Microscopic description of radiative pion capture in nuclei

Nuclear Physics A240 (1975)

493-520;

Not to be reproduced by photoprint

MICROSCOPIC

Institut

DESCRIPTION

de Physique

North-Holland Publishing Co., Amsterdam

@

or microfdm without written permission from the publisher

OF RADIATIVE PION CAPTURE IN NUCLEI

J. DELORME and M. ERICSON+ NuclPaire, UniversitP Claude Bernard Lyon-I and IN2P37f and

G. F;ILDT Institute for Theoretical Physics, Received

Lund, Sweden ttt

16 September

1974

Abstract: We have investigated the radiative capture of pions in nuclei by a microscopic method. We have shown that the elementary nucleonic amplitude is renormalized by two effects : (i) The effective field correction which takes into account the incoherent scattering of the pions before the absorption. This correction differs for the short-ranged and long-ranged part of the interaction. (ii) We have also introduced the meson exchange corrections where the pion interacts with a nucleon which exchanges a pion with the neighbouring nucleons. The introduction of this correction makes the microscopic description and the soft pion one eqqivalent for pions at rest.

1. Introduction The radiative capture of low-energy pions in nuclei or its inverse process pion photoproduction can be theoretically approached in two different ways: (i) For a vanishing pion four-momentum (soft pion limit) the partially conserved axial current hypothesis (PCAC) relates these two reactions to weak interaction processes such as p-capture or p-decay. The transition amplitude is found to be proportional to the nuclear matrix element of the axial current ‘). The extrapolation to the physical pion mass shows that the axial current matrix element is the Born amplitude. In the soft pion method no explicit mention is made of the nucleonic structure of the nucleus which is treated as an entity. (ii) The second method is a microscopic one. One relates the nuclear amplitude to the nucleonic one. An effective Hamiltonian is built from the elementary nucleonic amplitude and its matrix element is calculated with *apion wave distorted by the strong interaction. This usual impulse approximation description is only approximate since the presence of the other nucleons modifies the elementary nucleonic amplitude. These modifications are on the other hand contained in the soft pion approach but unfortunately the application of this method is restricted to pions at rest. For pions in flight, even at a few MeV, or for pions bound in a p-orbit, one has to resort to the impulse approximation. t Also at CERN, tt Postal address: ttt Postal address:

Geneva, Switzerland. 43 Bd. du 11 novembre 1918, 69621_Villeurbanne, Siilvegatan 14, Lund, Sweden. 493

France.

494

J. DELORME et al.

It is the aim of the present work to improve the microscopic description for low

energy pions by the inclusion of many-body effects which are not contained in the distorted wave impulse approximation. Although we have no systematic way of including all many-body effects, our description includes those which are expected to be the most important. For pions at rest they provide the link with the soft pion method. We have been inspired in part by the similar problem of the elastic scattering of pions on nuclei. In this case it is known that a large renormalization effect arises from the nuclear polarizability : the pion scatters on one nucleon exciting the nucleus and rescatters on another nucleon, bringing back the nucleus to its ground state. This leads to a renormalization of the elementary amplitude since the distorted wave description only includes the rescattering terms where the nucleus remains in its ground state in the intermediate stages. In the static approximation where the nucleons are taken to be massive, the renormalization can be conveniently described by distinguishing between the average pion wave in the medium and the effective pion wave which hits a scatterer “). For the elastic scattering of pions by nuclei this distinction leads to the Lorentz-Lorenz renormalization of the nN amplitude. It has been shown by Scheck and Wilkin ‘) and Faldt 4, that this correction is needed in order to satisfy the requirement that the n-nuclear scattering amplitude should be inde~ndent of the off-shell behaviour of the n-nucleon amplitude in the case of non-overlapping scattering potentials. A similar distinction should be made in the radiative capture of pions. However the Lorentz-Lorenz correction depends critically on the shortness of the xN interaction as compared to the two-nucleon correlation length. This may be a valid assumption in the scattering case but the situation is not as simple in radiative pion capture, for the photon may be radiated by the pion through a one pion exchange diagram as shown in fig. Ic (photoelectric graph). This graph represents an intera~ion of long range (Z m; “) in which case the pion interacts with the photon mainly outside the correlation hole. This interaction is not subject to the same renormalization as the short-ranged one and it will be given a special treatment. Our work is divided into three sections: Sect. 2 discusses briefly the nucleonic amplitude and the separation between the short- and long-ranged parts of the interaction. The rest of the section is devoted to the renormalization of the short-ranged part. It will be shown that the corresponding nuclear amplitude does not depend on the off-shell behavior of the elementary RN interaction. This part of our work is an extension of the works of Scheck and Wiikin 3, and FHldt “) for the scattering case. We shall make use of the effective field formalism in coordinate space 2*4). For simplicity we shall neither introduce the complications of the isospin notations nor treat the kinematical complications due to the difference between the n-nucleus and rt-nucleon c.m. systems 5). These aspects are treated in detail in a forthcoming publication by Delorme et d6). In sect. 3 we treat the effective field correction for the long-ranged photoel~tric

RADIATIVE

PION

495

CAPTURE

interaction. For this we need the expression for the effective field not only at the position of the interacting nucleon but also outside the correlation hole. This expression is derived together with the resulting modification of the nuclear radiative capture amplitude. As expected in this case the nuclear amplitude is sensitive to the off-shell behavior of the nucleonic amplitude. Finally in the sect. 4 it is shown that the effective field correction is not the only modification to introduce in the microscopic description. The radiative capture of the pion can occur on a nucleon which exchanges mesons with the neighbouring nucleons. The introduction of these contributions displays the equivalence between the microscopic description and the soft pion approach for pions at rest. 2. The nucleonic amplitude and the separation between the short- and long-ranged parts of the interaction 2.1. NUCLEONIC

AMPLITUDE

We consider the process of charged pion capture + on nucleons, TT*+N + y+N, and we denote by q and k the pion and photon momenta. The conventional decomposition of the c.m. transition matrix h(k, q) into rotationally invariant amplitudes is given by h(k, q) = ih, c. e+ h,(a . @)(a*(k x 4)) + ih,(a . &o( * E)+ ih,(a . ij)(tj . e),

(1)

where the amplitudes hi depend on the variables s and t. The photon gauge has been chosen so that e. k = 0. We have suppressed all isospin index dependence for 7c* capture. The amplitudes hi can be expanded into electric and magnetic multipoles and for low energy pions only the first few multipoles are retained. Their values are obtained from dispersion relation estimates which nicely tit the experimental data ‘). This expansion gives an effective low energy amplitude correct up to second order in the pion momentum, h(k,q) = iA,a*e+iA2(o~e)(q~k)+iA,(a*k)(q~~)+A,q(kxe) +iA,(a.qXq.&)+iA,a.e(q.k)‘+iA,(a.kXq.&Xq.k)+A,q.(kx&Xq.k),

(2)

where the coefficients Ai depend on the energy alone. The modification of the nucleonic amplitude in a nucleus depends critically on the range of the interaction. The physical amplitude cannot provide this information since the pion and photon momenta are restricted by the mass-shell conditions. However a description of the amplitude in coordinate space which is the Fourier transform of h(k, q) requires a knowledge of h(k, q) for all momenta. Here a model + We treat the case of radiative photoproduction.

capture

for definiteness,

although

the discussion

applies

as well to

496

J. DELORME et al.

Fig. 1. Born contributions to the radiatwe capture amphtude.

is needed. At low energies it is known that the amplitude is well described by the Born terms calculated with a pseudoscalar nN coupling (fig. 1). Diagram Ic is the photoelectric term which has already been discussed. Its contribution to the amplitude is given in the non-relativistic limit by: h

=

+

It

JZg,e,~kcl)a-(k-8)

-

(k-q)‘+mz

mN

where g, is the rcN renormalised mass (the 2 sign applies to rr* order in the pion momentum following part of the nucleonic

’

coupling constant, ef = 4x/l 37 and mN is the nucleon capture). This expression may be expanded to second q, and the expression thus obtained dominates the amplitude :

iA,(a - k)(q * e)+ iA,(a * q)(q * 8)+ iA,(a * k)(q * s)(q * k).

(4)

We shall therefore replace these terms (4) of the nucleonic amplitude by expression (3) which will be used on-shell and off-shell. In coordinate space the corresponding nucleonic amplitude is

J&r@,- i

hrrfr, r’) = + ~

mN

(2$6

s

ik.r’e-iq.r

d3qd3ke

(E-q)a(k-qf (k-q)‘+m,2

where r and r’ stand for the pion and photon coordinates. The modification of the photoelectric nucleonic amplitude by nuclear effects will be discussed in sect. 3. The remaining diagrams la and lb have the range of a xN vertex, i.e. a short range (the lightest object which can be exchanged between the pion and a nucleon has a mass z 3 m,). We shall make the assumption that this interaction is zeroranged. The exact range will be shown to be irrelevant provided it is smaller than the two-nucleon correlation length. In coordinate space the short-ranged part of the nucleonic amplitude is then: h$&., r’) = @A, d * e+iA,a

.

efv,. V,)+A,V,

-t i&a

’ E{V, - v,p+/i*

* (V, x &) v, . (V,s x 8)V#,’ V,,~~~)~(~‘)*

(6)

RADIATIVE 2.2. NUCLEAR

PION CAPTURE

491

PION CAPTURE

We consider the process rr + i + y + f, where i and fare the initial and final nuclear states described by the wave functions t,bi(r.l, . . ., rA) and ef(rl, . . ., rA) respectively. In the usual treatment of the impulse approximation one builds an effective Hamiltonian as a sum over the individual nucleonic contributions : If&r,

r’) = 1 h,(r - ri, r’- ri).

(7)

The transition amplitude is taken as the matrix element of this Hamiltonian with a pionic wave distorted by the strong interaction. In this case it can be immediately remarked that the nuclear amplitude depends strongly on the off-shell behavior of the rrN scattering amplitude used to build the optical potential. Different off-shell behaviors lead to different optical potentials and hence to different pionic wave functions. This sensitivity will be shown to disappear for the short-range part of the interaction once the distinction is made between the average pion wave in the medium IC/(r)and the effective pion wave cp(r; ri) which hits a nucleon located at ri . In a nuclear medium with correlations, the average and the effective waves do differ and a correct treatment of the nuclear process must use the effective Hamiltonian in conjunction with the effective wave q(r; vi). However what is known in practice is the average wave function t&r), solution of the Klein-Gordon equation with the strong interaction potential. One has then to express the effective wave in terms of the average wave. We defer until sect. 3 the general relation between these two waves needed only for the photoelectric term. For the point-like interaction (eq. (6)) the nuclear amplitude involves the effective field cp(r ; ri) or its derivatives for r = ri. One has therefore to relate cp(r ; r) and e(r) or the derivatives [V, q(r, ri)],=,, and V+(r) etc. Once these relations are established one may forget about the effective field and use only the average field. But the difference between the effective and average fields, which is not explicitly taken into account, has to be implicitly contained in the effective Hamiltonian i.e. one has to use a renormalized Hamiltonian which is not a simple sum of nucleonic amplitudes. The expression for the renormalized Hamiltonian will be given. A remark should be made at this stage: the differences between cp(r,r) and $(r), [V,cp(r; ri)lr=,, and V+(r) depend on the off-shell extrapolation of the rcN amplitude. Hence the renormalized Hamiltonian does depend on this extrapolation. It will be given here with the form that leads to the non-local Kisslinger potential *). Hence our renormalized Hamiltonian has to be used in connection with the pionic wave obtained from this potential. An alternate way which avoids the off-shell extrapolation problem is to express directly the nuclear amplitude in terms of quantities which do not depend on the

J. DELORME et al

498

off-shell extrapolation. This will be shown to be possible for the short-ranged part of the interaction. Both procedures are adopted in the following and they are of course equivalent. We shall make the assumption that the effective pion wave in the radiative capture is the same as in the elastic scattering. This identi~~tion is strictly valid when the wave functions & and \fiEare truly degenerate and as long as l/A corrections are ignored. Defining P(r)

= cP(r; 4,

E:‘)(r) = [VFcp(r;r’)JgEr, Dam)

= [VX$(r;

(8)

r’)lrscr,

the transition amplitude for the short-ranged part of the interaction is Xfi =

s * aJr)$“‘(r)

d3reFik”[iA,

- A,a,,(r)

* s(& * E(‘)(r))

s + iA,(k x 8) * E@)(r)pfi(r) - iA, E . an(r)D$(r)k”kS - A&k x 8)“k@D$(r)p,i(r)],

(9)

where pn(r) and a&) are the transition density and the transition spin density, Pfitr)

=

(fl

$5%r--rk)li>,

k=l

@r,(r) = (fit a,r:6(r-r,)li). k=l We shall now give the relations between @If, E(i), DoI and the average pion wave $. It will also be shown that the quantities $ (I) , I?‘) and L)(l) are independent of the off-shell behavior of the rrN amplitude when there is no overlap between the scatterers (due to nuclear correlations), whereas the average pion wave $ depends on the assumed off-shell behavior.

2.3. THE EFFECTIVE PION WAVE AND THE SHORT-RANGE

NUCLEAR ITERATION

The average wave I(I(r) is the sum of the incoming free wave Ipk(r) and of the waves scattered from all the nucleons i,&(r)= fpk(r)+ fd’r’$j(r, r’) Jd3rip(r$

jp3fy(r’

- ri; Y” - r$p(r”; ri),

(11)

where f(r’, r) is the pion-nucleon scattering amplitude, p(r) the one-nucleon density

RADIATIVE

499

PION CAPTURE

in the nuclear ground state i and g(r’, r) the pion propagator eiql*‘-rl dr’,

r)

=

m.

Similarly the effective wave cp(r, ri) on a nucleon at ri is the sum of the incoming wave and the wave scattered from all the nucleons but the nucleon i itself. The effective wave can be expressed in terms of the pair correlation function G(r, r’) defined such that the two particle density is given by

P2(r,r’) = hMr’)Cl + G(r,r')l,

G(r, r) =

cp(r; ri) = &r)+

-

1,

d3r”f(r’-rj,

r”-rj) x fp(r”; ri, rj)

= $(r)+

Jd’ig(r,

r’) fd3rjp(rj)G(ri,

rj) fd3r’y(r’-rj,

r”-rj)(p(r”;

ri, rj),

(12)

where rp(r ; ri, rj) denotes the wave incident on a nucleon at the position rj given the fact that there is another nucleon at ri. In the usual approximation cp(r ; ri, rj) is replaced by cp(r, rj) in eq. (12). For the zN scattering amplitude we use the off-shell form

Sh, 4’) = b+cq . (I’, or in coordinate

space f(r,

r’) = (b +

CV,, . V,)&r’)&r).

(13)

For the short-range interaction (9) we need the expression for E(r) and D(l). That of E(r) is well known and we refer to refs. 2*“) for the derivation. In the short-range correlation limit E(‘)(r)

=

1 ~ 1 -*u(r)

VIC/“‘(r),

where a(r) =

Furthermore wave +(I)

-4&r).

from eq. (12) we can derive a differential (A +q2)+‘l)(r)

where q(r) = -4&p(r).

= q(r)i+P’(r)-

equation for the effective

Va(r)E(‘)(r),

(15)

500

J. DELORME

et al.

has not been given previously and we give therefore some The expression for D (') details. Using eq. (12), D(l) may be expressed in terms of E(l) and $(l): o:;)(r) = V,E~‘(r)+VgE~‘)(r)-V,Vgtj(‘)(r) + [d’r!g(r,

r’) jd3rip(ri)VaVPG(r,

vi)

Sd)r’.f(r’-

ri, rr’- ri)

(16)

= V,E~)(r)+VBE~1’(r)-V,V,tj(“(r)+6,+6,, where, in the integral contribution, the amplitude f has been split into its s- and pwave parts, and the corresponding contributions are denoted by 6, and 6,. The calculation of 6, is straightforward and gives 6,‘= -~i3s,,q(r)ll/“‘(r). A partial integration of the p-wave part gives 6, = c

d3r’[V,Vg G(r, r’)][(V,&g(r, r’)]p(r’)Ey)(r’).

(17)

s

Due to the short-range nature of the correlation function G the integral only gets contributions from a region close to r. One may therefore expand the quantity p(r’)E;“(r’) around this point. The zeroth-order contribution in r’ - r vanishes upon angular integration. The first-order term contribution is the only one which survives when the correlation length/ goes to zero and it gives 6, = cV,[p(r)Ey)(r)]

s

d3r’[ViVbG(r’)][Vtg(r,

= -i{V,(a(r)EF’(r)) +i{V,(a(r)Eg)(r))+

r’)]rb

+ V&x(r)E~“(r))} V&a(r)Eb”(r)) + 8,,V(cr(r)E”‘(r))}.

(18)

The resulting expression for D(l) may be simplified by writing the differential equation (15) for @l)(r) and by using relation (14) between E(r) and II/(‘) D:;)(r) = V,V,$“)(r)-@aS~$“‘(r)+~{V,(a(r)E~’(r))

+VB(a(r)E~1’(r))-~~6,,V(a(r)E’1’(r))}-~6,8qZ~”)(r).

(19)

Taking the particular case a = /I the previous expression gives D$(r)

= [A,&;

rJ]r,=r

=

-4211/‘1)(r).

(20)

This result means that the effective wave at the scatterer obeys the free wave equation as it should since it is evaluated at a point which is free from nuclear matter. In the special case of a constant density p(r) = p,,, a(r) = cq,, D::(r)

=

RADIATIVE

501

PION CAPTURE

We are now in a position to discuss the expression for the nuclear amplitude. This amplitude is now expressed in terms of one single quantity $(l). It remains to relate I,@ and $ ; this relation depends on the particular potential ; in the Kisslinger potential this relation is t,P’(r) = $(r). However we have preferred to keep $(l) in the nuclear amplitude since it will be shown that II/(‘) is an invariant quantity, independent of the off-shell form. Another potential leads to a different average wave but the relation between $(I) and $ is affected in such a way that II/(‘) remains the same. The recipe for calculating the nuclear amplitude is therefore the following: (i) calculate the average wave function t+G with any potential; (ii) use the proper relation (depending on the potential) to obtain tj (I) from $; (iii) use eqs. (14) and (19) to give EC’) and D(l) and insert in expression (9) for the transition amplitude. The other possibility, as discussed previously, is to use a renormalized Hamiltonian. The first step is to choose a potential and calculate from it the average wave function. The second step is to associate to this potential a renormalized effective Hamiltonian (serf) to be used with the average wave in such a way that expression (9) for the transition amplitude is recovered, i.e. %ri =

d3rd3ri$f*(ri)2eff$i(ri)$(r). s

(22)

In the case of the Kisslinger potential for which one has +‘l’ = $ the renormalized Hamiltonian is readily obtained. In a medium of constant density the effective field renormalization procedure amounts to the following replacements in the expression for the Hamiltonian deduced from (6) and (7) : A, +A,

= A,,

(23) Az94

+

A,,,

=

A,,4

I-l&

3

0 ,

while for the terms of the effective Hamiltonian which involve D(l), in the parts which have a tensor character in the pion momentum: I+*, A6’8

+

‘6,s

=

A,,,

l_‘cr

, 3

I)

while in the scalar parts A, remains unaffected and the pion momentum has to be taken as the incoming momentum. In the case of a variable density there are additional terms in the expression for the effective Hamiltonian arising from the expression for D(l). Another question which may be conveniently answered at this stage concerns the energy dependence of the parameters A. Since we keep the terms up to second order in the pion momentum one should expand A,(q’) = Al(0)+q2A;(O). If the q2 dependence is treated as an operator dependence in eq. (9) it leads to a contribution

J. DELORME

502

et al.

proportional to [A,tp(r, ri)]r,=r which, due to eq. (15) is simply -q21C/“‘(r). We conclude therefore that the parameters A should simply be evaluated at the proper energy. We shall now show that the short-ranged part of the Hamiltonian gives rise to a nuclear amplitude which is inde~ndent of the off-shell behavior of the elementary rrN amplitudes. This is true as long as the correlations prevent the overlap of the potentials. We assume that the EN interaction f(r, r’) and the photoproduction interaction h(v, r’) are not point-like but have a finite spatial extension such that f(r, r’) and h(r, r’) are zero unless r < a and r’ c a. The non-overlap condition means that we require [1 + G(r, r’)] = 0 when Ir - r’l c 2~. We shall first prove the independence of the nuclear amplitude on the off-shell form of the ZENscattering amplitude. The variables of the off-shell amplitude are the momenta 4 and 4’ and the c.m. energy E. or the associated moments Q*, qg = Ez - m:. We can go off-shell by having q # qo, q’ f qh, or both. An expression for the general amplitude is _I-(@,4, ~3,) = b + cq * 4’ + Cd2- dM,,k, or in coordinate

q’, &J + (q2 - q;Mq,

q’, E,),

(24)

space

ffr’, r, E,) = [b -I-cV, - V,,]W&‘) + (A, + ~~)~~(~‘,r, &,) + (A, + q~)~~(r’, r, 44, (25)

where the functions fL and fRare zero unless jr\ < a, (r’( < a. We shall now prove two theorems concerning the properties of the effective pion wave cp(r, ri): (i) cp(r; vi) obeys a free wave equation (~~+~~)~(r;

ri) = 0

for

IV-Vi/ < a.

This is shown by applying the operator A, +qi to eq. (12) (fL+ 4Mr;

vi) = -4n

d3r’f(r - rj, r”- rj)cp(r”; rj). (26)

d3rjP(rj)[1 + G(r,, rj)] s

s

The finite extension off restricts the contributing region of r - rj to Ir - rjl < a. On the other hand, we also consider Ir - rJ < a, from which we conclude that Iri - rji < 2~. But according to the non-overlap assumption this implies 1 + G(r,, rj) = 0 and therefore proves our theorem. (ii) cp(r, ri) is independent of the off-shell form of the amplitude for \r- ril < a. With expression (25) for f, the effective field is cp(r; ri) = qPk(r)+

s

+ jd’fg(r,

d3r’p(r’)[l + G( r’, rJ][bs(r,

r’)lC/“)(r’)+ cV,,g(r, r’)E(“(r’)J

r’) ~d3rj~~r~~~~+ Gfr,, rj)j

RADIATIVE

x +

x

s

ld3r’dr,

f

r”-

d3r”[&+q~]fL(r-rj, r’) Jd'rjp(rJ[l

d3r”fR(r’-rj,

r"-

503

PION CAPTURE

rj)(p(r"; rj)

(27)

+ G(r,, rj)]

rj)[&+qi]cp(r";

rj).

The contribution of the functionf, can be rewritten by having the operator A,. + qi acting on the function g(r,r’) which gives a b-function &r-r') and the integral reduces to : -4~

s

d3rjp(rj)[1 + G(ri, rj)]

s

d3r”fL(r-rj,

r"-rj)cp(r"; rj).

(28)

For the same reasons as above Iv-ril < a and Ir-rj( < a lead to Iri-rjlc 2a for which [l + G(r,, rj)]= 0 and hence the contribution offr vanishes. In the contribution depending onfR, Jr”- rjl has to be c a otherwisef, = 0. But when this is realised (& + qb)cp(r('; rj)= 0. It has thus been shown that cp(r; ri)does not depend on fL nor on fR,i.e. of the off-shell properties off when lr - rilc a. Since the nuclear amplitude which arises from the short-ranged part of the Hamiltonian involves cp(r;ri)only for jr- ril< a, its independence of the off-shell properties off is thus proved. In a similar way it may be shown to be independent of the off-shell properties of h, by writing the expression h(r, r’) = h,(r, r’) + (A, + q&Jr,

r’) + (A,, + E&(r,

r’),

(29)

where h,(r, r’) is the point-like form used in (6). It is important to note that in practice this independence is achieved by the renormalization of the effective Hamiltonian. Each off-shell form implies a different renormalization which compensates the difference in the distorted pionic wave functions.

3. The effective field correction for the long-ranged photoelectric interaction It has been shown in the previous section that the part of the nuclear amplitude which is short-ranged is independent of off-shell effects. However this is not the case for the photoelectric term for which the incident pion interacts at a large distance from the nucleon, i.e. in general outside the correlation hole. In this case the nuclear amplitude does depend on the off-shell properties. The nuclear amplitude corresponding to the elementary amplitude (5) is

J. DELORME

et al. e-m,lr-ril

d3rd3riewik”cp(r; ri)[~ri(ri)

VJ[E . Kl 47+. _ r.1 I

s

(30) x

((8

’ V,X~fi(ri)

’ VA-

i(@fi(ri)

’ kh * V,))cpk

rib

In order to calculate the nuclear amplitude we need the effective pion wave q(r ; ri) not only at the nucleon position r = ri as previously, but also for r # ri.

We shall give approximate expressions for cp(r; ri) in the particular case of a squarewell correlation function G(r, r’) = - tl[< - Ir - r' I]. Since cp(r; vi) depends on the off-shell behavior its expression will be given for both the Kisslinger and the local interactions. For the Kisslinger one the effective field is given by cp(r; ri) = $(r)-

d3r’G(ri, r’)(q(r’)@“(r’)-

&

a(r’)lP(r’)

. V,}g(r, r’).

(31)

s The structure of the expression of q(r ; Vi) and of its derivative is rather complicated and we give them in appendix A. The resulting expression for the photoelectric transition amplitude is

(%*)fi = f i

e-m&-r,l

J&e,

d3rd3rie-

7

’ vr][e * V,] 4n,r _ r , t

ik’rd4r)[flfi(ri)

d3rie- ik”‘$JrJsB{(D$’ (ri) - V,Vp+(rJ)-

+

ik,(E:jb’(rJ- V,$(ri))}9

,

(32)

I

s where 3 = k

=p

- m,r dsre-ik’r P&z e r s 1

m,2+lk12

- I) cos k<+ 7 sin k<

,

The first part involves only the average wave t,b(r), i.e. it is the amplitude in the usual impulse approximation. The second part represents therefore the effective field correction to the impulse approximation and it contains the renormalization effect. However this effect cannot be simply accounted for by a multiplicative factor of the nucleonic amplitude. The importance of the range can be illustrated from the results of eq. (32). The pion mass m, fixes the range of the photoelectric interaction. In the long-range limit m,< 4 1 (which also implies kt 4 1) the quantity 9 --t 0. The nuclear matrix element is given by the usual impulse approximation, i.e. there is no need to renormalize the effective Hamiltonian as expected. In the short-range limit we have

RADIATIVE PION CAPTURE

505

instead m, 5 9 1, Y --) l/(m,2 +k2) and we obtain results similar to those of sect. 2 for the effective field corrections. In practice it is reasonable to assume m, 5 < 1, and one is close to the first situation. The photoelectric part of the effective Hamiltonian should not be renormalized by the effective field correction. However the photoelectric term is interesting in another respect. The nuclear photoelectric amplitude involves the average wave function which depends strongly on the off-shell behaviour of the rcN amplitude. For the local potential, for instance, the average wave function differs from the Kisslinger one by a multiplicative factor 4, 1 +$x x 0.5. An experimental study of the photoelectric term may therefore provide some information about the off-shell extrapolation of the aN amplitude. 4. Meson exchange corrections We have seen that the processes where the incident pion scatters incoherently on one nucleon before it is radiatively absorbed on another nucleon lead to a renormalization of the effective Hamiltonian. For the Kisslinger potential and in the short-range correlation limit it has been shown that this renormalization only affects those parts of the effective Hamiltonian which depend on the pion momentum. For pions at rest, the Hamiltonian reduces to the electric dipole, which is not modified (the Pauli correlations introduce a small correction which has been investigated by Ericson and Rho ‘)): = 1 iA,

I,,, However the soft pion approach pions at rest is instead ‘) Zen(r)

bi

.

e6(r - ri).

(33)

gives a different result. The Hamiltonian

= T S c’ * A,(r)+ O(mJm,),

for

(34)

f,

where f, is the pion decay constant (the T sign applies to rc* annihilation). It involves the nuclear axial current (plus small corrections of order m,/mN). It is only in the single-nucleon approximation where the axial current is taken as a sum of singlenucleon contributions that the two approaches coincide. For the Kroll-Ruderman theorem in the nucleon case gives A,--+&!?! - 2m, Then the effective Hamiltonian

*Ad

+ O(m,/md = L e, +t + O(mJmN). x of the impulse approximation

lefr1i

= + L

bi

*

aS(r - ri),

(33) becomes

(35)

506

J. DELORME et al.

which is nothing else than the single-nucleon approximation of the soft pion Hamiltonian. However the description of the axial current as a sum of nucleoni~ ~nt~butions is not complete. The many-body contributions to the axial current may be important g). This is an incentive to look for additional types of many-body contributions which are not contained in the effective field corrections so as to make the microscopic approach and the soft pion one consistent. It is very natural to introduce meson exchange effects of the kind extensively studied for the reno~al~tion of electromagnetic and weak vertices lo) : the pion and the photon can interact with a nucleon which is exchanging mesons with a neighbouring nucleon. These additional two-nucleon diagrams are of the following type (fig. 2). The heavy line represents all the objects that can be exchanged between the two nucleons. Here it will be assumed that the lightest exchange, i.e. pion exchange, gives the main contribution. At the upper vertex the incoming pion is producing a photon plus a pion which is subsequently absorbed by the second nucleon. The reason why these diagrams fulfill our aim of unifying the two descriptions will become clear in the following. 9

k

9

k

Q

i

--+i Fig. 2. Meson exchange diagrams.

k

Fig. 3. Effectwe field correction.

For comparison we have drawn (fig. 3) a two-body graph representing the effective field correction {only the incoherent part of this graph has actually been considered). One clearly sees that the meson exchange processes of fig. 2 are not contained in the effective field renormalization and have to be explicitly taken into account. The upper vertex of fig, 2, (NP2~krrPlNp, 7~~)= J%‘, for production of a photon and a pion by the incoming pion is evaluated in the soft pion method. This implies that the meson exchange corrections are not treated properly to order q, nor as will be seen later are those of order k. However we have kept systemati~lly the contributions of order qJm, or k/m,. Taking the incident pion momentum to be zero 4 = 0 we make an expansion in its energy q. = m,, keeping only the leading terms. The procedure is analogous to the one used in the ordinary radiative capture and we refer to the work of Ericson and Rho l), where a detailed derivation is given. The presence of an additional pion in the final state is the only difference from the radiative capture. The zero-order term in m, is ~j(e~/~=)(N~~~~l&~’ A,(N). Mention should be made of the possible contributions of the Born terms. Indeed, in

RADIATIVE

PION

507

CAPTURE

addition to the previous expression there may be contribution of zeroth order contained in the quantity iq~(N,,npyk~A,JNpl), which appears also in the PCAC prescription for the amplitude (see Ericson and Rho I)). These arise from the Born terms where the axial current is attached to a nucleon line (fig. 4a). k

k

_--p /A I

Fig. 4. Born terms

for the amplitude

(NqlA,IN).

The black current.

circle represents

the action

of the axial

In spite of the multiplicative factor qp these terms may contribute because the Born terms produce a singularity in l/q. However as in the radiative capture it turns out that their contribution is of order mn/mN [Ericson and Rho ‘)I and they will be omitted. The Born term where the axial current is attached to a pion line, though of order k-q, will be kept because it is enhanced by the smallness of the pion mass in the propagator (fig. 4b). Its contribution to the amplitude & is

E’. (k-q),

To

kc (k-q)Lm; where T, is the xN scattering amplitude is written therefore as A’=

T ~(N,,a,l.Y~A,IN

.A,

(npNPZ IN,, rrk_,J. The amplitude

P’

)+e -

6”. (k-q), r(/+_q)?._m,Z

J?

T

X*

(37)

An important conclusion may already be drawn from this result. The fact that this amplitude becomes proportional to the axial current matrix element in the soft pion limit shows that the two-body diagram of fig. 2 represents the meson exchange correction to the axial current, thus proving the equivalence between the microscopic and the soft pion descriptions. The weak pion production matrix element (NnlA,lN) on the r.h.s. of eq. (37) can be calculated, in the limit of low momentum transfer, by the Adler and Dothan theorem ’ ‘). We write :


= W~l~,W),,,

+
(38)

whereWC IA, INhornis the truncated amplitude defined in ref. ‘l). It contains the Born amplitude where the axial current is attached to a positive energy nucleon line. In the static limit this part does not give rise to a genuine two-body contribution

508

et al.

J. DELORME

since the exchange of pions is already contained in the nuclear potential which generates the wave function. We retain only the Born term where the axial current is attached to a pionic line. The remaining part (NxIA,IN) is given by Adler and Dothan ’ ‘),

where rc is the momentum transfer, IC= k-q, and the bar on T, means that the Born part has been removed. Thus putting everything together, we obtain

For the off-shell EN scattering amplitude we take the Kisslinger form’ r, = - 4rrclc. p, which gives (41) We shall first discuss the contribution of the first term of 2, T e, 4x2~ - p. In the static approximation where the nucleons are infinitely massive (pi),, = mN and the energy of the exchanged pion in the diagram (fig. 2) is p0 = 0. The twobody operator arising from this diagram is in the non-relativistic limit

Ojj = I!I

ierg, 3 _

2m

s

4nZe -i(k-q).r,

_

d3P

eip.

x

(8* PWai . P)

(W3

N

IpI +d (42)

where rj is the position of nucleon j, (TVthe spin of nucleon i and x is the relative distance between the two nucleons. This two-body operator leads to a transition matrix element : (~fi)twonucl

=

T

s

ie,g,J2_ 7 N

c

d3rid3rj[1 +G(r,,

rj)]e-ik’Y-$(i)(rj)

e-m&-~Jl

x P(rj)Cafi(ri) . vr,lCE ’ “r,l

Ir.

_

rjl

*

I

(43)

One would like to bring this expression into a form which can be compared to the single-nucleon expression so as to display the renormalization of the effective Hamiltonian. In order to do so, one has to perform the integration over rj. This can be done in specific cases: + This simpli~ed expression A? is given in appendix C.

is given so as to illustrate

our way of reasoning.

The exact expression

for

RADIATIVE

PION CAPTURE

509

(i) In the limit (uninteresting physically) of small nuclear radii mrrR < 1 the twonucleon contribution of order rni R2 vanishes. (ii) When ri is located at the center of the nucleus which is taken to have a constant density : p(r) = p. B(R - r). The corresponding results are given in appendix B. (iii) When p(rJ remains constant in a region around rj which exceeds the range of the Yukawa function so that we can replace p(rj) by p(rJ in the integrand of eq. (43). This assumption restricts the results to nuclei where the density is slowly varying over the range rn; ‘, i.e. to rather large nuclei. In this case

kg, (xfiLnucl

=

+

f

7

s

J2

d3rie-ik”$t(‘)(ri)

f!$ a,,(ri)

*8

N

d3rid3rje-ik”~$“‘(rj)E(rj)[a,i(ri)~

-m*lr,-r,l

V,j][~. V,J e

. Iri

-

ew

rjl

The first part of this expression arises from the correlation function and it renormalizes the electric dipole part of the effective Hamiltonian, i.e. it multiplies A, by the factor 1 +$(ri). The renormalization factor is the same as for the axial coupling constant 9), as expected. This correction, though formally resembling the effective field correction, is however well distinct since it involves an off-shell extrapolation of the 7rnNamplitudes, as discussed in appendix C, which is not required in the effective field correction. This result has been obtained in the limit of zero-range correlations. The Padi correlations, which are long-ranged modify this result, their effect is discussed in appendix C. The second part has the character of a photoelectric term (cf. eq. (30)) though the virtual pion is exchanged between two nucleons in the nucleus and not between a nucleon and the pion-photon system (see however appendix B). Neglecting in the two-nucleon expression the difference between the average and the effective field, the apparent photoelectric contribution is equal to the genuine one but for a multiplicative factor ii(r). Thus it renormalizes the photoelectric part of the effective Hamiltonian by a factor l+&?(r). (48) These two renormalization factors represent the corrections in large nuclei. The rate at which they grow from zero to these limiting values with increasing nuclear radius R can be inferred from the expressions of appendix B. Here we give the renormalization coefficient at a point ri located at the center of the nucleus for all values of the nuclear radius R. These results suggest that in light nuclei such as 6Li the meson exchange corrections can be neglected. The remaining part of the weak production amplitude (the term with the pion propagator in eq. (41)) gives a two-nucleon contribution which modifies the photoelectric term :

510

(~ndtwonucl

J. DELORME et al.

s

erg,J2 -

=Ti 4nm,’

d3rd3rid3rje -ik’r~(r; rj)p(rj)[l + G(r,, rj)]

Instead of calculating this contribution explicitly we shall show that it represents the effect of the scattering of the virtual pion in the photoelectric term. In the main photoelectric term the emitted pion propagates freely from yi to r. Here it has scattered at the point rj, i.e. e-m,lr-r,l @fi(‘i)

. ‘rx

iy _ yil

9

has been replaced by e-m&I-rJI CT;, %(ri)

. V,,

Iri - rjl

e-m&-r,l v’J

Ir-rl

(46)

’

The modification of the virtual pion field around a nucleon by the scattering on the other nucleons has been investigated by Ericson et al. 9). They find that the pionic field which is

in free space becomes

(47) in nuclear matter with a moditied mass such that mi2 = mi/( 1 + a). (This expression is valid only outside the correlation hole). With this modified field the photoelectric term has to be written as

v%Sr (l+@)

(xk)fi = f i mN __ where the In order expression transform

s

d3rd3rie +‘rp(r;

e-m’&-r,l

ri)[%(rJ * V,][a a VT]

,r_r,l I

,

meson exchange corrections (42) are included. to discuss the quantitative modification with respect to the single-nucleon (30) we note that the photoelectric amplitude involves the Fourier of the pion field at the momentum k-q. This is

RADIATIVE

PION CAPTURE

511

in free space and

in nuclear matter. Since a typical value is CEx - 1 and lk - q I2 z rnz the nuclear matter value exceeds the free space value by about 40 %. With the apparent photoelectric term which was derived previously, adding a contribution $E w -0.5, the resulting modification should not represent more than a 10 % correction. To summarize this section we have shown that the meson exchange corrections renormalize the elementary nucleonic amplitude. In large nuclei the electric dipole amplitude is modified by a factor 1 +$ which represents a decrease of w 30 %. The renormalization of the photoelectric term has a more complex nature, part of it arises from the modification of the virtual pion field but another part is derived from an elementary interaction which is basically an electric dipole. However the global modification should be small. In light nuclei the meson exchange corrections are not expected to play a significant role. 5. Conclusion

We have investigated the radiative capture of slow pions in nuclei. Our aim was to improve the impulse approximation description with the inclusion of the renormalization of the elementary nucleonic amplitude due to the presence of surrounding nucleons. Two kinds of effects were taken into account. The first one is the incoherent rescattering of the incoming pion before the absorption. It is described by the effective field correction which depends on the range on the elementary interaction. For the short-ranged part of this interaction the distinction between the effective and average field is essential and in this way the corresponding nuclear amplitude becomes independent of the off-shell behaviour of the elementary interaction. For the long-ranged photoelectric term, in contrast the distinction is not needed. This implies that the corresponding nuclear amplitudes does depend on the off-shell behaviour, which may be in itself an interesting fact. The second kind of renormalization comes from the meson exchange corrections where the pion and the photon interact with a nucleon which exchanges pions with the neighbouring nucleons. This type of correction is size-dependent. Though much more complicated in nature than the effective field correction, it may be ignored in light nuclei such as 6Li, since it vanishes for small nuclear radii. As mentioned, our method does not include all many-body corrections in a systematic way. However the fact that it is equivalent to the soft pion theory justifies it a posteriori for pions at rest. In general we have include-d the corrections which are related to the strong interaction of pions, real or virtual, with nucleons as is clear

J. DELORME ef al.

512

from the fact that all our corrections involve the aN scattering volume. This justifies our belief that we have included the main corrections. Experiments on the radiative capture of pions are interesting in many respects: in light nuclei where the meson exchange corrections may be ignored, the photoelectric term may open possibilities for exploring off-shell effects. In heavier nuclei evidence for many-body effects of the type mentioned would display the influence of the short-ranged correlations both on the scattering of pions and on the expression of the axial current.

A.1. USEFUL FORMULAE

In the first part of this appendix we list some mathematical formulae which are useful for the evaluation of the photoelectric contribution. We introduce the tensors T(n) = 1,

(A-1)

T,(n) = n,,

(A-2)

T,,m> = R&--5 lS q?>

(A.31

T,,,tn) = n,n,n,-3&~,,+

up%,+ +&J,

(A.41

T,,,,(n) = n,n,n,n,-3{n,n,s,,+n,n,s,,+n,n,s,, fn,n,6,,fn,n,6,,+n,n,S,,}+~5(6,,Byd+Sprdgpy+S.yBpd),

(A.51

where n is a unit vector, n2 = 1. These tensors have the property dl;2,e- ik’rX,. . .,,(P) = 47r(- i~~~~(~~)~~. . ..,(I& s

(A.61

RADIATIVE

PION CAPTURE

513

where

II=;;,

n =

r2h2+3h.

In the calculation of the outer parts in the nuclear photoelectric we meet integrals of the form

s

eemp 1 d3re-ik’+7&. ..,,(?) -4nr ~&---G

(AS) amplitude (30)

(A.12)

= (-VT,,....,

For short-ranged interactions (mt % 1) these contributions are proportional to em”‘
(A.13) where the integral has been performed by expanding the Bessel function. The integral 1, is performed by inserting the Poisson representation jl(Z) = z1&Re .

‘du(l - u2)‘eizU,

(A. 14)

s0

so that

4

=& ’

s 1

du(l-u2Y

m2

0

+mk2u2 ,

(ASS)

i.e. an elementary integral. In particular f, = &

1

fm2 +

I3

=

--&

A.2. THE EFFECTIVE

fm2 +

(m2+k2~~~~g~

-mk

k2)2 arctg z -$zk(3m2

,

(A.16)

+-5k2) ,

k2)3 arctg $ - rn~~k~ +$k2m2 -I-m4)

(A.17)

.

(A.18)

FIELD

We want to derive the expression for the effective field and its derivatives from eq. (31). The quantities @l)(r) and E(‘)(r) are slowly varying over a correlation length and we expand them around the point Fi

1. DELORME et al.

514

q(r+ri;

ri) = Y(rfrJ-

i

s

d’r’G(ri, r’){q(r’)ll/“)(r’)-a(r’)E(‘)(r’)

* V,}g(r+ri, r’) * V,)p(r)

= Y(r+ri)-t-(q(ri)$~i)(ri)-a(ri)E(l)(ri)

(A.19)

- V~(~(ri)E~“(~i))V~ Va44, where V’ denotes a gradient with respect to ri and d?‘g(r, r’)G(r’) = e(r-r){i5’-t2i+o(r-r)~,

p(r) = - -&

(A.20)

s

VA(r) = - &

d3r’g(r, r’)G(r’)r’ = V[c;Yc- r)(&<‘r’ --&r4ir*> + 0(r-&$~5/r]. s

One should note the existence of an irregular part outside the hole which arises from the wave that would be scattered if the correlation hole would be filled. When 1r 1 becomes large, both quantities p( 1r I) and Vl.( Ir I) vanish and the effective wave becomes equal to the average wave. Any interaction which takes place at large distances from the nucleon is therefore not submitted to the effective field renormalization. This is approximately the case for the one pion exchange interaction. There remains to calculate the derivatives of rp(r, pi). For this it is most convenient to use the systematic method given in the first part of this appendix. In the tensor notation introduced there we obtain V,q(r + Vi; Vi) = V, Y(r f rJ+ 0(5 - r)[$(YJE~‘)(rJ +,(A +

+ e(r- 5)

-~q(r~~‘~‘(ri)r~

$

+ q2)11/(‘)(ri)

3VC(a(ri)E~“(ri))(ry48 + raa,, - 3,6 rdl

- ~(r~~E~)(rj)~~(~~5

’

t5 -vl(a(r.)E~‘)(~~))?“a~~(P) Y r4

V,V,&

+ ri; ri) = V,V, Y(r+

rJ+e(<-

1 ’

(A.21)

r)[ -~~s,(a’+42)~“)(ri)+$(V~(M(rI)E~)(ri)

+ ~~(ff(ri)E~“(r~))- @s,,~(~(ri) *E”‘(ri))f] + @r- 4

[

driW’(rJT,,W

$ + 4ri#YriKp,fW

5

f V’,(~(ri)EP’(ri)Tapy4(P)7 5

1.

(A.22)

In spite of their complicated structure these formulae display the fact that outside the correlation hole rp z Y, V,u, z V, Y and V,V, rp z V,V, Y ; thus cp and its

RADIATIVE

PION

CAPTURE

515

derivatives depend on the shell behaviour. While inside the correlation hole cpx t,P, V,cp z EA” and similarly V,V,(p x D !$I; thus cpand its derivatives are independent of the off-shell behaviour. [The terms retained in expansions (A.21) and (A.22) are the minimum number of terms required in order to reproduce correctly the zero-order terms in eq. (32).] The calculation of the nuclear matrix element from expressions (A.21) and (A.22) presents no particular difficulty. Since our aim is to discuss general features we shall make a simplified evaluation which exhibits the effects of the finite range and we keep only the leading correction terms in m,< and kr. These corrections arise from the interior region, the differences between cp(r, vi) and e(r) and between Vq(r, ri) and V@(r) outside the correlation hole giving negligible contributions. Inserting the expressions (A.21) and (A.22) into eq. (30) we obtain eq. (32) given in the text. Since the photoelectric amplitude depends on the off-shell behaviour of the pionnucleon amplitude it is instructive to repeat the calculation for the local pionnucleon amplitude4). We denote here by a label NL (L) the quantities calculated with the non-local (local) potential. fL(r’, r) = (b + cq’ +*c(V,, + V,)“}i@‘)d(r)

= &(r’,

r) + &A, + 42)&r’)&r) +%A,, + s2P(r’P(r).

(A.23)

Now, eq. (A.19) for the effective wave must be replaced by

(Pt.@+ ri; ri) = tiL(r + ri) +

d3r’g(r + ri, r’) s

d3r’fL(r’ - rj, r” - rj)(pL(r”, rj).

d3rjp(rj)G(ri, rj) s

(A.24)

s

Since the effective wave cpL(r ; ri) is a free wave inside the correlation hole it follows that the term proportional to (&+q2) in eq. (A.23) disappears. The term proportional to (A,, + q2) can be integrated directly with the result: (PLO.+ ri; ri) = eL(r + vi) + $ct(r + ri)$(‘)(r + rJG(r) + 6,,(r,

where a,, is the correlation correction calculated previously interaction. In the limit r --f 0, we obtain the result k(r)

= (1 +Mr))P(r)

ri),

(A.29

for the Kisslinger

= (1 +MWNL(r),

(A.26)

i.e., the average wave functions for the local and non-local potentials differ by a factor 1 + h(r). Appendix B MESON

EXCHANGE

CORRECTIONS

We want to perform the integration

over rj of the two-nucleon

amplitude (43)

516

J. DELORME et al.

when ri is located in the center of the nucleus. We take the nucleus to have a square well density p(r) = po6[R- lrl]. We denote by q the local pion momentum t,6(rj) st Ifr(ri)ei*‘(‘~-‘J).We use the identity b’s > e - m+z

-i-m$--x

-2_+ T7tx x

(A.271

Performing the integration over pi the integrand in the eq. (43) becomes

11

(A.28)

where K is the local momentum transfer XC= k - q. This expression is valid only in the central region of the nucleus ri w 0. It has two parts : One contains a&J * E and therefore it renormalizes the electric dipole part of the effective ~~iltonian by an amount which depends on the nuclear radius. The second fiart contains (G&) *rc)(rc~E~/(K~+mz) and it has the character of a photoelectric term. It is interesting to remark that the pion pole appearing in both parts is not a true pole. Indeed, extrapolating eq. (A.28) to K = im, one finds the residue to be zero. The renormalization coefficients for the electric dipole and the photoelectric part of the effective ~amiltonian are, respectively,

and . (A.301 Their limiting values are 1 for R + 0 and for R -+ co 1 -I-$%(rJ and 1 +$@ri), respeu tively.

RADIATIVE PION CAPTURE

517

Appendix C WEAK PRODUCTION

AMPLITUDE

AND EFFECT OF THE PAUL1 CORRELATIONS

Cf. Weak ~~o~ct~on ~rnp~~~~e. With the following expression for the EN scattering amplitude c@, where a and B are the isospin indices of the initial and tinal pions, K and p their momenta and o the usual energy variable,

+ [br +cr K * p+ id,@ - (P x @]3[z,, Zal>,

(A-31)

I

(A-32)

the quantity (a/&) T, is

$[T,,z~] ,

+ 1

where mN is the nucleon mass and P and P’ the initial and final nucleon momenta. All the amplitudes have to be taken at o = 0 and rc2 = 0. We have made use in this expression of the crossing relations between the amplitudes. The quantity b; = (d&/do),,, is, according to the relations of Weinberg, b; = - 1/47cJlfwheref, is the decay constant of the pion (this formula is strictly valid forp’ = 0); &’ = --___ s,“(OI 1 8nmi

1 47Tf, *

On the other hand the consistency condition of Adler gives

For the extrapolation of the other ~plitudes lations of Chew et al. 12):

gco = 0) =

1 +2-

Qf2K2(0) [

ct,(o=

we have used the dispersion re-

0) = -$LT&

Or

=

)

mN1

O),

wheref2 is the aN coupling constant cf” = 0.08), K(0) the form factor of the nN vertex and w,, the value of o at the 3-3 resonance.

518

J. DELORME

et al.

Numerically, we find : Gm+

J&(O) G =

0.15,

N

Z,(O) = -0.035, ~~(0)/2~~ =

0.0036.

These values are consistent with the ones of ref. lo) which have been obtained with other extrapolation methods. These numbers lead to a value a = -4np(E,+ &,/4mi) x - 0.9 in eq. (44). The small term in S; can be neglected and the contribution of the term in 2, averages to zero in nuclear matter in the limit of short-range correlations. C.2. effect of the Padi c~rre~~t~~~. The hypothesis of zero range for the correlation function does not apply to the Pauli correlations. The effect of the twobody operator on the Pauli correlation function does not reduce to that of the Sfunction (see eq. (A.27) in appendix B): the Yukawa part also plays a role. We shall perform an explicit calculation in the free Fermi-gas model. We consider the transition i + f for rc- production. Here the nuclear states i (f) consist of a core plus a valence neutron (proton). The initial and final momenta of the valence nucleon are denoted pi and pr = pi + lit. The additionai contribution of the Pauli correlations to the expression (44) is

where pp and P,, are the proton and neutron densities of the core p = pp + P,, . The exchange integrals Z and F are defined by (A.34) The integration is performed over the momentum of the neutrons of the core, from zero to their Fermi momentum (Pr),,. In the expression for Ii,, the proton Fermi moments @r&, replaces (Pr),, and the integration is performed on the proton moments of the core. A similar definition holds for Fp,n with the final momentum pf replacing the initial one. The integrals with a prime differ in the expression for the integrand. For instance,

r:,=-L 4&J&

s

d3k * ’ (It:-PJE. (P-PJmd. E(k-Pi)2 ~~(k-_pi)2+~~ ’ n

and similarly for expressions for q, FL and Fi.

(A.35)

RADIATIVE

PION CAPTURE

519

In the limit 1~1-+ lprl and for an isospin syrmnetric core, @F)n=@F)p, a,,,ribecomes APauli

=

471[-~cCgP(I-~))-dlP(z’+~)l

>

,

xlog

(A.36)

( >I l+-

4Pi

.

4

Numeri~lly

with Pr = 1.95 pn, : Z = +a~~xO.125+(a~~~)(~~~~)xO.425, I’ = - 0 - e x 0.677 + (a . j$)(S

. fii) x

0.425.

(A.37)

The contribution of the terms in Zand Z’ depending on the nucleon momenta depends on the transition i -+ f. The terms in d +E produce a reno~~i~tion of the same type as was discussed previously. The additional reno~alization introduced by the Pauli correlation is therefore

’ + &auli

=

1+4+.104(E,+

&)

+o.oMJ.

(A.38)

With the values given previously for &, 6e and & this quantity is 1+4zp x 0.015 to be compared to l -47rp x 0.05 introduced by the short-range correlations. It should be noted that, in the low density limit (Pr + 0), the exchange integrals Z and I’ vanish like pi. Therefore the overall effect of short-range and Pauli correlations is, in this limit, 47V 50 “0+411;+Q 7 3 ( N> which nearly vanish in the Chew et al. model +. + We thank Dr. J. Blomqvist for suggesting to us the investigation of this limit.

520

J. DELORME

et al.

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