Backward deuteron electrodisintegration

Nuclear Physics A492 (1989) 529-555 North-Holland, Amsterdam

BACKWARD

DEUTERON

ELECTRODISINTEGRATION

E. TRUHLIK

and

J. ADAM,

Jr.

fnstitute of NM&W Physics, CS-25068 de.5 n. Prague, Czechoslovakia Received

30 June 1988

Abstract: The double-differential cross section for deuteron electrodisintegration at large momenta transfer in the chiral scheme containing TI;p and A, mesons is calculated. The effect of static relativistic corrections to exchange currents is studied. Other heavy-meson exchanges (w, 8, CF,7) present in the Bonn potential are also considered.

1. Introduction The reaction

of backward

deuteron

electrodisintegration

at threshold,

e+d-+e’+(np),

(1)

has intensively been studied for several years, since it provides rich information on, and clear evidence for, the presence of non-nucleonic degrees of freedom in nuclei ‘-‘). The experimental data concern the double-differential cross section of the highenergy electrons scattered at backward angles (typically 8 2 IW’), the (np) pair moving forward with relative energy I?,,,, = O-3 MeV. Under such kinematical conditions the square of the four-momentum transfer t can be very large 6-1”). The latest Saclay data lo) cover the region 6.6 fm-‘s t s 27.82 fme2( 0 = 155’). It has been known for long ‘II that the calculation of the electron spectra in reaction (1) based on the impulse approximation (IA) does not yield even a qualitative description of the experimental data. This discrepancy inspired a series of theoretical investigations based mainly on the standard picture of meson exchange current (MEC) models “-18). Together with the single-nucleon current j^,(l), the two-body MEC j:(2) is to be introduced so that the nuclear continuity equation k*j-[&(l)], j*,(k) =j^,(L

@-tjl,(2,

(2) k)= (3(k),

IL?+?,

i@(k)),

(3) (4)

is satisfied. Here the hamiltonian fi [with ?(?) the kinetic (potential) energy operator] enters the nonrelativistic Schroedinger equation from which the nuclear wave functions (w.f.) are obtained. In the non-relativistic limit only isovector MECs contribute to the left-hand side of eq. (2). The most important ones are those corresponding to the exchange of YT- and p-mesons. The model including the 037%9474/89/$03.50 @ Eisevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

530

E. Truhlik, J. Adam,

Jr. / Backward

Galilei-invariant n-- and p-MECs isovector transitions. The main

difficulty

the calculations.

encountered

At the momenta

(i) both the electromagnetic the shape of spectra.

deuteron

has become in interpreting transfer

electrodisintegration

the standard

for calculations

the data concerns

of the

consistency

of

considered,

(e.m.) and the hadron

form factors

The choice of these form factors

strongly

poses considerable

change problems

because of the absence of an elaborated fundamental theory. (ii) A nonrelativistic description not be sufficient. The standard may calculations “,“) are, in fact, not purely non-relativistic - the p-MECs used are of relativistic origin, the realistic NN potentials include selected relativistic terms. The problem frequently discussed in the past few years is that of the choice of the e.m. form factors “5~‘o~‘6~2s).For high momenta transfer, the difference between the Dirac form factor FT and the Sachs form factor Gg F;(r)-G;(t)=-& is no longer

negligible.

The standard

F;(t) > non-relativistic

calculations

‘6,‘7,‘y*20)agree

with the data “‘) only if the MECs are supplied with the form factor Fy, though the picture changes after using the potential generating smaller admixture Pu of the deuteron D-state “) - agreement with the data is obtained with the currents multiplied by Gg instead of Fy. An argument in favour of this step stems from considering a special form of eq. (2) with charge density p^(1) containing the static relativistic correction proportional to Gz. Then according to refs. ‘6,24) the MECs should also be multiplied by GE. However, such a solution of the problem is only of restricted value. More general schemes exist 1,22,:3,26,27 ) allowing to construct the MECs satisfying eq. (2) but not explicitly multiplied by Gx. The chiral model 22) yields the one-nucleon e.m. current This form of the current has the correct off-mass with the Dirac form factors FT,. shell behaviour “.I’). The constructed MEC 22) also contains these form factors unavoidably. Only its longitudinal part (proportional to by the continuity equation and can be redefined 27) so the one-nucleon current satisfy eq. (2). The e.m. form As has of ref. 22) are dictated by vector dominance.

Fy) is nontrivially restricted that the MEC together with factors entering the MECs been shown by Gross and

Riska “) within the framework of the two-body Bethe-Salpeter and Gross quasipotential formalisms, the e.m. form factors are present only in the transverse part of the MECs. We show in ref. 27) that our currents can easily be generalized to the form obtained by Gross and Riska. The most elaborated analysis of the electron spectra for reaction (1) has been performed by Leidemann and Arenhiivel 16). They considered several realistic NN potentials and the contributions from multipoles up to L = 4. They chose Gg as the nucleon e.m. form factor in the MECs. On the other side, Mathiot “) considered a somewhat simpler but still realistic model (contribution only from the ‘So np state, Paris NN potential, Ml transition) of the two-nucleon system. With the Dirac form

531

E. Truhlik, J. Adam, Jr. / Backward deuteron electrodisintegration factor

Fy

parameters

and for the specific values of the nuclear A,, A, Mathiot

A, = 1.25 GeV corresponds

monopole

has been able to describe to the nucleon

r.m.s. radius

form factors

(cut-off)

the data. The deduced

value

r. = 0.48 fm. The tail of the

and, in spectra for t 3 20 fmd2 was found to be sensitive to the meson parameters particular, to the short-range effects connected with the hadron form factors. In order to avoid the problem of the choice of hadron form factors, Riska “) constructed a generaliz;d isovector ~(0~)- and p(l-)-like MECs so that they satisfy eq. (2), where for V stand spin-spin and tensor components of the realistic NN interaction. Similar MECs were introduced also by Buchmann et al. *“). These authors keep also the MECs associated with the momentum dependence of the potential, which makes the construction inconsistent *‘). Besides, the Paris potential 29) used in calculations, contains explicitly a contribution from the A,(l+) meson. However, the procedure of ref. “) does not allow to construct the corresponding MECs. As we shall see later, these MECs affect non-negligibly the tail of the electron spectra. The process (1) has also been studied in the schemes 30) based on the quark bag ideas, which enable one to describe the short-range effects through the quark bag components in the wave functions. The calculations show that for the momenta transfer ‘“) considered the specific quark effects are negligible. In this paper we restrict ourselves to the conventional meson-nucleon description of both MECs and potentials with the form factors introduced phenomenologically. The presence of the relativistic terms in the NN potentials 29,3’) indicates that the nonrelativistic description of the nucleon e.m. current may not be sufficient in the region of the considered momentum transfer. In ref. *‘) we derived the MECs for the general scalar, vector and pseudoscalar exchanges that include leading relativistic corrections to the non-relativistic currents. The following features of MECs are present in our model: (i) The standard p MECs “) are, in fact, the relativistic correction to the leading spin-independent current. (ii) In the order -l/M’ the MECs due to the exchanges of neutral mesons (c, w, . .) contribute to the isovector exchange current y&(2). (iii) When the relativistic corrections are included, both Dirac e.m. form factors F;(t)

and F;(t)

appear

in MECs,

(5). The use of the Dirac relativistic e.m. amplitudes.

form

however

not in a combination

of the one-nucleon

current

of the form of eq. is enforced

by the

(iv) The z--MECs contain an amplitude of the pion electroproduction on the nucleon. This amplitude is properly approximated by a sum of the nucleon Born term, and contact and mesonic contributions (fig. 1) in the non-relativistic limit. In higher order the non-potential part is required by chiral invariance (low-energy theorem). The improvement of the calculations of the MECs effects in reaction (1) employing these currents is the objective of the present paper. To make the calculations feasible we consider the same model of reaction (1) as in ref. I’): (i) Ml transition between the deuteron and ‘S, np states.

532

E. Truhlik, J. Adam,

;:-;-:j*

Jr. / Backward

deuteron electrodisintegration

_&-+,~yj

b

c

d

Fig. 1. Meson exchange currents with the exchange of B = ST,p mesons in the gauge chiral theory electromagnetic interaction yp(k) with the (NA,pn) system.

(ii) The velocity-dependent

terms in the currents

and the relativistic

of the

c.m. motion

effects are neglected. (iii) The monopole parametrization of the strong form factors is used. The cut-off parameters A, are fixed 3’) by the fit to data, except AA. Unfortunately, the A, meson is not included in the Bonn potential. Comparing to ref. 17) we are left only with one free parameter AA. In contradistinction to ref. 17) we test the sensitivity of the spectra For comparison parametrizations of the nucleon e.m. form factors 17*32*33).

to various we calcu-

late the spectra with Paris 29), Bonn 31) and Reid soft-core (RSC) 34) nuclear w.f. The structure of our paper is as follows. In sect. 2 we analyze the electron spectra employing the V, p, and A, MECs *‘) in the leading order in l/M (figs. 1,2). Additional currents due to the reasoning of ref. 26) are -k (momentum transfer) and do not contribute to process (1). We find a large effect from the part of the non-potential A,n current demanded by chiral invariance. Without it, our interpretation of data is of the same quality as that of ref. 17), if the Paris or RSC nuclear w.f. are used. It is not possible to describe the data with the Bonn 3’) nuclear w.f. with the currents

used so far.

b Fig. 2. Operator

of the A, MECs; (a), (b) - pair term; (c) - mesonic current; the only non-potential MEC of our model).

d

(d) - A,T current

(this is

533

E. Truhlik, .I. Adam, Jr. / Backward deureron elecrrc~disinregrutitxn

In sect. 3 we discuss the MECs derived mesons

the problem

of the importance

in ref. “). Here we also include

(c, (~,a, 77) present

of relativistic

a contribution

in the Bonn OBEPQ

and QBEPR

a large effect from the pion retardation

current

calculations.

both the retardation

The parameter

ZJentering

corrections

to

from other heavy models ‘I). We find

and stress the necessity

of consistent

MEC and the potential

can be fixed only for the Bonn OBEPQ and QBEPR (v = 5). In comparison with ref. *7) we are left only with one free parameter - the cut-off AA entering the A,NN vertex. With /IA = 2.2 GeV and Gari-Kruempelmann’s (GK) parametrization of the nucleon e-m. form factors from ref. “) we can describe a whole set of data lo) satisfactorily. Our main

2.1.

results

and conclusions

are shown

in sect. 4.

GENERAL CONSIDERATlON Here we give the ingredients

needed

throughout

the paper to perform

the calcula-

tions. In describing the electron spectra for the reaction (1) we restrict ourselves to the Ml transition from the initial 3Sr -3D1 deuteron to the final IS, np state. In the laboratory frame the double-differential cross section for zero mass electrons is given by formula I’) (6) t = k2 fc is the relative

CO’ = 4ki kr sin’ $e ,

np momentums

(final) electron momentum. In this paper we carry EnI, = K’/M = 1.5 MeV. We normalize

d is the electron

out

the np radial

o = kf- k.I 9

calculations

scattering

of the

spectra

(6) for

0 = 155” and

w.f. W(K, r) as

W(K,r) -rem sin (Kr-&) and define the deuterons

angte, ki(k,) is the initial

~

(7)

w.f. by 0:

Id)=, J z>,.

q

[ q(i)OXllrm~~,

dr[u~(u)+u~(r)]=l. I0

(8)

The nuclear wr.f. used here are generated from the Bonn “), Paris “9) and RSC 34) NN interactions. The content of the deuteran D-state admixture E)b of the Bonn NN interaction is considerably lower (PO =S4.81%) than that of the Paris (P,= 5.77%) or RSC (PD = 6.47%) potentials. The Bonn NN interaction correctly reproduces the experimental value a,, = -23.748 *Oo.OIO fm “) of the singlet np scattering

534

E. Truhlik,

J. Adam,

Jr. / Backward

deuteron

electrodisintegration

yielding anp = -23.75 fm. On the contrary, the Paris (as= -17.61 fm) and RSC (as = -17.1 fm) potentials are suitable for application to the nn system in the

length,

‘So state at low energies.

A recent analysis

a nn = - 18.5 f 0.4 fm. Unfortunately,

r-space

potential

available

X6)of the reaction

there is no modern

which takes correctly

F + d + 2n + y provides

soft-core

into account

or super soft-core

the charge

indepen-

dence and symmetry breaking of the NN force in the ‘So channel, though the situation might change soon 37). Inclusion of the monopole hadron form factors_& into the potentials and currents, .I&?) has been discussed

= (A’,-

&)l(q2+

mZ,) 3

(9)

in refs. 23,27).Here we specify our choice of the cut-off parameters

Aa and of the other constants. We shall use two sets of A,, meson masses and coupling constants. The first one (SM) corresponds to the choice of ref. I’), rrq, = 776 MeV, A, = 1.25 GeV, A,, = 1.5 GeV, gE/4r = 2.83, Xr = 6.6. The second one (SB) is in accordance with the OBEPR potential and is taken from table 14 of ref. ‘I). We also need a value of AA. Our choice is AA = 1.85 GeV for both sets of parameters if not stated otherwise. The mass

mA and

the A,NN-coupling

are given

in our model 22) as rni = 2m:,

g:, = g;.

2.2. SINGLE-NUCLEON

CURRENT

The single-particle

reduced

matrix

element

(r.m.e.)

& &

(‘SolIf’?,lld)=ill

is drw( K, r)

x [j,(fkr)u,(r)-Jtj2(tkr)uz(r)].

Here k = lk[ and the isovector

magnetic

(10)

form factor

G;(t)=F:(t)+F,V(t). As is well known,

the interference

between

(11) S- and D-wave

contributions

leads to

at t = 12.5 fme2 for the RSC and Paris potentials, for the the dip is shifted to t = 14 fm-* [ref. “)I. In this region of

a dip in the cross section Bonn NN interaction, four-momenta transfer,

the impulse-approximation

cross section

underestimates

the

data strongly.

2.3. PION EXCHANGE

CURRENTS

Riska et al. “) were the first who realized that the data can be described by 38,39) which resolved the long-standing discrepancy introducing the same r-MECs of 10% between the precisely measured total cross section, oeexp= 334.2 * 0.5 mb, of thermal neutron capture by protons 40) and its IA value 4’), (T,~ = 302.5 * 4.0 mb.

E. Tr~hij~, 1. Adam, In

the leading

the continuity

order in I/M

535

Jr, / ~~~~~ard d~ufertin clectrodisintegratioII

and without

em.

form factors

the W-MECs

(I21

LL(2)=lLWH, with the static one-pion density

exchange

potential 4 and the one-nucleon point charge with refs. 26.27), the e-m. form factor FY is used in the

p^“(1). In accordance

pair current of fig. la, b, In the pion-mesonic current (fig. Id) form factor F,(r) [refs. ““,“)]. As we have already discussed situation allows us to employ FY in this case and also for the (which does not contribute in the Iowest order, however). vertices,

satisfy

equation

the contribution

x:‘,=

from these currents

i I

seagull term of fig. lc For the point TNN

to the r.m.e. is

1

dva,e -“‘~,(+qkr) )

0

x5=

I

drl e--Jj,(:qkr)

constant f’ = 0.0792. of the strong factors

il@

(9) leads to the following

in eqs. (151, (16)

a epczr + a ewe’ - a., e -a~r+trf.~42-pn2)(f

-a

,$)YO(X,J))

e PilT + e-“‘_e-““-~r*(~“-m~)Yo(X,}. FW the sake of simplicity 2.4. RHO-MESON

)

a;3’:k2(l-$)+m;f

The renormalized TNN coupling For the rr-MECs, the inclusion modi~cation

one may use the pion in ref. 27), the present

EXCHANGE

we suppressed

the subscript

(16a) B = T.

CURRENTS

In parallel with the contribution from the T-MEG one usually considers also those from the p-MECs (see fig. I), As we have already mentioned, our p-Ml33 diirer from the minimal currents “1 by a factor 2 in the transverse part ofthe mesonic

E. T~hIjk, J. Adam, Jr. / Backward deuteron elecrrodisintegrarion

536

exchange

cm-rent.

transverse

part is given by eqs. (13)-(16),

The contribution

to the r.m.e. (Isa),

from the p-MECs

$+&(f)2(!-j!$)2, m,,A, In order to satisfy the current

conservation,

without

this

(16a) with the change

the value

+ mp,A,,. of

K,”

(17)

should

be the same

as that in the anomalous pNN coupling entering the p-exchange potential. The commonly accepted “) value of K,” = 6.6 It 0.4 is about twice larger than the vector dominance prediction K\~ = 3.706. A simple explanation of this long-standing enigma has recently been proposed by Brown “). Let us write down the r,m.e. due to the transverse part of the p-mesonic p MEC,

(18) Here c = 1 for the minimal

current,

eq. (18) with the analogue of there is no reason to neglect a Moreover, model dependence can be clearly seen for t b 25 2.5. A, MESON

EXCHANGE

whereas

c = 2 in the hard-pion

model. Comparing

eq. (14) for the p-mesonic p-MEC we conclude that priori the contribution from this piece of the p-MECs. arises from it. Its influence on the electron spectra fmm2 (see fig. 3).

CURRENTS

These MECs consist of the pair (fig. 2a, b), mesonic current (fig. 2d). The current 21~ is transverse by itself,

(fig. 2~) and

of the A,rr

k,~l"=O.

It is the only non-potential range with the additional

MEC of our model. short range structure

is an analogue of the well known p-r relativistic form of the space component :'AIv=

J

-igg,

(19)

4Mfn

In fact, this current

due to the A, meson

weak decay axial of this current is

is of the pion propagator.

current

It

j3). The non-

(rl x 72)3F~(f)A~(q:)A~(q:)

x kx(a,xqJ+kX(a,Xkf 1 +$

(Ul * q,)[k x (k x a211 t-2 .42)+(1-a I

*

(20)

Only the first term (with incorrect sign) was given in our previous work “). For calculating the tail of the electron spectra the second term is much more impo~ant. The third one comes from the A, meson propagator and will be neglected.

531

E. Truhlik, J. Adam, Jr. / Backward deureron electradisintegration

i /I 3

Fig. 3. Electron spectra calculated with the GK e.m. form factors. Long-dashed line (lo), (13), (14) and (17); short-dashed line the p-MIX, eq. (18) is added with c = 1 (the Full line - full model contribution (without all these cases the hadron form factors eq. full line but

The origin

of the second

Paris nuclear w.f., the set SM of parameters ,‘I,, and with IA+ rr+ p standard current contributions according to eqs. [dash-dotted line] - contribution from the transverse part of standard minimal e.m. interaction) [c = 2, hard-pion model]. the model independent part of the A, 71 current), c = 2. In (9) are included. Dash-double-dotted line - the same as the with the point BNN vertices.

term in eq. (20) can be traced

to the existence

of the

corresponding fig. 2d without

part in the hard-pion photo-production amplitude (the current of the second nucleon line), which makes it compatible with the soft-

pion theorem

given in eqs. (41) and (44) of ref. 44). We shall call the piece of the

current

eq. (20) due to the term

kx (a, x k), the model

term. As we shall see soon, its contribution energies. The contribution

independent

(m.i.)

to r.m.e. is large even for relatively

of the first, model dependent

A,n small

(m.d.), term in eq. (20) to r.m.e. is,

(21)

538

E. T~~ii~ J. Adam, Jr. / Backward deuteron eieetrodisintegmtion +, q,t=

TX=.

I

--I

+,

yij,=

,

dn e-"'j,($qkr)

dn a e-“rj,(f$cr)

I

,

-I

dnC a

Wbr=

Cl+ V7)2j1&?W,

dn emar( 1 t q)j,(+$cr)

Introduction of the form factors eq. (9) into the BNN vertices the mass dependence in eq. (22) as follows,

,

(B = lir, A,) changes

f(%, m*) + fi%n mA)-f(kr,m,)-f(m,,AAl+f(&, /IA),

(23)

where f represents any of four combinations of ema” and a present in W;,. The model independent current jl”l” (m.i.) contributes to r.m.e. similarly,

The function

F,,

is defined

The pair and mesonic

currents

in eq. (22), whereas

satisfy the continuity

equation

for the point A,NN

vertices. Equipping the pair current and the A, exchange potential with the form factor eq. (9), the continuity equation demands an additional current ‘“),

(26) The non-relativistic

reduction

of the space part of this current

is

539

E. Truhlik, J. Adam, Jr. / Backward deuteron electrodisintegration

The contribution

from the current

(27) is

x(2%-)3’?

u1dr W(K, r)[XsoUg(r)-J~X52u2(r)lY I0

(28)

(29) where a I is given in eq. (16) with VI,+ AA. Besides Ai, in the leading order in l/M only the mesonic

current

j:T

contributes

to r.m.e., (‘SO]l?‘yAA,(mes) lld)=-i(~)‘F:(1)~(2”)-~‘2

(30) Here the functions

,Y?,, ,I& are given in eq. (16), the function

x$, is

L A x31

2.6. NUMERICAL

=

77

dn em’“Aj,($qkr).

(31)

RESULTS

Let us now discuss the numerical results order in l/M presented in this section. In fig. 3 we display

the spectra

calculated

obtained

with the MECs in the leading

with GK e.m. form factors,

Paris nuclear

w.f. and the set SM of parameters. The transverse part of the p-MECs tends to shift the dip in the spectrum to lower values of t. On the contrary, the effect of the model dependent A, MECs (cf. dash-dotted and full curves) acts in the opposite direction. The results of calculations with different nuclear w.f. are displayed in fig. 4. The spectra are strongly changed by the contribution from the model independent part of the current 3”~~. The medium part of data is well reproduced by the RSC w.f. and the SM set of parameters. A similar curve for the Paris nuclear w.f. follows the full line for t 3 20 fme2. Without the model-independent current, the quality of the interpretation of the data with the use of the RSC or Paris nuclear w.f. is the same as in ref. “) however, with a readjusted set of cut-off parameters. The OBEPR potential cannot describe the data with the current operator used so far. However, the situation changes with multiplying the currents with Gg instead of Fy [ref. >‘)I.

540

E. Truhlik, J. Adam, Jr. ,J Backward deureron elecirodisintegraiion

-4

,d” -

io

i0

3’0

4

LO

t [firi Fig. 4. Electron spectra calculated with the non-relativistic MEC operators of our model, c = 2, e.m. form factors - GK. Dash-dotted line - P, SM. N; tong-dashed line - P, SB, N; short-dashed line RX, SB, N; short-long-dashed line - RSC, SM, Y, full line - OBEPR, SB, N; long-double-short-dashed line - OBEPR, SB, Y. P/RS~/OBEPR - Paris/RSC/OB~PR deuteron and ‘S,, np w.f.; SMjSB - the set of parameters &, . . - see text. Yf N) - the contributjon (24) from the model independent part of the A,?r current (not) included. The data points are taken from ref. I*).

Effectively, it means an introduction of relativistic correction into the currents. A qualitatively new approach should be sought within the framework of a relativistic theory. Attempts to build up and apply such a theory to nuclear reactions in light nuclei have already been made 45*46).Here we study further the effect of one of its ingredients - static relativistic corrections to the currents derived in ref. “) - on the spectra of reaction (I). 3. Relativistic corrections to the currents Any relativistic theory of the electroweak interaction with the system of hadrons should submit to the following basic invariance principles; (i) gauge, (ii) chiral, (iii) Lorentz invariance. A consequence of requirement (i) is eq. (2) which we considered up to now to the order 0(1/M) in Friar’s classification 45).

E. Truftlik, J. Adam, Jr. / Backward deuteron eleetrodisintegration

541

III the next order in I/M we consider the static 71-MECs, satisfying the continuity equation up to the order 0(1/M’). Here we also include the important retardation FMEC. In the same order we treat also other heavy-meson MECs (v, w, S, cr). It is interesting to note that the heavy-meson retardation effect is of the same order in I/,V and should be taken into account as well as the commutator (external) terms demanded by the coniinuity equation “). The p MECs used so far are in fact the relativistic correction. Nevertheless, in the next order in I/ A/f we take into account a transverse term of the type considered by Mathiot and Riska 47), which non-negligibly deforms the tail of the spectra. We discuss the impact of requirement (ii) on our currents in appendix B. The ~orentz-~ovariant MEG were constructed in ref. *‘). Resides the static part, they contain also a velocity dependent part which is needed to maintain requirement (iii). Here we neglect this part of MECs for the sake of simplicity. Of course, in fully consistent calculations, it should be taken into account. 3.1. ONE-NUCLEON

CURRENT

The reduction yields

.i^o,=& Gm(l-&)(ujxk),

(32)

I

The change in r.m.e. eq. (IO) is evident,

(33)

3.2. PION

EXCHANGE

CURRENTS

The fulI T-MECs used here are given in appendix A. Numerically we estimated the contribution from the current

The current (34) is a sum of the first three terms at the right-hand side of eq. (A.2) and of the part of &L’. eq. (A.6) extracted from the piece proportional to

542

E. Truhlik,

J. Adam,

Jr. / Backward

deuteron

electrodisintegration

-(qf+q$)/8M*. This is also true when the form factor this case, however, ~~“’ = i m .I- *m( ( 37)

71 x d3h

- da,

(9) acts in the vertices.

In

. q,)(az . q&A %d)

xA~(q:)-A:(q:)A~(q:)-(A’,-m2,)A~(q:)

xA~(q:)[A:(q:)+A~(q:)I[l+(A/2M)*l.

(35)

Here A;(q’)

= l/(A’,+q2).

(36)

It can be seen from eq. (35) that the r.m.e. (14) is simply the relativistic

effects. The contribution

parts. In its turn, the part proportional eq. (13) by the substitution

equals

to (Fy+

can be incorporated

F:)k*ar

F;)k2/4M2,

& 3

dr

into

(37)

GE. The rest APT of the current

(‘SolIf~AA_k)ll4 = iFy(t)

to incorporate

(34) can be split into two

FT + FT-(F;+ which approximately

changed

from the current

W(K,

?r yields

r)

Y,(x) + Y2(x)) +j&-1 *{2[jo(%rl(-~

]

Y2(x)

xuo(r) -

(-z

+j,($kr)

Y,(x)+

In eq. (38), Y2(x) = (1 +3/x+3/x2)

0

+ 1 Yr(X)$Y,Cx)

”

X

m

Y*(x) + Y*(x) -

; 0

For the sake of simplicity

3.5. RETARDATION

Y*(x)

)I

I

u2(r) .

(38)

Ye(x). With the form factor eq. (9) included,

31 ;

Y,(x,,)+,

3 Y*(x‘,)+;

l[

l-

;

(“)‘I($

YoCX,+)

[ 1- ( A ,)‘](~)XJ1(X,~).

we put x, + x, A, + A, m,, +

>

(39)

m.

EFFECTS

Up to now, we considered the meson propagators in the limit of zero meson energy. The problem of avoiding this restriction has been frequently attacked in the effects study of reactions on the deuteron 44-46,48).Recently, the mesonic retardation have been treated in connection with the study of nuclear three-body force 49,50).

E. Truhlik, J. Adam, Jr. / Backward deuteron

There exist several methods tion between external potential.

All these

The difficulty

corrections

of taking into account

two nucleons

field.

depending

electrodisintegration

while

are intimately

is that the retardation either

the finite time of pion propaga-

one of the involved

methods

particles

related

interacts

with an

to the definition

effects are higher

on the method

543

of the

order (relativistic)

or on the parameter

v of a unitary

transformation (or both) and demand consistent calculations. This is still difficult to achieve, only for the Bonn OBEPQ and OBEPR I’) is the parameter v known, namely v = i. In this section the pseudoscalar

we consider the pion retardation (PS) nNN coupling *‘), (k x

(42x

effect in the static limit and for

U,))(T,

x 72Y-c

(k

x q2)d

1

x(m2,+dm2(u2.q2)(k. 4J+(l++2). The contribution

of the current

(40)

eq. (40) into the r.m.e. is

(‘SolI %‘&,(~)lld> 1-V = if*GL( t) 406

k2 KM*

dr

W(K,

r)

Y,(x) +xY2(x))j,(%r) -3xY2(x)j,(~kr)luo(r)

x{[7(-5

+J$(-23Y,(x)+lOxY,(x))j,($kr)+ Including

the hadron

12Y,(x)j,($kr)]u,(r)},

form factor eq. (9) leads to a change

x=x rr.

in Y,(x,)

(41)

given in eq.

(15a), whereas XV*(x)

3.4. RHO-MESON

-+ xY2(X)-

EXCHANGE

0 $

zx,Y2(x,)+~

[ 01 1-

1

* (l+x,)e-“1.

(42)

CURRENT

As we have already discussed in sect. 1, the spin-isospin dependent part of the p-MECs used in sect. 2 is already of the order l/ M2 relative to the central part of the p-meson mesonic current. However, in the next order in l/M, it is possible to extract

a transverse

piece,

from the p-meson

pair and contact

graphs,

the part of

which that is proportional to k* deforms non-negligibly the p-MEC contribution at the tail of the spectrum. This current is similar to that derived by Mathiot and Riska in ref. 47). For the transition considered, its contribution has the form,

xA:(d)[a,

x (~2

x

qdlC.r,x ~2)~+(1-2)

.

(43)

E. Truhlik, J. Adam, Jr. / Backward deuteron eiectrodisintegration

544

The contribution

from this current

to r.m.e. is obtained

simply

by substitution (44)

in the analogue

of eq. (13) for the p-meson

to Fy( t) is supressed

pair graph. The r.m.e. proportional

by the effect of the second

term in the right-hand

only

side of eq.

(44) by 9%) 20% and 40% at t = 8 fme2, 18 fm-2 and 34 fmp2, respectively. There is no similar effect from the A, meson pair term. Other heavy meson effects will be considered in the next section.

3.5. EXTENDING

THE MODEL

Our system of hadrons interacting with the external e.m. field consists of the nucleon and rr(J’, T = O-, I), p(l-, 1) and Ar(l+, 1) mesons. Now, let us add to it also other mesons present in the Bonn OBEPQ and OBEPR potentials, namely n(O-, O), ~(1~, 0), 6(0+, 1) and (T(O+, 0) mesons (the A, meson is in these potentials absent). The corresponding heavy-meson exchange currents were constructed in sect. 4 of ref. 27). Besides the pair terms, only for the &exchange the mesonic current of the type of fig. Id exists. These currents the corresponding

OBEP.

satisfy the continuity

Of these, the following

equation

contribute

(A.l) with

non-negligibly

studied reaction: (i) Pair terms of the S- and o-mesons. (ii) Commutator terms of the p-, w-, S- and a-mesons. (iii) Retardation effects due to the p-, o-, 6- and o-mesons.

to the

Let us discuss

the

contributions in more detail. 3.5.1. 6- and u-meson pair terms (‘S,,ll f&

(pair)

II4 a3

drw(K,

= i(-1)

x Yo(xB)[jo(~kr)uo(r) -4jj,(Sr)u2(r)l Here

T, is the meson

r)

J0

isospin.

,

With the form factor eq. (9) included,

A’,-m2, mB YO(XR) + ml3 YO(XB) - At3 Yo(x.4 ) - ____ 2A

I3

3.5.2. Heavy meson commutator

B=S,a.

XA Yo(x/l) .

terms

(‘Sol1 f~m&QlId) co =

dr W(K, r)

i(-1)

x Yl(xB)jl(%r)[udr) +Jb2(r)1

J0

,

B=p,o,&a.

(47)

E. Truhlik,

The presence

J. Adam,

Jr. 1 Backward

of the monopole

Also Ps is the intrinsic

parity

The term (47) corresponds the choice of the Lepton

deureron

form factor changes

kinematics

X

we would

To estimate

e-“B[j,($kr)u,(

of the strong e-%-e-x,

The heavy-meson parameters of the initial and final channels 3.6. NUMERICAL

to eq. (15a).

kinematics

have the same equation

to distinguish

between

5’). With

but with the

the two sorts of kinematics

the maximum

of the effect, we use eq.

r) +$(2j,(:kr)

-3.L(%r)Mr)l ,

+

according

to the choice of the free-nucleon

within the static approximation. (47) with 2Fy+2FY+ Fy. 3.5.3. Heavy-meson retardation

e -xl3

Y,(x,)

545

of the meson.

change 2Fy + - Fy. It is impossible

Due to the presence

elecrrodisintegration

(48)

B=p,w,&u.

form factor, _- :

[l-(zr]x,,eexl.

(49)

are taken from table 14 of ref. 3’). Since the isospins differ, we choose rnrr = 632.5 MeV and gi/4r = 12.

RESULTS

We present now the results implied by the currents treated in this section. In fig. 5 we display the r.m.e. of different currents. The shift of the interference minimum in the IA r.m.e. to t = 14 fmp2 is clearly seen. A similar dip can be observed also in the pionic seagull term at t = 23 fmp2. Then the contributions from other components of the current play an important role in smoothing the behaviour of the spectra as demanded by the data. Also the destructive interference between the contributions is evident. Inspecting the lower part of the graph we can also immediately see large contributions from the pion retardation and the A,n currents. The last one comes presumably from its model-independent part (24). On the contrary, the contribution from the correction to pion MECs (38) is rather small. From the upper part of the graph we conclude that the different effects from the heavy mesons are surprisingly

large. Fortunately,

valid for all three kinds of nuclear

they tend to cancel

mutually.

This tendency

is

w.f. used here. Let us also note that the contribution

(28) from the additional A, current (27) demanded by current conservation, is in the studied interval of t rather small. In fig. 6 we give the electron spectra calculated with different nuclear w.f. The inclusion of the relativistic correction into the current changes considerably the situation comparing with that seen in fig. 4. The spectra derived with the Bonn nuclear w.f. are much closer to the data than those corresponding to the Paris or RSC w.f. The difference between the three upper curves due to different Bonn

E. Truhlik, .I. Adam, Jr. / Backward deuteron electrodisintegration

546

3

t[fmd21 Fig. 5. The dependence of the r.m.e. of different currents on t. The deuteron w.f. is that of the Bonn full model 3*), the ‘S, np w.f. corresponds to the OBEPR “) interaction, GK e.m. form factors, the set SB, v = 0, strong hadron form factors eq. (9) included. Lower part of the graph: short-dashed line - IA; dash-dotted line - pionic pair term; dash-double-dotted line - pionic mesic current; full line - relativistic correction to the pion MEC eq. (38); long dashed line - pion retardation eq. (42); long-dash-short triple-dashed line - p-meson seagull; long-dash-doubledotted-dash-dotted curve - full non-potential A, rr current. Upper part of the graph: full line - the sum of heavy meson commutator terms eq. (47); dash-dotted line - A, meson mesic current; broken line - the sum of heavy-scalar pair terms eq. (45); dash-doubledotted line - the sum of heavy-meson retardation effects eq. (48).

deuteron

w.f. indicates

Unfortunately,

that consistency

the ‘So functions

of the calculations

corresponding

has not been achieved.

to the OBEPQ

and Bonn full model

are not at our disposal. Moreover, in the last case also the current operator should be constructed within non-covariant perturbation theory 52) with the retardation present in the meson propagators. But it is not clear to which value of v it corresponds. The agreement with the data of our calculations with the OBEPR nuclear w.f. can be achieved with v = f, which is, in fact, the correct choice of u for this case (see fig. 7). It is seen that the chiral filter hypothesis is not confirmed (short-dashed curve). Also the results obtained with the standard MEC operator of ref. “) (but with relativistic corrections retained) strongly underestimate the data (dash-dotted curve). The long-dashed curve represents the spectrum due to our model *‘). The difference between the long-dashed and dash-dotted curves comes mainly from the A,v model-independent MEC. The full curve corresponds to our extended model

E. Truhlik,

J. Adam,

Jr. / Backward

deuteron

electrodisintegration

541

0

Fig. 6. Double-differential cross section calculated with all contributions to r.m.e. displayed at fig. 5, GK e.m. form factors, Y = 1. With the Paris and RSC w.f. the set SM of parameters is used. In all calculations with the Bonn deuteron w.f. the OBEPR ‘S,, np w.f. has been employed. Dotted line - IA, Bonn full-model deuteron; long-dash-short-double-dashed line - RSC; long-dash-short-triple-dashed line - Paris; long-dashed line - OBEPR deuteron; full line - Bonn full-model deuteron; short-dashed line - Bonn OBEPQ deuteron.

(heavy-meson MECs including). It is seen that the effect of heavy-meson MECs is quite large. We achieve the agreement of our extended model with the data by reasonable adjustment of the only free parameter AA from the value AA = 1.85 GeV to AA = 2.2 GeV (dash-double-dotted for the model

dependence

curve). A slight change

due to the transverse

of AA can compensate

part of the p-MECs

in eq. (IS)] and also the effect of the A (1236) excitation included in the calculations.

current

[c = 1 or c = 2

which has not been

Generally, we should keep in mind possible dependence of the results on the e.m. form factor parametrization ‘7,32,33).We display it in fig. 8. It is seen that the dependence

of the form of the spectra on both v and e.m. form factors is impressive. 4. Results and conclusions

Here we sum up our main results and conclusions. In sect. 2, we started the study of reaction (1) by paying additional attention to the tail (TV 20 fmu2) of electron spectra in comparison with previous calculations “-2o). Using the T, p and A, MECs constructed in leading order in l/M earlier “2), we show that there exists a modeldependent transverse part in the p-MECs which deforms the tail of the spectra and

E. Truhlik, J. Adam, Jr. / Backward deuteron electrodisintegration

548

Fig. 7. The electron spectra calculated with the OBEPR nuclear w.f., GK e.m. form factors, Y = i, the set SB of parameters if not stated otherwise. Short-dashed curve - IA+ n MEG, point hadron form factors; dash-dotted curve - IA+ r and p-MECs without the transverse part eq. (18); long-dashed curve - IA+ r, p and A, MECs operators of this paper, c = 2; full line - as before+ heavy MECs effect added; dash-double dotted line - as before but A, = 2.2 GeV.

that the effect of the A, MECs is also non-negligible (fig. 3). With the GK parametrization of the e.m. nucleon form factors and using the Bonn nuclear w.f. our spectra overestimated the data (fig. 4). We also discussed the existence of a large contribution from the transverse non-potential model-independent A, rr current. The existence of this current is required by the low energy theorem (sect. 2.5, fig. 4). In sect. 3, we present the static relativistic corrections to the current. We consider in detail the corrections to the 7r-MECs. With these corrections included, the current satisfies the continuity equation up to terms of the order 0(l/M3) included (sect. 3.2). However, the resulting effective expressions for the e.m. form factors are neither Fy nor GE. Here we also extend our model to include other heavy mesons present in the Bonn model of the NN interaction (sect. 3.5). With the choice of the GK e.m. nucleon form factors, the best agreement of our calculations with the data (fig. 7) is achieved for the OBEPR nuclear w.f. and Y = $ (no retardation). Generally, the results are strongly affected by the contribution due to the pion retardation current and by the choice of the e.m. nucleon form factors (fig. 8). A similar dependence of the results on the e.m. nucleon form factor parametrization has been recently

noted

in ref. s3).

E. T~~lik, J. Adam, Jr. / backyard deutero~ efectrodisintegration

549

Fig. 8. The dependence of the electron spectra on the Y and e.m. form factors. Bonn full model of deuteron and OBEPR ‘S, np w.f. are used. v= l(M) - e.m. form factors from ref. “) (G;(t) =O); v = l(BZ) - em. form factors from ref. ‘I); other curves correspond to the GK e.m. form factors.

It is clear from our work that further progress can be achieved only when, (i) The problem of the parametrization of the e.m. nucleon form factors will be settled, which is connected with improving in our knowledge of the neutron electric form factor. (ii) The consistent realistic NN potential the order l/&f considered will be available with the Lorentz

invariant

including all relativistic corrections of (see comparison of the Bonn OBEPR

OBEP in ref. “)).

Although our calculations are not yet completely consistent we believe following conclusions are to be taken into account in any future study. (i) The non-potential MEG due to chiral invariance principle contribute and should (ii) When

that the sizeably

not be omitted. relativistic

effects come into play, it is equally important to consider as well as the short-range ones connected with the

the long-range pion MECs exchanges of heavy mesons.

Part of this work was done during the stay of one of us (E.T.) at TRIUMF and INFN Sezione Firenze. He thanks Profs. D.F. Measday, H.W. Fearing and 3. Mosconi for the kind hospitality. The grant of MPI of Italy is also acknowledged.

550

E. Truhlik, J. Adam, Jr. 1 Backward deuteron electrodisintegration

Appendix

The isovector

r MECs j1,(2) satisfying

A

the continuity

equation

k.L(2)=kml, up to the order 0( l/M’)

were constructed

(A.1)

in sect. 4 of ref. *‘) whereas

the potential

vv is given in appendix C of the same work. In ref. *‘), the currents are derived from the relativistic amplitudes corresponding to pseudovector (PV) nNN coupling. Here, we present the equivalent expression for PS rNN coupling,

(A.21

-2k*a,+al

x(q*xk)-q,(u,

xMd)br

. q2)+(1*2), ~

.4*)-3u,dl(?,

xd3 (A.3)

[(k x qr)d+(k

x (k x a,))(71 x 72)31

xA;(q:)(u, . q2)+(1++2), ~ ~“’

= j

. 42) + (l-2)

(k x qJr:AXq:)(o2

2FY(f)(.rlxT2)3(ql

(> mf

(A.4)

. 41)(u2.

-42)(-l

,

(A.5)

42)

m

-$

‘I

[(Ul . qt)(u* . pZ)(P2’~2)+(l~2)ljs(~~+~2)~~j=P:+P,~ (A.7)

b(I)=

i

j=l

$(l)=$‘(l)+k.

bj(l)=;i”(l)+k*

bj(l) 9

s;“,

F:(t) s;”= - z(k+iu,xP,):,

A;(q*)

(s^,P+$)=;;(l)+/?~‘(l), (‘4.8)

= l/(d+q2), $=--

F;(t) 4M2

(k+icrjxP,)2.

It is understood when calculating the right-hand side static terms are kept in the final result. In eq. (A.2), the of the pair current of fig. la, b proportional to Fy,*. The current (fig. lc) and its presence in the PS TNN-coupling

(A.9)

of eq. (A.l) that only the terms &“*’ are the parts current FW is the seagull chiral model is demanded

E. Truhlik,

J. Adam,

Jr. / Backward

by assumption

(ii) of sect. 3. It compensates

is the mesonic

current

deuteron

551

electrodisintegration

for the first term in _?G”. Further,

2.‘.

of fig. Id. In the lowest-order in l/M the currents &)’ and ‘7*2z,.38). The commutator in eq. (A.2) is are the same as the standard ones k’. required by current conservation (A.l) and corresponds to the current jC3)(2, k, ext) of ref. 27). Its influence However, fig. 5).

the situation

on the electron is different

spectra

for heavy-meson

is negligible exchanges

in the given

case.

(see sect. 3.5.2 and

Appendix B We derive the isovector e.m. r-MECs within the S-matrix method starting from the chirally invariant lagrangian of the NA,pc system. The arbitrary mixing of the PS and PV rrNN 9

NA,w

=

couplings

is admitted.

-Ny,a,N

- MNN + igN

1

E

03.1) The lagrangian

(B-1) together

with the vertex

(B-2) and the lagrangian

Z’,,,,,,, of the A,pn

system 22)

9 A,P?r=&P&L .n.xa,n-s,P,xP,*a,P,+g,(P,xa, -pn x $1

* a,% 2

fn

PW” ’ (a, xrl,?r+~Trxu,,),

03.3)

consists of all vertices necessary to construct the operator of the isovector netic z-, p and A, exchange currents in the tree approximation.

electromag-

Tn eq. (B.l), the notations the Dyson transformation,

h is that of

N=exp

are the same as in ref. 22). The parameter

-ih-

g ysT.n 2M

N’, >

Mg, = g.L 7

(B.4)

552

E. Truhlik, J. Adam,

which transforms

the PV nNN

Jr. / Backward

coupling

deuteron electrodisintegratiorr

(A = 0) to the PS one (A = 1). Besides the

terms -O(n), we write down also the terms of order -0( r2) necessary to reproduce the s-wave TN scattering lengths. Calculations with the use of the 3rd, 4th and 5th terms of eq. (B.l) give for these lengths

a result independent

of the nNN

coupling.

The parameter

K:= 3.706 when entering the one-body current, otherwise K:= the operator of the r MECs. The 6.6 f 0.4 4,42). Let us derive from eqs. (B.l)-(B.3) p and A, MECs are the same as in 22). The nucleon Born term is .,?,,(A)

= -Q(P’I)[+‘(q2,

A)1

A)S,(P)jl,(k)+~~(,(k)S,(Q)~“(q,,

xu(p,)A~(q:)r”(p;,p,,A)+(1~2), Tn(Ai72,

A)

=

c(A)M”(-q2,

WI

Ab4(~2),

y57”,

&(A-l)q2-U I y&)

=;

S,(a)= The mesonic

I

y,J?(t)

-T

F:(t)

a,&,

1

r3,

-l/(i@+M).

current

03.6)

yF;k is the same as in ref. 22) ,

em.t. J r,CL =

-i(q,-q2),F~(t)E3m”r”(P:,~1,A) xA~(q:)A;(q:)T”(P;,p2,

because

for the on-shell r”(Pi,

The contact

(B.7)

A),

nucleons, Pi, A) = -i@(P:)?‘s~“U(Pt)

= r”(Pi,

Pi)

.

(B-8)

term has two parts, j^‘,,~(A)=-iF~(t)&(1-A)~‘““P(p~)~,~~?’u(p~)

K; g

xA;(q:)rn(p;,p2)+A2M2MF~(t)U(p:)ys (B.9)

xa,"k,U(p,)AF"(s:)r'(P;,~2)+(1~2).

The first term is for A = 0 the well-known contact term demanded invariance, the second term ensures chiral invariance of the theory. It is a simple

matter

to verify that (B.lO)

k,(I~,,+j^~;~+~~,,,,=o. We now split the nucleon parts YE;(A) and calculate kj::;(A)

by gauge

Born term j^”“,~ (A ) into negative-

= -V,(P;,P:; x ?4(0,

P, Pz)iW p;; PI

and positive-frequency

Pl)+be(P:,

3P2) = -1 R,

iml

0)

.

(B.ll)

553

E. Tru~l~k, J. Adam, Jr. / Backyard deureraff efectrodisinfegration

Here the potential

c, and the charge

%(P:,P;;

P,P~)=

density

-C.G)fi”(qz,

G”(l)

are defined

the current

charge

density

(B.12)

h)u(P)A~(q:)jT’“tp;,p,),

~‘(l)=Cp”r(p:,pj)‘p^P(l)+k. The one-nucleon

as

(B.13)

?.

,?“( 1) is derived

from the longitudinal

part _$ of

j^,( k) eq. (B.6), i;(k)

for definition

(B.14)

=fiFY(t)r&J’,

of s^’ see eqs. (A.8) and (A.9).

It follows from eqs. (B.lO) and (B.11) that the longitudinal exchange current j^,,(A) satisfies the continuity equation k,j&(A)= The current

;:I:(

k,(~~~~(A)+~~;~+-~,,(A))

part j^c,,(A)

+ SF’(P)

Next we shortly discuss the impact some standard identities, we rewrite ~~~(A)=~~~(A

2

j*,(k)

of the chiral invariance

~~;~(A)=~~;~(A

on our scheme.

Using

=l)fi~(l-A)8(p:)[y,y,r”u(P)a(P)

jB.17)

=l)+i&(l-A)u(p;)[y,~J%(-P)0(-P)

xjl,(k)+j*,(k)u(-Q)u(-Q)y,y,T”lu(p,)d,”(q:) xr”(P;,P*)+(l+-+a * = 1) are just the positive-

Born current in the PS z-NN coupling. and eq. (B.18) is equal to

+i&

change (B.16)

+ .i:W.

xj^,tk)+j^,(k)u(q)ii(Q)y,y,7”lujp,)d;r(q~) x~“(P;,pJ+(l*2) 1

The terms ;%:(A

(B.15)

= [ CA, ;p(l)].

,,+ in eq. (AS) with obvious A ) is of the same form as J*N S,(P)

of the

(1 -A)Fy(t)e

and negative-frequency

The sum of the second

(B.18) parts of the

terms of eq. (8.17)

3”“~(~:)~~~s~m~(~~)A;(q:)~n(p;,~~)+(l~2).

(B.19) It follows

from eqs. (A.15), k,j^r,,(PS)=

(A.9), (A.17)-(A.19)

that

k,(~~~~(PS)+~~;~+~=,~(PS))

=Ibs,pA”(1H.

(B.20)

554

E. T~~~ik, J. Adam, Jr. / 3ackward

Similarly the

we could obtain

PV 7rNN coupling

has been shown operator nuclear

from (B.15) the continuity

equation

for

(A = 0). Eq. (B.20) is exact but not complete,

in ref. 27), this problem

(B.15) so that it would continuity

deutemn electrodisintegrrition

is related

satisfy together

to the redefinition with the one-nucleon

the

current

in

however.

As

of the MEC current

the

equation (B.21)

The corresponding current is given in eq. (A.2). It differs from the standard one by the third and the last terms. The current j^= comes from chiral invariance and is connected with the last term of eq. (B.l), whereas the last one, in the form of a commutator [e, it’), is due to the redefinition of the MEC. Existence of the current plW of eq. (20) is induced by the last term of eq. (B.3). It is this term which makes our model compatible with the soft-pion prediction for the pion photoproduction amplitude, as noted in sect. 2.5. In our discussion of the chiral model we supposed the nucleons to be on-mass and on-energy shell. Continuing the potential off-energy shell induces additional relativistic r MECs 54) which depend on h. The transverse part of these MECs can considerably deform the tail of the spectra. Here we neglected this effect, as well as the one due to the potential-dependent boost in transition to the c.m. variables s4) which is also h-dependent.

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NIKHEF

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7.2) 23) 24) 25) 26) 27) 28j 29) 30)

31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43)

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J. Adam, Jr. / Backward deureron efectrodisintegration

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555

forces,

below