The potential-like electromagnetic meson exchange currents in the obe approximation

Nuclear Physics A492 (1989) 556-594 North-Holland. Amsterdam

THE

POTENTIAL-LIKE CURRENTS

ELECTROMAGNETIC

MESON

EXCHANGE

IN THE OBE APPROXIMATION J. ADAM,

JR., E. TRUHLIK

institute qf Nuclear Physics, de; near Prague, CS 250 68, Czechoslovakia D. ADAMOVA Nuclear

Center, Charles university, V ~o~e~ovi~k~ch2, CS 180 Ot?,Prague g, Czechoslovakia Received 30 June 1988 (Revised 19 September 1988)

Abstract: The electromagnetic OBE nuclear currents including the leading relativistic corrections are derived in the framework of a modified S-matrix method. The problems connected with the introduction of the phenomenological hadron form factors are discussed. The resulting description is gauge and Lorentz invariant up to the order in v/c considered.

1. Introduction The studies of electromagnetic (e.m.) reactions on few-nucleon systems provide useful information about the nuclear constituents and their interactions. In the last two decades the existence of the e.m. exchange currents in the pion range has been firmly established ‘). For larger momentum transfer (k) the effects due to p-meson exchange were found to be important ‘). While, for moderate k, the r- and p-meson exchange currents describe the data short-range satisfactorily, in the intermediate region of k =S1 GeV the additional and relativistic phenomena are expected to play an important role. The quantitative this energy region are marred by the internal inconsistencies results *,‘) concerning in the calculations. UsualIy *), only lowest order pion exchange and selected static p-exchange currents (chosen so that nuclear continuity equation holds in nonrelativistic form) are considered. Other meson exchanges and non-static effects present in recent models of NN potentials are not accounted for. In a semiphenomenological approach 3), currents reflecting isospin dependence of the potential are constructed in analogy with the standard n- and p-exchange ones 2). Furthermore, additional contributions of the order a*/ c* are generated via minimal replacement in velocity-dependent parts of the potential. Some contributions of the same order in v2/c2 - namely, those compensating the commutator of the potential with the Darwin-Foldy and spin-orbit one-nucleon charge density and meson retardation currents - were not included, either in models of MECs in refs. 223), or in the numerical estimates of effects due to the spatial part of current therein. The 0375-9474/89/SO3.5~ @ Elsevier Science Publishers (Noah-HoIland Physics Publishing Division)

B.V.

557

J. Adam, Jr. et al. / Electromagnetic OBE currents

implied

violation

uncertainties

of the

The relativistic equation

gauge

of the numerical invariance

agreement

between

(approximate)

the predictions

invariance

to estimate

in the framework

approximations

suits the nuclear

Lorentz

that are difficult

is fully respected

“) or its quasipotential

of these formulations

and

results,

leads

to

reliably.

of the Bethe-Salpeter

“). It is not clear these days, which

calculations

best. Moreover,

of the basically

the qualitative

non-relativistic

calculations

2,3)

and the experimental data, indicates that the relativistic corrections (RCs) may treated perturbatively in the intermediate-energy region. The approximate (up to any finite order in V/C) Lorentz invariance of Schroedinger-like dynamical theory in the Hilbert space with fixed number nucleons, imposes some general constraints on the form of the hamiltonian and e.m. currents 6m8). Several techniques have been proposed for construction of

be the of the the

NN potential and MECs satisfying these constraints from the effective mesonnucleon lagrangian (see discussion in ref. ‘)). Common features of these methods are the elimination of the antinucleon degrees of freedom by means of the FoldyWouthuysen transformation and a subsequent use of various formulations of the non-covariant time-dependent perturbation theory. In the practical calculations only the leading RCs (- v’/c’) are kept. The adopted choice of the perturbation technique fixes the off-energy-shell dependence of the NN potential. The OBEPs following from the various approaches are, up to the order in v/c - p/m considered, connected by a one-parameter (v) unitary transformation ‘). The corresponding v-dependent retardation part of the OBEP is given by V’Bet(q,Q,

, Q2)

=$

I

4=41=-42, where

V’,“(q)

Qi =Pi-Pi

is the non-relativistic

tum of the ith nucleon

Q1, Q2)

(final)

$%r(Q,-QJ12}

potential,

i=l,2,

+ Vi?(q,

l/m’) QI

,

> (1.1)

pi(pj)

state, m is the nucleon

(up to the order = V’,“(q)

Q2+

Qi=PI+Pi,

7

Galilei-invariant

in the initial

is the meson propagator. The total OBEP is given V&q,

q * Qlq.

V’,“(q)A,(q)

is the momenmass and A,(q)

by Q2) + VG=Yq, 91392)

,

(1.2)

where Vc3’(q, Q,, Q2) follows from the RCs to the BNN vertices (appendix C). It should be pointed out that the OBEP (eq. (1.2)) can be obtained directly by the non-relativistic reduction of the irreducible kernel of the Bethe-Salpeter equation in the OBE approximation ‘). In this approach the meson propagator is reduced to the instantaneous form:

A,(q)=

’

m~+q2_q22A,(4)+A2,(9)q~, 0

(1.3)

558

J. Adam, Jr. et al. / Electromagnetic

where q,, is treated

as the RC, -p/

write qi as q$,

or -qloqzo,

linear

q:,,

combination,

particles,

symmetric

OBE currents

m. On the energy

shell (q. = qlo = -q&

from which only two are independent. explicitly

with respect

to interchange

one can

The general of the two

is:

d = -q*oq20+ y?%Uf 42J2 *Q,s* Qz+ +r Hence, the retardation Obviously, the second

(QI - Qd12}.

(1.4)

correction to the meson propagator reproduces eq. (1.1). term -(l - v ) in eqs. (1.1) and (1.4) vanishes on the energy

shell. Finally, it can be shown (appendix C), that this OBEP satisfies the Foldy constraint “) for an arbitrary v and consequently defines the correct Schroedinger-like dynamics. While the above-mentioned calculation of the OBEP is relatively simple, the derivation of the MEC operators using the Foldy-Wouthuysen reduction of the BNN vertices and the time-ordered diagram techniques, is rather complicated. The method has been applied mainly to the discussion of the time component of the e.m. MEC operator - the exchange charge density (see e.g. refs. 7*8*‘o)). In the present paper we would like to show that once the OBEP (eq. (1.2)) is fixed, the MEC operators can be determined unambiguously from the relativistic amplitudes in a way similar to the non-relativistic S-matrix method I’). We start from the standard potential-like OBE Feynman diagrams (fig. la-c) for arbitrary scalar, vector and pseudoscalar exchange. The non-potential contributions, such as the diagrams in the fig. Id, e, for which the additional assumptions about the form of the yBB’ (B # B’) and yNN* vertices are needed, are not considered here. A possible model for the non-potential interaction, incorporating the principles of the chiral invariance and vector dominance, is discussed in ref. I*). For the transition from the relativistic amplitudes to the nuclear operators we modify

the S-matrix

method

I’) so that

the relativistic

off-shell

and

retardation

effects are taken into account in a way consistent with the Schroedinger description of the nuclear states based on the OBEP (eq. (1.2)). The space and time components of the e.m. MECs are given up to the order l/m2 and l/m3, respectively. This paper deals only with the e.m. processes that can be treated in the one-photon exchange contained

approximation. in the transition

In this case, the nuclear amplitude

T,(k)=(ICIfJ_TA(k)lrLi)~

y*(k)= where 1qc.i)are the nuclear

J

part of the S-matrix

k=P,-Pi,

dxexp(ikx)j*(x,t=O),

element

is

(1Sa) (1Sb)

states (Pl$f,i) = P,il+f,i), ~1~f.i) = Ef,i(~,,i); F is the operator

J. Adam, Jr. et al. / Eleefro~ugneric

of the total

(j(x),

the gauge should

momentum

of the nucleus,

Q?(x)) is the e.m. current invariance

operator

fi is the nuclear in the Heisenberg

of the amplitude

satisfy the continuity

r,(k)

559

OBE currents

hamiltonian), representation.

( kAT, (k) = 0), the current

j*,(x, t) = To ensure operator

equation

k. j(k)

= [fi, ;(k)l,

ii=&?,

(1.6)

where g and 0 are the kinetic and potential parts of the hamiltonian. In the OBE approximation the potential ? is a sum of the pair contributions of the form (1.2). The current operator and the continuity eq. (1.6) can also be decomposed into the one- and two-nucleon parts. From now on we will consider, the two-nucleon system for which (g = $, + &):

for the sake of simplicity,

(1.7a) (1.7b) where, throughout the paper, ]( 1, k) stands for the one-nucleon current of the first nucleon (hence the factor l-2 on the r.h.s. of the eq. (1.7b)). As is usual in the OBE approximation, the term [q, @(2, k)] is omitted in eq. (1.7b). The effects due to the internal structure of the hadrons are not taken into account in the local meson-nucIeon lagrangians. They are usually simulated by form factors (FFs), introduced into the potential and current operators in a phenomenological way. The continuity equation (1.7b) has frequently been used to link the e.m. FFs attached to the one- and two-nucleon currents and the strong form factors entering the potential and j* (2, k) [refs. ““)I. H owever, it has recently been shown by Gross and Riska 13) in the framework of the Bethe-Salpeter theory, that arbitrary e.m. FFs may enter the nuclear currents. Then, to preserve the gauge invariance, the additional longitudinal currents (-k,) appear. We show that one can arrive at the same conclusion (appendix corrections

by making

a simple

extension

of the vector

meson-dominance

model

B). It occurs that, unlike the fully relativistic approach, the longitudinal cannot be omitted in calculation of T,(k) when the decomposition in

v/c is employed. The paper is organized as follows. Sect. 2 presents the relativistic OBE Feynman amplitudes. The vector meson-dominance prescription for the e.m. FFs is used discussion

of the more general

model being relegated

to appendix

B. The monopole

hadronic FFs are introduced in the Lorentz invariant way. In sect. 3 the S-matrix method is modified to treat the non-static RCs consistently. The gauge invariance of the theory is proven and the MEG contributions following from the nucleon Born amplitude are discussed in some detail. The nuclear e.m. operators including the leading RCs are presented in sect. 4. To make the results more transparent, individual contributions to the continuity equation are discussed and comparison with previous results is made. In sect. 5 our approach is linked to the convenient formalism, in which the one-nucleon current is redefined so as to include the part

560

J. Adam,

of the interaction

dependent

Jr. et al. / Electromagnetic

OBE currents

effects. Sect. 6 contains

the summary

and some final

T = 0) to nucleons

are given

remarks.

2. The Feynman amplitudes The couplings

of various

terms of their interaction

mesons

lagrangian

(with isospin densities

in

by: (2.la)

ZNNs = g&&Ps 3

(2.lb)

(2.lc) For mesons

with isospin

T = 1, indices

of the meson

fields and the Pauli matrices

7 are to be introduced. The adopted form of the BNN interactions that used in the derivation of the NN potential (see e.g. ref. ‘“)).

is the same as

The couplings with the e.m. field - yNN, yBB and yNNB - can be obtained by means of the minimal replacement of the charged-particle field derivatives I?, + a,, - ieA, in the corresponding kinetic and interaction lagrangians. To take into account the internal structure of the hadrons the interactions with the anomalous magnetic momenta are to be included and the vertices should be equipped with form factors. These days, the fundamental theory (QCD and Weinberg-Salam model) does not provide the description of the vertices on the precision level needed in the nuclear calculations. duced phenomenologically.

Therefore, the anomalous couplings and FFs are introTo simplify further discussion, we adopt here the vector

meson

of ref. I’), e.g. the yNN

dominance

vertices

1

,

e*,=;[F;(t)+F;(t)7:], where

t = k*, F,,2 are the Dirac e.m, nucleon

vertex:

x*,=;[F;(t)+F;(t)Tf],

(2.2b)

FFs. The off-shell

of the

continuation

vertices and possibilities to choose the alternative parametrization are discussed in appendix B. Now, for the point BNN vertices, the one- and two-nucleon Feynman amplitudes read (in the notation

of appendix %(L

3A (2, k, B-Born)

A): k) = U(P’)&(~,

k)G) ,

(2.3a)

g’, --ii(~:)b%(L k)W’)f:b,‘,(B, -4

= - (2V)3

+f:b,‘,(B,-q,)W)h,

k)}U(p,)AB(s*)rj”,‘,(BI P:, pz)+ l-2, (2.3b)

561

J. Adam, Jr. et al. / Electromagnetic OBE currents

P=pl+k,

P’=p;-k.

B=s,v,ps, 2

4,(2,

k, B-con)

= $$F?“(?)E

16, P,)

3hcG,J(B-con,

~Ag(q2)r~~)(B,p;,~2)+1~2

9, (2, k, B-mes) = -I ‘gZnF~rr(~)~3hc~~~~~~(B, (2rr)3

q,,

q2)

P:, ~,)%(b,

x &(qr)&(qX;vj(B,

(2.3~)

B=v,ps,

3

P;, ~21,

B=s,ps,v, (2.3d)

where FF”( t) = F’;;““(t) = F‘;( 1)

(2.4)

in the vector meson-dominance model ‘*). Eq. (2.3a) contains the one-nucleon amplitude. The nucleon Born amplitude (fig. la) is given by eq. (2.3b). This amplitude can be separated into positive and negative Born contributions corresponding to the nucleon (S(p) + S’+‘(p)) and antinucleon (S(p) + S’-‘(p)) propagation in intermediate states. For the neutral mesons the

B _--__---

------B / (0)

I B ----

\

ICI

B

.-B

------_ iI1 lb)

-.- 7 B’

IdI

Ni

N'

(e)

Fig. 1. Potential-like (a-c) and non-potential-like (d, e) OBE diagrams. The diagrams (la-c) are called nucleon Born, contact and mesonic diagram, respectively. Wavy line stands for the interaction with the e.m. field, solid lines represent nucleons, thick lines nucleon isobars, dashed and dash-dotted mesons.

562

Born

J. Adam,

amplitude

mesons

represents

the contact

isovector

amplitudes

the total two-nucleon

(fig. lb)

Jr. et al. / Efeci~offiag~etie

the only

potential-like

and mesonic

(2.3~) and (2.3d),

UBE

~~~r~~fs

contribution.

(fig. Ic) diagrams

respectively.

For the charged appear,

It should

yielding

the

be pointed

out that

k, B-con) + 41,,(2, k, B-mes)

(2.5)

amplitude

9h (2, k, B) = 4, (2, k, B-Born) t 9,(2, is gauge invariant:

k,$, (2, k, B) = 0 , only if, in accordance

with the strict vector meson

(2.6) dominance,

eq. (2.4) holds. The

additional contributions, that appear for more general choice of e.m. FFs are discussed in appendix B. General technique of the introduction of the strong BNN FFs into the relativistic amplitudes is described in ref. ‘3). We would like to point out that for the frequently used monopole parametrization (2.7) each of the amplitudes (2.3b-d) (except for the part of the u-mesonic one, see appendix B) with the BNN FFs can be expressed in terms of the amplitude with the point strong vertices in the following way:

(2.8) Eq. (2.8) represents covariant extension of the approach suggested in ref. “). The prescription (2.8) is not affected by the transition to the nuclear operators described below.

Hence,

Finally,

we can omit the hadronic

let us define the NN-scattering

BNN FFs from further amplitude

discussion.

in OBE approximation:

q=q1=-q2.

The derivation of the NN potential from 2’ is discussed in the introduction (1.2)-(1.4)); the OBEPs for B = s, v, ps are given in appendix C.

(2.9) (eqs.

3. The modification of the S-matrix method The S-matrix method “) is based on comparison of the S-matrix elements (or the corresponding transition amplitudes - see appendix A), following from relativistic field theory and quantum mechanical nuclear description. To avoid double

563

J. Adam, Jr. et al. / ~leclrarnag~ef~c OBE currents

counting, the contributions of the one-nucleon current ]:(l, k) to the transition amplitude T,(k) (eq. (1.5)) are to be carefully identified and separated from the genuine two-nucleon current effects. In the lowest order in v/c - pl m the transition amplitude

t:*(k)

due to Jlh(1, k) coincides with the positive 9,,(2, k, Born-l-) 10,‘6). Then, the nuclear currents in the momentum by: jh(lr

W=%(l,

&(2, k)=&.(2, .9h(2r ~)~~~~~=~~(2,

ko=

k,=P;-P,,

k),

k)P&,, k)-4,(2,

= indicates

convenient

the non-relativistic

to distinguish

(3.la)

,

(3Sb)

k, Born+) k, con)t-_Fa,(2,

reduction

k, =pi -p,

- E(P,)

k=P;+P;-P,-Pz,

= 9A(2, k, Born-)+9&(2, where

EWI)

Born amplitude space are given

k, mes) ,

up to the order

in the matrix

element

(3.la)

(3.lc)

considered.

It is

from the total

momentum transfer k (appendix A). The prescription (3.1) has become standard technique of the nuclear mesonexchange currents calculations. Although it is justified only in the non-relativistic limit, it has been used sometimes for derivation of the relativistic corrections (see (3.1) violates e.g. refs. ‘2*‘7)). H owever, when the RCs are included, the definition the gauge invariance. The continuity equation satisfied by the currents (3.1) can be obtained directly by the non-relativistic reduction of the following identities “f: k,,$,(I, k $A (2, k):&,

(3.2a)

k,) =O,

= Vt-(Pi, Pi, p, Pr)u+(w,4P,) -~+(PIl)e^,u(p’)“lr(p’,P;,P,,P,)+l~2,

where u+(p) = ii(p) y,,. Let us compare (1.7) in the momentum space: k&(1, k&(2,

k) = VP;,

(3.2) with the nuclear

continuity

equation

(3.3a)

k,) =o,

P:, P, p&(1,

(3.2b)

P,P,)

-P(I,p;,P’)V(P’,p;,Pl,pz)+I02,

(3.3b)

where the dependence of p( 1, k,) on final and initial momenta is restored (appendix A). In the non-relativistic limit eqs. (3.2) and (3.3) coincide. The relativistic corrections to eq. (3.2b) do not contain the “transverse” part of the charge density pll( 1, k) proportional to g,. Moreover, the energy carried by a meson is in Y fixed by the four-momentum conservation to a value different from that following from eqs. (1.4) and (3.3b). It was concluded in ref. “) from eq. (3.2b), that the part of MECF,(t) satisfies the continuity equation (3.3b) with p + pc, where ,Q( is the “longitudinal” part of the density, proportional to 8,. It is shown in the next section that this is correct only if the BNN vertex does not depend on the energy transfer q. and retardation effects are neglected.

564

J. Adam,

Several

approaches

to a consistent

of the non-covariant Here, current

we would operators

Jr. et al. / Electromagnetic

time-ordered

treatment

perturbation

like to suggest

of the RCs to the MECs techniques

current

amplitudes operators

in terms

have been proposed

a more straightforward

from the relativistic

Let us define the exchange

OBE currents

derivation

9*‘o).

of the nuclear

of sect. 2. as follows:

j, (2, k) = 4, (2, /c)” = 9, (2, k) - riA( k) ,

(3.4)

where tiA(k) is the first Born iteration of the one-nucleon current contribution to T,(k), calculated explicitly from the underlying nuclear theory (see below). The definition (3.4) has been used by Riska is) for the calculation of the e.m. exchangecharge density in the static limit k,+ 0, and by Thompson and Heller 16), who have investigated the non-relativistic exchange current j”‘(2, k). Obviously, the form of tlA(k) depends on the choice of the equation governing the dynamics of the nuclear states. In this paper, we adopt the Schroedinger description with the leading RCs included “): Al@) = El+), Bi z ($:+

J5=&+&,, where ? is the instantaneous (1.2), (1.4). The two-nucleon equation:

ci=i+G, m2)1’2,

hermitean energy-independent states Ipi, p2 +) satisfy the

lP,,P*+t)=~~+~0(~)~(~)1IP~,P*), f(E)

= ?+

i=l,2,

where Ip1, p2) is a free two-nucleon posing eq. (3.6) into the iterative

&(E)=[E-E+k-‘,

3

(3.6)

state (plane wave), l? (p1, p2) = E Ip1, pJ. DecomBorn series:

~P,,P~+)=~P~,P~)+~O(E)~~IP~,P~)+.

and keeping contribution

potential of the form Lippmann-Schwinger

E =E(P,)+E(P*)

E,(E)f(E),

(3.5)

* *)

(3.7)

only the terms of the first order in the potential G, one gets for the of the one-nucleon current to the transition amplitude TA(k):

~fp(k)=(p:,p;-I~~(1,k)Ip~,p2+)+1cf2 ~.~,(~,~,)~(~,+P,-P;)~(P;-P,)+~~~+~~~(~)~(~+P,+P~-P:-P;).

(3.8a) 1 llA(k) =_L(Lp;,

f”)

+v(P;,P;,RP2)p

P&--E(P)+

is

0

V(p’,

Pi, Pl

_E~p)+i~~*(l,~,P*)+Icf2,

3

P2)

(3.8b)

where P’ = pi -k = p1 - q2, P =pl + k =pi + q2 (see fig. 2). The meson exchange currents in the OBE approximation are in our approach determined by eqs. (3.4) and (3.8b).

565

J. Adam, Jr. et al. / Electromagnetic OBE currents

5’ --- *----_

P2l

q2 P -*-k

q2

9 Fig. 2. The kinematics of the nucleon Born diagrams. The same diagram represent the first Born iteration of the one-nucleon current contribution (for differences see sect. 3). This kinematic is relevant also for (anti)commutators of any one-nucleon operator with the NN potential.

The remarkable nuclear

description.

feature

of this definition

Indeed,

is an explicit

since for k = p’ -p

gauge invariance

the following

relation

of the

holds

(see

eq. (3.3a)):

khjh(l,p’,P)=[E(P’)-E(P)-k”lP(l,p’,P), the divergence k,j,(2,

of the current

j,(2,

k) (eq. (3.4)) is easily

(3.9)

calculated:

k) = -k,tfP(k) = V(P:,P;,P,P,)p(l,P,P,)-p(l,P~,P’)V(P’,P;,P,,PZ)+lt,2, (3.10)

which is to be compared with eq. (3.4). Only eq. (3.9), the definition of tfP(k) (eq. (3.8b)) and the gauge invariance of the relativistic Feynman amplitude (eq. (2.6)) have been used in the derivation of the continuity equation (3.10). The fact, that the exchange current (3.4) is transformed properly with respect the Poincart

group is less transparent.

j,( 1, k) are Lorentz

four-vectors

It is clear intuitively,

and our description

that since 4,(2,

of the nuclear

to

k) and

dynamics

is

approximately Lorentz invariant, the exchange current j,(2, k) (3.4) also forms a four-vector (up to the order in v/c considered). This can be verified explicitly with the help of commutator relations of j,(2, k) with the generators of the Poincart algebra, discussed in refs. 7,8). T o make our definition of the exchange current more transparent,

let us rewrite

it as follows:

jA(2, k) =_A@,kLtand+tjh(2,klposr j* (2, k),,, = -ah (2, k, Born+)

- t:^( k) ,

(3.11a) (3.11b)

where j, (2, k)stund is given by eq. (3.lb) and j,(2, k),,, represents the contribution of the positive Born amplitude to the nuclear MEC. Making use of the explicit form of S’+‘(p) (appendix A) and of the definitions of the amplitudes -ah (1, p’, p) and

566

J. Adam, Jr. et al. / Electromagneric

(eqs. (2.3a) and (2.9)), we express

‘V(p~,p~,pl,p2)

k, Born+)

4,(2,

OBE currents

=-a,(l,p;,

P’) p, _ &)

k, Born+)

4,(2,

+ iE VP’,

PL PI

in the form:

> PJ

0

+ VP:, Obviously, (3.8b)) appear:

Pi, P, Pz)

in the non-relativistic

and j,(2,

k)ros=O.

1 PO-E(P)+ie

k, Born+)

limit $,(2,

In the higher

9*(L

order

in l/m

(3.12)

P,p,)+l++2.

coincides

with t:*(k)

the following

(eq.

differences

(i) The nucleon in the intermediate state in the relativistic amplitude (3.12) is off-mass-shell: P,# E(P) = (P*+ m*)“*, Pf, f E(P’). The amplitudes $,( 1, p’, p) and V entering LJa,(2, k, Born+) are the off-mass-shell continuations of the quantum mechanical expressions j,( 1, p’, p) and K The different dependence of the ?NN and BNN vertices in eqs. (3.8b) and (3.12) on the corresponding energy transfer gives rise to contributions into j,(2, k)pos. (ii) The relativistic amplitudes ‘V in eq. (3.12) are on the energy shell (E,= E,), which is not so for the potential V in t:*(k). In the adopted definition of the NN potential (eqs. (l.l)-(1.4)) the off-energy-shell dependence appears only in the retardation potential V”‘. The different description of the meson retardation in Yf and V leads to the appearance of the retardation exchange current. Since 9,, and “I’reduce to j, and V in the non-relativistic limit, one can decompose the relativistic expressions into the Taylor series in l/m around the quantum mechanical values. Taking into account only the leading RCs, the contribution of the positive Born amplitude to the MEC is given by: j, (2, k)ros =j,(2,

k,ext)+j,(2,

k,ver)+j,(2,

The first term on the r.h.s. of eq. (3.13) follows P,(l, k): _L(‘Lk,ext)=

k,ret).

from the energy

(3.13) dependence

V”‘(q2)j~(1,~,~,)-j~(l,p:,~)V”‘(q2)+l~2,

a&(1,k)

A(l,p’,p)=U(p’)

ak

of

(3.14a)

U(P),

(3.14b)

0

The appearance vertex function

of the current j, (2, k, ver) is due to the dependence Plb,‘,(B, 9) (appendix A) on the energy transfer qo:

of the BNN

2

.h (2, k

ver)= -[c2:,lJo (1, P;, -

%?(R P;,

P’)rjb,‘;(B,

P)j,(L

P,

P’,P,)

pl)lAB(q2)~~b,‘~(B, A, p2) + l-2

r’(B, P’,P) = U(P’)

ah4 9) i3q, U(P)

,

(3.15a) (3.15b)

567

.I. Adam, Jr. et al. / Electromagnetic OBE currents

The last contribution meson

retardation

j,(2,

k, ret) reflects

in V and

V,S(P’, A Pl, P2) - V&P’, = V’,“hk&&h)

V. Making

the difference

- G?p’,

Pi, P, Pz) - v&p;,

=

of the

pi, PI 3 pz) P;, p, , PI)

=~[E(P’)-pblv’B”(q2)A,(q,){(l+v)q20-(1 ~B(P:,

in the description

use of

p;,

-u)[E(P’)-E(p,)lI,

(3.16a)

v)lE(~:)-E(P)l1,

(3.16b)

p, Pz)

V(BI~hb&hdq~) - GYP:,

P;, P,

PI)

=~[po-E(P)lvk”(q*)AB(q2){(1+V)q~g-(lone gets for j, (2, k, ret): 1 _L+ (2, k, ret) = (2r)34m

&(q,){&LL(Lp’,,

P’)%%z,)+

-R,[j,(l,p:,P’)V’,“(q,)Rk=(l-v)k.qZ,

V’,‘,(qJ_L(l,

V’,“(qJ.L(L

P,P,)I

P,~,)l)+l*2,

Ro=(l-v)Q,.q,+(l+v)Q,.q,.(3.17)

Explicit expressions for the current operators scalar exchanges are given in the next section. decomposition of the individual vertices in l/m

with the scalar, vector and pseudoThe calculation is now reduced to and evaluation of the (anti)commu-

tators in eqs. (3.14)-(3.17). Each of the exchange contributions, including those coming from the positive Born amplitude, has a transparent physical interpretation.

4. The explicit First, let us briefly

form of the nuclear

survey the currents

The one-nucleon

e.m. operators

following

currents

from eqs. (3.1).

are given by: p’(“(1, k,) = Z,,

4+2x^, ~(‘~(1, k,) = -8m’[k:+

(4.la)

ia, x K, . k,] ,

(4.lb)

(4.lc)

jC3,(1, k,) = -&

,{i?,(K~+k:)K,+i[&(K;+k:)+x^,k:]a,xk,

+[e^,(k, . K,)+~,4mk,,J(k,+ia, -Gk,(k,

x K,)),

x K,)-i$(a,

. K,)k,xK, (4.ld)

568

J. Adam, Jr. et al. / Electromagnetic

where k, = pi -pl, in l/m

K, =p{ +p, . The superscripts

[ref. ‘)I. In eq. (4.ld)

in accordance

with eq. (3.la). l/m3.

The non-relativistic lowest order:

reduction

choice of k,, is discussed

of the two-nucleon

k, s-mes) = -iFr’“(t)(~i

j”‘(2, j”‘(2,

amplitudes

equation

(3.3a)

in the next section. (3.lb)

yields in the

B=s,v,ps,

(4.3a)

X 7~)‘~(,~)~A.(~,)A~(q&qi

- 42)

iC”eS(t)(~lx 72)3~~‘)(q2)A8(q1)(q, - 42),

(4.3b)

= iFreS( t)(r, x T2)3 f?‘(q2)A,(q1)(q1

(4.3c)

k, v-mes)

k, ps-con)

(4.2)

Then, it is easy to verify the continuity

An alternative

=

the order of the operators

= (pi’ -pf)/2m,

~“‘(2, k, B) = 0, j”‘(2,

indicate

we put

k,,, = k, . K,/2m

up to the order

OBE currents

= -iFg”(

- q2) ,

t)(~, x 72)3 (2~;\m$&Jo,(0..

42) + l-2

9 (4.3d)

2

j”‘(2,

k, ps-mes)

= iF’r(f)(r, .

where cc)(q) is the non-relativistic (appendix C).

x TJ3 (25P;~m2d,,(q,)d,,(q2)

(a, * qd(u2 . (I2)(41potential

(4.3e)

q2),

with the isospin

dependence

excluded

The non-relativistic currents with the pseudoscalar exchange (eqs. (4.3d, e)) are the standard ones 1,2). S ca 1ar and vector exchanges in the lowest order contribute only to the mesic currents (fig. lc). Therefore, the calculations ‘) with the additional p-meson currents are not strictly non-relativistic. This is also clear from the fact that the NN potentials, which generate wave functions used in refs. ‘*2), contain contributions

of the relativistic

origin

(e.g. L- S-terms).

One should

keep this in

mind in any comparison of the individual MECs contributions and estimation of the relativistic effects. In the next to leading order the negative Born (“pair”) amplitudes yield the following contributions to MECs (let us recall, that the coupling with derivative is used in the psNN vertex of the relativistic amplitude (2.1~)): ~‘~‘(2, k, B-pair) = 0, jc3’(2, k, s-pair) = -5

B=s,v,ps,

pi1’(q2) {F:(

-F;(k+iu,xQ,)}+l++2,

(4.4a)

Q1 + ial x k)

(4.4b)

569

J. Adam, Jr. et al. / Electromagnetic OBE currents

-F;[(l+K,)q,+iu,XQ~-(1+Kv)U,X((f~X42)1}+1~2,

(4.4c) jC3’(2 3k 3ps-pair)

= - ~2~~6mnA&Jb~

. 42)

.{F:[ikxq,-u,(q,.

Qku,x(Q1xqdl

+F;[u,q:-u,x(q2xk)-iQ1xq21}++lf,2, where

r = k2 and Ft are defined

for an exchange

(4.4d)

as follows:

F:={e^,,~,.72}=F~(f)~l.12+FT(r)7:,

(4.5a)

F,=[~^,,T,

(4.5b)

of the meson

. TV]= -iFY(f)(T,

T = 1, and

with isospin = F;(t)+

F:=2e^,

XT,)~,

F\;(t)7;,

(4.6a)

F;=O, for an isoscalar exchange. The exchange densities given by:

(4.6b)

following

from the contact

and mesonic

diagrams

~‘~‘(2, k, v-con) = 0, ~‘~‘(2, k, ps-con)

= -iFr(t)(~, * (~1 .

~‘~‘(2, k, v-mes),,, ~‘~‘(2, k, B-m=)

(4.7a)

x 72)3(2r;&&m3 Uq2)

Q,)(uz

.

e)+

l-2,

= ~(~‘(2, k, v-mes) +pc2’(2,

= iFi”“(t)(T,

are

(4.7b) k, v-mes),,,

xT2)‘~~‘(q2)A8(4,)(q10-

h),

(4.7c)

B=s,v, (4.7d)

~‘~‘(2, k, v-mes),,=

-iFF”“(

t)(~,

X T2)3

cy)(q2)A8(q,)

x~{k.(Q,-Q,)+i(l+K~)[U,~ql.k-u2Xq2.kl}.

~“‘(2,

k, ps-mes)

= iFFses(t)(ll * (at . qdu2

X

(4.7e)

72)3 (2~~~~2A’,(q,)A”(q2)

. d(%o- 920),

(4.7f)

570

where

J. Adam, Jr. et al. / ~lect~omag~etjc

qio = qi * Qi/2m. The second

the vector k,S,,)

of the amplitude

The general

contribution

~‘~‘(2, k, v-mes),,,

exchange,

OBE currents

to the mesonic

follows

charge

from the transverse

density

for

part -(k,6,,

-

(2.36).

form of the RCs to the non-relativistic jc3’(2, k, B-E;)

currents

=jc3’(2, k, B-::S),er+j(3)(2,

(4.3) is

k, B-“,O,:),,,

(4.8)

where the first term follows from the RCs to the BNN and yNNB vertices, the second one - from the RCs to the meson propagation according to eq. (1.3). Moreover, there appear the contributions from the contact diagram with the vector exchange (j’3’(2, k, v-con)) and from the transverse part of the mesonic vector amplitude (j”‘(2, k, v-mes),,) in the order lfm3. From the contact amplitude (2.3~) one gets the relativistic j”‘(2,

k, v-con)

x T?)~S

= iFy”(t)(~, *[q,+(l

+~)a,

currents:

?$‘)(q2)

x (ozx

ia,x(Q, - QJl+ 1++2,

qd+

(4.9a)

2 g,s iFz”( t)(?, x ?2)3 ~~~~~32m~Aps(92)

jt3’(2, k, ps-con),,,=

~(((72

Q:+ Q:)+qh

- qAIod&+

.qA

- QI(~, . QJ+ 41 x QJ +a,(~*

Qd(Qz. qJI+lf)2,

jc3’(2, k, v-con),,,

=j”‘(2,

R ;” = $-&qz

The mesonic jc3’(2,

k,

amplitudes

s-mes),,,

.

(4.9b)

k, v-con)RE”,

Qd2A,,(qd .

(2.3d) yield the following

= -iFTeS (f)(?, x Q)~$

(4.9c)

contributions

to j’3’:

~~“(q*)A~(q,)

~(~,-q2)[Q:+Q:+~~,~~,~Q~+~~,~~~~Q~lt jc3’(2, k, v-mes),,,

= -iFyes (t)(~l x d3-& X {(i+

K,h&+4t)+

-t i&-

K~)[o/

-

i(l+Kv)‘ic~‘~X

?)(qJAvCq,)

Q,

x QI

* Q2-(1+da,

* %+u2X

Q2 * %+=2X

(4.10a)

* (4, - cd X41

’ azX92

Q2 * 921 Q,

. d,

(4.lOb)

571

J. Adam, Jr. et al. / Eleciromagnefic OBE currents

f3’(2,

k, v-mes),,=

-iFY(r)(r,

x TJ~-~-$

q?‘(qJA,(q,)

* {kx[(1+K,)2(a,xq,)x(a,xq,)-Q,xQ,

+ i(1-c JG)((u~x ((2)x Q, -(a, x 4,) x Qdl (4.1Oc) L

= -iF$:‘“(t)(~, x 72)3 (25rg2

jc3’(2, k, ps-mes),,,

(q1- qzH(u1 . 4d(u2. +(uz.

q2)(u,

.

m~Aps(qAAps(qd.

Q:+ Q:)+(a, . q,)(uz. Qd(q2.

q2h:+tq:+

Q2)

(4.10d)

Qd(q, . Q,)l ,

jC3’(2, k, B-mes),,t =j”‘(2, R Y=

q:o&(q,)+

k, B-mes)RFes, (4.10e)

doA,(

When the retardation corrections Rf;;” are neglected, our currents for the scalar and vector exchange are in complete agreement with the recent calculations by Blunden I’) and Towner 18) (the scalar exchange with T = 1 was not considered in these papers, Towner has used the non-minimal ypp coupling “)). According to equation holds (our discussion of the continuity refs. 17,‘8), the following continuity equation in this and next sections is limited to the simplified choice of e.m. FFs me.5=FTnZ Fy, for a more general case, see appendix B): FB k. jc3’(2, k, B),,, =

(p’l,p:I[~“‘,p^~‘(l,k)l+[~‘3’,~‘0’(1,k)llp,,~~)+l~2, (4.11a)

jC3’(2, k B),,, --jC3’(2, k, B-pair) +jC3’(2, k, B-mes),,,+j’3)(2,

k, v-con)&V,

B = s, v. (4.11b)

The equation

(4.11a) corresponds

to eq. (3.2b) with the meson

qIo, q20, k,) neglected. Let us point out, that an analogous exchange does not hold. This is due to the energy dependence (see below).

The retardation

corrections

((eqs. (4.7d),

(4.9c), (4.10e))

retardation

(terms-

equation for the ps of the psNN vertex

satisfy:

k. jC3’(2, k),,, - k,pC2’(2, k, mes) = iFy(f)(T,

x T2)3 %‘)(q2)&(q2)&+

jC3’(2, k),,, =jC3’(2, k, mes),,,+jC3’(2, For the transverse

v-mesonic

contributions

k. jC3’(2, k, v-mes),,-

l-2, k, con),,,.

(4.12a) (4.12b)

(eqs. (4.7e), (4.10~)) one gets: k,pC2’(2, k, v-mes),, = 0 .

(4.13)

572

J. Adam, Jr. et al. / Electromagnetic OBE currents

In the actual calculations terms in the current transverse

p-mesonic

(4.4~) are usually proportional

of the MEC effects ‘) all velocity

jc3’(2, k) with the p-meson current

dependent have been

(4.10~) and the part of the p-pair

also neglected.

The remaining

to (1 + K,,)~. This selected treatment

by the anomalously

exchange p-meson

constant

current

exchange

of the p-meson

large tensor pNN coupling

(-Q,,

exchange

(( 1-C K~)~

=

tion of the classification of the e.m. currents by the large value isovector magnetic momentum of the nucleon xv (connected dominance) has been pointed out by Friar ‘).

Q2)

omitted.

The

F:

in eq.

currents

are

is justified

50). The distor-

of the anomalous to K~ via vector

Let us now turn to the contribution of the e.m. exchange operators following from the positive Born amplitude (eqs. (3.11)-(3.17)). Here, we omit for simplicity the dependence of e^i and &, (2.2b) on t = k2, that can be treated more conveniently in the framework of an alternative description discussed in the next section. Then, the derivative of the 7NN vertex (3.14) equals to: A A j~(l,~‘,~)=S,,U(~‘)~~~~U(~)~-~[~’-~+iCrx(~’+~)]i, iZ4, (4.14) and the current

j,(2,

k, ext) is given as follows: ~‘~‘(2, k, ext) = 0 ,

jc3’(2, k,ext)=

(4.15a)

F;$[Fg’(q2)k-iQ,x

-F;--&[iQ,x v: = ${a* 3 6?(42)1

Vz+iq,x

Vi-iq,x

3

Vi]

Vz]+l++2,

K = +[a,,

W(42)l

(4.15b) (4.1%)

,

where F; are given by eqs. (4.5) and (4.6) with 8, + Gi. The explicit show that k.jc3’(2,

k, ext) =(P:,P;~[%“,

where p*{:’ is the part of the relativistic x^,. For the individual jc3’(2, k, B-ext) = &

meson

exchanges

v(B1)(q2)[F,(k+

&?(l,

one-nucleon

k)llpl,pJ+l++2, charge density

calculations

(4.16) proportional

to

eqs. (4.15) yield: ia, x Q1) - Fiia,

x q2] f l-2,

B=s,v, (4.17a)

2 j’3’(2,

k, ps-ext) = c2TPIP;6m4Aps(r2)(

-2 . 42)

x {F;[qz x (42 x a,) - (a, . qdk + iQ1 x 421+ F:Q, x (a, x 42)) t-l-2.

(4.17b)

J. Adam,

As to the current exchange

j,(2,

contributes

Jr. ef

41. ,J

Electromagnetic

k, ver) (eqs. (3.15)),

in the order considered. &PS,

4) =- 1 2m

acre

9MP) =

~(P’P(P% one gets for the corresponding

513

OBE currents

it is easy to show that oniy the ps Since

Y4Ys

9

--&u,.(p’+p)

,

(4.18)

e-m. operators:

2 ~‘~‘(2,

k,

ps-ver)

=

~2T~~~m3Aps(tJ(~2

. qJlIF:(a,

+ k)

-

C(al

*

Qdl+ l-2, (4.19a)

jC3)(2, k, ps-ver) = (2a$..z,A&I;i(oz

-

- ez)

F;[Q,(ul~ Q,) +d-1 * k)]-G-[ikxQ,+kx(kxa,)]}

+1-2,

(4.19b)

where G’ = F: + Ft. Let us point out, that for Fy = FE” the part ~‘~‘(2, k, ps-ver) proportional to F; cancels the contact charge density (4.7b). The non-retarded e.m. currents with the ps exchange satisfy: k - _t3’(2, k, ps),,, - k,p’*‘(2, =(~:,~;l[~‘b’s’,

,$‘(I,

k ~s)ver

k)l+[~b3,‘,$*‘U, Whd+l-2,

jc3’(2, k, ps),,, =jC3’(2, k, ps-pair)

(4.20a)

+jC3’(2, k, ps-mes),,,

+jC3)( 2, k, ps-c0n)~,~+j(~)(2, ~‘“‘(2, k, ps)“_ = ~‘~‘(2, k, ps-con) +p’*‘(2,

k, ps-ver) ,

k, ps-ver) .

(4.20b) (4.20~)

These equations are to be compared with eqs. (4.11). Presence of j,(2, k, ps-ver) in eqs. (4.20b, c) is vital for the gauge invariance of the total current (the situation is even more dramatic in the case of the axial current 18*19).It is, of course, possible to use the psNN coupling without the derivative (ZZ’;p6sr.,N = -jg~s~~~~~), but the calculations of the “pair” current is more cumbersome in this case and for pions the additional contact term should be added to maintain the chiral invariance I*). The unitary freedom in the choice of the form of the psNN vertex is discussed in appendix D.

J. Adam, Jr. et al. / Electromagnetic OBE currents

574

Finally, making use of the explicit form of j?‘(l,p’,p) the retardation current j,(2, k, ret) (eq. (3.17)):

(eq. (4.lc)),

one gets for

pi2’(2,k,B-ret)=~~k”(q,)A,(q2)[F:R,-~;Ro]+1tt2,

.i’3’(2,k, B-ret) = &

(4.21a)

+Roq2)

~'B"(c12)A,(q2)[F:(RkQl

- F,(Rkq2+

RkQI)]+

jc3’(2, k, B-ret),,,

1-2-t

(4.21b)

j”‘(2,k,B-ret),,=-$A,(q2)kx{Gf[&V~-RoVJ +G-[&I’,-R,V;]}+lo2,

(4.21~)

where Rk, R, and V: are defined in eqs. (3.17) and (4.15c), divergence of the retardation current can be written in a form: k. jc3’(2, k, B-ret) - k,p’*‘(2,

respectively.

The

k, B-ret)

=(P~,p;I[~‘,p^‘“‘(l,k)(p,,P2)-iF~(f)(

71 x r2)3 V(B’)(q,)A&)&+ a

I-2, (4.22)

The second term on the r.h.s. of eq. (4.22) cancels the divergence of the retardation corrections to the standard currents (4.12). Collecting the equations (4.11), (4.12), (4.13), (4.20) and (4.22), one gets all relativistic corrections - l/m3 to the nuclear continuity equation (3.3b). The explicit form of the transverse retardation current (4.21~) is: jc3’(2, k, B-ret)tr=

c$,“(q2)A,(q,)(

&

GfRk

- G-Ro)a,

x k + l-2,

B=s,v, (4.23)

for the scalar and vector exchange,

and: 2

f3’(2,

k

ps-ret),,

=

c271$2m4A&(q2)(n2

* (12)

x ik x { G+[ Rkq2 + R&a, -G-[R~q2+Rkia,xq,]}+lt,2,

x q2] (4.24)

for the pseudoscalar exchange. It is noteworthy, that the retardation contributions (4.21)-(4.24) do not vanish for any choice of ZJ.Some of them are proportional to k* and may contribute sizeably into the amplitude T,(k) for the intermediate k [ref. *‘)I. It is essential for the consistency of the calculations to fix the parameter Y in j,(2, k, ret) in accordance with the NN potential used.

J. Adam, Jr. et al. / Electromagnetic OBE currents

575

Let us now compare our relativistic corrections to the MEC operators jf3)(2, k), p@‘fZ, A) with the previous results 17*18322-26).

In the calculations of jc3)(2, k) for the scalar and vector exchange the current j’“‘(2, k, ext) - s, has been missed ‘7*‘8322V23) and the retardation effects have been omitted 173.23). Let us point out, that the isoscalar part ofjc3’(2, k, ext) is suppressed by the small anomalous &scalar nucleon magnetic momentum x, = -0.12, but the isovector part is rather large. Friar’s currents “) are related to our results as fotlows:

=j”‘(2, k,s-ext)fF:~Pl”(q,)o,xq,+lt,2,

=jc3)(2, k, v-ext) + FT-&

(4.25a)

~~‘(q,)[u~x~*+(l-t2~,)a*xkj+1~2, (4.25bf

The retardation currents and potentials of ref. 22) correspond to the choice Y= 1. It can easily be shown, that the divergence of the currents on the r.h.s. of eq. (4.25) compensates for the commutator I:Q(l), $‘)(I, k)]. Hence, the currents of ref. 22) satisfy the continuity equation only with the nou-relativistic charge density $“‘(l, k). omitting p”‘2’(1,k), one violates the one-nucteon part of the continuity equation. Therefore, Friar’s currents do not form a consistent theory. It is interesting to note, that both our and Friar’s exchange currents conform with the requirement of the approximate Lorentz invariance. Indeed, the r.h.s. of eq. (4.25) are velocity independent (do not contain QI, Q2), white the corresponding constraint (see ref. “)) concerns onty the velocity-dependent currents. In the order considered, the exchange charge density operator p”“‘(2, k) for the scalar and vector exchange is given just by the retardation term p^‘*‘(2,k, ret) (eq. (4.2la)). The isoscalar part of Gt2’(2, k) has been studied in ref. 24), where also some contributions of the order I/m” are included. The operator p1’2’(2,k, t-ret) of ref. ‘“) corresponds to our result (4.21a) with Y= 0. It is questionable whether the higher order contributions l/pn’ can be used in the calculations with the Schroedinger wave functions. The pion-exchange charge density is discussed in detail in refs. R*25*26). In accordance with our choice of the OPEP (appendix C) and the treatment of the chiral contact contributions (appendix If), our results for sg2’(2, k, ?r> agree with that of refs. 8*25 ) with G,(t)+ F,(t), y = -1, and with that of ref. 26) with GM(r)+ F,(t), c = -1, v = 0 and d = 1. We are not aware of any previous consistent derivation of the pion-exchange current jc3’(2, k, T) in the framework of the Sehroedinger description.

J. Adam,

576

Jr. et al. / Electromagnetic

OBE currents

5. The alternative formulation The discussion

above

was based

on the definition

according

to eqs. (2.2) and (3.1). The operator

structure,

all corrections

the NN interaction

connected

were included

of the one-nucleon

J: (1, k) had a strictly

with the modification in the exchange

of the nuclear

current

j,(2,

current

single-particle current

by

k). The alternative

description has been suggested in refs. 27-29): the total e.m. current j* (1, k) +j^h (2, k) may be rearranged so that some interaction effects are transferred into the onenucleon-like operator and the exchange current is modified correspondingly. In this section we will describe briefly the effect of such a redefinition on our derivation of the MECs. Let us rewrite

the relativistic ~“‘(1,

one-nucleon

e.m. operators

(5.la)

k,),, ,

k,) = ~‘~‘(1, k,),+p’*‘(l,

d2)(L

as follows:

k,)e = 4k, * 41, k,) ,

~‘~‘(1 9 k 1) tr =2x* 11 k . s(1 91,k ) (5.lb)

s(l,k,)=-&[k,+icr,xK,], jc3’( 1>k 1) =jc3)( 13 k&”

+ Ajc3)( 17 k,) k,) ,

=j(‘)(l, j”‘(l, f3’(L

k,)z&+j’3’(1, k,)~“+Aj’3’(1, 1 k,)z&= -&(Xf+k:)K,,

kA% =jc3’(1, kl)/+jc3)(1,

(UC) (5.ld)

k,),,

1

(5.le)

41, k,),

where j,(l, kl)! and j,,(l, kl),, follow from the Dirac and Pauli parts of 3,( 1, k) (eq. (2.2)), respectively. The current j”‘(1, k,) is separated into parts, satisfying the continuity

equation

with the non-relativistic

and relativistic

charge

density:

k . ;(‘I( 17,)“.,con= [ g’p’, ;‘*‘( 1>k)] 3

,ip = -p^;/gm3 9 1

(5.2a)

k.~‘3’(1,k)~“=[~!“,~‘2’(1,k)],

l?!” =p^f/2m,

(5.2b)

and the transverse klO= E(pi)-E(p,),

part Ajc3’(2, k,), that is not important the operator j^“‘(l, k)& is given by: j^“‘(l,

k)gn=

(e^,+2x^,)[&“,

here.

s^(l, k)].

Obviously,

for

(5.3)

Delorme 27) has suggested to replace k ,. in eqs. (2.2a) and (3.la) by the time component of the total momentum transfer k,,, which is equivalent to the following modification of J:?( 1, k): j*‘3’(1, k)t,+j’3)(1,

k);=2x^,[& =jc3)(1,

s^(l, k)] k),,+6j’3’(1,

k):,

(5.4)

571

J. Adam, Jr. et al. / Electromagnetic OBE currents

where

& is the full nuclear

total e.m. current,

hamiitonian

the exchange j’3’(2,

(eq. (3.5) in our case). To maintain

operator

k)43’(2,

should

k)D=j”(3’(2,

where jC3’(2, k) is the exchange

current

be modified it) -

$“‘(l,

of the previous

the

as follows: k)&

(5.5)

section.

It is easy to see

Sj^“‘( 1, k): = Zx^,[ t s^(l, k)] =jC3’(2, k, ext) .

(5.6)

from eqs. (3.14a) and (4.14), that:

Therefore, $3’(2

, lp=j(3’(2

, k)stand+;C3y2

j^“‘(Z, k, po~)~ =23)(2, where the exchange currents are defined can be obtained also from the modified

9 k)” pas 7

k, ver)+jC3’(2,

k, ret) ,

(5.7a) (5.7b)

in eqs. (3.lb), (3.15) and (3.17). This result S-matrix method (eq. (3.11)):

jY’(2, k)Dpos=:$h(2,

k, Born+)-rlA(k)“,

(5.8)

where tlA(k)” is given by eq. (3.8b) with the Delorme one-nucleon current (5.4). Since in this definition of j, (1, k,) the dependence of the y NN vertex on the energy transfer k. is the same for both terms on the r.h.s. of eq. (5.8), the corresponding current does not appear (j,(2, k, ext)D = 0) and eq. (5.7b) is reproduced. tinuity equations (--l/m”) for the Delorme currents read:

The con-

k~j1’~‘(1,k)D=f~.13~,~~0~(1,k)]+[~.j1f,p”~~~(1,k)r]+[~j”~,~~2~(1,k),,J,

(5.9a)

k~~~3~(2,k)D=[~~3~+~,~~0~(1,k)]+[~~’~,~~2~(1,k)~]+1t,2.

(5.9b)

It should be pointed out, that the Delorme redefinition does not remove all terms 2, from the exchange currents; the transverse parts of j”‘(2, k, ver) and j’j’(2, k, ret) (eqs. (4.19b) and (4.21~)) contain G’ = F:+ F:. The same redefinition can be applied Then, according to Ohta 28): j^“‘( 1, k)Oh =

to the remaining

j^“‘( 1, k),,+

jC3’(2, k)oh==;‘3)(2,

(- e^,) part of j^‘“‘( 1, k)$,,.

Si^“‘( 1, k)Eh,

k)-i$(“‘(l,

k)Eh,

s~“‘(l,k)eh=(e^,+2~,)[~s^(l,k)],

1

k)Oh = [ $43) + j+

(5.10b) (5.1Oc)

k . i^“‘( 1, k)Oh = [ ii;3’, ;‘O’( 1, k)] + [ ii(l), $‘)( 1, k)] , k . _$3’(2

(5.10a)

,p^‘“‘(l,k)]+1++2.

(5.1Od) (5.10e)

To reproduce these results in the framework of the modified S-matrix method one should perform the non-relativistic reduction of $*(l, k) in the positive Born amplitude before comparing it with the corresponding tfP(k)O”. The point is that

578

J. Adam, Jr. et al. / Electromagnetic OBE currents

the factor

k, . K,/2m

+ k. in eq. (5.le)

does not come from the k. dependence

4, (1, k), but from the combination

of the nucleon

after

Nevertheless,

the non-relativistic

S-matrix

calculations

There are several

reduction. reproduce

reasons

spinor when

momenta

of

that appear

performed

carefully,

the

currents

as described

Ohta’s results.

for redefinition

of the nuclear

in

this section. First of all, the Ohta definition is convenient for discussions of the low-energy theorems 28). Furthermore, the calculations of the transition amplitude T,(k) (eq. (1.5)) are usually being made in the configuration space. Then, in the standard approach (eqs. (2.2) and (3.1)):

(5.11) which complicates

the calculations

unnecessarily.

In the alternative

approach

k,, +

k,, where k. can be treated

as the c-number parameter, fixed by the energy conservation law. This is particularly convenient when the k,, dependence of the e.m. FFs in eq. (2.2) is considered (up to now this dependence has been omitted). In the standard approach one should decompose the form factors F,,*(k*) into the Taylor series and use the replacement (5.11) 30). Additional contributions to j,(2, k, ext) (eq. (3.14)) would appear (compare to eq. (4.14)). Obviously, it is preferable to treat this k, dependence according to the Ohta prescription. It should be clear from the discussion above, that the total e.m. current remains the same.

6. Summary and conclusions A simple

and straightforward

modification

of the S-matrix

method

is given, that

makes possible to get the MEC operators with the leading relativistic corrections included. The e.m. nuclear currents are derived for the exchange of arbitrary scalar, vector or pseudoscalar mesons. Our results for the exchange charge density reproduce the previous ones, while the full expression for the spatial part of MECs, including the retardation effects and the contributions -x^,, are new. Arbitrary e.m. form factors may enter the MECs operators. Unlike to the fully relativistic theory 13), the additional longitudinal MECs should be included into calculations in the framework of the Schroedinger formalism. In the actual calculations, our currents should be used with the wave function generated from the corresponding potential. Unfortunately, none of the recent realistic parametrizations of the NN potential coincides fully with our prescription. However, the Bonn OBEP(R) i4) is quite close to it; in particular the form of this potential fixes the retardation parameter v ( vBonn= $). Even if a consistent realistic potential is available, the inclusion of the total e.m. operators into the numerical calculations would make them rather complex. Our results provide a solid ground for the systematic approximations. We have made a

J. Adam, Jr. et al. / Electromagnetic OBE currenlv

first attempt

in this direction

the deuteron

*I).

Without

any principal

in the study of the backward

difficulties,

our results

of systems of two particles

with different

The possible

cover the studies

applications

579

electrodisintegration

can be extended

of

to the description

masses and to few particle of the e.m. properties

systems (N 2 3). of deuteron,

3H,

3He and heavier nuclei, and the calculation of e-m. meson and baryon transitions within the potential model. The general expressions for the exchange currents (sect. 3) can be used for treatment of interactions with other external fields. In this way, we have obtained the relativistic corrections to the axial MECs and used them in the calculation of the tritium P-decay 20). The authors on the subject

would like to thank Prof. P.U. Sauer for many stimulating

discussions

of this paper. Appendix A

NOTATION

Throughout

the paper

the

Pauli

metric

(with

fi = c = 1) is used:

A,(A,

iA*),

A,B,=A.B-A~B,,A=A,~,,{~,,~“}=~S~”,Y~=Y,Y~Y~Y~,O~~=(~/~~)C~~,~~I-

Free nucleon

spinors

are normalized

as follows:

~+(~)~(P)

= ~(~)~~~(~)

(A.la)

= 1,

(A.lb) The nucleon (S”‘(p))

propagators

S(p)

and antinucleon

is separated

into parts corresponding

(S’-‘f p)) propagation:

S(p)=(@+m-i+‘=s’+‘(p)+S’-‘(p),

where the summation use:

to the nucleon

S’+‘(P) = -(PO-

F(p)+

s’-‘(p)

E(p)

= -(p*+

over spin indices

(A.2a) (A.2b)

i&)-‘uCP)u(P),

- is)_‘?J(-p)U(-p)

is suppressed.

A,(z)=(m’,+z)-‘,

)

In the meson

(A.2c) propagator

we

(A.3)

where mB is the meson mass and z = q2 or q2 (after the non-relativistic reduction). The field-theoretical S-matrix etement for the reaction hi( Pi) + y(k) + h,( Pr) in the lowest order in e.m. charge e reads: S ~“d=.“,(h,(P,)Ihi(Pi)y(k))i, s

i(2a)S’4’(k+Pi-P,)ea,(k)$,(k),

(A.4a)

580

J. Adam, Jr. et al. / Electromagnetic

OBE cwrenis

%ih->= ~~lA~(~)l~(k)) > where hi,f are the initial paper),

r(k)

and final hadron

is the photon

state, j,(x)

(A.4b)

states (one- and two-nucleon is the hadronic

e.m. current

states in this operator

in the

Heisenberg representation, Ah(x) is the e.m. potential; for an electron scattering ah (k) is to be replaced by the Moeller potential generated by an electron current j7)< The corresponding nuclear S-matrix element equals (A,(x) is treated as a classical external field):

=i(2w)6(fc,S-Ei-Er)eah(k)T,(k),

(A.%) (A.5b)

Th(k)=th(k)~ifk+Pi-PF)=(tlrfl~~(k)l~i),

where IJli,r) are the nuclear states, j*,(k) is defined in eq. (1Sb). is chosen so that .9,(k) and t,(k) coincide in the non-relativistic normalization and sign convention was adopted in ref. “)). In a similar way we put:

The normalization limit (a different

field

sli

=,,t(N(P~)N(P;)(N(Pr)N(Pz))in

(A.64

~-jf2~)s’4’Cp;+p;-pl-Pz)~(p’I,pl,pl,p,),

S Z=‘=W,P’-l&P+> ~--ii2p)s(4)(P:+P;-P1-P*)v(P;,P;,P1,P2) (A,6b)

=-i(2~)S(E(p’,)~E(P;)-E(p,)-E(p*))(p:,p:l~lp*,p,), for the NN-scattering S-matrix in the Born OBE approximation. coincide in the non-relativistic limit. The nuclear

e.m. operators

in the momentum

space are defined

Again,

V and

V

as follows:

(P;ijlh(l,k)fP,)=jn(l,p;,p1)6(kt+pl_pj) (A.7a)

=j,(2,

k)W-+-p~-t-p~-p:

-24,

(A.7b)

where in eq. (A.7a) we put k + k, = p; -p1 to distinguish it from the momentum k in eq. (A.7b) fixed by the two-nucleon momentum conservation law. Obviously, in a matrix element (pi pi lj*h(1, k) Ip,pJ and k 1-3 k (since pi = p2). As indicated in eq. (A-7), we omit the explicit dependence of j, on the final and initial momenta where possible and use the shortened notation j, (1, k,), jh (2, k).

581

J. Adam, Jr. et al. / Electromagnetic OBE currents

From the interaction and their matrix

lagrangian

elements

densities

(2.1) the corresponding

vertex functions

are defined:

(N(P’)li~gNNIN(p)BI~:(q)) =g,W)Pj;:(B, = gJ$:(B,

s)u(p)&~~‘,(q)S’“‘(p+q-p’) q -P’) ,

P’, p)e:;:(q)6’4’(p+

(A.8a)

r^(a) (s) = i(F),

flp’k

the following

k 4) = Y, -g*,,q, [

vertex functions

for the mesonic

vertices,

THE

yh y5rc,

= (a-

~~~;(v,

91, q2) = (4, - 42)*&p + q*J*,

(A.lOa) (A.lOb)

-q&,,

yBB vertices.

VECTOR

MESON

B

DOMINANCE

F:(t)

= &Uk),

is the isoscalar

I?;(P,

= %,vW(

(isovector)

and with the anomalous

= F;(t)

F:(r)

F:,“(t)

into Feynman

amplitudes

via the

model. In this case, the OBE amplitudes

FT”( 1) = F;‘“(t)

nucleon;

B = s, ps ,

qJA\,

qr,qd

In ref. ‘*), the e.m. FFs have been introduced

xS(xv)

(A.9a)

CYB,

vector meson dominance (VMD) dynamical equal to those of sect. 2 with (t = k*):

where

(A.8d)

and

Appendix BEYOND

(A.8c)

= &rr,,P,

fC,(ps, con) = -k yNNB

CT”),

are used:

fC,,(v,con)

for the contact

1

q) = -&ByS(TO).

P(ps, Further,

(A.8b)

(B.la)

= &L(k),

(B.lb)

t) ,

(B.lc)

anomalous

y~yppcoupling

)

magnetic

(compare

momentum

of the

to eq. (A.lOb)):

41, q2)=(4,-42)nsv~++*~(q2+k)v_Shv(ql+k)~L,

(B-2)

582

.I. Adam,

Jr. et al. ,J ~le~tro~a~netic

UBE currents

It has been pointed out in ref. I*), that only eq. (B.la) invariance of the theory; for Fs;l( t) a phenomenological the VMD form (B.lb, w propagators comparison

c). Further,

can be omitted.

the full form of the VMD vertices

~~e*(~)~~~~(B, the full amplitude

k) ,

k) + (a,, f k&,J~2,)&(1,

D = p, w for the isovector ~~(~)~~~~~(B,

Then,

Indeed,

in the p and

holds, the terms -k,k,

is (in

with the form used in sect. 2): &(l,

where

if eq. (B.la)

is important for the gauge fit can be used instead of

03.3)

part of 3, (1, k), respectively,

and isoscalar

con) -+ ~~A~(k)(~*~+ 41, qJ-$ +k$AW%,

~~~~/~~)~~~~~(B, + ~~~~/~~)~~~~(B,

con), a,&

and (B.4a)

-

Wb)

$A (k) reads: (B.5)

~,(k)-,(6,,+k,k,lm2,)4,(k)=$,(k). The equality in equation above follows from the gauge can be verified with the help of the identities:

invariance

of 9, (k), that

kh~~(l,k)=i&,~=Z,[S-‘(p’)-S-‘(p)],

k=p’-p,

(BAa)

kJ’$,~(~, ~i’,,(B,

con) = &(B,

k) ,

(B.6b)

B=s,ps,

k,T~fB,q,,q,)=q:-q:=A,‘(q,)-A,‘(q,), 91, qA = 6&+

kJK;(v,

s:, - evq+

= &(qd R,(q)

= &,A+-

(B.&f

-q2) = i=;I,,(B, st) ,

k)+ r”;JB,

- R,(q,)

+ qzuqze

9

q2) - qz& = K;(q)

&s,(q) = A,(qN&,

(B.6d)

+ sssph’v)

3

.

(B.6e)

Let us now refrain not only from eqs. (B.lb, c), but also from the constraint (B.la). We wifl suppose, that the VMD predicts the correct Lorentz structure of the photon-hadron vertices, but that the scalar r-dependent functions may be replaced by arbitrary

phenomenological

ones: 1 -F(t)

&A,(k) where F(t) Substituting

--, F(t),

Adkby,

is Fs:$(t), Fz”“( t) or FC,““(t), according (B.7) into eqs. (B.3) and (B.4) one gets: &(l,

k)mod = &(l,

k)+ii+,),

to the vertex

?=+(l+T:f,

03.7) considered.

(B.Sa)

1 - FF(

P:(,,(B,con),,,=F’,O”(t)j’“,(,,(B,con)+k,

t

t) * &,(I%

con), (B.8b)

1 - FF’“( t) TY(B,

91, &mod = KY”(~)~Y”(B,

91, q2) + k,

t

C&422),

B = s, ps , (B.8c)

Divergence

of these modified

vertices

is the same as that of the bare ones. Hence,

the total amplitude is gauge invariant for arbitrary FFs F(t). The fact that the relativistic amplitudes may contain the arbitrary e.m. FFs has been discovered recently by Gross and Riska 13). In their paper the photon-hadron vertices had been modified phenomenologically to a form satisfying the WardTakahashi (WT) identities 31). Several points are to be stressed to clarify this result and its impact on the actual MEC calculations. (i) The WT identities are satisfied in any field theory based on the gauge invariant lagrangian. However, Gross and Riska 13) have not sought for the most general form of the vertices, conforming with the WT constraint. Therefore, some models may lead to an alternative description. In what follows, we will compare the results of ref. r3) with our prescription (8.8) and discuss for which vertices more (or less) general parametrization is suitable in actual calculations. (ii) The ?BB vertices, given in ref. 13), seem to be rather general.

For scalar and

pseudoscalar exchanges their form fully coincides with that of eq. (B.8c) (apart from the effects due to the strong BNN FFs, that are discussed below). For a vector (U # p) case eq. (B.8d) agrees with eq. (6.22) of ref. 13) if: p*=1,

Q*=O, K(r)

for p-meson

comparison y”=2,

=O,

F,(f)=F,(r)=F:““(t), K(f)

= I ,

(B.9a) (B.9b)

leads to: Q*=-1, 5(f)

= 0,

F,(t)=F*(t)=F;‘“(t) F4(f) = I 7

F2(t)(~*-l)-sF~es(t)-l.

(B.lOa) (B.lOb) (B.lOc)

The eq. (B. 10~) concerns the part of the ypyppvertex proportional to [ S,,k, - S,,k,]. It seems that the new FF F5( t) [ F5( t) = FFes( t) - 1 in our model] should be attached to this term, rather than F2( t>( p* - 1) used in ref. I’). Being transverse, this term does not affect the WT identity.

584

J. Adam, Jr. et al. / Electromagnetic OBE currents

Since little is known

about the individual

e.m. FFs entering

the yBB vertices,

VMD ansatz (B.9) and (B.lO) may be useful in actual calculations. only the pion

FF Fr’“( t) is available

(iii) The contact

yNNB

vertices

the

To our knowledge,

these days 32). are in ref. 13) modified

occurs to be the same as eqs. (B.8a, b). Again,

little is known

in a minimal

way, that

about the FFs FT”( t),

with the exception of the pion vertex yNN+r. In the soft-pion limit, current algebra predicts the axial FF FA( t) at this vertex (see e.g. ref. “)). However, in the lagrangian realization of the hard-pion method this FF arises from the non-potential rA,p diagram 34) (this fact was overlooked in ref. “)). Chiral invariance of the OPE amplitude demands F”,““(t) = Fr( t) [see ref. “) and appendix D]. (iv) In this paper, as well as in ref. 13), the Dirac form of the ?NN vertex (2.2) with the minimal modification (B.8a) is used. A suitable choice of this vertex has been discussed extensively (see e.g. refs. 3,‘2727735,36)). Particularly, it was shown 36), that e-p scattering can for f s 1 GeV* be described satisfactorily by a (non-relativistic) current: P(L

k,) = Gzw

(B.lla)

3

(B.llb) where the Sachs form factors

are given by:

h(r) =;[Gs(t)+

ova:],

(B.12a) (B.12b)

‘S.,(t)

=

F,(f)

+

F*(t)

.

(B.12~)

Then to maintain the continuity equation, the exchange currents were multiplied by G’;(t) 36). Let us inspect this approach on a level of relativistic amplitudes. The current (B.ll) follows from the lowest order non-relativistic reduction of

g(t) = (1 + t/4m2)_’

)

(B.13)

where G(t) are considered to be of the order (l/m)“. It is easy to show, that the Born amplitude (2.3b) with the one nucleon operator (B.13) does not contribute into j”‘(2, k) for any meson exchange (not even for the PS rrNN coupling). Since the operator (B.13) does not satisfy WT identity, an additional non-minimal contact amplitude must be postulated for saving gauge invariance. This amplitude yields j"'(2,k) only for the PS qNN coupling, the current being given by eq. (4.3d) with FF( t) + G;(t).

J. Adam,

Ahernatively,

Jr. et al. / Electramagnetic

OBE

585

currents

one may add to the operator (B.13) an off-shell longitudinal term (B.14)

that restores the WT identity. After the non-relativistic reduction these terms yield onty the fongitudinal part of the pion exchange current (4.3d) (with FF(t) + G;(r)) and a longitudinal correction to the Born current (corresponding to eq. (Bt6b) below with Fy( t) + Gg( t)). To sum this point up, the theory with &( 1, k) of eq. (8.13) does not allow independent choice of the contact and mesanic FFs. The non-minimal contact term has to be added. The corrected WT form (B-13) and (B.14) does not reproduce the important nosy-relativistic pion current (4.3b). Moreover, it is difhcult to formulate an effective hadron Iagrangian Ieading to (B.13) and fB.14) and respecting the chiral symmetry. Consequently, we consider the Dirac form of 9, (1, k), appearing in a natural way in the VMD model, to be a more suitable one. (v) With the help of the identities (B&), the relativistic amplitudes with the modified vertices (B-8) are cast into the form: (B.15)

3, (k)mod =&(k?+A.YAaA(k), where $n (k) are given in eqs. (2.3) and A9,(1,

x ~(p:)f;,,(B, 64, (2, k, B-mes) = - kA

l-FT(t) I

(B.16a)

k)=O,

k)ut~W:,,(B,

P;,P~-+

l*2

t

(B.16c)

ig2,8’CA,(q,)

x f,‘,,(B, 14,PW:~B, pitPZ>+ l-2 .

jB.lt;d)

Due to eq. fB.l&) the one-nucleon current remains the same as before. Hence, the calculations of the MEC operators made in sects. 3-S are unchanged, only the non-relativistic reduction of the amplitudes (B.16) is to be added to j,(2, k). The longitudinal ampfitudes A.FAP,f2,k) - kh feq. (B.16)) can be omitted from the fully relativistic calculations 13). However, this is not so in the Schroedinger description based on the decomposition with respect to l/m. Since k, - k/m, in each order (n) of the non-relativistic reduction A.9A{2,k) yields Aj’“‘(2, k) and Ap”“+“(2, k), whereas the nuclear continuity equation (1.7b) connects Aj’“‘(2, k) and Ap’““““j2, kf.

586

.f. Adam,

Jr. et ni. / Eleetromagneric

Ui3E ctirrems

means, that neglecting A4,(2, k) one violates the gauge invariance of the nuclear amplitude T,(k). Thus, adding the lowest order non-relati~sti~ reduction

This

of eqs. (B.1B) one gets*:

Ap”‘(2, k) = 0 r Aj”‘(2,

k, B) =

i(~,

X ~~)~k/k’[F”(

t) -

F”(t)]

(B.17a) 9g’(q2)

i-

l-2,

B=s,v, (B.17b)

It is easy to show, that these currents recover the continuity equation in the order I/m when the arbitrary Fr and FF are used in eqs. (4.3). Let us point out that

being longitudinal (-k) these currents do not contribute to the magnetic multipole operator. Their contribution to the electric transitions is non-zero only if the electric multipole operator is (with the help of the continuity equation) rewritten into the Siegert form (see e.g. ref. “‘)). (vi) Gross and Riska have introduced the strong BNN FFs into the relativistic amplitudes via a straightforward generalization of the WT identities 13).To simplify our derivation of the MEC operators we have used the monopole parametrization off,(g) (2.7) and the d~~orn~osition (2.8) of the OBE amplitudes into the sum of the terms with the point vertices. This decomposition follows from the results of ref. 13) with the off-shell FFs given by: y(C

4:, SZ) = &I:,

42)

?5(C s:, 4:) = B(q?, s:)

for B=s,ps,

(BlSa)

for B=v.

(B.l&b)

For a general choice of FFs &(q) the Taylor decomposition of the BNN vertex functions around qz is to be made (see e.g. ref. “5))a From the additional q. dependence of the BNN vertices new currents j,(2, k, ver) (3.15) arise. Let us point out that since (see ref. I’)): C(&,

cr:, f I%& 4:) 7

(B.19)

in the v-mesonic amplitude, the decomposition (2.8) does not apply to the part of $,, (2, k, v-mes) C: or yi-, In the OBE approximation these terms become transverse and no constraint on the FFs can be obtained from the demand of the gauge invariance. The approach of ref. 13) proves to be more powerful at this point. * Eqs. (6.17) were obtained independently by P.U. Sauer and collaborators 39) using an extension of the definition of MECs given in appendix C of ref. 36) and non-relativistic reduction of the Gross-Rkka approach 13).

J, Adam, Jr. et al. / Electromagnetic

587

OBE cwrenis

Appendix C

THE OBE POTENTIALS

In this appendix as described

the explicit

expressions

in sect. 1. For completeness,

for the OBEPs the commutators

are presented,

the charge density, needed for the verification of the continuity equation The connection of our OBEP with the Bonn OBEP(R) is discussed. In the non-relativistic limit the OBEPs read:

=

with

are given.

2

2 V(4)

derived

of the potentials

-&“h),

?‘(d

=+4(l)

(C.la)

,

i

q&d = - (2n;:;,,A,s(q)(u,

1 q)(uz.

4).

(C.lb)

The non-relativistic potentials are Galilei invariant and depend only on q = q, = -qz (qi -pi-pi), hence V”‘(p;, pi, p1 ~p2) = V(‘)(q) (see eq. (A.6b)). For exchange with isospin T = 0 V = q, for mesons with T = 1: V = (TV . Q) c (Anti)commutators of the potential ?(I) with a one-nucleon operator a^(1, k) are given by:

where

{Q}=f{i,7,

.7*}{~~}-~[i,7,,~,][~a’],

(C.2a)

[C (;]=;{?,I,

.1*)[~~]-~[i,ll’121(~,~3,

(C2b)

a^= i%, f contains (Pi, PiI ci*

the isospin

dependence

&Qlp,, P2) = [Ictp:,

of a^. In the momentum

P’, pi 1PMl,

space:

P’, PI)

~a”(l,~;,P)~ii(p:,p:,P,p,)16(k-q,-q,), where P’=p, -q2=p; - k, P=p;+q2=p, + k (see v”‘: i’yp:, p’, p,, pJ = iyp;, p;, P, pJ = ?‘)(qJ. For 6(1, k)+p^(l,

fig. 2).

In

particular

(C.3) for

k) one gets with the help of eqs. (4.la, b) and (C.3): (P;, P:/[ cc’), pl’“‘(L

Wlh, pz)= -F; +%zJ ,

- I=:[ ik x Q, + V, - ik x q2 . V:])

(C.4a)

,

(C4b)

where F: = F: + 2F; (eqs. (4.5) and (4.6); 6, + Gl for F:), Vl: are defined in eqs. (4.1%) (Vi = 0 for B = s, v) and the &function from eq. (C.3) will be omitted from now on. The (anti)commutators from eqs. (3.14)-(3.17) are calculated analogously.

588

J. Adam, Jr. et al. f Electromagnetic

In the next order in l/m

the non-relativistic

OBE currents

reduction

of Y( pi, pi, p, , pJ yields:

Fi3’(q, Q1, Q2)=-~Qllr(q)(0:+Qi+iulxq.

Ql-icr,xq+

Q2). (CSa)

~tl’(q,Q,.Q*)=-~~hl’(q){(1fw,)(rlXq.uzX4-(l+2K,)q2 +Q1.Q2-i(S+K,)[u,Xq.Q1-u2xq.Q21 +i(l-+-K,)C~lxq* gC3)(q PS 3 Q,

, Qz) =

Qz-uzxq. Q:+ Q:)(u,

~Z~~2m~“B’(V)i(~~‘i

(CSb)

Qtl}, * rI)(oz ’ 4)

+ (a, 9 q)(az . Q&q * QJ + (~1 * QI)(w

. q)(q.

QAI

, (C.5c)

where we have used the momentum conservation and expressed VC3’( pi, pi, PI, pJ in terms of q, Qi = pf tp,. The commutators of $‘-(3)with p(‘)( 1, k) are given by (C.3): (Pl,

PllW3’,

Wllf-6, P2)

b’“)(l,

= Ft[ c”3’(-qz,

Qi+ k Qz) - q’3)(-qz,

- F;[ p1(3+q2, (P:, p;i[%‘,

i+“‘Cl,

Q, + k QJ+

+3’(-q,,

QI - 4 QAI 91 -k, Q&l,

(C.6a)

W/p,, PZ>

=&~~“(q,){F;(-2Q,~k+iu,~q~*k) +F~(Qf+Q~+k’+iulxQ,*q,-iuaxQ2*q2)}, (P:, P&IL%3’, ;‘“‘(l,

k)lb,

(C.6b)

,PA

=-~~~‘(q~)~F~[k.

Qzi-i($+Kv)a,xq2.

k+i(lfK,)CTZXq2.

k]

-F;[(1+2K,)q:+(l+K,)“U,Xq,‘u,Xq,+Q,’Q, +~(~+K,)(ET,

Xq2.

-i(l+K,)(u,xqz* (P:,

pzj[ +b”s’, 6’“‘(L

k)lbl,

9, --~xq2 Qz--uzxqz.

Q,)l),

(C.6c)

A * qa)(az * qz)(k . QI)+ (a, * k)(oz . qz)(qz * QI)

= (2~~~;,m,A,,(a){F:[(u~ + (-I . Q,)(uz * q&k. - KE(@l

+ Q2)

41

- qz)(uz . q&&z: + k2+ Q:+ 0:) + (~1 1 qz)(az - Q&q2 s Qz)

-c (~1 * Q,)(uz . a)(qz

. 91) + (a, 1 k)(oz - qz)(k * 41)

.

(CAd)

J. Adam, Jr. et al. / Electromagnetic OBE currents

The retardation commutator M,

potential

p

follows

immediately

589

from eqs. (1.1) and (Cl).

Its

with $“‘( 1, k) equals:

P;l[I 69,

WllPI>PJ

$“)(l,

Qd+ $%,z

- fTL(e. QJ(e.

*(QI - Qd2+W

qd*)l . I

(C.7)

For use in the Schroedinger equation, it is necessary to separate from the hamiltonian the centre-of-mass (c.m.) motion. In terms of cm. variables P= P’=p,+p?=p;+p;, P’=tcP;-P;L the potential

Q =p’+p. P=!dP,-Pd

(C.8)

9

Vc3) reads: V’3’(q,

Q, >QJ + V’3’(q,Q, PI = V’3’(q,Q) + AV’3’(q,Q, PI .

cc.91

The potential V’“‘(q, Q) enters the Schroedinger equation in the cm. reference frame (A V’“‘( q, Q, P = 0) = 0). For individual meson exchanges: iir8et(q, QJ=$-$

%3’(lr,

~%z, Q)=$

Q)

=

B s(q)(q’ Q12, ~%)A

-$-g

%“(qWQ2+

?“(q){(l+2K v

Y

-t- (1 + Ky)‘[ m;(U,

.

Q)

q”‘(q ps

1

=-.+‘l)(q)( 2 4m

PS

+02)

. q x

Ql

(ClOa)

,

(C.lOb)

)m2+ Q' ”

a*)+

+i(;+2K,)(U,+u>)

i(u,

B=s,v,ps,

(ut

- qX

* 9Ha

*

a)1 (C.lOc)

Q},

* _ Q")

mF

i -+-(2$Y2m

,A,Aq)(q.

Q)L(o, * 41. (~2

-

Q)+ (~1. Q)(oz . q)] . (C.lOd)

The P-dependent

parts of the OBEPs

A %“k, Q, PI

read:

=& ~‘(B1’kMdq)(q~ P)‘,

B=s,v,ps,

(C.l la)

590

J. Adam,

Ap”,)(q,

Q, P) =

Jr. et al. / Electromagnetic

----&i33q)[P2+;i(u,

OBE currents

-u2)

. q x P] ,

B=s,v, (C.llb)

2

Q, P) = ~2~~;,mz,~,.M~~

A+$%,

+

(~1.

f’Xa2

+ (~1.

QHu2

+ (~1

To define the

the proper

potential

Foldy

constraint

. Wu2

* q)(q

. q)(a,. * PI

. q)(q.

. q)(q

approximately

-(a,

PI

*

-

Q))

qP2+(-, * rl)(a,

(~1.

q)(u2

. q)(uz. P)(q- P)

*

Q)(q. PI

. PNq.

Q)

(Cllc)

.

Lorentz-invariant

Schroedinger

A cc3’( q, Q, P) = A cc3’(q, Q, P) + A Qref(q, Q, P) should

dynamics, satisfy

the

“): (C.12a)

xo=-~[(r.p)(p.P)+(p.P)(r.P)+(u,-u,)xp.P], where xv is related

to the interaction-dependent

(C.12b) part of the Lorentz

boost.

It can

be shown that the potentials (C.lla, b) with the scalar and vector exchange satisfy (C.12) with xv = 0. The form of xv for the pseudoscalar exchange has been discussed extensively by Friar 8722*25). The presence of AVet is essential for the validity of eq. (C.12). It should be pointed out that neither AF’, nor xv depend on the retardation parameter v. Our OBEPs for B = s, v are the same as those given by Olsson and Miller “) (for v = l), Towner

“) (who has neglected

retardation)

and Friar 22) (for v = 1). As given

above, the pseudoscalar potential corresponds to that of Hyuga and Gari 2”) with v = 0, c = - 1 (PS rNN coupling; as well as Friar 8), we have got a different coefficient of ref. 26): 3/402+ at the term -(3/402)[ Tr - T2, [T, - T2,. . .]] in the potential 1/4w2), and to Friar’s potential 8,25) with p = -1. In our definition the pseudoscalar potential does not depend on the type of the psNN coupling (pseudovector (PV) or pseudoscalar (PS)) used for the calculation of Y The point is, that the relativistic amplitude v is continued off the energy shell only with respect to the q. dependence of the meson propagator. The non-relativistic reduction of the BNN vertices is being made for the nucleon spinors on the mass shell and for the full four-momentum conservation at each vertex. One may also continue the dependence of the BNN vertex on the energy transfer off the energy shell (in a way similar to eq. (1.4) with the independent off-shell parameter). In the order considered, only the pseudoscalar potential would be affected by this additional off-shell dependence. Instead of making the off-energy continuation of the psNN vertex for the arbitrary mixing of the PS and PV couplings (appendix D), one can use Friar’s results and

591

J. Adam, Jr. et al. / Electromagneiic OBE currents

introduce

this additional

unitary

freedom

with the help of the p-dependent

transformation i’, 25). Let us now compare our OBEPs with the Bonn ones 14). Our Schroedinger for the two nucleons

equation

in the c.m. frame reads: W

where kinetic

unitary

= [2($-c

m2)“2+

mj

= L.l~) >

(C.13)

c is a sum of the potentials (C.10). The non-relativistic reduction term in eq. (C.13) leads to the RC - p4/4m3 that makes the spectrum

of the of the

hamiltonian unbounded from below. It is, therefore, more suitable to transform eq. leads (C.13) as described in refs. 25738).In order l/m3 considered this transformation to the following equation: (C.14a) (C.14b) c.m. energy. Let us point where E,., is now the non-relativistic function 14) in eq. (C.14) is the same as that in eq. (C.13).

out that the wave

It is the transformed potential qtl(3), that is to be compared with the Bonn OBEP(R) [in ref. 14) the kinetic RC-p4/4m3 has been transformed out with the help of the “minimal relativity” redefinition]. Since oh’s’ is omitted in ref. 14), the comparison can be made only for the scalar and vector potentials. Apart from the last quadratic term in eq. (C.14b) our transformed OBEP for B= s, v and for the retardation parameter v = i coincides with the Bonn one. It would be interesting to refit the Bonn OBEP(R) with the ( ?1))2/4m and es’ included. Then, the MECs derived in this paper would be fully consistent with the realistic NN potential. Nevertheless, the comparison shows that the conventional NN wave functions contain already a certain part of relativistic corrections.

Appendix

PS AND

PV COUPLING

D

FOR THE psNN VERTEX

It has been shown in ref. i2), that an arbitrary coupling can be obtained from the lagrangian redefinition of the nucleon field +‘= exp (i*% d=(7*@)

for T=l,

mixing of the PV and PS psNN (2.1) with the help of the local

y,&)*, 6=@

for T=O.

(D.la) (D.lb)

592

_l. Adam,

Jr. et ai. / Ei~ctro~nag~~t~c UBE curreents

The relativistic OBE amplitudes non-minimal transverse contact

do not depend on the parameter terms are included “):

A, provided

the

2

A9,(2, k, ps, A, con) = -iAc2T;&m2

(D.2a) Fb(t)=F~(t)7:,+F~(1)63b

for T=l,

~(~)=~~(~)+~~(~)~~

for T=O.

(D.Zb)

In ref. 12), the transformation (D.1) was discussed from the point of view of chiral invariance of the OBE amplitudes. The A-independence of the relativistic amplitudes can be, however, used to simplify the calculations of the MECs for an arbitrary ps meson. In fact, this property has been used in the calculations of sects. 3,4. Indeed, the relativistic amplitudes (2.3) were obtained from the lagrangian (2.1~) with the PV psNN coupling. On the other hand, the pseudoscalar potential derived from the amplitude (2.9) corresponds to the PS psNN coupling (see appendix show, that our MECs for the ps exchange are nevertheless correct. The transformation

(D.l)

leads to the following

i”“‘(P%A,q)= tW)WPs,

[

A, q)u(p)

psNN

vertex:

1

A%-&(1-A)&

W),

= r’“‘(Ps,P’,

P@)‘(ps, A) = ap”‘“‘(ps, A, 4)/a&

C). Let us

P), l-h

= 2m

9==P’-P, yz+ys(Ta)

=(l-A)P’a)‘(p~,A=O). The matrix element vertex.

Since

dependence

(D.3b)

q. = E(p’)

(D.3c)

is taken for the full four-momentum

- E(p),

the Dirac equation

of the vertex, transIating

(D.3b)

conservation

can be used to remove

it to the (A-independent)

in the the

q.

PS form. To get the

relativistic amplitudes for arbitrary A, one should insert (D.3a) into eq. (2.3a), multiply the contact term (2.3~) by (1 -A) ( see eq. (D.3c)) and add the transverse contact amplitudes (D.2). Let us point out that these amplitudes are now added for arbitrary ps meson just to simplify the calculations of MECs. For the pion, their presence (as well as eq. FC,O”(t) = Fy( t)) follows from the chiral invariance principle. For other mesons (e.g., n-meson) one is free to use the theory with PS psNN coupling (A = l), A4,(2, k, ps, A, con) = 0 and Fp( t) # F;(t). However, there is no information available these days, that would fix the correct form of the vertices with heavy pseudoscalar mesons. Hence, the pion form may be used; if necessary one may easily pass to any alternative description.

593

J. Adam, Jr. et al. / Electromagnetic OBE currents

With the help of the identities ~(~~~~~b~(ps, h,

-q,)P’(P)

l)S’+‘(P)+i?‘b”(ps,

= u(p:)[i?‘h’(ps,

S’+‘(P’fP’h’(ps

A -&(p

= ~s”‘(d;f~~‘(ps,

(D.4a)

h)u(P)ti(P)],

I) 1) - ~~~~)~(~~~~~~~~(ps,h)]u(p,)

,

(D,4b)

the positive Born amplitude 9,,(2, k, ps, h, Born+) with the general psNN vertices Pch’(ps, h, -q2) can be cast into the form: 4,(2, k,ps,h, Born+)=&(2,

k,ps, 1, Borr+)+(l-h)j,,(2,

k,ver),

(D.5)

where j,(Z, k, ver) is given by eq. (3.15) and = stands for the non-relativistj~ reductions up to the order considered. Let us point out, that j, (2, k, ver) in eq. fD.5) appears when the go dependence of the psNN vertex is removed. Furthermore, it follows from sect. 3: $,(I& k, ps, 1, Born+) - tlA(k) =&(2, k, ext) +j,+(2,& ret) ” Finally, from eq. (D.5) and the ~-independence

fD.ftl

of the total amplitude r*), one gets

9, (2, k, ps, A, Born-) + (1 - h ),a, (2, k, ps, con) + A9,+ (2, k, ps, h, con) =.9,(2,

k,ps, l,Born-)-(l-h)j,(2,

k,ver)+A.9a,(2, k,ps, I,con).

Therefore, the Born and contact pseudoscalar

meson-exchange

(D-3

currents equal:

j, (2, k, Born + con) = .9x(2, k, ps, h, Born-) + (1 -h )Jp&(2, k, ps, con) -I-A$* (2, k, ps, A, con) + (1 - A)j, (2, k, ver) + j,(2, k, ext) +jk(2, k, ret)=9$,(2,

k, ps, 1, Born-)-t-A&(2,

+ j, (2, k, ext) i-j,+(2, k, ret) =

k, ps, 1, con) @W

The last equation, containing the MECs for the PS psNN coupling, show, that j,(2, k, Born 4 con) does not depend on A. Hence, an arbitrary A can be used, while the non-relativistic reduction is being done. The choice h = 0 simplifies the ealculations considerably. The ~-independence of the MEG is due to our definition of the pseudoscalar potential. When the off-she11 continuation of the psNN vertex is taken into account, the potential with PV coupling differs from that with PS one. To change the MECs accordingly, one can again use the unitary transformation given by Friar 25).

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594

J. Adam, Jr. et al. / Electromagnetic OBE currents

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