields a propagator roughly propor-

tional to l/r and this in turn gives a two-body nuclear operator for the relativistic

correction with the leading term A( l/ r) a S(r), the short-range correlations reduce its contribution drastically I) [see also ref. )I.

Let us ask ourselves why we obtained the zero-range operator. The reason is that the nucleon was regarded as a structureless point particle. But actually it has a finite

extension. Inclusion of this effect will modify the zero-range operator into one with

T. To~oda et al. / ~eui~~o~ess &3 decay 593

a range of the order of twice the radius of the nucleon. The impo~ance of the effect of the finite nucleon size had been pointed out by Vergados 26) considering the contribution of heavy neutrinos to the Or@/3 decay.

In the present paper we calculate the Oupp and 2@3 decay rates of the transition 76Ge(0:) + %e(O:). We show that the combined effects due to the short-range NN correlations and the finite nucleon size reduce the nuclear matrix element originating from the relativistic correction only moderately. This matrix element still gives the dominant contribution for the type of the Oz$@ decay caused by the admixture of the right-handed Ieptonic current. Therefore we can deduce from an experimental lower limit on the halflife of the 0~~~ decay of 76Ge an upper limit on the admixture of the right-handed leptonic current which is by two orders of magnitude more stringent than the corresponding limit obtained by Haxton and Stepehenson ). Our limit is more stringent by an order of magnitude than that obtained recently by Doi et al. 8, who used the nuclear matrix elements calculated by Haxton and Stephenson ) and took into account an enhancement of the electron P-wave com- ponents due to the Coulomb field.

In sect. 2 we describe the formalism. Basic formulae for the Oz@ decay and the forms of the two-body transition operators which are suited for nuclear physics calculations are given in subsects. 2.1 and 2.2. The effect due to the finite nucleon size is considered in subsect. 2.3. Subsect. 2.4 summarizes the formulae for the 22$3p decay. In subsect. 2.5 a method of calculation of the nuclear matrix elements using the VAMPIR approach *) is described. Subsect. 2.6 deals with the short-range NN correlations.

In sect. 3 we present the results of the numerical calculations for the O@p and 2 u&3 decay of 7hGe (0:) + %e(O:). In subsects. 3.1 and 3.2 the phase-space integrals and nuclear matrix elements are given. In subsect. 3.3 we compare our calculation with experimental data and deduce upper limits on the neutrino mass and the right-handed currents.

A summary is given in sect. 4. Appendices A, B and C give formulae for the electron phase-space factors, the neutrino propagation functions, and a method for the evaluation of the radial integrals of the two-body 0~~~ transition matrix elements, respectively.

2. Formalism

2.1. 0@3 DECAY RATE

We employ the following effective weak-interaction hamiltonian density 6*24):

H,(x) = 4 G cos &[ jL,Jft + Kj,+J$+ + vjRrJCLt + Aj,,.J,t] + h.c., (2.1)

where G = 1.16637 x 10F5 GeV2, cos & = 0.9737 [ref. **)], and with the left- and

l In this subsection we follow in principle the notation of refs. *0*24). The units h = c = I are used.

594 T. Tomodn et al. f Neutrinoless pp decay

right-handed leptonic currents

X(x) = F(x)Y(I - YS)%L(X) ,

j,(x) = @x)Y(l+ ~~)Gdx),

and the current neutrinos

V,L(x)= c ueiNiL(x) 3 ,=I

(2.2)

2n

YLR(X) = C veiNiR(x) . (2.3) i=l

Ni is the eigenstate of the Majorana neutrino mass matrix with the eigenvalue mi,

n the number of generations. The nuclear current is assumed to be

(JLt(x),JXx))= ? rLs(X-rr,)(gv-gACnr -g,a,+gvR), n=1

(2.4)

where (pl~In) = 1, gv = 1, g, = 1.254 [ref. 28)], and the right-handed current Jk(x)

is assumed to be obtained by replacing g, with -g,. C,, and D, are the relativistic

correction terms 2) [see also refs. 8,23*24)]:

with

C, = (pn +P:) * an/2M, (2Sa)

D, = [pn +A - G.+a, x (P, -PL)I/~M, (2Sb)

Pp = t(g, - gn) = 4.7 , (2.6)

where pn and p: are the initial and final nucleon momenta, M the nucleon mass,

and g,, g, the spin g-factors of the proton and neutron. Expressed in operator form

p,, and p: are actually -iv,, standing to the right and left, respectively, of any

function of r, (8(x - r,,) in the above case) in the transition operator. Keeping this

in mind, we use the simplified notation as in eqs. (2.4) and (2.5). In eq. (2.6), the

part pup - 1 = 3.7 originates from the weak magnetism analogous to the isovector

anomalous magnetic moment. It will turn out that the term proportional to the

nucleon recoil momentum p,, -pL in eq. (2.5) contributes much more than those

proportional to the average momentum $( pn +p,). Eqs. (2.4)-(2.6) are correct to

order v/c [ref. )I, where v is the nucleon velocity, if we assume conservation of

the vector current and the nonexistence of second-class currents. The effect of the

term porportional to K in eq. (2.1) will be neglected in the following because K

contributes to the Or@@ decay amplitude only in the combination 1 k K and we

expect IK[ Q 1.

We calculate the decay amplitude in second-order perturbation theory and employ the closure approximation ) in taking a summation over intermediate excited nuclear

states. This approximation means that the energy of the intermediate nuclear state,

EN, in the energy denominator is replaced by some average value (EN), i.e.

(m+E,+&,-E,)-+(w+Ej+(EN)-El)-, (2.7)

T. Tomoda et al. / Neutrinoless /3p decay 595

where W, Ej, E, are the energies of the neutrino, the jth electron (j = 1,2), and the

initial nuclear state, respectively. This approximation is expected *) to be good for

the Or@ decay because the neutrino exchanged between two nucleons is virtual and

its typical energy is ) w 2 k = l/r,, = 200m, (where k is the neutrino momentum

and TN,., the mean internucleon distance), which is much larger than the typical

nuclear excitation energy (EN) - E, - 20m,.

The wave functions of the electrons emitted in the OV@ decay are expanded in

terms of the solutions of the Dirac equation in a spherical basis. The leading

contributions for a O++O+ decay come from the S- (g_,,f,) and P-wave (g,,f_,)

radial wave functions with j = 4. These will be included 24) in the present work (see

appendix A). These radial wave functions are then expanded in powers of r, and

the leading terms (a constant for g_,,f,; a linear term forf-,, g,) will be retained 24)

(using by this the long-wavelength approximation).

Irrespective of a concrete radial dependence of the spherically symmetric potential

for an electron, the probability of observing a O++ O+ Ovpp decay with the energy

of the first electron El and the angle between the momenta of the two electrons

e12, per unit time, unit energy and unit solid angle, is given by*

d* Wo, dE, da,, =

(a+acos el*)Wo,

where

W ov = (g,G ~0s Q4dp

32~~ 1 2 1 p E E

2 (E,+E2+E,=E,),

(2.8)

(2.9)

~(~=zf: Re [X,X:]. (2.10)

Here Pj is the asymptotic momentum of the jth electron, EF the energy of the final

nuclear state. The summation in eq. (2.10) runs over 1, 3, 4, 5, 6. The phase-space

factorsfj: ( =fg) represent various combinations of electron radial wave functions

and are given in appendix A. Xi are the following combinations** of nuclear matrix

elements:

Xl = ((mJlmAxF- l)W#? ,

x3 = uA)i-+(di+m%),

x4= wX+(11)X:Me 3

x6 = (~>d&@?, (2.11)

l Eqs. (2.8)-(2.10) with (A.l)-(A.4) can be readily derived from eqs. (B.l) and (B.16)-(B.26) of ref.). l * X,, which contains the factors (A f 7)2x: miVte, is neglected. X,,4.6 = 4X:., .J R, X5 = ;Xpm,,

X3 =4X?/ R + (contribution from Y,), where the superscript D denotes the qua&es defined by Doi et al. 24).

596

where

T. Tomoda et al. / Neutrinoless &3 decay

(T)=rj C uetv.i. (2.12)

The summation in eq. (2.12) should be taken over light neutrinos (mi /W? , (2.13h)

M ~O=(gV/gA)(-meH(r,,)~~,, * (a, x&H, (2.13i)

xk =: M;/@;, (2.13j)

and their combinations

/C*=/l+)7G7,

x: = -xi=*(fx&l--2xlr) 3 (2.14)

where

* &-= i -ARxgiD, &=,$-.&Rx:O), xk =xkD/m,R; xc- and ,&T.F.T.P are the same as the xos, where the superscript D denotes the quantities defined by Doi et al. 2*).

T. Tomoda et al. / Neutrinoless &3 decay

with ]1> and IF) being the initial and the final nuclear states;

r,!?l = r, - r, , r+nm = r, + r, )

597

The relativistic correction term C,, (eq. (2Sa)) does not contribute under the present assumption of the long-wavelength approximation. The neutrino propagation func- tions H(r) and 2(r) are given in appendix 3. One obtains from eq. (8.8) the following useful relations:

%X=2-A&,

XF=2XF-X;. (2.15) *

In the limit of A 4 0, xGT = X& = I and ,& = & = xF (see appendix B).

M!$3, xF and ,&T,F (see eq. (2.13)) are the nuclear matrix elements for the iallowed* OvpP transitions where the two electrons are emitted both in S-waves and the leading contributions of the exchanged neutrino come from S-wave radial wave functions squared

598 T. T~~~dff et at. / N~u~ri~~le~s pf3 decay

The half-life for a Ov&3 decay ~7; z is obtained by integrating eq. (2.8) with eq.

(2.16) over E, and RIz:

where

a(i)w dE OV 1 * (2.19)

(2.18)

Replacing a in eq. (2.19) with c(j) or fi: of eq. (2.17), integrated quantities Cti)

or F$ are defined analogously to A.

2.2. NUCLEAR TRANSITION OPERATORS FOR THE RELATIVISTIC CORRECTION TERM

The nuclear matrix element M$ (eq. (2.13i)) which originates from the relatvistic

correction (eq. (2.5)) to the nuclear current, can be decomposed into four parts:

M Lioy) = ( VRm) , (2.20)

with

where *r)

V Rnm = ( vRC+ vR,4 vRPe VRLS~nm 3 (2.21)

(2.22a)

V JJ

RTnm = -* H(r,,)[[~~,,0q,,]2O[a,0a,l2l~o, (2.22b) e

V I

RLSnm=-2m,Mr,, wr,)L * (a, +-In1 -

(2.22c)

q,,,,,, P,,, l,,, are the relative-momentum transfer, the cm. momemtum, the relative

orbital angular momentum, respectively, of the two nucleons n and m:

4 nm =2(Pn-P*)-(PiI-PLJlr

pnrn=P+P?n,

L=rn,x4(Pn-p,). (2.23)

Remembering the argument following eq. (2.5), any function f(m,) of r,, times

qnm actually means the commutator [f(r,,,,,), -ia/ar,,,]. Thus eqs. (2.22a),and (2.2213)

can be rewritten as

V (01 RCnm = URCnm + ff &, + vgnm , (2.24)

T. Tomoda et al. / Neutrinoless pp decay 599

with

and

with

vw RCrn =s ~(hn)(u- urn), (2.25a)

e

(1) UfKnrn = - 2/+4

3 qrneMr2nm (a - a,), (2.25b)

(2.25~)

V (O) RTnm = URTnm -I- Da;, + tgnm , (2.26)

(2) !4= -- ~(~m?l)S, * vRTnm = 2mJf

(2.27a)

(2.27b)

(2.27~)

Of the six operators (2.25) and (2.27), the four (u@, r$&, ug: and t@) are contribu- tions from a(&,,,,) (see eq. (B.7)).

2.3. EFFECT OF THE FINITE NUCLEON SlZE

We obtained the zero-range operator eq. (2.25a) using the relation V s (f/r) = 4&(r). If the short-range NN correlations are taken into account, the matrix element of v:; vanishes ~ompIetely. This follows, however, from our assumption of a point nucleon in eq. (2.4). If we also take into account the fact that the nucleon has a finite extension, we shouid obtain an operator which has a fnite range and the matrix element of which is not affected so drastically by the short-range correlations. Thus we replace* the vector and axial-vector coupling constants in momentum space with the dipole form factors 17S26)

where A = 850 MeV. The b-function in eq. (2.25a) is then modified to

(2.28)

(2.29)

l This means a replacement of 6(x-r,) in eq. (2.4) with (A3/8s) exp (-A/x-r,i).

600 T. Tomoda et al. / Neutrinoless pp decay

Similarly, l/r* in eq. (2.25b) is replaced by

(2.30)

For all the other transition operators we neglect the effect due to the finite nucleon

size since either they are relatively long-ranged or their matrix elements do not

contribute very much to the total decay rate even without the short-range NN

correlations. The form factor treatment adopted here is a crude approximation.

However, we do not think that any other more trustworthy approach would give

very different results.

2.4. 2vflp DECAY RATE

We calculate the 2vj?/3 decay rate also in the closure approximation. The half-life

for a O+ + Ot 2 vpj3 decay r:/z is given by 24)

($*)- = F*(&@;))* ) (2.31)

with

ME; = (a, a,), (2.32a)

x~(u,+w~+E,+E~+E~-E,)~u,~~~~E, dE,,

where f,: is given by eq. (A.2) and

1 1 K=

w,+E,+(EN)-E,+wz+El+(E&E,

L = K (E, t, E2 interchanged) ,

(2.32b)

w2, =(gAG OS %I4 k,k2w,w p,p_E E 8rr

2 12. (2.33)

In these equations ki, wi (pi, E,) are the momentum and the energy of the ith neutrino

(electron). The contribution from the double-Fermi process is unimportant ) and

neglected in the present work.

2.5. CALCULATION OF NUCLEAR MATRIX ELEMENTS BY THE VAMPIR APPROACH

We describe the initial and final even-even nuclear states of the pp decay by the

recently developed VAMPIR approach *). A method of calculating the nuclear matrix elements for the Ovpp and 2v@p transitions between O+ ground states* is

given in the following.

* The present method is applicable also for transitions to other spin states by obvious modifications.

X Tomoda et al. / Neuthoiess pp decay 601

We define a state I@) to be a quasiparticle vacuum,

a&Q=0 (for any CX) , (2.34)

with respect to the Hartree-Fock-Bogoliubov-type quasiparticle operators

aL =C (A,CT + Bi,Ct) 3 (2.35) 8

where Cl is a creation operator for a nucleon in a spherical shell-model basis. We assume 27) that the quasipa~icle transformation eq. (2.35) does not mix (i) protons and neutrons, (ii) states of different parities, and (iii) states of different angular momentum projection on z-axis (axial symmetry). From the HFB vacuum state I@} we project out a state 1 P) which has a good proton (2) and neutron (N) number, and angular momentum I (=0 in the present case):

I~,=J+%kw>, (2.36) where N = (~jl~$~I@)-~~ is a normalization factor, and the projection operator is given by 27)

(2.37)

with

~(d)=exp[-i(yl,~~+cp,~)]~(~),

dji = dq, dpp, da. (2.38)

Here kP (fin) is a proton (neutron) number operator, I?(L?) and L&k(fi) are a rotation operator and a D-function 34) in the ordinary three-dimensional space. For a given pair of even numbers 2 and N, we can obtain the O+ ground-state wave function by minimizing the expectation value of the nucfear hamiltonian, (V/H] Q). The matrix element of any two-body operator 0 which is involved in the O+ + Ot ,&? decay can be written as

x[(2J+1)/(1+Sj~,j~)(l+Sj~,j~)]2

X(~U,{[[Cjfp~Cjtzl~J~[~j~O~j~]J]o~~~), (2.39)

where (~~~~~l~l~~~~~} is an antisymmetrized two-body matrix element, and cjm = (-)JCj_,. All the quantum numbers except the z-component of angular momen- tum necessary to specify a single-particle state are implied by the symbol j?, etc. in eq. (2.39). The third factor on the right-hand side of eq. (2.39) is the matrix element

602 T. T~rn~~a et af. / ~eu~~inoIes~ f3p decay

of the two-body transition density, and is calculated as

(?C,J[[Cj~OC~~],~6j,o~~]]*I~~)

X C [(A* + Bg)Bljprn~,j~mJI(B* +&)ATlj;m;,j;m~ mpm$ mym;

M

x(j~m~j~m~~JM>(j;lm~j~m~lJM)/(25+ 1)12, (2.40)

where*

g*, = (~,,la,a*~(~)lsP,}/{~~~~(fZ}I~I) (2.41)

The inverted relation

c~=C(A~&t-&&, (2.42) c(

obtained from eq. (2.35) was used in deriving eq. (2.40), where A and B are the quasiparticle transformation matrices for 1 CD,).

2.6. SHORT-RANGE NN CORRELATIONS

Our nuclear wave functions lack the short-range repulsive NN correlations. Their

effect is especially impo~ant for an evaluation of the nuclear matrix element Mz

since the operator vg& (eq. (2.25a)) is still relatively short-ranged even after the

modification (eq. (2.29)) due to the finite nucleon size. We multiply677*3) the

two-nucleon wave functions in eq. (2.39) by f(lr, - r,l), where 37)

f(r) = 1 - e-or( 1 - br2) , (2.43)

with a = 1.1 fm-* and b = 0.68 fme2. This means the replacement

(.C_EJl~li~.ZJ) + (~P.~PUVlflj~j~J) ,

of the two-body matrix elements.

All the two-body matrix elements will be calculated in harmonic oscillator basis

using standard shell-model techniques. A method of evaluation of the radial integrals

for these matrix elements is described in appendix C.

3. Numerical calculation for the p/3 decay of Ge

3.1. PHASE-SPACE INTEGRALS

We calculate the phase space integrals for the OV@ decay (cf. the statement

following eq. (2.19))

(3.1)

* For a more explicit expression of g,,, see ref. ?.

T. Tomoda et al. / Neutrinoless /3fl decay 603

and that for the 2vp/3 decay F* (eq. (2.32b)), wheref$ are defined by eqs. (A.2)

and (A.3) in appendix A. Concerning the electron radial wave functions g;(E, r)

and f;( E, r) of the large and small components used in eq. (A.4), we consider

the following four possibilities:

(WFl) The Dirac equation with a potential generated by a spherical uniform

charge distribution of radius R is solved. The solution with the property eq. (A.l)

is then expanded in powers of r and the leading terms are retained.

(WF2) Same as ( WFl ) but the exact solutions are used.

(WF3) The exact solutions of the Dirac equation with the Coulomb potential for

a point charge are used in eq. (A.4).

(WF4) The leading terms of the solutions for a free electron (plane waves)

multiplied by the square root of the following Fermi function 38):

&(Z, E) = 4(2pR)2tY- ewy rtY+Q) *

I I F(2y+l) (3.2)

with y = m, y = crZE/p are used.

The expressions for the decay rates eqs. (2.8)-(2.14) and (2.31)-(2.33) have been

derived assuming the long-wavelength approximation for the electron wave func-

tions. This is well justified when a plane wave is involved (case (WF4) above)

because the typical electron momentum (p = a few m,) is much smaller than the

inverse of the nuclear radius R- = 80m,. In the case of the Coulomb wave function

with the finite nuclear size effect (case (WFl) above), however, the effective electron

momentum inside the nucleus is of the order of p = aZ/ R = 20~ and the higher-

order terms may give a non-negligible ~ont~b~tion. Fig. 1 shows the electron wave

functions g_;( E, r) and fL

604 T. Tomoda et al. / Neutrinoless flp decay

t

(WF4) : _-.- _-..--------!---

-2 3 L 5 6 7 r(fmJ

Fig. 1. Electron radial wave functions g_;( E, r) andf!;(E, r) with Z = 34, E = 1.5 MeVand R = 5.08 fm. The four lines correspond to (WFI) leading finite-size Coulomb, (WF2) exact finite-size Coulomb, (WF3)

exact point Coulomb, and (WF4) (F,J times plane wave. See text for further explanation.

In the case of the point Coulomb wave function (WF3) we are forced to take the exact solution because it cannot be expanded into a power series.

Table 1 gives the phase-space integrals F,$ (eq. (3.1)) and F (eq. (2.32b)) for

the &3 decay $Ge(O:)-+ :iSe(O:>. We used Z =34, R = 5.08 fm, Qpp =

E, - I&-- Zm,c* = 2.0407 keV [ref. )I and for Fc2, (EN) - Et = 7.88 MeV [refs. 6=36)].

F() and F:zf with j, k f 4, 5 involve only S-waves and their values calculated by

the four alternative methods coincide with one another within ~10%. This is also

the case for F$) with j and/or k = 4 since the enhancement due to the Coulomb

distortion is canceled 7*8,20,24) in combinations such as f- +f_,, (see eq. (A.211

although P-waves are involved. The difference in the P-wave radial functions demon-

strated in fig. 1 is directly reflected in the values for I=:,, Fit and F:z). A plane wave multiplied by ( F0)2 underestimates 7z8*2o*24) F\: by two orders of magnitude.

60.5 T. Tomoda et al. / Neutrinoless &3 decay

TABLE 1

Phase-space integrals I$ and F* for the OV and 2@3 decay of 76Ge(0:)+76Se(0:)

ik WFl) WV (WF3) (WF4) LFC EFC EC P

(4 Fj? [ 10ei3 y- fm] Z

44

55 66 13 16

14 34 15 56

(b) I=:: [lo- y- . fm*]

44 55 66 13 16 14 34 15 56

(c) I=(*) [ 10-2 y-11

6.697 6.131 6.249 6.681 11.11 10.17 10.36 10.95 3.363 3.176 3,299 3.352

6241 3864 6044 39.69 59.90 54.84 55.92 60.24 -1.838 -1.683 -1.714 -1.890 -9.843 -9.010 -9.199 -10.18

2.581 2.390 2.445 2.574 -4.927 -4.602 -4.757 -4.909 97.61 72.98 92.95 5.148

-611.3 -460.1 -581.2 -46.60

-5.460 -4.999 -5.092 -5.440 9.131 8.360 8.515 8.981

-0.2104 -0.1622 -0.1250 -0.2109 4695 2932 4541 38.59

43.26 39.61 40.34 43.52 0 0 0 0 0 0 0 0

-1.837 - 1.694 -1.725 -1.813 -3.044 -2.855 -2.966 -2.994

3.605 3.324 3.385 3.626 -450.9 -341.1 -428.2 -43.47

1.567 1.446 1.471 1sso

The four columns correspond to the four alternative choices of the electron wave functions: (WFl) leading finite-size Coulomb, (WF2) exact finite-size Coulomb, (WF3) exact point Coulomb, and (WF4) (Fe) times plane wave. See text for further info~ation.

It should also be noted that the higher-order terms in the finite-size Coulomb wave function (WF2) reduce FE) by ~40%. F:: shows tendencies similar to those of F$) except for F&j and Fi:.

3.2. NUCLEAR MATRIX ELEMENTS

The ground-state wave functions of 76Ge and & have been calculated through the method described in subsect. 2.5. We assumed a model space consisting of active l~~,~, Of,!,, IP,,~ and Og,,, orbitals and used as an effective NN interaction the modified surface delta interaction 40) with the strengths 4) A, = 0.43 MeV, A, = 0.35 MeV and B =0.33 MeV. The single-particle energies were also taken from

606

ref. 4); they are

T Tomodn et al. / Neutrinoless /3p decay

~(lp~,~) = 0.0 MeV,

F( lp,,,) = 2.20 MeV ,

s(Of,,,) = 1.75 MeV,

.s(Ogg,J = 3.39 MeV.

Using the initial and final nuclear state wave functions obtained, we calculated

the various transition matrix elements for the Ovpp and 2v@ decay with the

harmonic-oscillator parameter v = Mu/h = 0.228 fm- and

A=(E,)- E,+m,c2+$Qpp =9.411 MeV, (3.3)

(see subsect. 3.1 for (EN)- E, and Qpp).

Table 2 gives various contributions to the matrix element Mz (eq. (2.13i)) due

to the relativistic correction term D, (eq. (2Sb)) in the nuclear current. Here m(i)

- (&&nm), MRC = (V RC - RCnm), etc. (see eqs. (2.20)-(2.27)). The matrix elements

rnp,& and ma& were calculated using eqs. (2.25a, b) for the columns (1) and (2), and

eqs. (2.29), (2.30) for the columns (3) and (4). For all the other matrix elements

the effect due to the finite nucleon size was neglected as was stated in subsect. 2.3.

The values in the columns (2) and (4) were obtained through the method described

in subsect. 2.6. As the line (a) of this table shows, rnf,& which gives the dominant

contribution to MG in the case of a point nucleon without the short-range NN

correlations, vanishes completely after inclusion of the correlations. However, if one

TABLE 2

Various contributions to the nuclear matrix element M$

(1) Point (2) Point no SRC SRC

(3) Extended (4) Extended no SRC SRC

(a) rng; [fm-1 -71.60 0

(b) m$ [fm-1 3.10 1.629 (c) r?$;. [fin] -0.20 -0.163 (d) M,, (a+b+c) [fm-1 -68.70 1.466

(e) m E+ [fm-1 -0.647 -0.535 (f) m,: [fm-1 0.003 0.003

(g) rng+ [fm-1 -0.004 -0.004

(h) M,, (e+f+g) [fm-I -0.649 -0.537 (i) M,, [fm-I 0.653 0.576 ci) MRLS [fm-l 0.116 0.097 (k) M; (d+h+i+j)[fm-1 -68.58 1.603

(1) xk 78.82 -2.271

-46.40 -24.26 2.35 1.77

-0.20 -0.16 -44.25 -22.66

-0.649 -0.537 0.653 0.576 0.116 0.097

-44.13 -22.52 50.72 31.92

The short-range NN correlations were taken into account for the values given in the columns (2) and (4), and the finite extension of the nucleon for mk02_ and m$ m the columns (3) and (4). (a)-(k) are given in units of fm-; xk is dimensionless.

T. Tomoda et al. / Neutrinoless pp decay 607

takes into account also the finite extension of the nucleon, rnko:: is reduced only to

-4 of its original value given in the column (1).

The operator uCLnm is relatively short-ranged and rnt,$ = 1.629 in the column (2)

shows a reduction by a factor -i relative to that in the column (1). Inclusion of

the finite nucleon size and the short-range correlations, however, does not change

the value of the column (2) appreciably. It enhances this value only by ~10%.

Table 2 shows also that the matrix elements other than rnf& and rng& cancel one

another almost completely, i.e.

(3.4)

Table 3 gives together with xk the other nuclear matrix elements (eq. (2.13)) and

their combinations (eq. (2.14)) for the Ovj3p decay. The values in the column (2)

were calculated with the short-range correlations and in the case of xk also with

the finite nucleon size taken into account. By inclusion of the short-range correlations

the matrix element MgT is reduced by -20% and xF, iGT,+ x&~,~,~ remain almost

unchanged. The numerator in eq. (2.13g) for x; is not affected very much because

the two-body matrix elements in which both the initial and the final states are in a

relative S-state do not contribute. As a result its ratio to M$T increases. The ratio

of Mz to Mgq, xk, becomes ~40% of its original value.

In order to check the validity of the closure approximation we tested the sensitivity

of the nuclear matrix elements to the value of A (eq. (3.3)). The calculated matrix

TABLE 3

The nuclear matrix elements for the Or@/.? decay of %e

(1) Point (2) Extended no SRC SRC

(3) Extended SRC

half A (4) Haxton

M!$ [fm-1 -0.694 -0.563 -0.609 -0.41 I XF -0.217 -0.219 -0.222 -0.200 i, -0.186 -0.182 -0.198 -0.200 XL 0.875 0.857 0.910 1.000 X; -0.249 -0.255 -0.245 -0.23 1 X&T 1.125 1.143 1.091 1.141 X; -0.021 -0.026 -0.027 -0.013 XL -0.223 -0.218 -0.205 0.269 Xk 78.8 31.9 30.5 0 x+ 0.689 0.675 0.712 0.800 L -1.061 -1.039 -1.107 -1.200 x+ 0.665 0.687 0.663 0.637 X- -0.168 -0.178 -0.172 -0.175

They are calculated (1) for a point nucleon without the short-range correlations, (2) with the short-range correlations (for Xk also the finite extension of the nucleon is included), (3) with the same approach as in (2) but with A = 4.706 MeV, and (4) by Haxton and Stephenson 6). A4!$ is in units of fm-; the other quantities are dimensionless.

608 T. Tomoda ei al. / Neutrinofess @@ decay

elements with A = 4.706 MeV which is half as large as its original value (eq. (3.3))

are given in the column (3) of table 3. One sees by comparing the columns (2) and

(3) that the dependence on A (therefore on (I?,)) is weak. This is the case also for

xk because the contributions from rn$ and rn& with i = 1 or 2 which contain a factor A or AZ (see eqs. (2.25) and (2.27)) are small. The insensitivity to /i supports

the closure approximation in the O&I decay.

The column (4) gives the values obtained by Haxton and Stephenson6). They

assumed the approximation*

&=xFt X;cX=It

and neglected the relativistic correction term, i.e.

(3.5)

x&=0. (3.6)

While most of their matrix elements except xfi are not very different from ours given

in the column (2), they obtained xp with a sign opposite to ours**. In our case the

process (Og9,* (neutron))2 + (Of,,, (proton))2 gave a dominant contribution 0.207 to the numerator A4$ = 0.153 of ,& resulting in J&CO.

For the 2z@ decay matrix element, we obtained (eq. (2.32a))

h4g;: = 1.727 . (3.7)

while Haxton et al. 6*36v43) gave IwgG = 1.278.

3.3. LIMITS ON NEUTRINO MASS AND RIGHT-HANDED CURRENTS

The half-life for the Or.@ decay T$ is related to (m,), (A) and (q) by (see eqs.

(2.16)-(2.19))

(7$- = C ((m )/ WlM y m, )2+ C(A)+ CO( Ah VI T )2

+2C~~((m,)lm,)(h)+2C,o~((m,)/m,)(77)+2C:0,(h)(17), (3.8)

where Cgk, etc. are defined analogously to eq. (2.17) withf$ replaced by F$) (eq.

(3.1)). Table 4 gives the calculated values of the coefficients CE&, etc. using the

nuclear matrix elements in the column (2) of table 3 and the four different sets

of the electron phase-space integrals of table la. From these coefficients and the experimental lower limit r$ > 1.7 x 1O23 y (la) which resulted 16) from the best

combination of the data of Avignone et al. 16) and Bellotti et al. 13), we obtained the upper limits for I(m,)l, l(A)1 and l(n)/ listed in table 4. Here the limit for a parameter on axis was deduced under the condition that the remaining two

parameters are put equal to zero (i.e. (A) = (r]) = 0 for (m,), etc.). The absolute

* The approximation eq. (3.5) was used also in our previous work I). There the matrix elements other than i,.,, were the same as the present sets (1) and (2) of table 3, where in the set (2) & = -2.273 from the column (2) of table 2 was taken instead of & = 31.9. ** A4$* is related to I%#; of ref.6) by Mu= -fih4&

T Tomoda et al. / Neutrinoless pp decay

TABLE 4

609

The coefficients C and the upper limits on the neutrino mass (m,) and the right-handed current

coupling strengths (A) and (7)

WFl) WF2) V-3) WF4)

cc,ol. [lo-2 y-11 0.315 0.288 0.294 0.314

c,: [lo-* y-11 -0.0560 -0.05 11 -0.0520 -0.0582

c,l, [lo-2 y-11 12.9 11.7 12.1 12.6

c:o,)[lo-* y-11 0.326 0.297 0.302 0.320

cc,ql [lo-2 y-11 2210 1980 2070 1960

c:; [lo-* y-11 -0.129 -0.117 -0.117 -0.126

A? I(m,)I [evl

(

g I(A)I

8 l(9)l

c2.21 12.31 ~2.29 c2.21

610 T. Tomoda et al. / Neutrinoless pp decay

TABLE 5

The upper limits on the neutrino mass and the right-handed current couplings

&=O

(WFl)

xk=O

(WF4) Haxton

half A

(WFl)

.2 Ih)l [evl C2.21 C2.21 13.14 ~2.03 5 I(A)I

i

13.58 x 1O-6 ~3.61 x 1O-6

T. Tomoda et al. / Neutrinoless &3 decay 611

Fig. 2 shows the allowed regions on a (q)-(m,) plane for a few fixed values of (A), where the coefficients C,ok, etc. of the column (WFl) of table 4 were used.

Integrating eq. (2.8) only over E,, we obtain (see eq. (2.19))

dW,, In2 - = 417 (A"' + A(') cos 012) . don,,

(3.12)

The angular correlation coefficient A()/A() for the cases in which only one of the three parameters (m,), {A} or (3) is nonvanishing is equal to C~~/C~~, C~~/C~~ or C$!J CT!, respectively. The values of the C*s in the column (WFl) of table 4 and the Cs calculated in a similar way using F$ in the column (WFl) of table lb give for these the values -0.815, +0.849 and -t-0.724, respectively. In the OvpP decay caused by a Majorana mass of the neutrino, two electrons are likely to be emitted in the opposite direction to each other because their helicities tend to be both negative. On the contrary in the Ov/3/3 decay caused by an admixture of right-handed currents a parallel emission is enhanced because of the mainly opposite helicities of the two electrons [see ref. )I.

Fig. 3 shows the angular correlation coefficient a(Er) as a function of the electron energy Et defined by (see eq. (2.8))

cr(E,) = u(I)(~,)/u(O)(~,) ) (3.13)

together with the single-electron spectrum,

dwov -=4rra(E,)wo,(E,)) dE,

= 0

(eV)

3

(3.14)

Fig. 2. The allowed regions (inside the ellipses) deduced from the experimental lower limit of the half-life for the Ov&3 decay of Ge, T$~ > 1.7 x 1O23 y [ref. 16)], for a few fixed values of (A).

612 T. Tomoda et ol. / Neutrinoless pp decay

*&+nkT-& (,_AJW ,-A ,z_OI) 3P/ OMP

T. Tomoda et al. / Neutrinoless /3p decay 613

for the limit values (on axis) of (m,), (A) and (77) given in the column (WFl) of

table 4. The phase-space factors fj: which give upon integration F$) of the column

( WFl) of table 1 were used. While figs. 3a and 3b look similar to the corresponding

figures of ref. O), fig. 3c has a shape different from theirs because of our large value

of ,&. The difference in the single-electron spectrum and the angular.correlation

for the three limiting cases enables *) one in principle to distinguishi between the

Ou@ decay processes caused by (m,), (A) or (7). The time-projection chamber

experiment 14) would give such a possibility if Or@ decay events should be observed

in this method.

The 2vpp decay half-life is calculated to be T:/Z = 2.14 x 10 y from the matrix

element MEG (eq. (3.7)) and F (*I listed in the column (WFl) of table lc using

eq. (2.31). It is about half of the value 7:y2 = 4.15 x lo* y calculated by Haxton and

Stephenson ) but still compatible with the experimental lower limit T$* > 2.8 x lOI y

of Avignone et al. 45) cited in ref. ). For the relation between the magnitudes of MgG) and M$$), see the recent works by Grotz and Klapdor 19*&).

4. Summary

We have calculated the Or&I and 2u/3/3 decay rates for the transition 76Ge(0:) +

76Se(O:). In the case of the OV@ decay we calculated also the nuclear matrix element

,& (eq. (2.13j)) originating from the relativistic correction to the nuclear weak

currents. Since the relevant two-body transition operator has a very short range, the

effects due to the short-range NN correlations and the finite nucleon size are

important. These effects, if taken gotether, reduce xk only by a factor ~0.4 of its

original value. xk has a large value because the neutrino exchanged between two

nucleons is a virtual particle and the momentum transfer to it is limited only by the

amount which a nucleon in a nucleus can provide.

We have shown that the upper limit on the parameter (7) describing an admixture

of the right-handed leptonic current is determined dominantly by the second-

forbidden matrix element ,&. Comparing our calculation with the experimental

data of ref. 16), we obtained I( ~)1< 6 x 10e8 which is more stringent by two orders

of magnitude than the limit deduced from the same data and the calculation by

Haxton and Stephenson ). The effect of the weak magnetism is very important since

the upper limit on l(q)1 is roughly inversely proportional to pUa (eq. (2.6)). If we neglect the weak magnetism completely (i.e. if pB = l), the upper limit on l(n)1

would be less stringent by a factor -5.

It has also been shown that the correction of the electron Coulomb wave functions

due to the nuclear finite size is of the same order as the error associated with the

long-wavelength approximation. Therefore we can use in practice the much simpler

point Coulomb wave functions without spoiling the accuracy of the calculation.

Furthermore even if we approximate the electron wave function by a plane wave

multiplied by ( Fo)*, it does not cause an appreciable error. The enhancement of

614 T. Tamadu et al. / ~eut~noless &3 decay

the P-wave components relative to the S-wave components of the electron wave functions due to the Coulomb field [called the P-wave effect by Doi et al. 7*8*20)] does not affect the fip decay rates very much because the contribution from the matrix element XL (eq. (2.13h)) is much smaller than that of xk and plays no important role. (This would not be the case if xk% lo,&)

The closure approximation is expected to be good in the case of the O$p decay. This approximation is strongly supported by the insensitivity of the final results to the assumed average excitation energy of the intermediate odd-odd nucleus.

Wu et al. 47) have recently calculated the nuclear matrix elements for the Or@3 decay of 48Ca caused by heavy neutrinos using an effective operator approach. They showed that the usage of the correlation function eq. (2.43) overestimates the effect of the short-range NN correlations (i.e. underestimates the nuclear matrix elements). If this holds true also in the case of the 76Ge decay, the matrix element ,I& becomes larger and an even more stringent limit on I(n)] will be obtained.

In the present work we assumed a relatively small model space and used a schematic effective NN interaction to calculate the initial and the final nuclear state wave functions. We believe that these wave functions are good enough for the purpose of discussing the relative magnitudes of the contributions of the various nuclear matrix elements to the total OV@ decay rate. As for the absolute magnitudes of the matrix elements, it is probably necessary to improve the present nuclear wave functions. The present method can be applied without di~cu~ty to a calculation which involves a much larger model space and a more realistic NN interaction. Such an improvement is in progress and we hope to be able to obtain more reliable upper limits on the lepton-number violation parameters (m,), (A) and (7). An analysis of the geomchemical data ) for the p/3 decay of 28*30Te is also under way.

The authors are grateful to Professors F.T. Avignone, H. Ejiri, W.C. Haxton, B. Kayser, T. Kotani, T.T.S. Kuo and M.K. Moe for sending their results prior to publication. We thank also Professor A.H. Wapstra for information about the Q-value of the p/3 decay of 76Ge. This work was supported by the Bundesministerium fur Forschung und Technologie.

Appendix A

ELECTRON PHASE-SPACE FACTORS

We solve the Dirac equation for an electron in a field generated by a spherically- symmetric charge distribution of total charge Z. The radial wave functions gj;( E, I) and fL-( E, t) [ref. 48)] of the large and small components satisfying the boun- dary condition a plane wave plus an incoming spherical wave are normahzed in

T. Tomoda et aL / Neutrinoless &3 dewy 615

such a way that

1 X-

Pr

sm (pr+y In 2pr-&I, +A,)

, 64.1)

2E cos(pr+yln2pr-fz-/,+A,)

where K = A( j +$), l, = jrtf, y = aZE/p, and AZ is the phase shift. E, m,, p and j are the energy, the mass, the asymptotic momentum and the angular momentum of the electron.

The electron phase-space factors f$ (=f$)) appearing in eqs. (2.10), (2.17), (2.32b) and (3.1) are defined by

fit, = (yyw + Ifl-111

f~=(~)2~if-+f_,,~2+IX_~+f~-~~21 1 c

.C, = 4( &J2W1-I +L;l12+lf11 +_m ,

fi? = WtS--I*+ k,11 ,

E, -.E2 fgL- m2R [f -Af -11 +f-iI)* -f*-ul-1 t-f ?*I, e

fp:= -5 Ef*,(.h+f II)*+f --(f -I-I-+f-*-)*I , e

f&- -& MAfi+f a* -f --(f--l +f-,-)*I , c (44.2)

616

and

T. Tomoda et al. / ~euirino~ess &3 decay

!I: = -2 Re [f--f;, +f-J,-*I ,

2 Re [f-,f,-*I ,

f5-_2 Re [If-"+f-ll)tfi-l+f'-')*l,

f%= -f$--&)* Re [(f-"-l+f-l-')(fI'+f',)*l,

f,+- & Re [f-l(fi-l+f-)*+fi-(f-+f-l,~*lf e

fo&_y Re [f-t(fi-t+f-)*-fi-(f-+f-ll)*], e

f;:= --& Re LAl(f --, +f-l-)*+f--(fi+f ,)*I, e

f$= ~Re[f,,(f-_l+f_l-)*-f--(f,+fl,)*]. (A.31 e

All other f_$ are zero. Here R is the nuclear radius and

f Kx = g-( E,, R)gj;;( E;, R) h- ,

fec= d-(5, R)f i+(&, RI , etc. , (A.4)

i.e. a superfix K (a suffix K) in fKKT, etc. indicates that g;( f ;) should be taken; a left (right) superfix or suffix in f K1(., etc. refers to the electron of the energy E, (&). The radial wave functions g;( E, r) and f ;( E, r) are expanded in powers of r and the leading terms are retained in eq. (A.4). Although eqs. (2.8)-(2.10) with (A.l)- (A.4) have been derived originally assuming the long-wavelength approximation, we use also exact radial wave functions in eq. (A.4) for some purposes (see the cases (WF2) and (WF3) described in subsect. 3.1). All the phase-space factors (eqs. (A.2) and (A.3)) become independent of R for a free electron (Z = 0).

T. Tomoda et al. / Neutrinoless $fl decay 617

Appendix B

NEUTRINO PROPAGATION FUNCTIONS

The neutrino propagation function H(r) which appeared in eq. (2.13) is given by

H(r) =t[H(r, A,)+ H(r, &)I, = H( r, A) ,

4T

=m I dk exp (ik* t) k(k+A) _ #Gr)

r f

where (cf. eq. (2.7))

Aj=Ej+(E,)-El,

A=~(A,+A,)=(E,)-~(E,+E,).

The function 4(x) is defined by 33)

~$(x)=i(sinxci(x)-cosxsi(x)),

(B.1)

03.2)

03.3)

where

ci (x) = - s

m m t-r cos t dt si(x)=-

I t-sin tdt,

X X

and it has the property

&O)=l, ~_~C$(x)=o. 03.4)

In going from the second to the third line in eq. (B-1) the neutrino mass was neglected in comparison with the typical neutrino momentum k = 2OOm,. Taking the derivative of H(r), we obtain

a(Ar) -rW(r) = -r; H(r, A) =-

r fB.5)

where

a(x) has the propery

Q(X) = 4(x) -x$(x) * (B.6)

cw(O)=l, lim a(x) = 0, x-00

a(x)=xf$(x)-;, 03.7)

618 T. Tomoda et al. / Neutrinoless pp decay

which is used in subsect. 2.2 to rewrite M$. Another type of neutrino propagation function k(r) which appeared in eqs. (2.13b, e) is defined by

f?(r) =-& [AIH(r, 4) -A,H(r, A2)1, 1 2

=H(r,A)+A~H(r,A)l,_,,

=2H(r,A)+r$i(r,A). (B.8)

It should be noted that in the limit of A + 0, H(r) = - rH( r) = g(r) = l/r.

Appendix C

RADIAL INTEGRALS FOR THE TWO-BODY MATRIX ELEMENTS

The radial integrals for the two-body matrix elements in a harmonic-oscillator

basis, can be reduced to the Talmi integral 49)

I,[V(r)] = z (2~~~)!! j: V(r) exp (-$r2)r2f+2 dr, (C-1)

where v = Mu/h is the oscillator parameter. The Talmi integral is related to the

integral in momentum space 50),

J,o[dk)l= 2?r2(;v) jam u(k) exp ( -$)k2,+ dk, (C.2) where

u(k) = I

V(r) emikdr, (C.3)

through the relation 50)

1J[V(r)]=mC_o(2m+1)!! m m (-2) l Jo[v(k)]

0 . (C.4)

In the case of the function H(r), eq. (B.l), the following recurrence relation for J,0[47r/k( k + A)] can be obtained:

J!G

J@ = J 2v 2J2v 0 ---u@(u) ) ?r rr (C.5)

T. Tomoda et al. / Neuirinoless pp decay

where u = A/X& and*

Q(u)= I m exp (-t*) 0 t+u dt, [ I

=exp(-u2) J, exp (t) dt -iEi( u) 0 1 ,

where Ei( x) = P I, t- e dt (a principal-value integral).

-rH(r), eq. (BS), is evaluated through the relation

619

(C.6)

The Talmi integral for

A[-rWr)l= (21+3HZdH(r)l- Z,+l[H(r)ll. (C.7)

Other radial integrals can be obtained using the above results or standard Talmi

integrals for the Yukawa potential, etc. combined with the relations

21+3 Z,[r*V(r)]=- ZI+,[ V(r)1 ,

V

1+3/2

Z,[V(r); v+2a]. (C.8)

Finally it should be noted that the radial integral of the type

I

CC R,,( r)Z-Z( r)R,.,.( r)r* dr (Z+Z=odd), (C.9)

0

which is necessary for the calculation of XL, eq. (2.13h), or ( VRPnm), eq. (2.22c), can

be reduced to the Talmi integrals of -rH(r) by a slight modification (of the

B-coefficient) of the usual method described in ref. l).

References

1) P. Langacker, Phys. Reports 72 (1981) 185 2) H. Primakoff and S.P. Rosen, Rep. Prog. Phys. 22 (1959) 121; Proc. Phys. Sot. 78 (1961) 464 3) H. Primakoff and S.P. Rosen, Ann. Rev. Nucl. Part. Sci. 31 (1981) 145 4) D. Bryman and C. Picciotto, Rev. Mod. Phys. 50 (1978) 11 5) T. Kirsten, in Proc. Workshop on science underground, Los Alamos, 1982, AIP Conf. Proc. 96, ed.

M.M. Nieto et al. (New York, 1983) 6) W.C. Haxton and G.J. Stephenson Jr., Prog. Part. Nucl. Phys. 12 (1984) 409 7) T. Kotani, Proc. 4th Moriond Workshop, La Plagne, 1984, ed. J. Tran Thanh Van (Editions Front&es,

Gif sur Yvette, 1984) p. 397 8) M. Doi, T. Kotani and E. Takasugi, Osaka Univ. preprint OS-GE 84-09 (revised version, 1985) to

be published in Proc. 3rd Telemark Meeting on neutrino mass and low energy weak interaction, Telemark, 1984, ed. V. Berger and D. Cline

9) T. K&ten, H. Richter and E. Jessberger, Phys. Rev. Lett. 50 (1983) 474; Z. Phys. Cl6 (1983) 189 10) A. Forster, H. Kwon, J.K. Markey, F. Boehm and H.E. Henrikson, Phys. Lett. 138B (1984) 301

l Both the Dawson integral and the exponential integral in the second line of eq. (C.6) are standard library functions on modern computer facilities.

620 T. Tomoda et al. / NeNeuWroless &3 decay

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