Quasi-elastic electron scattering and nucleon-nucleon correlations

Nuclear Physics A532 (1991) 301c-312c North-Holland, Amsterdam

NUCLEAR PHYSICS A

Quasi-elastic electron scattering and nucleon-nucleon correlations C. Marchanda*, G. Salmèb and S. Simulab a DSM/SEPN, CEN Saclay, 91191 Gif-sur-Yvette Cedex, France b INFN, Sezione Sanità, Viale Regina Elena 299, I-00161 Roma, Italy 1 . INTRODUCTION The mean field approach, based on the assumption that single particle (sp) states below the

Fermi level have occupation probability equal to one and sp states above the Fermi level have zero occupation probability, can describe a large set ofexperimental data concerning low en

nuclear physics. However, theoretical many-body calculations using realistic models of the nucleon-nucleon (NN) interaction have pointed out that the nuclear ground state contains an appreciable amount of high momentum components due to the short range correlations (SRC) among nucleons. As a result, the sp occupation probabilities below the Fermi level are depleted

and, moreover, the calculated momentum distribution at large values ofnucleon momenta (k > 2 fm-1) is almost entirely exhausted by SRC effects . In this contribution we briefly review some recent results, illustrating the possibilities ofobtaining information on SRC by the quasi-elastic

(QE) electron scattering, where the reaction mechanism is expected to be governed mainly by one-body electromagnetic nucleon currents and multi-nucleon final states can be excited. The paper is organized as follows: in Sec. 2 the results of recent experiments on inclusive 4He(e

e)

and exclusive 4He(e,e'p)3 H reactions performed at Saclay with the aim of checking the one-

body nature of the reaction mechanism, will be presented; in Sec. 3, the y-scaling analysis of SLAC inclusive cross sections for complex nuclei and the extraction of the nucleon momentum * In collaboration with: M. Bernheima, M.K. Brussels, G. P. Capitanid, E. De Sanctisd, S. Frullanie, F. Garibaldie, A. Gérarda, H.E . Jacksonf, L. Lakehal-Ayata, J.M. Legoffa, A. Magnona, Z.E. Mezianig, J. Morgensterna, J. Picarda, D. Reffay-Pikeroena, S. Turk-Chiezea, P. Vernina and A. Zghichea . Department of Physics, University of Illinois at Champaign, Urbana, IL 61801, USA. Laboratori Nazionali di Frascati, INFN, I-00044 Frascati, Italy. e Laboratorio di Fisica, ISS, and INFN Sezione Sanità, I-00161 Roma, Italy. f Argonne National Laboratory, Argonne, IL 60439, USA. Department of Physics, Stanford University, Stanford, CA 94305, USA. 0375-9474/91/$03 .50 © 1991 - Elscvier Science P4bl shcrs B .V. All rights reserved .

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C. Marchand et al. / Quasi-elastic electron scattering

distribution will be illustrated; in Sec. 4 the effects of SRC on the QE response function will be analyzed in terns of different models of the nucleon Spectral Function. . ELECTROMAGNETIC RESPONSE FUNCTIONS FOR INCLUSIVE AND EXCLUSIVE QUASI-ELASTIC ELECTRON SCATTERING ON 4 He AT EN U TRANSFER q BELOW 1 GeV/c (C. MARCHAND*) Due to its main one-body nature, QE electron scattering has been used e::tvaâsively in the last 20 years to obtain detailed information on the nuclear structure. In particular, assuming Plane

Wave Impulse Approximation (PWIA) in the final state, proton occupation probabilities and momentum distributions were measured on a variety of nuclei [11 in (e,e'p) experiments. However, at the precision now attained in these experiments (few %), the validity of the PWIA is questionable and must

checked. Apath followed extensively at NIKHEF is to get an idea

on the sensitivity to Final State Interactions (FSI) by measurements at various kinematics for the (eel) action. Another approach is to study separately the electric (longitudinal) and magnetic (transverse) components of the cross section. In the latter case, one may disentangle the effects

due to MEC

Experiments where the longitudinal and transverse response functions were obtained have been performed at Bates, NIKHEF and Saclay. We present here recent results on the (e,e7) [2) and (e,e7p) [3] response functions for 4He obtained at Saclay and we will discuss the limits they put in our knowledge of QE scattering below 1 GeV/c.

With the use of the 700 MeV Saclay Linear Accelerator electron beam and the fully equipped

two magnetic spectrometers of the HEI end station [4] we recently measured the cross sections for the reactions 4He(e,e') and 4He(e,e'p)3H. These experiments have been performed with the control of all known parameters to the % level (charge measurement, beam direction,

spectrometer alignment, cryogenic target density, detection acceptance and efficiency, ...) in

order to obtain absolute measurements with less than 5% systematic errors . From them, we

have obtained the longitudinal and transverse response functions in the following kinematical range: 4He(e,e')

: RL , RT

250 < q < 600 MeV/c with q II k' and (e - 2M -100 MeV 2M 0 < km < 190 MeV/c (QE kinematics)

4He(e,e'p)3H : WL, WT 250 < q < 830 MeV/c

C. Marchand et al. / Quasi-elastic electron scattering

303c

where w and q are the energy and the momentum transfer, respectively, k' the ejected . nucleon momentum, km the missing momentum = q - k'.

In PWIA, the cross section of these reactions can be written as the product of the electron-

nucleon cross section times a nuclear response. In the (e,e'p) reaction, the latter is the so called Spectral Function, and in the (e,e') reaction, it is the integral ofthe Spectral Function over some

kinematical phase space. Therefore, if PWIA strictly holds RL(,e)

ZRL+NRL

_

RT(e,e') zR1+NR1,

«

F(q,w)

p) W ( ) = W ( = S(E km,q c6) WL

WT

The (e,e') ratio F is known as the scaling function and is discussed in the following section. In the limit q-)oo, F(q,w) = F(y), where y is the minimal longitudinal momentum of the initial

nucleon for a given (q,w). The (e,e'p) ratio is just the Spectral Function and does not depend upon the projectile variables (q,(b) in PWIA . Any deviation from one for the ratios FLJFT and SL/ST and non flat (q,w) dependence of SL and ST are indicative of the non validity of the PWIA.

Fig. 1 shows the response functions for the reactions 4He(e e') and 4He(e,e'p)3H plotted in

terms ofthe above discussed ratios. We stress here that this is the first time that absolute X10-3

6.0

4He(e,e') ', F,, Saclay ® OF,, Saclay F,, Laget FT Laget

r

q=500 MeV/c le'?, ô:

40 40 40

0-

0.0 -400 -300 -200 -100 0 100 200 300 y (MeV/c) Fig. 1. Longitudinal (FL) and transverse (FT) scaling functions for the reaction 4He(e,e') at three-momentum transfer q = 500 MeV/c. Experimental data [2] are compared to calculations [6] by J.M. Laget including FSI and MEC.

1.0 0.8 0.6

4He(e,e'p)3H Saclay

-

Ak.30 McVM ek.9o Mevic t90 Mewc

0.4 0.2 0.0 200

400

600 q (MeV/c)

800

Fig. 2. Ratio of longitudinal to transverse response functions for the reaction 4He(e,e'p)3H. Experimental data [3] are compared to PWIA (dashed line) and Laget's calculation [6] (solid line).

C. Marchand et al. I Quasi-elastic electron scattering

30

response functions on the same nucleus have been measured in both (e,e') and (e,e'p) reactions with the same experimental apparatus. It is striking to notice that for QE kinematics, FL/FT - SL/ST - 0.65. This reduction of the LfT ratio in both (e,e) and (e,e'p) reactions indicates that

- for the two reactions 4He(e,e) and 4He(e,e'p)3H in the energy range < 1 GeV, PWIA does not hold;

- the origin of the breaking is of the same nature in both reactions (the p-3H channel is dominant in the (e,e') reaction [2]) .

The most evident candidate to explain this deviation from PWIA are FSI and Meson

Exchange Currents (MEC). It is clear that MEC affect the IJT ratio (MEC contribute to order I/M2 to RL and 1/M to RT), and recent calculations, based on an infinite nuclear matter model [5], show that the spin and isospin dependence of the p-h interaction leads to different FSI in the longitudinal and transverse channels . The result of the most complete calculation by J.M.

get [+6] fbr these two reactions are shown in figs. 1 and 2. Although there is a trend to obtain a LJT ratio smaller than one (- 0.87), this calculation underestimates the observed breaking of P

A. As a possible explanation ofthis attenuation, it has been suggested that the nucleon size

is different in the nucleus[7] . The results for the 4 He(e,e'p)3 H reaction, namely the q dependence of SL and ST, rule out a change in size of the nucleon greater than 2% for the mass radius rm and 10 and

for charge radius re. In our opinion, other consistent calculations of FSI EC are called for, before one may ascribe to some exotic effects such a 1,/T reduction.

To summarize, we have observed a breaking of PWIA of the order of 35% in the ratio 14T in both 4He(e e') and 4He(e,e'p)3H reactions, of which only halfis yet accounted for by recent calculations of the reaction mechanism including FSI and MEC. This gives some indication on how well QE scattering may be interpreted by one-body operators in the energy range below 1 GeV. It is usually assumed that FSI dies out at high q and that there the breaking of PWIA should be less relevant. The following section addresses the importance of FSI at SLAC

energies in inclusive reactions. But what about MEC ? Clearly, L,rT separations and (e,e'p) data are demanded for at higher energies to see if the concept of QE scattering still holds at q 10 GeV/c for xg - 1 .

T

E CAN LEARN ABOUT SRC

T

-SCALING (G. SALME')

In this section the possibility of extracting reliable information on the nucleon dynamics inside the nucleus, even in presence of effects violating the PWIA description of the QE electron

scattering by nuclei, will be addressed. In particular, the analysis in terms of y-scaling of the QE peak allows one to single out the kinematical region where the elastic scattering of an

C. Marchand et al. / Quasi-elastic electron scattering

305c

electron by a bound nucleon is dominant and therefore it is possible to study the nucleon dynamics without being greatly concerned about the inelastic channels. On the other hand the presence of FSI, not negligible also at high momentum transfer (q >_1 GeV/c), prevents a direct interpretation of the data in terms of PWIA, namely in a framework where a transparent link

between the cross section and the Spectral Function can be established. In what follows the analysis presented in Ref. 8 will be briefly reviewed and applied to a real case, the 56Fe nucleus, in order to illustrate the procedure for extracting the nucleon momentum distribution (i.e. the missing energy sum rule for the Spectral Function) from the experimental cross sections, even affected by FSI.

As it is well known, the scaling variable y and the scaling function F(q,y) are respectively defined by w + MA= M2+(q+y)2 +

MA 1 + y2

where MA is the mass of the target and MA-1 the mass of the ground state of the spectator system, and a2 _ F(q'y) - (Z Sep + N sen)

aW

k acosa

where a2 is the inclusive electron scattering cross section and sep(n) the relativistic, offshell electron-nucleon cross section.

In PWIA framework, it turns out [9,10] that: i) the scaling function is related to the Spectral

Function as follows

00

00

FPWIA(q,y) = J dE J kdk S(k,E) Emin kmin(q,y,E)

(3)

where E is the missing energy and k the nucleon momentum inside the nucleus, and ii) FPWIA(q,y) scales in y; within relativistic kinematics, at high q, kmin(q,y)-)~ ly - (E - Emin)l

and any dependence upon q disappears . Thus, if PWIA holds, we should observe a scaling behaviour of the experimental F(q,y) ; as a matter of fact, FXP(q,y) seems to approach a scaling

behaviour, as shown in Fig. 3 for 56Fe (the same feature appears to hold for all nuclei investigated by high energy electron beams [I 1 ]). In order to analyze in more details the

soundness of the scaling behaviour, a linear plot of Fex P(q,y) at fixed values of y and various momentum transfers is very useful ; in such a way, it is possible to put in evidence that the

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C. Marcizand et al. I Quasi-elastic electron scattering

present data at high q are still affected by effects that violate PWIA, such as FSI, that seems to be the dominant one for y<0, also for q ranging from 1 to 2 GeV/c. Therefore, since even at the highest available value of q FexP(q,y) cannot be considered as the asymptotic one and on the other hand for increasing q the effects of inelastic channels become relevant, a new procedure for obtaining the experimental asymptotic scaling function FexP(y) from the present data has en introduced . This procedure is based on the assumption that at the available momentum

transfer Fex P(q,y) is not largely affected by inelastic channels and therefore it can be represented by a power series of inverse q, i .e. F(q,y) = F(y) + F~q

+

F(-2)(y)+ F(3)3Y) q + ... q

If such a series holds, then it is possible to obtain the asymptotic scaling function F(y), from the intercepts of linear fits of FexP(q,y) at the highest values of q, for fixed y . In Fig. 4, where Fe-X-P(q,y) for y<0 is shown as function of 1/q, for fixed values of y, it can be seen that the data, with a g approximation, approach a linear behaviour at the highest values of q. The linear behaviour of FexP(q,y), at very high values of q, should be also recovered by plotting the data versus 1/qß (with = q2 - 0), as heuristically expected, since from eq.(1) one gets q2L q1j- (q- w) (q+W) = 2 q (MA - y +

Mp1+ y 2 ).

At low values of y the presence of inelastic channels is shown by the increasing behaviour of F(q,y) for small values of 1/q; such an effect has been taken into account in our fitting procedure by adding to the linear part of eq.(4) a term growing with q from a given value qn(y), that

represents the kinematical threshold for the pion production. By inserting or disregarding such a tern, the obtained intercepts are affected by changes of the order of 10%, at most. If we identify the asymptotic scaling function FexP(y) (i.e. the intercepts of the linear fits shown in Fig. 4) with FPwI A(y), then it is possible to extract the nucleon momentum

distribution n(k), since in PWIA one obtains n(k) = -

1 2ny

[dF(y) + d B (y) dy dy ]

where

°°

B (y) = 2n

j dE Emin

kmin(y,E) Sex(k,E) k dk J lyl

C. Marchand et al. / Quasi-elastic electron scattering

16-2r

Fig. 4. The experimental scaling function (eq. (2)) of 56 Fe vs. the inverse momentum transfer. The data points represent the average values of Fex P(q,y) in a range of ± 50 MeV around fixed values of y (eq. 1). The straight line represents a least square fit of the data at highest momentum transfers (see text). The data are the same as in Fig. 3. (After Ref. 8)

56Fe 0

3995 30 3595 39 3595 30

>,

Cr v_

a

w 10 5

3595 25

v " y~

0 F

-700

-500

56Fe

3595 20

0

-300

-100

y (MeV/c)

0

3600 16

e

2020 20

I<

2020 15 1

100

307c

21 10 -3

1

y=-100 MeV/c

1x10'3

Fig. 3. The scaling function FexP(q,y) (eq. (2)) of 56Fe obtained from the experimental data of Ref. 11, using a relativistic off-shell electron-nucleon cross section. (After Ref. 8).

1 , 10-3 6X10_4 v

2 X 10-4

a 1

y=-200 MeV/c

3 X 10 -4 200 - 4

v

1

10

X

10 -4

y=-300 MeV/c

6 x 10 -5 4 x 10 -5 k (h-feV/c)

Fig. 5. The nucleon momentum distribution n(k) in 56Fe, obtained from eq. (5) using the intercepts of linear fits (i.e. the experimental asymptotic scaling function FexP(y)) shown in Fig. 4. Solid squares represent n(k) obtained from eq. (5), calculating the binding correction dB(y)/dy with the realistic Spectral Function of Ref. 12. The error bars include the statistical uncertainties of the asymptotic scaling function . Solid line : n(k) obtained from the Spectral Function of Ref. 12 . Dotted line : n(k) obtained within the Hartree-Fock approximation. The normalization of n(k) is I n(k) d3k = 1 . (After Ref.8).

y=-400 MeV/c

2 x 10 -5

2 x 10 -5

1 x 10 -5

.00

y=-500 MeV/c .10

.20

1/q (fm)

.30

.40

30Se

C. Marchand et al. I Quasi-elastic electron scattering

is the contribution arising from the Spectral Function Sex(k,E) at E > Emin and it is related to the relevance of the nucleon binding. The correction term dB(y)/dy has been evaluated for 56Fe by using the Spectral Function of Ref. 12, which includes the effects of the short range correlations, while dF(y)/dy has been obtained by the intercepts of the linear fits shown in Fig. 4. It is worth noting that dB(y)/dy is positive for lyl>kF, due to the general asymptotic structure of S(k,E), and therefore the first term in eq. (5) represents a lower bound for n(k), in this momentum region. In Fig.5, n(k) for 56Fe extracted by using eq. (5) is shown, and it is comp

with the corresponding theoretical quantity obtained both by taking into account the

SRC 112] and by using a mean field approach. The orders of magnitude difference for k>kF between the experimental n(k) for 56Fe (that follows the same trend shown by all nuclei we have analyzed, such as 2H, 3He, 4He, t2C, see Ref. 8) and the mean field prediction stresses

the relevance of such a region for obtaining valuable information on SRC, as mentioned in the

introduction . Therefore a realistic description of n(k) and S(k,E) for k>kF are prerequisites for a theoretical analysis of all those erects which will be investigated by the future high energy accelerators and involving bound nucleon with high momentum k.

4 .E

F SRC

N THE NUCLEAR SCALING FUNCTION (S . SIMULA)

As explained in the previous section, the calculation of inclusive QE electron scattering both at finite and infinite momentum transfers does require the knowledge of the nucleon Spectral

Function S(k,E). At present, calculations of such a quantity within many-body approaches and using realistic models of the NN interaction are available only for the three-nucleon system (see e.g. Ref. 13) and nuclear matter [14]. In this section, different models of the effects of short-

range NN correlations on the momentum and removal energy dependences of S(k,E) are briefly

outlined and the corresponding results of the calculations of the nuclear scaling function for sorrR; finite nuclei are presented.

When SRC are present, the nucleon Spectral Function can be represented in the following

form

S(k,E) = So(k,E) + S I fk,E) where So includes the ground and one-hole states of the residual (A-1)-nucleon system and S1

more complex configurations arising mainly P,3m 2p-2h states generated in the ground state of the nucleus A by SRC. The hole part of S(k,E) can be written in the following form [12]

C. Marchand et al. / Quasi-elastic electron scattering SO(k,E) =

1

4nA

309c

X Aa na(k) S(E + I ea ~) a

where the sum over a runs only over hole states, na is the momentum distribution ofthe shell model sp state a, with sp energy ea and number of nucleons Aa (A=Ya Act) . Within the mean field approach approximation S1(k,E) = 0, and the usual Hartree-Fuck (HF) Spectral Function, i.e. eq. (8) with na=na~ is recovered SHF(k,E) =

41

nA

Y, Aa nQHF(k) S(E + 1 Eoc ~)

(9)

The main effect of SRC on the hole part So(k,E) concerns the normalization of the sp momentum distribution n,(k), namely j dk k2 na(k) < 1 in eq. (8), whereas jdk k2 nOF(k) =1 in eq. (9), for sp states below the Fermi level.

The most simple model for SI(k,E), introduced in Ref. 15, is a 8-type model in which the strength of S 1(k,E) is concentrated at a fixed value of the removal energy E= El independent on k, i .e.

S1(k,E) « S[E - El ]

(10)

The value of El can be determined through the energy sum rule according to Ref. 15.

Such a model will be referred to as the mean excitation model. However, the analysis [16] of the perturbative expansion of both the NN interaction and the nucleon momentum distribution,

for potentials decreasing at large k as powers of k, has shown that the Spectral Function, at high values of k and E, should be governed by ground-state configurations in which the high momentum k 1= k of a nucleon, is balanced mainly by the momentum k2 = - k of another

nucleon, with the remaining (A-2) nucleons acting as a spectator system with total momentum kA-2 = 0. Such a picture [16-17], supported also by recent experimental (e,e'p) results on 3He

and 4He (see Ref. 18), predicts a new 8-type link between the removal energy and the momentum, i.e.

with

S1(k,E) a S[E - E1(k)]

(11)

)

(12)

k2(A-2 E1(k) = Ethr + 2(A-1)M

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C. Marchand et al. / Quasi-elastic electron scattering

where Ejhr = I EA I -1

EA-21

is the two-particle break-up threshold.

We will refer to eqs. (11) and (12) as the naive Two-Nucleon Correlation (naive-2NC)

model. By allowing the spectator (A-2) system to share momentum with the hard nucleons, the Spectral Function will acquire a removal energy and momentum dependence for E*E1(k). In Ref. 17 a model Spectral Function which takes into account the finite width due to the center-of-

mass motion of the correlated pair, has been proposed. Such model is able to correctly reproduce the results of the existing many-may calculations of the Spectral Function for the

three-nucleon system [ 13] and nuclear matter [14] in a wide range ofvalues ofE. We will refer to such model as the realistic Two-Nucleon Correlation (realistic-2NC) model.

10-s

10-5 10-6

10-7

-600

-400 -200 y ( eV/c)

-600

-400 -200 y (MeV/c)

Fig. 6. The asymptotic nuclear scaling function of 4He and 56Fe obtained within different models of the nucleon Spectral Function. Double-dotted line: mean field approach (eq. (9)). Dot-Dashed lines: mean excitation model (eqs. (8) and (10)). Dashed Lines: naive-2NC model (eqs. (8), (11) and (12)). Solid lines: realistic-2NC model [ 17] Black dots and error bars represent the extrapolated values of the nuclear scaling function obtained in Ref. 8 from the SLAG data [11] .

C. Marchand et al. / Quasi-elastic electron scattering

31 1e

In Fig. 6, the nuclear scaling function of 4He and 56Fe calculated in the asymptotic limit q->oo using the Spectral Functions of the mean field approach (eq. (9)), the mean excitation model (eq. (8) and 10), the naive-2NC model (eqs. (8), (11) and (12)) and the realistic-2NC model [17] are reported and compared with the corresponding values extracted from the SLAC experimental data [11 ] following the extrapolation procedure of Ref. 8 described in the previous

section. The details ofthe theoretical calculations can be found in Ref. 12. From Fig. 6 it can be seen that for large negative values of y (< -250 MeV/c): i) the predictions of the mean field approach are several orders ofmagnitude smaller than the corresponding ones obtained within models which include the effects of SRC; ii) the predictions of both the naive-2NC and the realistic-2NC models are significantly different

from those obtained within the mean excitation model, and are not in disagreement with the values of the nuclear scaling function extrapolated from the SLAC data. In conclusion, a meaningful interpretation of experimental data concerning scattering processes off nuclei involving high values of nucleon momentum and removal energy, requires

the use of a nuclear structure model which should include, in a realistic way, the effects of SRC among nucleons.

5. CONCLUSIONS Information on SRC can be obtained from both inclusive and exclusive electron scattering experiments. For the latter, as shown in Sec. 2 for the case of the reaction 4He(e,e'p)3H, FSI and MEC effects violating the simple framework ofPWIA, are found to be important. Detailed calculations of FSI and MEC are required in order to reliably extract S(k,E) over a wide range

of values of k and E. On the contrary, as shown in Sec. 3, in the case of the inclusive data, from the asymptotic scaling function one can extract the nucleon momentum distribution, and

therefore one can gain information on SRC in a more model-independent way, since it is possible to take under control the effects of FSI, exploiting the approach to the scaling of the nuclear scaling function. The main outcome of this analysis in terms of y scaling is that the experimental momentum distribution for complex nuclei largely differs, for k>2 fm-1, from the one predicted in a mean field approach. In Sec. 4, some theoretical models of S(k,E), which explicitly take into account SRC, are illustrated. The detailed comparison between theoretical calculations and the available inclusive data for 4He and 56Fe shows that a meaningful

interpretation of experimental data at high values of y requires the use of realistic models of S(k,E) .

In conclusion, we have pointed out that a realistic description of both the initial and final

nuclear states is necessary for a deep understanding of the electromagnetic response of the

C. Marchand et al . I Quasi-elastic electron scattering

312c

nucleus over a wide range of the kinematical variables. On the other hand, in view of the future experimental activity, such as the investigation of the xB > 1 region and the search of colour transparency effects, a detailed analysis of the nucleon dynamics is highly desirable in order to

keep under control the unavoidable nuclear background on which the subnucleonic degrees of freedom will show up.

F 1 2 3 4 5 6 7 8 9

10 11

12 13 14 15 16 17

18

NCES

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