Virtual nuclear excitation corrections to elastic electron scattering from 40Ca

2.L

[

Nuclear Physics A199 (1973) 14--22; (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

VIRTUAL NUCLEAR EXCITATION CORRECTIONS TO ELASTIC E L E C T R O N SCATI'ERING FROM 4°Ca WING-FAI L1N

Institute of Theoretical Physics, Department of Physics, Stanford Unicersity, Stanford, Cali]ornia 94305 t Received 1 September 1972 Abstract: The virtual nuclear excitation (dispersion) corrections to elastic electron scattering from

4°Ca have been calculated in second-order Born approximation. A model that can reproduce the whole inelastic response surface is employed for the nuclear currents that enter. The dispersion correction, expressed relative to the first Born result, is dependent only on the nuclear Fermi momentum. When the intermediate nuclear states are summed (without using closure) through the quasi-elastic scattering region, the corrections are found to be small (about 3 %). However, these should be included in analyzing future data with better experimental accuracy. For the first time, the effect of isobar excitation on the dispersion correction has been examined; because of the questionable validity of the approximations in this case, no definite conclusion can be drawn.

1. Introduction Our present knowledge of nuclear charge distributions comes mainly from elastic electron scattering and muonic X-ray spectra. With the substantial improvement in experimental accuracy and theory in recent years, the agreement between the two methods has, in general, become extremely good. Occasionally, the electron and muon experiments give results which appear to differ by an amount exceeding experimental error. This is often attributed to the higher-order corrections to the usual way of analyzing the data in terms of the static Coulomb potentials generated by the charge distributions. Of these corrections, the least well calculated one is the virtual nuclear excitation (dispersion) correction to the electron scattering 1,2). This concerns the modification of the elastic sacttering by the existence of the inelastic channels. Since the electromagnetic fields of the electron and the muon are expected to distort the same nucleus differently, the dispersion corrections for the two cases should be rather distinct. The present practice is to correct the muonic data, but omit such effects for electron scattering, in the belief that they are negligible. Previous dispersion correction calculations have either included only some of the intermediate nuclear states, for example, the giant resonance and quadrupole states, believed to be most likely excited by high-energy electrons; or they have neglected the nuclear excitation energy so that all intermediate states can be summed by using Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, US Air Force, under AFOSR contract no. F44620-71-C-0044. 14

VIRTUAL NUCLEAR EXCITATION CORRECTIONS

15

closure. And, usually, only Coulomb excitation is considered. Such approximations are eliminated here. Furthermore, the model for the nuclear electromagnetic vertices and the excitation energy can reproduce the whole inelastic response surface. This capability is significant because the domains of relatively large inelastic cross section will be important in the dispersion calculations. Within the second-order Born approximation, the dispersion correction to the elastic electron scattering, expressed relative to the first Born result, is shown to be dependent on the nuclear Fermi momentum only. Numerical computations have been carried out for ~°Ca. When the intermediate nuclear states are summed through the quasi-elastic scattering region, the corrections are found to be small [about 3 %, ref. 3)]. However, these should be included in analyzing future data with better experimental accuracy. The inclusion of isobar excitation in the intermediate states has been attempted, but no definite conclusion can be made. Details and further discussion are given in sects. 4 and 5.

2. Second-order Born approximation Electron scattering can be treated in the S-matrix formalism, where a perturbation expansion in powers of the fine structure constant, ~, is done. The usual basis for analyzing the data consists of retaining only the first term and is known as the Born

I 0 ~ ~ k2

II>~ 1 0 > ~

,,/2" \k, Fig. I. Diagrams contributing to dispersion corrections in lowest order. approximation. Remaining terms constitute corrections: the lowest-order processes involving virtual nuclear excitations are shown in fig. I, and the corresponding expression is

( S '2~) = e 2 f d'*x day d az d4co ~k_,(x)T,S~-(x--y))',. Ok,(Y) × {O~(~ - x)O~(z - y) e , ( ~ ) r ' s ~ ( ~ -

z)r'v,(z)

+ O F(O)-- .y)DF(Z -- X) ~f(O.))r vsN(o)

-- z ) r I'tII/i(z) }.

(I)

Here $ and ~ are the electron and nuclear wave functions respectively; S F is the Feynman propagator while D F is the electromagnetic propagator. The nucleartransition vertex is F. In the non-relativistic limit for the nucleus,

S~(co- z) = - iO(t% - Zo) E ~,(c°)eT(z) •

1

(2)

16

W.-f. LIN

This is "non-relativistic" in the sense of omitting particle-antiparticle pairs in timeordered perturbation theory, where they are high-energy excitations (twice a particle mass), and are correspondingly unimportant. Use of the representation of the 0function,

dse-"

O(z) = lim (-S-)| °° ~-.o 2ni d-oo s + i6 '

(3)

allows the time integrals to be done; Fourier transforms of the nuclear currents result after spatial integrations. In fact, the direct term can be simplified to = e 2

IH e

2no(E,+e~-Ei-et)U,.~(k2)y, .-. : d4t

X, oji

1 (/~, --,¢) - ',,, .

.

.

.

.

.

.

.

.

.

.

.

dfUt(q-t)d;i(t) (to - AEN)(q -- t)et2 . .

.

.

.

.

(4)

Here A E , is the nuclear excitation energy; the I = 0 term is excluded because it corresponds to Coulomb correction. The cross term can be obtained by replacing the electron propagator with [(~2 + a Q - m , ] . Note that the nuclear transition currents that enter here are similar to those to which appear in inelastic electron scattering. Hence it should be emphasized that any reliable estimates for virtual nuclear excitation corrections can only be obtained with a model that is able to fit the whole inelastic excitation spectrum 4). The regions of momentum transfer and energy loss where the cross sections are relatively large will also be dominant for dispersion.

3. Inelastic electron scattering As stressed previously, a good model for inelastic scattering is vital to dispersion calculations. The one applied here is essentially a Fermi gas model, with which Moniz has been able to fit the quasi-elastic scattering data well 5). The electromagnetic vertex used corresponds to that of a free, relativistic nucleon (neutron and proton):

j~ = ( k ' = k +tl3,1k) = [~EE'] M 2 ] t e p U(k')[YuF,(t2)+ia..P'F2(t2)]U(k);

(5)

the standard "dipole fit" is utilized for the elastic form factors F I and F2. Therefore the nuclear part of the cross section has a tensor T,v that can be written t2

]+

k,- -~.-t, kv--Tt,

where

1"1 = -½rE(F, + 2 M F 2 ) 2, T2 = 2M2(F~- t2F~).

,

(0)

VIRTUAL NUCLEAR EXCITATION CORRECTIONS

17

Here, t is the four-momentum transfer to the nucleus and k is the nucleon fourmomentum. For large momentum tranfer (greater than twice the Fermi momentum kr), the recoil nucleon is described by relativistic kinematics and an average binding energy is included for the initial bound nucleon. In the case of small momentum transfer, non-relativistic kinematics and effective mass are employed, both for the target and the recoil nucleon; an average behavior of the low-lying resonances (e.g., that due to the collective state) in the inelastic excitation spectrum can be reproduced t. The calculation of isobar excitation [N*(1236)] has also been done in the same spirit 5). For the nucleon-isobar transition current, the completely covariant, gaugeinvariant coupling of Gourdin and Salin 6) is employed:

j.

=

Here the isobar (mass

M')

EE'/

5

C3(t2)

(7)

is treated as a discrete state, described by a Rarita-

Schwinger wave function og(k') [ref. 7)]; the inelastic form factor C3(/2) comes from the relativistic N/D calculation of Walecka and Zucker a). Such modifications cause only a change of the form factors T~ and 7"2 in the tensor T~,. :

Z, = \--~x/(63) 2(M'-M)2-t22 7"2 =

+~/~/~ M')-, {(M''I-M)2"]- 3I I-M(M L

2] " } '

\m~, 3

Resulting cross sections resemble that of the quasi-elastic scattering. Therefore. in contrast to previous practice, the effect of isobar excitation should also be included in estimating the dispersion correction. The Fermi gas model illustrated here is particularly appropriate for regions of large momentum transfer and energy loss, where the inelastic cross sections are relatively large. Although the details of the low-lying resonances cannot be realized, the ability to satisfy the inelastic sum rules serves the purpose here 9). Note also that the gauge terms in T,~ can be omitted, owing to electron current conservation. 4. Virtual nuclear excitation corrections 4.1. INTERMEDIATE EXCITED STATES WITHOUT PIONS

Virtual nuclear excitation corrections have been evaluated with a similar combination of approximations as indicated above. The nuclear ground state is described by a simple shell model: a Slater determinant of single-particle wave functions, which, in principle, can be expanded in plane-wave basis. With all the intermediate states rept Since calculations emphasizing some of these low-lying resonances have been performed [see refs. 1.2)], the more satisfactory approach of employing a different and better model for this rcgion of the excitation spectrum is not followed.

18

W.-f. L I N

resented by plane waves, the current matrix elements in the second-order Born term [cf. eq. (4)] are to be replaced by expressions like that of eq. (5). Then q in j u ( q - t ) is dropped giving rise to the tensor Tu,.(k, t) [cf. eq. (6)]. Such a forward scattering approximation for the elementary vertices is justified for values of q small compared with the t-values that dominate the integration. This is found to be the case for q less than about 250 MeV/c. Pauli exclusion is inserted; for simplicity in calculation, an average of the 0-function over the Fermi sphere, giving the exclusion factor S(t), is used:

S(t) = 1

t >=2kv

=~

~

t <2kF.

That this is the correct correlation factor to be included in the Fermi gas limit can be demonstrated, if only the Coulomb interaction is kept 2). However, for realistic nuclei, if only the Coulomb interaction is kept,

S(t) = Cs t2,

for small t.

(10)

This just expresses the orthogonality of the excited states to the ground state. The proportionality constant, Cs, can be obtained by using the experimental nuclear photoabsorption cross section 1o). In view of this, the computation has been repeated with S(t) as given by eq. (10) until its value became equal to that ofeq. (9). Numerical results were essentially unchanged. The second-order Born term is a seven-dimensional integral. The to integration is done by contour integration. To avoid inessential numerical difficulties, the nucleon momentum is neglected, i.e., k = (M, 0). Hence the k-integration is trivial and, after summing over the occupied shell orbitals, the ground state charge form factor can be identified. Within this model, the dispersion correction to elastic scattering, expressed relative to the first Born result, is of the order ~¢, the fine structure constant, and is only dependent upon the nuclear Fermi momentum. Remaining integrals are of dimensions up to three. Principal value integrals and those that involve rapidly varying integrands, arising from the electron being highly relativistic, are handled by the subtraction procedure:

I

=f_N(x)

dD(x) dX

=f[N(x)-N(Xo)]

.J

~(xi

N x " F dx dx+

( o)jo~ ) .

(ll)

Zeros in the denominator of an integrand are separated from each other by the trick:

ab

a+b

Numerical computation follows the Gaussian-Legendre quadrature method. Most of the contribution comes from one-dimensional integrals and rapid convergence can be seen by varying the number of nodal points.

VIRTUAL NUCLEAR EXCITATION CORRECTIONS

19

The results for 4°Ca are displayed in fig. 2. It shows that the dispersion correction is energy dependent and is only slowly varying with scattering angle. The magnitude of the effect is small (about 3 ~), but it should be included in analyzing future data with better experimental accuracy. The common but objectionable approximation of using closure (or partial summation of intermediate states) and neglecting transverse current interaction is removed here. 4.2. ISOBAR EXCITATION EFFECT

Following the approach described above, intermediate nuclear states involving isobar excitation [N*(1236)] can be taken into account, with some modifications. SCATTERING 0 0

30 I

60 I

ANGLE 90 I

(DEGREES) 120 I

150 I

180 I •

~ 2 5 0

MeV

i

C O4 0 0 0 x ',

|

z

g

/

5 0 0 MeV

v

'~'-3

--

~

~

I 0 0 0 MeV

Fig. 2. The dispersion correction relative to the Born approximation cross section for electrons elastically scattered by 4°Ca. Intermediate nuclear states are summed only up to the quasi-elastic scattering regions.

Firstly, Pauli exclusion is omitted, as an isobar is distinct from a nucleon. Secondly, the elementary vertex is replaced by the expression in eq. (7); the corresponding form factors for tensor T,v are given in eq. (8). In dispersion correction calculations, an integration over all t (intermediate momentum transfer) occurs. Since the mass of the isobar is greater than that of the nucleon, form factors for the unphysical region (t 2 = t ] - t 2 > 0, i.e., time-like) are required. Devoid of experimental guidance, these are obtained here by invoking threshold arguments. To do this, it is noted that the form factors may be expressed in terms of the helicity amplitudes 11), f± and f¢, Tl(t 2) = M'2[If+lz+lf_12], 7"2(t2) = MZ[(2t'/t*a)[fcl2 -(t2/t*2)(If+ 12+ IJ-12)],

(I 3)

20

W.-f. LIN

where t,2 iS the magnitude squared of the three-momentum transfer in the isobar rest frame. That is, t .2 = {[(M' + M) 2 -

t2]/2M'}{[(M'-

M) z -

t2]I2M'}.

(14)

Furthermore, the helicity amplitudes are related to the relativistic form of the vertex [cf. eq. (7)] by f+ = _ C~ ( M ' + M) [ ( M ' - M ) 2 - t z ] rn, 2M'

~

(~)½ C3 [(M'-M)2-t2]i[M(M + M')-t 2] f_

~

re=

--

mn

4M '2

'

(~)' C3 [(M'-M)2-t2][(M' + M)2-t2]½ m~

4M '2

(15)

Then, threshold (angular momentum and parity) arguments give, for the unphysical region,

f

+ ~ Ct+ t*~

f_ = ~_t*, f~ = ~ct*2;

(16)

the proportionality constants are determined by matching the helicity amplitudes at

t2=O.

Calculations have been performed for 4°Ca, at a scattering angle of 30 °. For the three values of energy considered, the dispersion corrections coming from isobar excitation alone are - 3 . 7 % at 250 MeV, - 6 . 9 % at 500 MeV, and - 10.1 ~o at 1 GeV. These corrections are much bigger than those reported in subsect. 4.1. Unfortunately, a major portion of the contribution to the integrals comes from the unphysical region. This is where the form factors are least reliable and the forward scattering approximation is not well justified. In contrast to real inelastic scattering, electromagnetic vertices off the energy shell are also needed for dispersion corrections; these follow conveniently from the Lehmann representation [see, e.g., ref. 12)]. The gauge terms in T.~ are dropped because they do not contribute to inelastic scattering. As it stands T,v(k, t) violates current conservation off the energy shell. This could be remedied by using a model in which to in the gauge terms is allowed to go off-shell; again, in such a case, these terms will not contribute when contracted with the electron current. However, the large results here are related to the possibility of photoproduction of isobars, and may be strongly dependent on how current conservation is satisfied off the energy shell. An assessment of this sensitivity is done schematically, by observing that the elementar3" vertex of eq. (7) can conserve current everywhere if the to in the "interaction term" (tpT,-~gp,) becomes a variable. The algebra, involving the introduction of a

VIRTUAL NUCLEAR

EXCITATION CORRECTIONS

21

projection operator for the isobar [see, e.g., ref. 13)] and the evaluation of traces, is repeated; a new tensor for the nuclear part of the second-order Born term emerges. After discarding terms that will vanish on contraction, the new tensor can be written as

. T~ k~.k.+T~(k,~+k.Tj.)+Tj.'i,t.. T£.(k. t) = -- Ti. g~.~+ ~-2

(17)

where

to = ( M ' 2 + I 2 ) * - M , c = (t;, t), 7 = [(t~--to), 0], T; = -

2 (C312{[(M + M , ) 2 _ t , 2 ] t 2 ( t o

~/M,2 + t z _ t 2 ) i M , 2

x [2MM'2t 0 + (t o v/M '2 + t 2 - t2)(M 2 + M 'z - t'2)] }, T~=

4 "w

T~= T~=

-

'

2

2 -3

2- M m ( M

\taxi

2

t2(M2+

-

\mxl

Numerical computations are repeated with the results +0.5 % at 250 MeV, + 1.0 % at 500 MeV, and +1.6 % at 1 GeV. Contributions coming from the physical and unphysical regions are about equal. Hence no conclusion can be drawn about the effect of isobar excitation on dispersion corrections t. It should be noted that the difficulties encountered here are not present in subsect. 4. l.There the integration over all t involves only the physical region (t2= t ~ - t 2 < 0), where the form factors are well known. And since the energy denominator is large whenever the photon propagator is small, the effect of current conservation off the energy shell is not expected to be important. 5. Discussion

The present investigation of the virtual nuclear excitation correction to the elastic electron scattering stresses the importance of the domains of large momentum transfer t If one considers the nucleus as a collection o f essentially free nucleons, with the correlations between t h e m just permitting an overall m o m e n t u m transfer to the nucleus, then the second-order Born t e r m for 4°Ca s h o u l d equal the product o f the nuclear charge form factor and twice the singlenucleon dispersion effect. (This is true even without neglecting antiparticles in the intermediate states; the factor o f two arises because there is a dispersion effect for both p r o t o n s a n d neutrons). By m e a s u r i n g the ratio o f positron-nucleon scattering to electron-nucleon scattering, corrected for the difference arising from radiative corrections, experimental estimates for the single-nucleon dispersion corrections have been obtained. D e p e n d i n g on the f o u r - m o m e n t u m transfer, the limit can be as high as 1 2 ~ . For a c o m p i l a t i o n o f experimental results, see ref. ~,L).

22

w.-f. LIN

a n d large energy loss o f the inelastic response surface. U n d e r such c o n d i t i o n s the transverse c u r r e n t interaction should be included and the f o r w a r d scattering a p p r o x i m a t i o n for the e l e m e n t a r y vertices is a p p r o p r i a t e . T h e latter can be s u b s t a n t i a t e d numerically when all the i n t e r m e d i a t e nuclear states are s u m m e d (without using closure) t h r o u g h the quasi-elastic scattering region. T h e c o r r e s p o n d i n g dispersion corrections, expected to be reliable in general, are small ( a b o u t 3 %), but o u g h t to be included in a n a l y z i n g future d a t a with better e x p e r i m e n t a l accuracy. Neglecting surface and long-range collective effects, the s e c o n d - o r d e r Born term becomes p r o p o r t i o n a l to the nuclear charge form factor 2). Hence the c o m p u t e d corrections at diffraction m i n i m a c a n n o t be useful; this s h o r t c o m i n g is shared by all second Born a p p r o a c h e s , i n d e p e n d e n t o f the vertex model used. F o r a light nucleus like 4°Ca, the omission o f C o u l o m b d i s t o r t i o n o f the electron waves is reasonable. Even t h o u g h s h o r t - r a n g e h a r d - c o r e c o r r e l a t i o n s have not been a c c o u n t e d for, the e r r o r is insignificant, since these correlations d o not affect the inelastic sum rule a p p r e c i a b l y 9). F o r the first time, an a t t e m p t has been m a d e to e x a m i n e the effect o f i s o b a r excitation on dispersion corrections. However, the a p p r o x i m a t i o n s used c a n n o t be justified here and the p r o b l e m s o f the unphysical region arise. N o definite conclusion can be d r a w n a b o u t the size o f the i s o b a r effect. M a n y o f the difficulties met m a y be resolved only after m o r e extensive studies, b o t h theoretical a n d experimental, have been perf o r m e d on p r o b l e m s p e r t a i n i n g to the e l e c t r o p r o d u c t i o n o f isobars. F u t u r e i m p r o v e d calculations on this subject would certainly be very worthwhile. It is a pleasure to t h a n k Professor D i r k W a l e c k a for suggesting this investigation and discussions. I a m also grateful to Professor Bill Bardeen for his c o m m e n t s .

References 1) D. Drechsel, Prec. Adv. Institute on electron scattering and nuclear structure, Cagliari, Italy, 1970, to be published 2) H. Bethe and A. Molinari, Ann. of Phys. 63 (1971) 393 3) Wing-fai Lin, Phys. Lett. 39B (1972) 447 4) R. K. Cole, Jr., Phys. Rev. 177 (1969) 164 5) E. Moniz, Phys. Rev. 184 (1969) 1154; E. Meniz, I. Sick, R. Whitney, J. Ficence, R. Kephart and W. Trower, Phys. Rev. Lett. 26 (1971) 445 6) M. Gourdin and Ph. Salin, Nuovo Cim. 27 (1963) 193; 27 (1963) 300 7) W. Rarita and J. Schwinger, Phys. Rev. 60 (1941) 61 8) J. D. Walecka and P. A. Zucker, Phys. Rev. 167 (1968) 1479 9) T. de Forest, Jr., and J. D. Walecka, Adv. in Phys. 15 (1966) 1 10) L. Foldy and J. D. Walecka, Nuovo Cim. 36 (1965) 1257 I I ) J. D. Bjorken and J. D. Walecka, Ann. of Phys. 38 (1966) 35 12) A. L. Fetter and J. D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, 1971) 13) H. Pilkuhn, The interactions of hadrons (Wiley, New York, 1967) 14) J. Mar, B. C. Barish, J. Pine, D. H. Coward, H. DeStaebler, J. Litt, A. Minten, R. E. Taylor and M. Breidenbach, Phys. Rev. Lett. 21 (1968) 482