Inclusive electron scattering by nuclei and free nucleon form factors

Volume 195, number 1

PHYSICS LETTERS B

27 August 1987

I N C L U S I V E E L E C T R O N S C A I T E R I N G BY NUCLEI AND FREE N U C L E O N F O R M FACTORS O. B E N H A R a, E. PACE a, b and G. SALMl~ a " Istituto Nazionale di Fisica Nucleate, Sezione Sanitgl, Physics Laboratory, Istituto Superiore di Santia, Viale Regina Elena 299, 1-00161 Rome, Italy b Dipartimento di Fisica, Universit?z di Roma "La Sapienza", P.leAldo Moro 2, 1-00185 Rome, Italy

Received 16 April 1987; revised manuscript received 10 June 1987

The effect of using different free nucleon electromagnetic form factors in theoretical calculations of (e, e') longitudinal and transverse responses has been investigated, both in three-nucleon systems and in a complex nucleus. Large discrepancies have been found between the results corresponding to different models of nucleon form factors, mainly due to the uncertainty of the neutron form factors at high momentum transfer.

In recent years, a number of experiments on quasielastic (q.e.) electron scattering by nuclei, involving Rosenbluth separation of the longitudinal and transverse responses, has been carried out [ 1,2]. Conventional nuclear models, although able to yield a satisfactory description of the measured total cross sections, are generally inadequate to simultaneously explain the separated response functions in complex nuclei. Among the different "exotic" effects which have been proposed to account for the disagreement between theoretical predictions and experimental data, the hypothesis that free nucleon structure might undergo significant changes when nucleons are embedded in a nuclear medium aroused great interest. It has been suggested [ 3 ] that the existence of "swollen" nucleons in nuclei could explain at least part of the significant quenching exhibited by the observed longitudinal responses with respect to the results of standard theoretical calculations, where the electromagnetic structure of nucleons is described in terms of free form factors. Investigations have also been carried out aimed at extracting quantitative information on the possible change of the nucleon radius in nuclei from the analysis of q.e. scattering data at high m o m e n t u m transfer [4]. All of these investigations rely on the assumption that essentially no ambiguity is involved in the description of free nucleon properties. However, it

has recently been pointed out by Gary and Krumpelmann [5 ] that the behaviour of the free nucleon form factors at high momentum transfer might be considerably different from that predicted by the existing models, particularly as far as the neutron is concerned. Within the approach of ref. [5], vector meson dominance accounts for the nucleon structure at low momentum, whereas QCD prescriptions are included to describe the asymptotic region. The proposed expressions yield a good simultaneous fit of the experimental data, including those at high momentum, both for the proton and neutron electric and magnetic form factors. The most striking feature of this model is that, for q~ > 0.5 (GeV/c) 2, the neutron electric form factor turns out to be very large compared with the predictions of previous approaches. The present letter is aimed at investigating whether different free nucleon form factors significantly affect the results of theoretical calculations ofq.e, inclusive electron scattering cross sections. The nucleon form factors obtained in ref. [ 5 ] are compared with those resulting from the widely employed models of Blatnik and Zovko [ 6] and Hoehler et al. [7], yielding fairly good descriptions of the experimental form factors in the range of data fitted. However, even in the low-qu2 region discrepancies appear with respect to the Gari and Krumpelmann model, involving all

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V o l u m e 195, n u m b e r 1

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of the four form factors. The study is carried out by computing the (e, e') longitudinal and transverse response functions at the top of the quasi-elastic peak in different nuclear systems: the three-body systems, 3H and 3He, and 4°Ca, as an example of a complex nucleus. Our analysis has been performed in a wide region of four-momentum transfer 0 < quz ~< 3(GeV/c)2), whose systematic investigation could represent one of the relevant issues for future experimental activity [ 8,9 ]. The inclusive electron scattering cross section a(q, to), in Born approximation, is given by the expression

a( q, tO)=aM[(qZ/q2)2RL(q, tO) + (q~/2q 2 +tg 2 ½0)Rx(q, O9)1 .

(1)

In eq. (1), aM is the Mott cross section, q and to are the momentum and energy transfer, qZu=q20) 2, and 0 is the scattering angle. RL and RT denote the longitudinal and transverse responses, respectively, which have been computed within the plane wave impulse approximation (PWIA), thus neglecting the final-state interaction between the emitted nucleon and the residual nucleus and the antisymmetrization of the final state. Using relativistic kinematics and a relativistic expression for the elementary e-N cross section, the PWIA reads as follows: Emax

RLcr)(q, to)=2n f dE Emin

kmax(E)

f kmin(E)

kdkEfq

× [ZPp(k, E)R~_(x)+NP.(k, E)RP_(T)] .

(2)

In eq. (2) k and E are the nucleon momentum and removal energy, respectively, whereas Pp(n)(k, E) denotes the proton (neutron) spectral function and R PL(V) (n) are the off-shell longitudinal (transverse) response of the proton (neutron), given by the expressions [ 10]

R [(" ) = ( 1/ei Er) { [ ½(ei + El) 12 × [F2p(~)

-2 2 2 + (qu/4M)F2p(n)]

- ~ 1q 2 (F,m,)+F2m,))2},

R~(,,)=(1/EiEr){kZsin2a[F2p(n) + (qu/4M-2 -2 + ½qu(Flp(.) +Fzp(,))2 },

14

(3a) 2 )F2p(.)12

(3b)

27 A u g u s t 1987

where F1 and F2 are the Dirac and Pauli form factors,

Ei=(M2 +k2)m,

G=[M2 +(k+q)2]la,

q2=

q2 _ (Ei - E r ) 2 and, finally, cos a=/~'0. The integration limits in eq. (2) are determined by relativistic energy and momentum conservation (their explicit expressions can be found in ref. [ 11 ]). The three-body system has the merit of being "exactly" solvable within the framework of the nonrelativistic potential model, so that reliable results of microscopic calculations of the spectral functions appearing in eq. (2) are available. In the present work the proton and neutron spectral functions of 3H and 3He obtained in ref. [ 12 ] from the variational wave function of ref. [ 13 ], corresponding to the Reid soft core (RSC) interaction [ 14], have been used. Unlike the case of the three-nucleon systems, realistic many-body calculations of the full nucleon spectral function for complex nuclei involve prohibitive difficulties, so that, besides PWIA, further approximations are required to evaluate the inclusive cross section. In the present paper the response functions of 4°Ca have been obtained using the closure approximation to carry out the energy integration involved in eq. (2). As a result, RL(T)(q, tO) turns OUt tO be expressed in terms of the nucleon momentum distribution n(k). Numerical calculations have been performed using the nucleon momentum distribution of 4°Ca obtained in ref. [ 15 ] from a realistic many-body wave function including central, tensor and spin-isospin correlation and corresponding to the so-called V6 form of the RSC interaction [ 16 ]. As far as the validity of the employed approximations is concerned, it should be pointed out that: (i) PWIA is expected to work well at high momentum transfer, at least in the region of the top of the quasi-elastic peak; (ii) the closure approximation, which amounts to taking into account one-nucleon knock-out processes through the nucleon momentum distribution, seems to yield reasonable results in the kinematical region relevant to the present study, when a realistic n(k), including high-momentum components generated by the core of the nucleon-nucleon interaction, is employed [ 17 ]. To give an idea of the quality of PWIA, in fig. 1 the theoretical response functions of 3He, obtained using the form factors ofref. [ 5 ], are compared with the experimental data from Saclay [ 2 ] correspond-

Volume 195, number 1

PHYSICS LETTERS B

.008 ,;

.006-

Y %~ q.=O.3(GeV//c) ,~,

o:'2'9' o~,,o/

.004-

% ,,~,~

27 August 1987

10 0.8 0.6 0.4

-----

-.

0.2 1.2

.002-

j

, ""',-.

1.0

.000

8'0

'

120

'

11~0 '

200

'

2~-0 '

2'80

~(MeV)

O.8

3He

0.6

"""'""..

1.2 Fig. 1. Longitudinal (solid line) and transverse (dashed line) responses of 3He at q~, = 0.3 (GeV/c) 2 obtained using the spectral function of ref. [ 12] and the nucleon form factors of ref. [5 ]. The experimental data are from ref. [2].

ing to the m a x i m u m measured value of fourmomentum transfer: qu2 = 0.3 (GeV/c) 2. A similar comparion in the case of 4°Ca would probably require more caution, due to the fact that the final-state interaction is likely to play a more relevant role in complex nuclei [18]. However, we would like to emphasize that our work is intended to focus on the effect of using different free nucleon form factors rather than on a comparison with existing experimental data. The employed approximations are expected to be quite reasonable for this limited purpose. Moreover, our analysis of nuclear responses has been performed at the top of the quasi.elastic peak, corresponding to the energy loss O)----(.Oq.e.m(M2-}-q2)I/2WMA_I--MA where these approximations are at their best. In fig. 2 (3) the ratio 0L(-c~ between the longitudinal (transverse) response evaluated at ~=O~q.e. using the Blatnik and Zovko form factors [6] and the same quantity obtained with the form factors of ref. [ 5] is shown, as a function of the four-momentum transfer, for 3H, 3He and 4°Ca (solid lines). Dashed lines represent the same ratio QL(T), but with the Blatnik and Zovko form factors replaced by the form factors by Hoehler et al. [ 7 ]. It clearly appears that using different models for the free nucleon form factors appreciably affects both the longitudinal and the transverse responses over the whole range of fourmomentum transfer. While the relevance of the dif-

1.0 ...,,

0.6~~0"8 ""',,

q;(GeV/c) 2 Fig. 2. Ratio between the longitudinal responses evaluated at the top of the quasi-elastic peak using the Blatnik and Zovko [6] ( solid lines) and Hoehler et al. [ 7 ] (dashed lines) nucleon form factors and the corresponding quantities obtained with the form factors by Gari and Krumpelmann [ 5 ].

ferences is not large in the region of existing experimental data for the separated responses q2< 0.3 (GeV/c)2, there are dramatic discrepancies in the ~T 1.4

.//

3H I

15

f/

1.C i

1.2

3He

J

I

............

1.G

O.E 1.2

i

i

i

4°Ca ..........

1.0 0.8

!

2 3 q~(GeV/c) 2

Fig. 3. The same as in fig. 2, but for the transverse responses.

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Volume 195, number 1

PHYSICS LETTERS B

high-q~ region, particularly in the longitudinal response. It should be noticed that the effect o f different form factors o m the total inclusive cross section, although not negligible, is m u c h smaller t h a n on the separated responses. The similar b e h a v i o u r exhibited by the curves corresponding to 3H a n d 4°Ca indicates that the neutron form factors play a m a j o r role in p r o d u c i n g the disagreement between different models. In o r d e r to obtain quantitative information on this point we have separately evaluated the p r o t o n a n d neutron contributions to the nuclear responses in 4°Ca. T h e p r o t o n c o n t r i b u t i o n turns out to be almost i n d e p e n d e n t o f the choice o f the form factor, except in the case o f longitudinal response, where, for the Blatnik a n d Z o v k o model, it becomes 30% larger than the results corresponding to the other form factors already at q2= l(GeV/c)2. On the contrary, much m o r e significant discrepancies are displayed by the neutron contributions: ( i ) the neutron longitudinal response for the m o d e l o f ref. [ 5 ] is a factor o f four larger than the corresponding quantities obtained using the other form factors already at q2 _ 1 ( G e V / c ) 2 a n d the difference m o n o t o n i c a l l y increases with q2; (ii) the neutron transverse response for the form factor o f Hoehler et al. is larger than the responses corresponding to the other form factors by a factor o f two at q f2, _- 3 ( G e V / c ) 2. The results o f the present study strongly suggest that a better knowledge o f the electromagnetic structure o f free nucleons at high q~, particularly o f the neutron electric and magnetic form factors, is needed to perform a meaningful c o m p a r i s o n between theoretical calculations a n d e x p e r i m e n t a l inclusive electron scattering cross sections, when a R o s e n b l u t h separation is performed. An u n a m b i g u o u s description o f (e, e ' ) processes in terms o f free nucleon p r o p e r t i e s should be regarded as a prerequisite for the study o f possible m o r e " e x o t i c " effects, such as the m o d i f i c a t i o n o f the nucleon size in the nuclear m e d i u m , whose inlcusion has been p r o p o s e d to explain the long-standing p r o b l e m s o f the quenching o f the C o u l o m b sum rule a n d the simultaneous theoretical i n t e r p r e t a t i o n o f the longitudinal a n d transverse responses. F o r this reason, a systematic measurement o f the neutron form factors over a wide range o f f o u r - m o m e n t u m transfer is strongly called for [ 19 ]. In this regard, is should be pointed out that, 16

27 August 1987

due to the high sensitivity o f its response functions to the choice o f the nucleon form factors a n d to the availability o f reliable realistic calculations o f the spectral function, the three nucleon isodoublet and especially 3H represent good candidates, besides the deuteron, to p r o v i d e i n f o r m a t i o n on the electromagnetic properties o f the neutron. The authors are grateful to Professor C. Ciofi degli Atti for calling their attention to the p r o b l e m o f nucleon form factors.

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