The effect of meson exchange on the forward cross section for d(γ, p)n

Nuclear Physics A365 (1981) 477-504 @ North-Holland Publishing Company

THE EFFECT OF MESON EXCHANGE ON THE FORWARD CROSS SECTION FOR d(r,p)n’F W. JAUS

Institut fiir Theoretische Physik der lJniversit&, Schiinberggasse 9, 8001 Ziirich, SwiherZand and W.S. WQOLCOCK Research School

ofPhysicalSciences,

The Australian National Urtiuersity, Caabema, ACiYWO,

Australia

Received 24 November 1980 (Revised 30 January 1981) Abstract: We have investigated the effect of meson exchange on the theoretical calculation of the cross section for the photodisintegration of the deuteron in the forward direction, in the hope of reducing the present large discrepancy between theory and experiment. Two recent papers reporting a significant reduction in the discrepancy were found to have a sign error; when ~se~dosculur ?rNN coupling is used the effect of one-pion exchange is to increase the discrepancy. We have calculated the one-pion and two-pion exchange effects on the El transitions using pseudovector ?rNN coupling and the resulting correction is small and in the right direction. Thus, assuming the reliability of both theory and experiment, our calculation provides a strong argument in favour of using PV rather than PS ?rNN coupling in calculating meson exchange effects in nuclear processes. We have found that the effect of the exchange of p- and o-mesons is very small. Meson exchange effects change the normalization of the deuteron wave function and cause a further small reduction in the calculated cross section. Since the corrections to the Ml transitions are expected to be very small, it seems unlikely that meson exchange effects can account for the discrepancy between theory and experiment.

1. Introduction The photodisintegration of the deuteron in the forward direction has been measured by Hughes et al. ‘) and a substantial deviation of the measured cross section from the cross section calculated with standard phenomenological potentials has been reported. For photon energies between 20 MeV and 120 IvfeV the theoretical predictions based on the Ramada-Johnston potential, the Reid soft-core potential and the one-boson exchange (OBE) potential of Bryan and Scott are systematically higher than the measured values by about 30 to 40% [ref. ‘>I. In this paper we investigate in some detail whether the inclusion of meson exchange effects reduces this discrepancy. We note first that meson exchange currents are not expected to give an important effect 3, since they affect only the Ml transitions, while the dominant contribution to the cross section for photon energies i’ Work supported in part by the Schweizerische Nationalfonds. 477

478

W. Jam,

W.S. Woolcock / d(y, p)n

between 20 MeV and 70 MeV comes from El transitions. On the other hand, as Foldy “) pointed out a long time ago, meson exchange effects may influence even the long wavelength El matrix elements by producing a two-particle term in the charge density operator. We consider only the El transitions in this paper; we plan to check the importance of the effect of meson exchange currents on the Ml transitions later. Radiative transitions in the limit of zero photon energy are determined exactly but the matrix elements of the commutator [H, D] of the electric dipole operator D and the nuclear hamiltonian H. The operator D should relate only to the excitation of the internal degrees of freedom of the nucleus and is therefore given by L) =

fW

J

XQ(X)d3X,

where x is the position vector relative to the c.m. of the nucleus. The charge density operator p(x) for a two-nucleon system can be expressed as the sum of one-body and two-body terms: p(x)

=Pl(~)+Pz~~x)

-

0.2)

In the long wavelength (zero photon energy) limit, pl(x) is (1.3) where xi is the position vector of nucleon i relative to the c.m. of the nucleus and r,, is the z-component of the isospin vector TV. The result given in the previous paragraph, if used with the charge density for point nucleons given in eq. (1.3), is commonly known as Siegert’s theorem 5). The theorem is modified by meson exchange effects which give rise to the term P&Z) in the charge density operator (1.2). The selection rules for the matrix elements of pz are not the same as those for the matrix elements of pl, with the result that even small meson exchange effects could be considerably enhanced. This is why the charge density associated with the so-called one-pion pair current in the case of pseudoscalar vNN coupling, which vanishes in the static limit and is a relativistic effect of order nC3 (m being the nucleon mass), gives rise to a correction to the cross section for deuteron photodisintegration which has been reported to be quite large 6,7). There is, however, a substantial discrepancy between the numerical resuls of ref. 6, and ref. 7), which led us to reconsider the one-pion exchange (OPE) calculation. We then discovered that the negative correction reported in refs. 677),which was in the right direction to reduce the discrepancy between theory and experiment, should in fact have been positive. We have checked this conclusion carefully and will discuss it in detail in sect. 4. In the work of refs. 6,7)pseudoscalar (I%) GTNNcoupling is used. The main correction then comes from the pion pair current. We argue in sect. 2 in favour of using pseudovector (PV) ?rNN coupling, in which case the one-pion pair current is replaced by the so-called seagull contribution, which arises from the y_NN

W. Jaw, W.S. Woolcock / d(y, p)n

479

contact interaction. The seagull correction is quite different from the one-pion pair current correction. In addition, for both PS and PV aNN coupling there are other OPE effects which we consider and calculate. In this calculation we shall use a different and, we hope, clearer method than the usual transformation approach. We therefore present in this paper a complete calculation of the OPE contribution to the El cross section for low-energy deuteron photodisintegration in the forward direction. We also give the results of a calculation of the contribution of two-pion exchange (TPE) processes to the same cross section. The intermediate states may be nucleons or nucleon isobars (4(1232), . . .). The effect of the exchange of p- and ~-mesons is also estimated and turns out to be very small. Sect. 2 gives the general formalism, sects. 3 and 4 present the details of the two-pion and one-pion exchange corrections, respectively, to the cross section for low-energy deuteron photodisintegration in the forward direction and sect. 5 gives the results and conclusions of our study. 2. General formalism We consider the photodisintegration in the laboratory frame, in which the deuteron is at rest. This enables us to avoid the difficulties of considering the wave function of the deuteron in a frame in which it is not at rest. In fact we are going to calculate the correction to the forward differential cross section due to meson e.xchange as a percentage of the cross section obtained when meson exchange is neglected. We shall therefore take ratios of matrix elements and can work in the most convenient frame. We take the incoming nucleons to have four-momenta pi, ph and define p’= $(P’l -Pi)

*

Also p=p;

3

+p;

where P = (AI, 0),M being the mass of the deuteron. nucleons have four-momenta p;‘, p$, we define p”=;(p;

Similarly, if the outgoing

-p;>.

Moreover, pytp;

where 4 is the four-momentum

=p+q,

of the photon, 4 = (WY4) 3

@=ld*

The matrix element for the absorption of a photon of momentum 4 and polarization E, which produces a transition of the two-nucleon system from the deuteron

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480

W.S. Woolcock / d(y, p)n

state ii) to the continuum state If), is given in terms of the internal current density operator j(x) by T = -(2W)-1’2(f]e * j j(x) ei4’%d3X]i) . The final-state wave function is taken to represent the internal or relative motion only of the proton and neutron, the effect of the overall motion being incorporated into the operator, Following Foldy 4, the matrix element T can be decomposed into an electric matrix element and a magnetic matrix element,

T=T,+T,,,, where

T

e

=

-i&

T,= -i&

(f(e * Pji) , P=

(f[e’ * Mli) ,

(2.1)

xp (x) e iSq’xd3x ,

M=

(2.2)

x X](x) eisq’xd3X,

In the long wavelength limit (o small) the operator P of eq. (2.2) reduces to the electric dipole operator D of eq. (1.1). We are going to work in this limit and will consider the electric matrix element T,of eq. (2.1) with P replaced by D. Now from the decomposition (1.2) of the charge density operator, the electric dipole operator D is the sum of a one-body and a two-body part. Using eq. (1.3), for our two-particle system the former is given by

=~e(Tlz-72z)X12

(2.3)

9

where x12=x1-x2.

Again following Foldy 4), who uses general invariance and symmetry principles, the two-body part D2 of the electric dipole operator can be written as the sum of the eight operators listed below. DI= &I= I&II=

eGdxl2)(n, eGrdx12>(~1,

--T~~)xI~, -~2r)(u1

eG111(~12)(71r -72*)[(%

* u2h2, * x12bz+(u2

* x12)

ml,

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481

W.S. Woolcock / d(r, p)n

DIv = eGw(x~d(nr - 72,)[3(a1 * fduz Dv = Dvr &II DVIII

eGv(xl2)(~1

x 72Mu1+

= eGvdxl2)(~1,

+ 72r)(m

= eGdx12)(n

* 72)(m

= eGv&12)(m

432) x x12 x ~2) X ~2)

x ~2)

x x12

*&) - ~1 * ~21x12 , ,

x x12

Xx12

,

,

(2.4a)

.

In eq. (2.4a), xl2 = 1x12,1 P12=x12/x12. We have followed ref. 4, exactly as far as DvI. Beyond that we have used the charge independence relation given near the end of ref. “) to write only two more invariants. The scalar functions G depend only on the distance between the two nucleons and are determined by the dynamics of the various meson exchange processes. The list (2.4a) exhausts the momentum independent terms. As we shall see in sect. 4, certain OPE contributions to D2 are momentum dependent, so that we need to enlarge the set (2.4a) by adding momentum dependent operators. We now write down just those operators which we shall find it convenient to use in sect. 4. DIX = eGx(X12)i(Tl DX = eGx(xl2)iCn 17x1= eG&di(Tl

X 72b1

- u2d2V,

X ~2k%2~:2V

(2.4a)

,

XT&XI~(UI * ~12~2 * V + (~2 * ~12~1 * 8) ,

where sr2 = 3(Ul * &)(u2

(2.5)

* $12) - Ul * u2

and the gradient operator V involves differentiation with respect to the components of xl2 and acts on the initial state (the deuteron wave function in our case). As we shall see, it is always possible to split the momentum dependent terms which arise in the calculation into the sum of a momentum independent part and a part which involves only differentiation of the initial wave function. The initial wave function is the usual deuteron wave function ki(X) =

@S(x)

+ @D(x)

,

where

--

Xlm,

with ,ylm the spin wave function and Sl2 defined in eq. (2.5). For the matrix elements of the operators DI, . . . , DxI between the initial deuteron S- and D-states and the various possible final scattering states, the angular integrals were calculated in closed form and the radial integrals were evaluated numerically.

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W. Jam,

W.S. Woolcock f d(y, p)n

To obtain the corrections given later, we have calculated the quantity

(2.6) where Te,x and Te,2 are obtained in the long wavelength limit from Dx and D2 respectively via eq. (2.1), with the final state having a proton in the forward direction. The symbol 2 in eq. (2.6) indicates that the appropriate spin sums and averages are taken. The operator L$ is given in eq. (2.3); the two-body operator D2 has contributions from many meson exchange processes which we shall consider in the next two sections. The final corrections are given in tables 1 and 2 as percentages of the El contribution to the theoretically computed cross sections and do not take account of the fairly small contribution of the magnetic transitions and the higher electric transitions?. The general formalism for the discussion of meson exchange effects in a twonucleon system has been given in sect. 2 of ref. *). This formalism is easily adapted to the particular application of this paper. We are interested only in the component A0 of the two-nucleon current AP, which is related to the amplitude M, by eqs. (2.9) and (2.10) of ref. “). In our case M, is obtained from the amplitude for the process ~(~)+N(~P+p’)+N(~P-p’)-*N(~~+:q+p”)+N(~~+~~--p”). To obtain M, for any particular diagram the overall delta function expressing four-momentum conservation is omitted, as is the factor (2rr)-3’2(20)-“2~p. The nucleon spinors are included, but a non-relativistic reduction is performed and the initial and final spin functions are absorbed into the initial and final state wave functions. In this way one arrives at a function A($‘, p’, q) which depends also on the spin vectors ul, u2 and isospin vectors CT~, 12 of the two nucleons. For the impulse approximation {no mesons exchanged) for example we have A’“‘(p”, p’, q) = -e$(ii -e&U

+ ~~=)(2~)3~‘3’~~~

-a’-iq)

+ 72z)(2rr)3S’3’(P”--p’+~q)

.

(2.7)

From i(p”, p’, q) the electric dipole operator in momentum space is determined by the relation J&p”, p’) = -iV,A(p”,

p’, q)14=0

(2.8)

and the matrix element of D [which appears in the expression for T, in equation (2.1)] is (f]Dli) = (2~)~~ 1 m

&(p”, p’)+i(p’) d3p’ d3p” .

The nice thing is that for the small photon energies with which we are concerned the approximation that intermediate baryon states can be treated non-relativisti~lly ?

The El contribution

to the forward cross section at 20 MeV is about 76% of the whole.

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483

W.S. Woolcock / d(y, p)n

is certainly valid [see appendix B of ref. *)I. Then many of the meson exchange processes give rise to a local electric dipole operator in configuration space. For one-boson exchange processes which give a local operator, & (p”, p’) depends on p”, p’ only viap = p” -p’ and then the electric dipole opeator D(X) in configuration space is given by D(X) = (27r)-3 1 G(p) e-“‘* d3p,

(2.9)

where x is the relative coordinate x12 introduced earlier. It is easy to verify that Ai in eq. (2.7) leads via eqs. (2.8) and (2.9) to the one-particle operator D1 of eq. (2.3). The expression for the impulse approximation in eq. (2.7), together with eqs. (2.8) and (2.9) and the resulting operator D1 in eq. (2.3) serve as standard reference expressions. By using them the interested reader can consistently apply the Feynman rules to check all the results given later in this paper. For calculating T,,J in eq. (2.6) we have only the operator Dr with Gr = 4. All the G-functions used in calculating the many contributions to Te,2 must then be correct relative to this choice for the impulse approximation. For two-boson exchange processes each component &(pN, p’) is of the form &p”,

p’) = (27r)-3 C 1 j(p”-k)&(g’-k) I

d3k.

Then, using the techniques in appendix 1 of ref. ‘>, it follows that (2.10)

D,(x) =;fi(M-x), where

1i(p) e-ip’Xd3p,

h(x) = (27r-3

and similarly for gi(x). For those OBE processes which do not lead to a local electric dipole operator in configuration space, it is possible to split ti(p”, p’) into the sum of a term depending only on p and a second term whose rth component d?(p”, p’) is of the form J%(P”, P’) = -&(P)P:. with a sum on the repeated ‘quasilocal’ form

,

index S. Then in configuration

space 0,

has the

(2.11) where d,,(x) = (2~)~~ 1 &(p) e-ip’Xd3p .

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W.S. Woolcock / d(y, p)n

A field theoretic approach to pion exchange effects needs to be based on a realistic model for the pion-nucleon interaction. A simple and consistent phenomenology of low-energy pion-nucleon scattering has been given by Olsson and Osypowski. The effective lagrangian is given in eq. (1) of ref. lo) and uses pseudovector coupling for the rNN interaction. Then gauge invariance requires an additional contact interaction which can be derived from 9 lrNNby the principle of minimal substitution. This gives the term LZYmNN in the effective lagrangian. Olsson and Osypowski lo) show that low-energy pion photoproduction can be described very well when only this contact term is used for ZY7FNN. There is no evidence for another yrNN interaction [the ‘equivalence breaking’ term of ref. ‘l)]. In the calculations reported in this paper we used pseudovector 7rNN coupling, so that in evaluating diagrams the effect of negative energy intermediate nucleon states is unimportant. For completeness we now list the various lagrangians used in calculating exchange processes involving pions, together with the corresponding vertices. We use the conventions of refs. 8*g),wh’rc‘h are exactly those of Bjorken and Drell i2). 2i?TNN

*qinY5ra,

V=fY T

52rNN = (charge only)

-&(a

+ T~)Y*NA@,

V=-ie$(ll+7,)y,,, f

v

=

-iem,

V

=

-e&ab(clh

E3abTbYwY5

+ 40~3~

,

.

In the last case the index a is for the outgoing pion, b for the ingoing. When the nucleon isobar A (1232) appears in intermediate states, we need expressions for the A-propagator and for the yNA and GTNAvertices. These will be given in sect. 4 when they are needed.

3. Two-pion

exchange corrections

We now consider the contribution to the electric dipole operator from the diagrams of fig. 1. The calculation is based on the model for the GTNinteraction discussed above, so that the intermediate nucleons are taken to be in positive energy states. For the box diagram of fig. la one finds that A(P”,

p’, q) = &prf,

the two terms corresponding

p’, 4) +A2(p”,

P’, 4) ,

(3.1)

to the cases where particle 1 and particle 2 respectively

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W.S. Woolcock / d(y, p)n

Fig. 1. The two-pion (TPE) diagrams which contribute to the electric dipole operator.

absorb the photon. Then in the non-relativistic

d ~l(la;p”,p’,q)=-~[~-Tl

3

x (2?+3 xule

limit

- T2+~(~1r+~22)-~(~1r-~22)1 f

or * (p”-k

(p’-k)az.

-$q)az

(p’-k)Jl(la;

* (p”-k-k) kp”,p’,q)d3k,

(3.2)

where f2/4r = 0.08 with sufficient accuracy for our purpose. The expression for A2(la; p”, p’, 4) is obtained from eq. (3.2) by making the changes urf,u2, TI~TZP_, p’_, -p’, p”_, -p”. The function JI(la; k, p”, p’, 4) in eq. (3.2) is given by

Jl(la; k P”, P’, 4) 1

dko ,

(ko-XN+i&)(ko-yN+i&)(ko+XN-iE) x(k;-d2+ie)[(kc.,-S)2-;“2+is]

(3.3)

where w t2=,.r;+k’2, g,=kL$4,

(3R*=,;+k;r*, k”=p”-k,

xN=E(k)-m, E(k)‘=m*+k’,

k’=p’-k,

yN=E(k+4)-m, S=q*p”/2m.

To understand the form of eq. (3.3) one has to note two points. The first is that the propagator for the internal nucleon after the photon is absorbed is different from that for the nucleon before the photon is absorbed, and so we have yN instead of xN in the second factor in the denominator. The second is that the two nucleons in the final state are not in their centre-of-momentum system, so that the correct Blankenbecler-Sugar (BBS) prescription is not p6 = 0 but pZ = $[E($4 +p”) -E($4

-p”)]

==4 - p”/2m

.

(3.4)

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486

Now close the contour

in the upper half -plane

and carry out the kc integration

to

obtain J1(la;

-1 k, P”, P’, 4) = 2XN(XN+yN)[&!r2_ (XN+s)2](*;

_/)

-1 +2o)(w’+y,)(X;

-w’2)[(W’+S)2-(3”2] -1

+2;n(;.+yN-8)[CJ2

-(3”-S)2][(;rr-6)2-X;]’

The individual terms just given are singular, but they can be rearranged explicitly non-singular expression. Making an expansion in the small (xN - yN) and 6, we obtain after a lengthy calculation

Jl(la; k, P”, P’, a) =

to give an quantities

W12+(p

1 2x N (XN+ y&0’2G”2 -

4(J4&M4

We shall defer until sect. 4 discussion of all the terms on the right side of eq. (3.5) except the second since, as we shall see, they need to be considered as part of the OPE contribution. For the moment we take the irreducible part only, W12+(yt2 J&la;,

We have already

noted

k

P”,

P’,

4)

=

-

4w,4G,,4

(3.6)

.

that

&(la;

k, p”, P’, q) =Jl(la;

k, -P”, -P’, q) .

It has the same formal expansion as in eq. (3.5) but now the meanings 3” and 6 are changed. In particular, Jz,&la;

k, P”, P’, 4) =Jr,Ala;

k, -P”,

-P’,

(3.7) of k’, k”, &‘, w’,

9) .

(3.8)

For the crossed diagram of fig. lb, A is again of the form given in eq. (3.1) and the contribution to & is

leading

x (27r>-3

J ul.(p”-

k -+q)a2 . (p”- k -44)

x 01. (p’- k)az *(p’- k)Jx(lb; k, p”, P’, q) d3k,

(3.9)

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487

where

Jl(lb; k P”, P’, 4) = -2Jl,dla;

k P”, P’, 4) .

(3.10)

Moreover, Az(lb; . . .) is obtained from /il(lb; . . .) in the same way as Rz(la; . . .) is obtained from A1(la;. . .>. Collecting all the information contained in eqs. (3.1), (3.2), (3.6) and (3.8)-(3.10) and using eqs. (2.8) and (2.10), we find that the total contribution from the irreducible part of the box diagram and the leading part of the crossed diagram to the electric dipole operator is given in terms of the operators of the set (2.4a), with

where

Yo(x,) =

xZ1em”*,

Yl(X,) = (1 +x3KJ(xd *

(3.12)

A form factor can be introduced at the rNN vertex in the same way as described in ref. *) but, as we shall see, it has very little effect on the electric dipole matrix elements. An analogous calculation of the function Gr(x) has been published in ref. 13) for the exchange of two scalar mesons. In that work the standard method [see ref. ‘“)I for the treatment of meson exchange in a -nucleus was used. This method involves splitting the Feynman diagrams of fig. 1 into time dependent graphs and leads to an enormous increase in the complexity of the calculation. There is a numerical error in the final result of ref. 13),which is probably due to an incorrect reduction of the box diagram. In our formalism this reduction corresponds to the separation of Jdla; . . .> [eq. (3.6)] from J(la; . . .) [eq. (3.91. There are other two-pion exchange contributions which we also considered. In addition to the leading part of the crossed graph there is another part whose contribution is large enough to need to be taken into account. It comes from the /co/m term in the spinor reduction; its form is very complicated and we do not write it here. To complete the two-pion exchange contribution, we also calculated the diagrams in which d (1232) appears instead of N in intermediate states. The total TPE correction to the El forward differential cross section is given in table 2 for photon energies of 20,30 and 40 MeV.

488

W. Jam, W.S. Woolcockj dfy, p)n

Among the various TPE contributions the tensor part Drv of the electric dipole operator allows the transition 3S1to 3P and gives most of the TPE contribution to the El matrix element. The El matrix element was calculated with Reid soft core wave functions i5). One sees from table 2 that the contribution to the electric dipole operator from TPE processes gives a very small reduction of the forward cross section. The reason for this is that the El transition matrix elements for soft photons are rather insensitive to the behaviour of the integrand for small values of the integration variable x, since the p- and f-waves are small there. Wowever, that is the region where TPE operators are most important. For the same reason the inclusion of form factors at the vertices does not modify the matrix elements significantly. 4. One-pion exchange corrections The exchange of one pion between two nucleons gives rise to an electric dipole operator which has been discussed in several papers 16-18).The dominant contribution for pseudoscalar 7rNN coupling is due to the pair current and the resulting dipole operator is

x[(l + ‘d71

’ %)(UI x Uz) XX + (1 f Kv)(~~WI

*X -T22UlU2 ’ X)1,

(4.1)

where 1+Kv=pp-pn=4.70, I+ KS= JQ,+p, = 0.88 and Yi is given in eq. (3.12). In terms of the set of operators (2.4a),

,‘:,2

GUI(X)= Gvdx I= Gvdxl =

2m -& 6

(1 +Kv)&‘%r),

(I+ K&r1 YI(&) .

(4.2)

The operator &Ii gives the dominant contribution; it allows transitions out of the 3S1state of the deuteron to 3P and 3F continuum states, which are enhanced relative to the main transition 3D1 to 3P, 3F because of the smaI1 D-state probability of the deuteron. The result is a substantial increase in the forward El cross section due to the pair current. This conclusion is in direct contradiction to the results reported in refs. 6*7).The calculation of the electric dipole operator given in eq. (4.1) is relatively straightforward and we have checked its sign carefully. Hadjimichael ‘) uses our convention for the electric dipole operator arising from the impulse approximation [eq. (2.3)] but his expression for D(x) arising from the pair current [his eq. (9)] has the u~~~~i#esign to the one we give in eq. (4.1). On the other hand, using eqs. (5), (6)“f and (9) of ref. 6), we find that, for the cont~bution to L)(X) arising from the absorption of the photon t

Eq. 16) of ref. 6, has an error; (m”, + ?c;)-~ should be (&

+k:)-‘.

See ref. “).

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W. Jaw, W.S. Woolcock/ d(y, p)n

on nucleon 1, Gari and Sommer agree with our result [eq. (4.1)] when account is taken of their extra factor -i (which appear also in the impulse approximation). There is thus a contradiction between the results of refs. 6,7) for the electric dipole operator arising from the pion pair current. The other possible source of error in the calculations is in the angular integrations involved in obtaining the matrix elements. For the matrix elements of DnI we have checked the angular and radial integrations very carefully and conclude that the correction is positive. It would appear that HadjimichLael ‘) has used the wrong electric dipole operator but has performed thie integrations correctly, while it is not clear how Gari and Sommer 6, arrive at their result that the correction is negative. In comparing our results with those already published, it is worth commenting on a recent paper by Cambi et al. lsa), wh’ic h considers the effect of the two-body charge and current densities on the deuteron photoabsorption sum rules. The expression for the electric dipole operator due to the pair current (with PS ~TNNcoupling) found by Cambi et al. [their eq. (2)] is exactly the same as we have given in eq. (4.1), and after performing the angular integrations involved in obtaining the matrix elements, we use an expression corresponding to eq. (15) of ref. 18a).Moreover, Cambi et al. find only a small dependence upon the particular model used for the NN interaction, and this is consistent with our result, formulated in the preceding section, that the El transition matrix elements are not sensitive to short-range effects. We shall show in this paper that non-local one-pion exchange terms, which are neglected in all the quoted references, are important. Besides that, we find a large difference between the results obtained with PS and PV rrNN coupling. It would be interesting to perform an analogous investigation of meson exchange corrections to the deuteron photoabsorption sum rules. The big effect in the wrong direction which results from the use of PS rrNN coupling makes it even more imperative to carry through a full calculation of OPE: effects using PV rNN coupling. The first contribution comes from the yrNN contact interaction given at the end of sect. 2, which generates the seagull current; this, replaces the pair current of PS coupling theory. Using our earlier notation, when nucleon 1 absorbs the photon, the function ir(SG; p”, p’, q) is /i&G;

p”, p’, q) =

-& $7;;;yf;;2 UI

*

(p +$q)az . (p”+p’-t-q)

.

(4.3)

2

Recall that ,p = p” -p’ and w (k)2 = rnt + k2. In obtaining eq. (4.3) we used V(yrNN) given at the end of sect. 2. The function A2(SG; . . .) is obtained from /I^,(SG; . . .) by making the now familiar changes (~1eu2, ~~-TV, p” + -p”, p’ = -p’. Then using eq. (2.8) we have

15(sG;pr~,p~)=&

blX72)rrW-2(w2

*P'+uz%

.P'>

?r

-20J-4p(ul ~pu2~p+ul~pa2'p'+u~~p'a2

*pII,

(4.4)

490

w. his,

W.S. Woolcock I d(y, p)n

o2 = m2, +p2. In writing eq. (4.4) we have separated a piece which depends only on p, in such a way that the remaining piece (which gives a non-local electric dipole operator in configuration space) depends only on p and p’, but not on p”. As explained in sect. 2, evaluation of the matrix elements of the electric dipole operator then involves differentiation of the initial (deuteron) wave function only. Note that the isospin dependence of the seagull current electric dipole operator is different from that of the pion pair current operator of the PS theory [eq. (4.1)]. However, both i(~i x ~2)~ and (71~- rzr) have the same action on an isosinglet state, so that in our problem we may replace one by the other. The local part of &SG; p”, p’) in eq. (4.4) will appear again. The dipole operator in momentum space, where

(4.5) leads via eq. (2.9) to an operator in configuration space specified by the functions

GW

The non-local pieces of &SG;

p”, p’) in eq. (4.4) are of the form

with

Using eq. (2.11), the operator in configuration space is given in terms of the operator Dxl of the set (2.4b), with

f” m,

G&x) = - - x ,I 4~ 2m

Ylbhr~

*

For both types of rrNN coupling there are other OPE contributions to the efectric dipole moment, which come from the diagrams of fig. 2. We consider first the diagram of fig. 2c, which gives rise to what is usually called the pion current (&) contribution. To evaluate this, we use the vertex V(ynr) given at the end of sect. 2 and the correct BBS prescription for p6 given in eq. (3.4), namely p6 = 4 *p”f2m.

W. Jam, W.S. Woolcock / d(y, p)n

2a) Fig. 2. The one-pion

exchange

?

(OPE) diagrams

491

c)

b) which contribute

to the electric

dipole operator.

Then ief2

1

A(7rC;p”, p’, 4) =-y

mn

(T1

xT2)rp2u1*y

h

x 72)r

*p,

and so, from eq. (2.8),

&rC;p”,p’)=

ef’

---y mm,

P +p’ 4Ul u

(4.8)

*puz*p.

Inspection of eqs. (4.4) and (4.8) shows that in fact the local pieces of &SG; B(7rC;. . .> exactly cancelt. For the non-local piece we have

. . .>and

so that, using eq. (2.1 l), the operator in configuration space is given in terms of the operators of the set (2.4b), with

(4.9)

We come now to the contributions of the diagrams in figs. 2a, b. In order to obtain the correct result for the remainder of the OPE contribution we must exercise very great care, and we shall try to make the argument as transparent as possible. In evaluating the contribution of the diagrams in figs. 2a, b the intermediate nucleon is taken to be in a positive energy state. If we denote the positive energy part of. the propagator for this intermediate nucleon by G+, then it is not difficult to check that the total contribution from these two diagrams, when the photon is absorbed on nucleon 1, may be written in the form

M:” (p”,g’, 4) = (27r)-3 j [?*‘(,“,

k)G+(k, dli?(k

P’, 4)

+&“(p”, k, q)&+(k, g) t”“(k, p’)] d3k, 1- We emphasise use the convention

(4.10)

that the separation into “local” and “non-local” pieces is an arbitrary one. We always (convenient for calculation) that the “non-local” piece depends only on p and p’.

W. Jam,

492

W.S. Woolcock / d(y, p)n

where u1

’ (&‘-$)a2

* (p”-p’)

-w(p”-pJ)2

*

(4.11)

The exact forms of the propagators G+(k, 4) and C?+(k, q), which belong to the intermediate nucleon in figs. 2a and 2b, respectively (with the diagrams read from right to left), are not required, since they disappear from the final result. The quantity /i?‘(. . .) is just the first term on the right side of eq. (2.7). In writing the amplitude M we use the notation of sect. 2 of ref. “); it is understood that we consider only the component MO of M, and are suppressing the subscript 0. On referring to eq. (2.10b) of ref. ‘), we see that the first-order OPE expression for Ar(p”, p’, 4) is not given by eq. (4.10) but by &‘(p”,

p’, 4) = (2~)-~ ] [?‘)(p”,

k){G+(k, q) - g(k)}&“‘(k, p’, 4)

+&@(p”, k, &+(k,

q) -g(kN~%,

101d3k,

(4.12)

where 4m2

g(k) = E(k)[M2-4E(k)2]

(4.13)

’

the BBS Green function [see eq. (2.2) of ref. “)I. This, however, is not the end of the story. At this stage we need to consider the first, third and fourth pieces of the box diagram kernel in eq. (3.5), for it turns out that they give effective OPE contributions. For the moment we take only the first term. If we had been very careful with our factors, we would have found that this term has the exact form

K&J2A&(km+q) -tx;+ydl ii&y where xN = E(k) -&f, yN = E(k + q) - $‘i!f.The second factor in square brackets may be written, using an approximation which is valid for our purpose, m W+q)

l -(XN+YN)

==L-(AJ2~~

with ZN=E(k+~q)-$f.

Then, using eq. (3.2), the contribution to the box diagram amplitude coming from the first term of eq. (3.5) may be written as M&Y’, p’, 4) = (27#

1 ?‘)(p”,

x g,(k’)e”)(k’,

k”)gr(k”>&“‘(k”, p’) d3k’ d3k”,

k’, q)

(4.14)

W. Jam,

493

W.S. Woolcock / d(y, p)n

where

2 gr(k)= (&

1 M-2E(k)

(4.15)

*

However, to obtain the correct second-order expression we need to subtract from the expression in eq. (4.14) the zeroth-order contribution iterated twice and the first order contribution iterated once, as given in eq. (2.10~) of ref. ‘). The result, written in terms of symbolic matrix products, is given by Ai”’ =

jpgrpgpl

_

z

_

p”l’(G+

_

p’ng&J’(&+

g)&@gc(l’

_

p(l’(g,

g)

pm

~“l’g&o’g~“l’

(4.16)

Q(l)

F”‘)(g, - g)&D’(g, -g)

+

_

_ G+)il”~‘g~(”

+ +l’g&@(g,

- &+) p(l) .

In writing eq. (4.16) we have consistently omitted terms of the type ?*)G+&‘), where CT(*)is the TBE part of the potential. Now from eqs. (4.13) and (4.15), (4.17) where B is the binding energy of the deuteron. Since we are working in the non-relativistic limit, it isthe first term on the r.h.s. of eq. (4.17) which we shall use. The contribution of the first term on the r.h.s. of eq. (4.16) to the electric dipole operator is then down by a factor of order mfJm2 compared with the contribution from J&la; . . .) considered in sect. 3, and is therefore completely negligible. For the second and third terms on the r.h.s. of eq. (4.16), we take matrix elements between the initial and final states. One can then use the BBS equation for the vertex functions [cf. eq. (2.1) of ref. 8>],with the potential approximated by v(l), in the case of the second term for the initial state and in the case of the third for the final state. The result is that the matrix elements of the second and third terms on the r.h.s. of eq. (4.16) are equal to the matrix elements of the operator (2~)-~ 1 [t”‘(p”,

k&(k)

- G+(k, &:“(k,

+/i:o’(p”, k, q){g,(k) - d+(k, q))?‘)(k,

$9 4) p’)] d3k.

On adding this operator to the OPE operator of eq. (4.12) and using eq. (4.17), we see that the resulting effective OPE electric dipole operator due to the processes shown in figs. 2a, b and the first term on the r.h.s. of eq. (3.5), when the photon is absorbed on nucleon 1, is

W. Jags, W.S. Woolcock/ dfy, p)n

494

The operator &(OPE); . . .) is obtained from Ar(OPE; . . .) by making the same changes as before. Adding the two, and using eq. (2.8), we have

&OPE;

pfJ, p’) = -&

II

(71r - 7&)

Finally, from eq. (2.9), the electric dipole operator in ~on~guration space is of the form D&r) + &v(x), with

2 m GnW-f-&

Yob,)

9

(4.19)

We note that the results obtained in the previous paragraph are the same as those obtained by Hyuga and Gari 17) (except that we find effectively only an isovector contribution) as a consequence of the recoil and wave function renormalization processes peculiar to their formalism [see also ref. ‘“)I. It is worth emphasizing that the OPE electric dipole operator obtained from eq. (4.19) is the same for both PS and PV 7rNN coupling. However, it does depend on the choice of the Green function g or, in other words, on the way in which the Bethe-Salpeter equation is reduced to a three-dimensional equation. The derivation which led to eqs. (4.18) and (4.19) shows that the contribution given there is entirely due to the difference between the Green function g#) and the BBS Green function g(k). Consequently it would appear that we could have obtained a zero contribution by using g, instead of g in reducing the four-dimensional Bethe-Salpeter equation to a three-dimensional equation. But we know that, while the BBS reduction scheme leads via g to the Schriidinger equation (2m -M + p2/mM = - V& the corresponding reduction scheme with g replaced by g, leads to the relativistic equation (M - 2E(p))$ = V&. Since we calculate matrix elements in terms of the wave function 4, the correction arising from the electric dipole operator given by eq. (4.19) would then reappear because of the difference between the wave functions (b and J/. Next we consider the contribution arising from the third and fourth terms on the r.h.s. of eq. (3.5). The origin of these terms is the motion of the c.m. of the two nucleons in the final state, which induces a relativistic correction whose form is again that of an effective OPE matrix element, as we shall now discuss. Taking account of

W. Jaus, W.S. Woolcock / d(y,

495

p)n

eq. (3.2) and the definitions after eq. (3.31, the contribution

x (27r>-3

in question is

(ptqc-$q)a~. (f-k) Jal.(P~~-k-$q)u2. (p’-k)az

.

d3k. At the level of approximation on which we are working, - 1/2xN may be replaced in this case by the BBS propagator g, by eq. (4.17). When matrix elements are taken between initial and final states, the approximate BBS equation may be used in exactly the same way as before to replace the operator just given by the effective OPE operator R,(CORR; p”, p’, q): &(CORR;

ef” ~‘1,p’, 4) = m2 Tr

~1

.PU~.P to4

The operator &.(C~RR; . . .) is obtained from If I(CORR; . . .) by the usual changes. Again using eq. (2.8), the ~orrespon~ng electric dipole operator in momentum space is

(4.20) When the initial state is an isosinglet state we can replace i(lr ~1~)~ in eq. (4.20) by (T$=- ~~~1.The piece of &CORR; . . .) which depends only on p is then just 6 of the expression given in eq. (4.5), for which the G-functions are given in eq. (4.6). Moreover, the second piece of &CORR; . . .>, which involves p”, is just 8 of I%(&; . . .) given in eq. (4.8). Thus we have already given all the results necessary for the calculation of the matrix elements of D(CORR) in con~guration space. We now collect together the results we have obtained. The total OPE contribution to the electric dipole operator from diagrams in which only the nucleon occurs in intermediate states has been written as the sum of four pieces. The first is the seagull (SG) piece, which has a “local” part whose form in configuration space is given by the G-functions of eq. (4.6) and a “non-local” part which is specified via the G-functions of eq. (4.7). Next there is the pion current (T&) piece, whose “local” part is exactly minus the “local” part of the SG piece and whose “non-local” part is specified via the G-functions of eq. (4.9). The third piece is local; we have called it the one-pion

W. Jam,

496

W.S. Woolcock / d(y, p)n

exchange (OPE) piece, and it is specified via the G-functions of eq. (4.19). Finally there is the relativistic correction (CORR), which has a “local” part which is 2 of the “local” part of the SG piece and a “non-local” part which is - $ of the non-local part of the TC piece. Next we consider the contribution to the electric dipole operator from the OPE diagrams of figs. 2a, b, with the intermediate state a nucleon isobar. The main contribution comes from A(1232) and we shall discuss it in a moment. The role of higher resonances such as N”(1470) in a model of low energy TN scattering is somewhat uncertain. There is a discussion of this point by Olsson and Osypowski 19). Fortunately, in our case, because of its very small radiative width, the contribution of N*(1470), even if it needs to be included, is negligible. Our estimate of the correction to the El component of the forward cross section is about 0.1% (in magnitude) at a photon energy of 40 MeV. This estimate uses the result [see ref. ““)I that r(N* + Np) = 0.07 r,,, and then uses vector meson dominance to obtain r(N* + Ny). Now we consider the contribution from A(1232) intermediate states. The first remark is that the operator (7rr + 72r) annihilates the deuteron state, while, as we remarked earlier, the operators (rlr - r2=) and i(ll x TV)= have the same action on the deuteron state. From this it follows that the diagram of fig. 2a, in which the photon is absorbed after the pion is exchanged, gives zero matrix elements of the electric dipole operator with an initial deuteron state. It remains to evaluate the diagram of fig. 2b, in which the photon is absorbed before the pion is exchanged. The isospin factor for this diagram, when the photon is absorbed by nucleon 1, is $r2= - $i(q X ~2)~ . For an initial deuteron state, we replace this factor by -$(rlZ - ~2~). When the photon is absorbed by nucleon 2, we make the interchange ?1 tsar, so that the isospin factor changes sign. For the calculation we need the forms of the rNA and yNA vertices and of the A-propagator. For these we use the expressions given by Olsson and Osypowski lo), altered to conform to our metric. The isospin indices will be suppressed. The propagator for an intermediate A with four-momentum PA is i(pz -mf,

+iE)-lAcLv(pA),

with A&“(pA)=+(y ‘~d+mA)[-3gC”“+yC”y”+2md2p~~~+md1(y’”p~-y”~~)]. The yNA

vertex carries the index v associated with the A; for y(~~, q)+N(p)+

A,(q +p) it has the form -iemilfy~A[(%qv-y

- q&>+Y%(Y

* wA-%Y

The TNA

’ q)h.

vertex carries the index p associated with the A; for N(PA + k) it has the form m,‘frnNA

(kp

+

27

’ hw)

.

A,(PA)

+r(k)+

W. Jaus, W.S. Wookock / d(y, p)n

497

For comparison with ref. lo), our constants y, z are related to the Y, 2 found there by JG-y-+

z=-z-’

,

2,

while our fyNdand frNd are called there C and gA,respectively. We shall discuss the values of these constants later. The information given in the previous paragraph enables us to write down the amplitude MO (A = 0). The non-relativistic reduction is performed as usual. Since the amplitude depends linearly on the four-vector q, it is sufficient to take ~0”= 0 in this case, as well as pb = q. = 0. It is also sufficient to take pi = m and to replace pi - rn,: in the denominator of the A-propagator by its static limit m2 - rni. In fact the correct dependence of the denominator of the propagator on pA( “p’) can be derived onky via the corresponding TPE diagram, in exactly the same way as for intermediate nucleon states. It turns out that a good approximation to the p’ dependence of the denominator is to multiply the static limit by the factor

with Ai = $(rni -m*> = (564 MeV)*. This modification has a very small effect on the correction to the cross section arising from the diagram. Adding together the contributions to /i(A; p”, p’, q) from the diagrams in which the photon is absorbed by nucleon 1 and by nucleon 2, we obtain, in the static limit, ef?rNNfyNAfvNA

&W’,P’,~)= m3,(mi

-m2)(m2,

+p*)

iq ~.(p’xp)(~l+~z)

* $(nz

- 72r)

‘P

+=

2mA+@‘mA

(-

+ (a1

* quz

* p +a1

* pa2

* 4NP

X

+(4

+ (a1

x

-+3iy

3m

4

3m

* P’) +2(-g-$y)

-Zfl)

.p)(01.PN32~P)(

* 4u2

’ p + u1*

mA -----

1

m

3m

6

6mA

pa2

* a)(p2)

mA 3m

1 +4YmA --2 3

3m

3y

*

As before, p = p" -p', and we have expressed the result as a function of p and p'. The

W. Jam, WS. Woolcock / d(-y, ph

49%

last two terms give a local operator in config~ation space, while the first three terms give a quasilocal operator involving differentiation of the deuteron wave function. Using eq. (2.8) to calculate &A ; p”,p’) and then eqs. (2.9) and (2.11) to go to config~ation space, we come to the following expression for the electric dipole operator: D

=

(712

-

72r)

(4.21) where

d = -(l-t- m/m,,)-r

,

Eq. (2.5) defines S12, while

Y&,) and YI(x,) being defined in eq. (3.12). The last three terms in eq. (4.21) are local and can be readily expressed in terms of functions Grr, Grrr and Grv. One of the non-local terms can be expressed in terms of GxI, but the other terms involve operators which we did not write down in sect. 2.

W. Jaw, W.S. Woolcock / d(y,

p)n

499

There is substantial uncertainty in the constantsf yNd,f,,.NA, y and z. The analysis of pion photoproduction from nucleons by Olsson and Osypowski lo) yields the values f =NA

=

1.84,

fyNA

=

0.30,

f?rNNfdAfyNA

2 = -0.21

y = -1.28,

=

o-550

, (4.22)

.

These values were used to make one estimate of the correction to the forward cross section. The coupling constants are tightly constrained by the photoproduction fit; the values of y and z are more uncertain. Thus changes in the low energy photoproduction data led to changes in y from - 0.75 f 0.25 to - 1.28 f 0.30 and in z from -0.50~~ 0.25 to -0.21 *O.lO in the analyses of refs. 21S10).Moreover, the analysis of low energy rrN scattering in ref. 19) gave z = -0.05*0.20. If is thus reasonable to say that the values y = -1 and z = 0 lie within the range of y, z values permitted by experiment. There is in fact some theoretical preference for the value z = 0 [see ref. “)I. With this choice of y and z, we can use the values of the coupling constants f=NA and fYNAobtained from the experimental decay rates for the A(1232) resonance to make a second estimate of the correction. For frNA we have the relation r(A++

+p?r+j=f:NA

-

&%+m)

3m$mA

4lr

with EP = (rni +m*- mf,)/2ma, qr = m. The 116 MeV then gives f,,NA = 2.14. For fvNA we have e

r(A++pr)=-

=

width

of

’

From the analysis of pion photoproduc-

= O.O053I’,,, = 0.6148 MeV ,

F(A++py) fyNA

experimental

1 +&,)

%&A

4?l

with k, = (rnz - m2)/2md, E, = my tion from nucleons,

so that

3 8k,(&

22 fvNA

’

0.34. Combining these, we use

f =NA

=

2.14,

fvNA

=

0.34,

y=-1,

f?rNNfnNAfyNA

z=o.

=

0.725

,

(4.23)

Using the expression for I) in eq. (4.21) and the values of the constants given in eqs. (4.22) and (4.23), we calculated the correction to the El forward deuteron photodisintegration cross section arising from one-pion exchange with a A(1232) intermediate state. 5. Results and conclusions

We now give the results of the calculations using the expressions for the electric dipole operator for OPE and TPE processes obtained in sects. 3 and 4. What is

500

W. Jam, W.S. Woolcock / d (y, p)n

calculated is the percentage correction to the El cross section for deuteron photodisintegration in the forward direction, which is obtained from the electric dipole operator as discussed in sect. 2. The corrections were obtained at photon energies of 20,30 and 40 MeV, using Reid soft-core wave functions ls) and the average masses m, = 138.12 MeV, m = 938.82 MeV, mA = 1232 MeV. The first two masses are the values chosen by Reid to calculate the deuteron wave function; they differ slightly from the most up to date masses. Table 1 gives the corrections arising from the four pieces of the electric dipole operator which come from OPE processes with a nucleon intermediate state. These results do not include a form factor at the aNN vertex. When the form factor F(q2) = (At-

m2,)l(X

- q2) ,

At = 72m2,,

is used, there is a very small change in the corrections, since it modifies the radial integrals only at short distances, where the scattering wave functions are very small. It should be noted that the use of such a form factor destroys the gauge invariance of the calculation. This defect may be remedied by the inclusion of a further correction of seagull type which could be calculated. But in view of the extreme smallness of the change due to the form factor, this is not necessary. TABLE 1 OPE corrections to the El forward cross section for pseudovector wNN coupling (expressed as percentages) Ey:

20 MeV

30 MeV

40 MeV

SG TC

-1.32 0.08

-2.01 0.20

-2.71 0.32

OPE CORR

-0.54 -0.04

-0.75 -0.03

-0.94 -0.10

total

-1.74

-2.59

-3.43

Inspection of table 1 shows that the rrC, OPE and CORR corrections nearly cancel and that the main contribution is the SG correction. The net correction reduces the theoretically calculated cross section. This is in sharp contrast to the correction which arises when PS rNN coupling is used. It is then f 6.3%, + 8.2% and + 9.8% at photon energies of 20,30 and 40 MeV respectively. In calculating these numbers we used the results in table 1 but replaced the SG correction by the pair current correction whose G-functions are given in eq. (4.2). For the corrections arising from one-pion exchange with a A(1232) intermediate state, we find, using eq. (4.22), the values - 0.85%) - 1.54% and - 2.40% at photon energies of 20, 30 and 40 MeV, respectively. Using eq. (4.23) the values are - 1.16%, - 2.04% and - 3.13%. We take these two estimates as reasonable lower and upper limits for the correction and therefore adopt as the final result for the

W. Jam, W.S. Woolcock/ d(y, p)n

501

correction the values - l.O%, - 1.79% and - 2.77%, with uncertainties of 0.15%, 0.25% and 0.36%, respectively. To complete the calculation of all the corrections to the El cross section which might be important, we considered processes involving the exchange of p- and o-mesons. We calculated the contributions of diagrams like those of fig. 2a, b with r replaced by p and w, and like that of fig. 2c with one of the pions replaced by p or w. The result is remarkably small. Even with generous changes in the coupling constants, the result for the total OBE correction varies from 0.0% to, 0.1% at 20 MeV, from - 0.1% and 0.0% at 30 MeV and from - 0.2% to - 0.1% at 40 MeV. The extra mass of the heavier exchanged bosons is clearly decisive in giving a very small result. The correction arising from TPE processes (which is given briefly in table 2) is also very small. The smallness of the OBE and TPE effects and the insensitivity of the results to form factors at the vertices are all due to the fact that they affect the two-body electric dipole operator L)z only at short distances, where the scattering wave functions are very small. TABLE

2

Corrections to the El forward cross section for pseudovector PNN coupling (expressed as percentages) 20 MeV

30 MeV

40 MeV

OPE (with N intermediate state) “) OPE (with A intermediate state) OPE TPE

-1.7 -1.0 +0.1 -0.2

-2.6 -1.8 0.0 -0.3

-3.4 -2.8 -0.1 -0.4

Total Estimated uncertainty

-2.8 0.2

-4.7 0.3

-6.7 0.4

E,:

“) OPE correction from table 1.

Table 2 lists all the corrections to the El cross section when pseudovector rrNN coupling is used. The total correction is also given, together with a rough estimate of the uncertainty in theresults. The correction to the whole forward cross-section is less, and is given in table 3 and represented graphically in fig. 3. This table also gives for comparison the correction due to the meson exchange charge density when pseudoscalar mNN coupling is used. Another meson exchange correction also appears in table 3. Meson exchange effects change the normalization of the deuteron wave function and cause a reduction of the cross section by about 1.5%. This correction is the sum of the number given after eq. (5.3) of ref. “) and the contribution mentioned after eq. (4.4) of the same reference. The uncertainty in the sum of the corrections due to the meson exchange charge density and to the change in the normalization of the deuteron wave function is difficult to estimate reliably, but 0.4% is a reasonable estimate at 40 MeV. The uncertainty is certainly less than 10% of the

502

W. Jam,

W.S. Woolcock / d(y, p)n

TABLE 3 Corrections to the full forward cross section for both pseudovector (expressed as percentages) E+

correction due to the meson exchange charge density (PV ?rNN coupling) correction due to the meson exchange charge density (PS rrNN coupling) correction due to change in the normalization of the deuteron wave function

and pseudoscalar

nNN coupling

20 MeV

30 MeV

40 MeV

-2.1

-3.5

-4.9

+4.0

i-4.1

+5.1

-1.5

-1.5

-1.5

32-

(duldQ)f$

[pb

Isrl

1

I

l-

,

I

I

.

10 20 30 LO 50 60

Er [MeVl

Fig. 3. Full cross section for the photodisintegration of the deuteron in the forward direction. The solid curve (RSC) represents the cross section that was calculated in ref. ‘) in the impulse approximation (i.e. without meson exchange effects) with the Reid soft-core potential. The dashed curve (RSC + ME) shows the cross section which results when meson exchange effects on the El transition (calculated with PV rrNN coupling) and on the normalization of the deuteron wave function are included. The experimental points are taken from ref. ‘).

correction itself and that is sufficient for our purpose. Even though the correction due to the meson exchange current remains to be calculated, this correction is expected to be quite small, so that it seems most unlikely that meson exchange effects are responsible for the large discrepancy between the experimental results of ref. ‘) and the theoretical calculations of ref. “). We see from table 3 that there is a big difference between the results obtained using PS and PV TNN coupling. Assuming the reliability of both theory and experiment, our work provides a strong argument in favour of using PV rather than PS coupling in calculating meson exchange effects in nuclear processes. However, it is well known [see for example ref. *“)I that the NN potential itself, if derived from an underlying meson theory, depends on the form of the TNN coupling. As an example one can take the potential due to one-pion exchange, which is given by VP’ = vo,

+

CV&(Ul . Lu*

* L + a 2 * La1 . L) - L2u1 * 1721+ non-local terms,

W. Jam, W.S. Woolcock / d(y, p)n

503

where VopEis the usual static OPE potential and the quadratic spin-orbit term is determined for PS coupling by c = -1 and for PV coupling by c = 1. This is a rather academic example, since the additional quadratic spin-orbit potential VQ is very small and the uncertainty connected with it is correspondingly small. The TPE potential, which also depends on the form of the rrNN coupling, would be expected to give a larger effect. The important point is that in a consistent calculation (which we have not made in this paper) the NN potential and the meson exchange operators need to be derived from a single underlying meson theory. Since it seems very unlikely that meson exchange effects can account for the discrepancy between experiment and theory, it follows that the experiments need to be repeated and, if the experimental results are correct, that the NN interaction needs to be better understood. The forward photodisintegration of the deuteron seems to be particularly sensitive to the details of the NN force. The role of the tensor force and especially the value of the D-state probability PD was discussed in ref. “). However, a rather low value of PD would be required in order to achieve agreement between theory and experiment. Moreover, the tensor part of the potential is not the only part to which the theoretical calculations are sensitive. The analysis in table 2 of ref. *) shows how sensitive the El matrix elements are to interference effects, which depend also on the spin-orbit force. It seems that more progress in our understanding of the NN interaction is required before the discrepancy can be resolved. We wish to thank G. Scharf for his generous assistance and advice in all questions concerning the numerical solution of the Schriidinger equation.

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