firas (482)

7/25/2019 FIRAS (482)

1/21

Pergamon

Prog. Puri. Nucl. Phys., Vol. 34, pp. 323-343, 1995

Copyright 0 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved

0146+410/95 52 ,.00

01466410(95)00029-l

Inclusive Electron Scattering

1 ICK

Instifurfiir Physik, Universitiir Basei, CH-4056 Basel, Switzerland

Abstract

We use inclusive electron scattering to study the short-range aspects of nuclear wave functions. The

spectral functions P(k,E) of nuclear matter and Iinite nuclei are calculated using correlated basis

function theory and the local density approximation. The cross sections for inclusive electron-nucleus

scattering are obtained using the factorized propagator approximation in which the recoil-nucleon

final state interaction is treated using correlated Glauber theory. Comparison to (e, e) data at large

momentum transfer allows detailed studies of P(k, E) at large momenta, N-N correlations and colour

transparency.

1. Introduction

In this paper I want to discuss a topic that seems

a

bit old-fashioned at

a

point in time where much

emphasis is placed on ezclwiue scattering and studies with

polotied

electrons. Inclusive scattering,

however, still has a great deal to offer, and

a

study is worthwhile for three main reasons. 1. Before

we can extract information from the more exclusive channels, we obviously need to understand the

inclusive one.

2. Inclusive scattering is the cleanest of all knock-out processes, as the final state

interaction is the smallest; accordingly a quantitative understanding is easier to achieve. 3. Most

importantly, inclusive scattering under selected kinematical conditions offers access to a number of

physical observable6 of very high interest, all related to the short-range properties of nuclear wave

functions.

The kinematical range of interest here is the one of large momentum transfer 9 and small energy

transfer w. To be precise, we are interested in momentum transfers of order of several GeV/c, and

energy transfers of several hundred MeV to 1 GeV, The piece of the inclusive response studied then

corresponds to the low-w side of the quasi-elastic peak.

The basic reaction we are interested in is the one of elastic electron-nucleon scattering, where the

nucleon is ejected from the nucleus. The Fermi momentum of the nucleons leads to the finite width of

the quasi-elastic peak. Nucleons with very high initial momentum i, with k roughly antiparallel to &

are responsible for the tail at the lower w. This is obvious when considering the kinematics in impulse

approximation (IA). The momentum transfer and energy loss w of the electron basically is transferred

to the recoiling nucleon, which after the reaction has a momentum i+ . Small WN (i+g*/2 rnN

+

at large

7/25/2019 FIRAS (482)

2/21

324

I. Sick

Recent calculations of the inclusive response have shown that, in the region where the IA cross

section becomes very small duejo the small probability to find in the nuclear momentum distribution

components of extremely large k, the final state interaction (FSI) of the knocked out nucleon becomes

important, to the point where, in the extreme tail of the quasi-elastic peak, FSI completely dominates

the inclusive response. In this region of w, one can use inclusive scattering to study the recoil-

nucleon FSI. This topic has acquired particular interest recently with the pQCD predictions that

stipulate that an effect called colour transparency leads to a reduction of the final state interaction

at high momentum transfer. Inclusive scattering turns out, much to the surprise of everybody, to be

particularly sensitive to this phenomenon.

The present paper deals mainly with inclusive scattering from nuclear matter, rather than with

scattering from finite nuclei, as nuclear matter offers a number of advantages. In the second part of

the paper, we then generalize the results to tinite nuclei.

2. Data for nuclear matter

As mentioned above, much of

this

paper deals with infinite nuclear matter. The study of nuclear

matter is preferable for a number of reasons:

For nuclear matter the S linger equation for nucleons bound by the nucleon - nucleon

interaction (a modem potential deduced from N-N scattering) can be solved with very few

.

approximations. The inllnite nature of the medium, where solutions can all be written in terms

of plane waves, simplifies the calculation very much. As a consequence, the quality of nuclear

matter wave functions is comparable to the one for the A=2,3,4-nuclei. Due to the exact

nature of the nuclear matter wave function, both he long-range and short range properties are

well under control. This is in contrast to finite nuclei, where calculations, that are designed to

do well on the long-range properties (mean-field calculations), usually do badly on the short-

range aspects.

Nuclear matter is a system that has a high average density. Accordingly, it is an ideal system

to study the short-range aspects of the wave function, related to short-range N-N correlations.

For applications to inclusive electron scattering -

a tool we will show to be sensitive to the

short-range aspects -

the main ingredient needed, the spectral function P(k,E) is available

and has been calculated for several N-N interactions.

The f inal state interaction between the recoil nucleon and the (A-l) system, which is important

for a quantitative understanding of inclusive cross sections, can be calculated more reliably for

the infinite system.

Nuclear matter also has an obvious drawback: only very little data is available. Up top now, studies

of nuclear matter properties have been limited to the two traditional observables binding energy

per nucleon and density, both extrapolated from ilnite nuclei. These two observables, which mainly

concern long-range properties, have for decades been the main playground of nuclear matter-

calculations.

In this section, we show that for nuclear matter one also can derive the cross sections for inclusive

electron scattering. The availability of such data greatly enhances the value of nuclear matter as

a vehicle for testing our understanding of nuclear wave functions. In particular, studies of nuclear

matter via (e, e) at large momentum transfer q all ow us to study the shortronge properties of nuclear

matter wave functions.

Before deriving the cross sections, we should realize that inclusive electron scattering at large q is

sensitive only to rather local properties of the wave function.

The spatial resolution of (e, e) is

7/25/2019 FIRAS (482)

3/21


325

of order

l /q, which

at large

q

is small. In particular, the scattered electron is not sensitive to the

interactions of the recoiling nucleon outside this range.

In this paper we will be interested in cross sections for inclusive electron scattering at large

q.

We

have measured the corresponding

data for finite nuclei a

number of years ago

at

SLAC [I], using the

BGeV spectrometer in ESA.

The

data have been measured at energies between 2 and 4GeV, and

scattering angles between 16 and 39. The data taken for He, C, 27Al, Fe and Au cover the

region of the quasi-elastic peak, and for some of the kinematics extend to large energy loss w, into

the region of deep inelastic

scattering.

we

first

discuss

the qualitative idea used to extrapolate from finite nuclei to nuclear matter. We

start from the following consideration: To a good approximation, the nuclear response function is an

incoherent sum over the contributions horn individual nucleons. The volume piece, proportional to

the nuclear mass number A, is the one we are interested in when discussing nuclear matter. Effects

of the nuclear surf ce re proportional to

A2j3

given the

AP

dependence of the nuclear radius. The

ratio of surface to volume contributions thus is proportional to A-j3. Extrapolating the response

function

per nucleon

to

A- l /3 = 0 as a

linear function of

A-II3 gives

the nuclear matter response.

This

A-f3 dependence can be derived more formally using the local density approximation (which

we

discuss in

more detail in

a

later section of this paper). In the local density approximation, we may

consider P(k,

E,p)

to be a quantity that, in addition to the

usual

dependence on initial momentum

k and removal energy E, also depends on the local nuclear density p(r). We then can write the

inclusive cross section

a(q,w ) = / P(p(i)) - F -

d;dE .

p(t)dr

(1)

The

factor

F, which contains the nucleon structure functions and all the kinematical factors, depends

on

k, E,

q, w,

and is of no interest for the following discussion.

To make explicit the dependence of

u(q,w )

on the nuclear mass number

A we

split the density

into two terms pc + P, see figure 1). The former corresponds to the idealiz.ed hardsphere density

Figure 1: Schematic representation of nuclear densities. Density of finite

nucleus (full line), idealized hard-sphere density pc (dashed line), surface

density p, (difference between solid and dashed line), density for lighter

nucleus (dotted line).

(p t < I ) = po, p=(~ > I ) = 0), the latter corresponds to a surface peaked distribution (with

total volume zero) that describes the difference between pc and the real density p(r). From elastic

electron scattering we know that pc is largely independent of

A,

with = ro -

A13.

We also know

7/25/2019 FIRAS (482)

4/21

326 I. Sick

that the quantity p(r) - po is a nearly universal function of I& - r, which for A210 haa a shape

largely independent of

A.

These two contributions to the density give different contributions to the nuclear response: The

nucleons n the constant-density

region

of the nucleus give the contribution we are interested in. The

correeponding contribution to eq. 1, integrated over the constant-density region, gives

u w) = A. / P(po) . F - dgdE

The quantity u,/A in the limit A -B 00 is the nuclear matter response per nucleon.

The nucleons in the surface region contribute differently due to the change in P between densities

of po and p = 0. Given that the radial dependence of p(r)

in the surface region is a

near-universal

function of r - I&, and that for large A the region where p(r) - po 0 is small compared to I&, the

angular part of the integral over p can be calculated

u,(q, w) = A213 - 4xri

J

P(p(r)) - F -d dE - p,(r)dr

Thin contribution repreeente the difference between nucleone with the idealized density dietribution

pc

and nucleone with a density having finite surface thickness.

The total nuclear response, divided by

A,

then reads

4q,w)/A = 4w)/A + 4wW

=

J

P(po) - F - didE +

A-13

J

P(p(r)) - F - d;dE .4x$ -p,(r)dr

with all of the fir&-order

A

dependence explicitly shown.

(4)

o.ooo n I , m.I s s

0 0.8 0.4 0.8 0.11

1

*-l/8

Figure 2: Ftesponee function per nucleon M a function of A-P, for

E=3.6GeV, 8=16 and w=lBOMeV. Only the points for A-j3 less than

0.5 (Ar12) are used for the extrapolation.

Eq. 4 shown hat, in agreement with the simple argument made above, the nuclear responeeu(q, w)/A

is expected to be a linear function of A-j3. Extrapolating the data

u q, w)/A = X(q,w) aa a

linear

function of A-i3 to 0 (A = oo) yields the nuclear matter response.

To fix these ideas, we show in iig 2 the extrapolation for a speciftcq,w. Disregarding the nucleus He

(A- j3 = 0.63),

the nuclear response

I l(q,w) for A =

12- 197 is well described by a linear function of

7/25/2019 FIRAS (482)

5/21


ISlTllllll-

o.olm -

-_

327

Figure 3: Same as figure 2, but plotted 88 a function of A. The arrow

indicates the nuclear matter value.

A-i3. When extrapolated to zero, this gives the nuclear matter result. The corresponding plot for an

extrapolation as a function of A is given in fig. 3. This figure shows that extrapolation a8 a function

of

A is

not practical, although the curve seems to better convey the idea of saturation. Due to the

large fraction of eurface-nucleons (>50% even for the largest A) even heavy nuclei significantly differ

from nuclear matter. Figure 4 shows an additional example for A-II3 extrapolation. We find that

at all q and w the data for AL12 can successfully be represented by a straight line a8 a function of

A-j3, with a slope that changes sign and size, depending on the region of the nuclear response.

Figure 4: Response function per nucleon ae

a

function of A-i3, for

E=3.6GeV, 8=20 and w=645MeV. Only the points for A-/ lese than

0.5 (Az12) are used for the extrapolation.

We note that the same type of extrapolation can be applied to other observable8 that relate to short-

range properties of nuclei and nuclear matter. In ref. [2] we have ehown that the ratio8 of nucleus

to deuteron cross section8 for 0 < z < 1, the quantity of interest for the EMC-effect, can also be

extrapolated to nuclear matter. The data, for all nuclei and all momentum transfers measured, can

actually be represented by the single ratio of nuclear matter to deuteron as a function of z. As for

nuclear matter we do have a much better understanding of the short-range behaviour of the wave

7/25/2019 FIRAS (482)

6/21

328

I. Sick

function, it would seem advisable to study the EMC effect in nuclear matter rather than in finite

nuclei.

3.

Spectral Function of Nuclear Matter

Benhar et ol[3] have calculated the one body Greens functions of nuclear matter

at

saturation density

PNM = 0.16fm

-3. They use the non-relativistic nuclear hamiltonian with the Urbana u14 +

TNI

interaction, and Correlated Basis Function (CBF) theory. The spectral function is proportional to

the imaginary part of the Greens function which describes the propagation of hole states, and can

be written in the form

P(k,E) = c )< Olo;JNA- >I2 6(E - Ef + Et),

N

where z: creates a nucleon with momentum k, ]O > represents the nuclear matter ground state with

energy eigenvalue

Et,

and INA- > are intermediate excited states

of

the

(A -

1) - particle systems

with energy eigenvalues

Es-.

100

10-l

E (WI)

Figure 5: Spectral function of nuclear matter at saturation density (full

line) for k/kF = 0.75. The dashed and the dashed-dotted lines corre-

spond to PJk, E) and P,,(k, E), respectively.

The calculation includes one hole (lh) (N = k > and

two hole - one parti cle

(2hlp) IN = hihi,pi >

intermediate states. The calculation is carried out using

CBF

perturbation theory within the set of

Correlated States (CSI

,

n>cs=

W >

< q p, >112

where ]@ > is the generic eigenstate of the

Fermi gas

hamiltonian, 4 is a many-body correlation

operator of the form B =S ni. The width of the peak provides a measure of the lifetime of the hole state and goes to

zero as it approaches the Fermi surface. The integral of P,z(k, E)

over the energy gives the strength

Z(k) of the hole state, which is quenched with respect to unity 151,due to N-N correlations. The

7/25/2019 FIRAS (482)

7/21

Inclusive

Electron Scattering

329

100

10 l

27

x

10 z

10 s

10-I

Figure 6: Momentum dis-

tribution of nuclear mat-

ter at various densities.

For each density the lower

curve at k k F .

The integralof F (k)

E )

over

the energy gives the so-called continuous part of the momentum distribution n,(k) [5]. Figure 5

shows an example for the distribution in E. For applications to finite nuclei (see section 8) we have,

in ref.[6] computed the spectral function and the related sum rules at five different nuclear matter

densities, namely p = 1.25, 1.0, 0.75, 0.5 and 0.25p~~, using the Urbana ~14 +

T N I

model of the

N-N interaction (7, 81. The resulting momentum distributions are shown in fig. 6. The calculations

confirm the naive expectation that the height of the quasiparticle peak increases when the density

diminishes, whereas its width becomes smaller. The extension and importance of the background are

much larger at higher densities. The calculated strengths of the quasi-hole pole at the Fermi surface,

2(e~), are between 0.6 and 0.7 for p between l/4 and 5/4 of nuclear matter density. For

a

purely

repulsive interaction, one would expect 2 to approach unity for low densities; the attraction given

by the empirical N-N interaction is causing the correlation function to overshoot 1, which simulates

bound states of pairs of nucleons and explains why 2 does not approach 1 for low density.

4.

Inclusive cross section

In order to calculate the inclusive e-nucleus

cross sectiona, we

employ the approach developed in

ref. 191 or nuclear matter; here we give the relevant equations already generalized to finite nuclei [4],

where the local density approximation can be used to calculate both the P(k,

E )

and the FSI, starting

from the nuclear matter results. For nuclear matter, the R-dependence can simply be ignored.

In Born approximation, the inclusive cross section is given by:

where a = l/l37 is the fine structure constant, t and s are the energies of the incident and scattered

electron, and q

is

the four momentum transferred by the virtual photon: q E k. -

k i ,

with k E (c, ke)

and

k ; , z t ,b ) . L

and IV, are the lepton and nuclear tensors, respectively.

7/25/2019 FIRAS (482)

8/21

330

1 Sick

The PWIA expression of the nuclear tensor is given by:

W ,(q)= / ~W,b,,,(R,

,

with

W$+,(R, q) = J dkdEP(R,,E) z*,fuk, 4 +N*,l, k,4 4) t

(8)

where @$l is the electromagnetic tensor of an off-shell proton (neutron); its expressions can be

found in ref. [9]. In the calculation of W, we employ the appropriate relativistic expressions as

imposed by the high recoil nucleon momenta.

We include the final state interaction (FSI) of the knochked out nucleon by using the

factonied

propagator approzimation and Corr elated Glauber Theory (CGT), BSdeveloped in ref. [9]. In this

approach, the effects of the FSI appear in a quantity that is analogous to, but more complicated

than the usual optical potential

V.

The factorized propagator approximation is a direct consequence

of neglecting the dependence of the complex

potential V = U + iW,

felt by the struck nucleon in

the nuclear medium, on the energy release w in the scattering process. This approximation has been

successfully used in a number of studies of inclusive scattering on quantum liquids [lo, ll] and on

nuclei [12].

In the factorized propagator approximation, the nuclear tensor is given by:

where p = k + q is the momentum of the recoiling nucleon and where the folding function

F

is given

by:

F (R, p,w - w) = k32

I

*dte

i(u-w)le-iV(E,p,l)t

0

(11)

The folding accounts for the interaction of the knocked-out nucleon with the (A-1)-particle system,

which couples the initial lp - lh state to more complicated 2p - 2h states, etc. Due to this, the

initial lp - lh state acquires a width and does not have to be on-shell. As a consequence the folding

function

F(R,p,w)

extends to both negative and positive w-values.

The imaginary part of

V

has the most important effect upon the inclusive

cross

sections; it is found

to be [9, 131:

WR, P; t) =

(12)*y) lo (II / $

eiknw(-PE)gNM(p(R),

t)%fp(k,vN),

where g(r) is the N-N pair correlation function taken from the nuclear matter calculation, and where

fp(kNN) is the free N-N scattering amplitude taken from experiment [14, 151. The k-dependence of

fp(kNN) accounts for the finite range of the N-N interaction. We neglect the effect [16] of the change

of the scattering amplitude off-shell, aa too little is known about it; according to ref. [17] only small

effects are expected at high q.

We emphasize that it is important to include the N-N correlation function g(t - v), i.e. to use

correlated Glauber theory. This accounts for the fact, that a nucleon in the nucleus is surrounded by

a correlation hole; therefore the probability of a collision with one of the (A-l) nucleons, during the

first 2

fm

of its trajectory as a recoil nucleon, is reduced.

In the above treatment of the FSI, we do not include the spectator effect which wan claimed in

ref. [18] to produce a sizeable reduction of the effect of correlations between hit nucleon and the

nucleons of the (A-l) system. We have recently [19] included this effect by extending our correlated

Glauber treatment to include the correlations between the spectator nucleons, employing three-body

distribution functions, the functional form of which is taken from the nuclear matter calculation of

7/25/2019 FIRAS (482)

9/21

Inclusive Electron Scattel-ing

331

[3]. We find very small effects; in the range of q,w of interest here, the spectator effect changes

the cross sections by < 4%.

In the calculation of the inclusive response, we also include the effect of colour transparency, an effect

that is discussed in more detail in section 7. In order to account for this predicted phenomenon, we

employ in our calculations a standard expression for the cross section [20], and the evolution of the

small Fock state back to the normal nucleon is treated as proposed by [21].

Furthermore, we also include the contribution due to excitation of the nucleon. This contribution

is calculated using P(k,E) and the inelastic response of the proton and neutron, as measured by

(e,e) and parameterized by Bodek et al [22, 231. Beyond the scope of the present calculation are

the contributions due to meson exchange currents,

5. Results for nuclear matter

In order to test the calculated P&E), we compare to the data on inclusive electron scattering

at high

q

[24, 251. Fig. 7 shows the calculated PWIA cross-sections for incident electron energy

e =

3.6GeV

and scattering angle B = 30. The PWIA calculation and the data are in close agreement

10-2

10-2

10-a

10-5

10-e

Figure 7: Inclusive cross sections for nuclear matter in PWIA. The

quasielasic (dashed) and inelastic contributions (dash-dot) are shown sep-

arately.

at w > lGeV, whereas s&able discrepancies occur, both in magnitude and shape, at lower energy

loss. The theoretical curve lies a factor of 3 to 4 below the data at w = 0.6 to

O.SGeV,

and exhibits a

pronounced kink at w N

O.SGeV,

reflecting the threshold for the three-body break-up processes. The

origin of this kink lies in the discontinuity of the nuclear matter momentum distribution at

k = kF.

When including the final state interaction, as done in figure 8, a very different picture emerges. The

full calculation is in nearly perfect agreement with the data over the entire region of energy loss w.

Figure 8 also shows that a calculation that does not account for the correlations in the nuclear ground

state does not agree with the data; the dash-dot line, which corresponds to the use of uncorrelated

Glauber theory for the description of FSI, is much too high at low energy loss. Figure 8 also shows

that it is important to include the effects of colour transparency, which reduce the FSI; the dashed

curve, which is calculated by omitting colour transparency, is significantly too high.

The results obtained for c =

4GeV, 0 =

30 are shown in figure 9. The theory is in good agreement

with the data at these higher momentum transfers. The PWIA response is again much too small

at small w, the FSI is sufficient to raise the response to the data. At these higher q, the colour

7/25/2019 FIRAS (482)

10/21

332

I. Sick

Figure 8: Crosclsection for nuclear matter. Full calculation including FSI

(solid), calculation omitting colour traneparency (dashed), calculation

using uncomrkatedGlauber theory (dash-dot).

Figure 9: Inclusive reeponee of nuclear matter. Full calculation (solid),

calculation omitting the effect of colour transparency (dashed).

transparency has an even larger effect on the response, and it is clearly necessary to include it in

order to obtain agreement with the data.

6. Effect of correlations

We have emphssized above that it is important to include in the calculation of FSI the fact that

the recoiling nucleon in the initial state wee surrounded by a correlation hole, Ce. that one haa to

use corrcloted Glauber theory for the description of FSI, with a correlation function g(r - r) taken

from the CBF calculation of the initial-state wave function (see figure 10). The pair distribution

function g(r) is very small at small r; therefore the motion of the struck nucleon is little damped

at distances 5 lfm from where it has interacted with the electron. The sensitivity of the cross-

section to g(r) actually is quite pronounced.

In fig. 11 we show the inclusive crose+ections at

3.6GeV,

25, calculated for both the normal nuclear matter g(r) and a modified g,&(r). In grd(r)

7/25/2019 FIRAS (482)

11/21

Figure 10: Pair distribution function of nuclear matter at the empirical

saturation

density. The hole at small r results both rom Pauli correla-

inclusive Electron Scattering

tlll j

333

tions and the effect of the repulsive core of the N-N interaction.

Figure 11: Inclusive cro8s

sections for nuclear matter,

using the normal (solid)

and a modified (dashed)

correlation function

we have artificially reduced by 20% the hole in g(r) around r = 0, due to short range correlations, by

simply compressing the radial scale. The effect on the cross-section ie significant. This sensitivity to

g(T) is most welcome, as in most observables the effects of N-N correlations are hidden and indirect.

This sensitivity provide8 a strong motivation to study (e,e) at large

q in

more detail in the future.

7. Colour transparency

We have discussed above the effect of the nucleonic FSI on the inclusive cross sections. When

calculating the cross section in PWIA, the predicted response is much too low at low values of the

energy lose w. When incorporating the FSI of the recoil nucleon using correlated Glauber theory,

the response gets significantly to high. This shows that, in the framework employed, the calculated

FSI is too strong.

In order to improve upon this, we have incorporated in the calculation the much-discussed effect of

7/25/2019 FIRAS (482)

12/21

334

I .

Sick

colour transparency [26, 27, 21, 28, 201, which leads to a reduction of the strength of FSI.

According to the predictions of pQCD, elastic electron scattering from nucleons at large

q

selects

nucleons in a small Fock state. Only for such a configuration the large

q can

be split among the

valence quarks by gluon-exchange, such as to keep the quarks together in the nucleon (as required

by elastic e-N scattering). Such a state, with the

valence

quarks very close together and the colour

interaction well shielded, is predicted to have a smaller interaction cross section with the rest of the

nuclear medium, i.e. the (A-1)-nucleus is more transparent for the recoiling nucleon. This reduction

of the FSI of the recoiling small nucleon is effective until the small state has evolved back to the

ordinary nucleon.

This small configuration can be regarded as a coherent superposition of different excited states of

the baryon which evolves into a nucleon; the superposition gets out of phase in a time called the

hadronieation length

l ,, = 2EJAM .

(13)

Various estimates of AM can be found in the literature and we have used AM2 =

0.7GeV2

in the

present work, as suggested by ref. [21]. The struck nucleon interacts with the other nucleons with

the free QNN only after travelling a distance Zs. At I< Zhthe interaction cross-section depends upon

z and is estimated to be:

l-

where &r is the transverse momentum of partons in the nucleon (< Zr >l12m

350M eV).

It is trivial

to include such a time dependence of the Glauber cross-section in the classical (z =

ut) limit.

One

then obtains:

where

c(q,l)=l+@(Zh-+l) (I-=).

(16)

As shown in figs. 8 and 9 colour transparency has a sign&ant effect on the inclusive

cross

section.

The calculation of the colour transparency effect - which has no free parameters -is in much better

agreement with experiment. This comes somewhat as a surprise, as inclusive scattering was not part

of the canonical set of observables advocated by the proponents of colour transparency. Generally,

the ezcltiue process (e,

ep) w as

emphasized; in (e,

ep)

colour transparency leads to the same effect

discussed above for (e,e), a weakening of the recoil-nucleon FSI relative to the one predicted by

standard Glauber theory, with the consequence of a bigger transparency of the (A-l) nucleus for the

recoil proton.

As a matter of fact, the effect of colour transparency in (e, e) is much larger than for (e,

ep)

at similar

momentum transfers. While our calculation predicts a major effect in (e, e), the same calculation

[13] predicts very small effects for

(e,ep), in

accordance with the

(e,ep)

data available [29]. This

finding, which at first sight is somewhat counter-intuitive, is explained by the fact that in

inclusive

scattering one does observe the juZZeffect of colour transparency, as l/q, the distance over which

(e,e) is sensitive to the interaction of the recoiling system, is comparable to or smaller than the

distance within which the small 3-quark state evolves back to a normal nucleon. For (e,

ep) much

larger

q

is would be needed to observe the full effect of colour transparency; to be specific,

q

would

have to be large enough to increase (by time dilatation) the lifetime of the small state to the time it

takes for the recoiling system to traverse the

entire nucleus.

The concept of colour transparency obviously has a number of difficulties. Several models have been

proposed for the evolution of the small state and its cross section, and for the size of the initial

7/25/2019 FIRAS (482)

13/21


335

small state a8 a function of

q.

The small state ie far off-shell (z500MeV when considering it 88 a

coherent superposition of nucleon resonances, and an additional N 300MeV for the (e, e) caee where

only the FSI gets the nucleon back on shell). While th

e evidence for colour transparency at present

is not entirely convincing, the effect8 observed in (e,e) at least provide a first real signal.

8. Finite nuclei

The data for nuclear matter are much more limited than for iinite nuclei, an it takes a

set

of data

for many nuclei at the 8ame

E,q

and w in order to extrapolate to nuclear matter. It therefore is of

interest to extend our approach, described above, to finite nuclei. This can be done, although thi8 is

only poesible at the expense of introducing additional approximations.

For the description of the initial state of the nucleus (AL12) we calculate the spectral function

P(k,E) aa in [6]. The approach is based on the separation of P(k,E) into it8 single-particle and it8

correlated parts. The eingle-particle part can be calculated ueing the independent particle shell model

or, alternatively, can be derived from the experimental results on (e, ep) reactions. The correlated

part, which is of prime interest here, is evaluated employing the reeults obtained for nuclear matter.

It is calculated at varioue nuclear matter densities, and the corresponding quantity in finite nuclei

is obtained using local density approximation (LDA). The calculation of P(k,E) ie expected to be

quite reliable at high

k,

since the short-range properties of nuclei can be expected to be properly

treated in LDA.

The inclusive cro88 section8 for nuclei AL12 are calculated using the

factorized propagator approai-

mation for the treatment of the FSI, a8 described above. The FSI occurring at different densities in

the nucleus is again treated in LDA.

We show first that the calculation of the P(k,E)

in LDA produce8 a result for the momentum

distribution that agrees very well with exact calculations. Figure 12 give8 a comparison of our

lb

0 I P 3 4

k (fm-)

Figure 12: Momentum distribution of Oxygen: LDA (solid line), Varia-

tional MontoCarlo calculation [30] (squares), nuclear matter momentum

distribution, normalized to 16 nucleon8 (dashed).

momentum distribution (labeled LDA) and the one of Pieper

et al

(301, calculated using the Vari-

ational MontoCarlo (VMC) approach. This calculation wa8 baaed on a realistic nucleon-nucleon

potential, the Argonne ~14 interaction supplemented by the Urbana VII threonuclcon potential. In

spite of the fact, that the two results have been obtained using different nuclear hamiltonianr, we

observe a remarkable agreement, indicating that LDA is likely to be a good approximation for the

7/25/2019 FIRAS (482)

14/21

336

I. Sick

description of the correlated part. The small diRerences at low

k

probably reflect a shortcoming of

the variational wave function employed in ref.[30], whereas the differences at large

k

come from both

the statistical fluctuations of the VMC calculation and from the differences in the hamiltonians and

in the correlation operators. For illustration, fig. 12 also gives the

n(k)

of infinite nuclear matter.

400 600 600

moo

1i?oo

-aw~ow

Figure 13: Inclusive cross sections for 0 at 3.6GeV and 25. LDA re-

sult (fuh Iine), calculation employing the mean-field piece of the spectral

function only and no FSI (dotted), calculation using the nuclear matter

spectral function and the corresponding FSI for the empirical nuclear

matter density (dashed).

In figure 13, we show the cross sections for inclusive electron scattering from *C at 3.6 GeV and

25. The solid curve shows the fuII result, obtained using the spectral function calculated in LDA,

and the FSI treated in CGT. The calculation agrees very we-h with the data, both in the region of

the quasielastic peak (w N lGeV), and in the tail at smal.l energy loss. The dashed curve shown

10 e

10 7

10 8

10 0

10-10

400600

600

1000 1200

-

lo= WV

Figure 14: Inclusive cross

sections for Fe for 3.6

GeV and 25.

in figure 13 uses the nuclear matter spectral function for the fulI nuclear matter density and the

corresponding FSI. Due to the excess of high momentum components and a final state interaction

7/25/2019 FIRAS (482)

15/21

Inclusive

Electron Scattering

337

which is too strong, the cross section becomes too large at low energy loss. The dotted curve uses

the mean-field part of the spectral function and no

FSI

(th

e on

g-

range part of which has a small

effect). This calculation also clearly

disagrees

with the data.

In figure 14, we show data and the calculation for the same momentum transfer, but for ssFe. Again

we observe excellent agreement with the data.

From this, we conclude that the LDA

allows

to

correctly treat the evolution of both spectral function and tinal state interaction as a function of

nuclear mass number A.

9. Ratios of nucleus-to-deuteron cross sections

Ratios of cross sections - we here consider those between nuclei with A>12 and the deuteron, in

analogy with the ratios studied in deep inelastic scattering in connection with the EMC-efect - off er

a particularly sensitive tool to study the short-range properties accessible via inclusive scattering.

At the same time, a number of ingredients

such as e.g.

nucleon form factors) cancel, thus allowing

for a better separation of the

various effects.

Experimental values for the nucleus-to-deuteron ratios have been determined in refs. [31,32] starting

from the cross sections for nuclei [24,1] and the deuteron [33,34,35]. These ratios have been evaluated

as a function of the Bjorken scaling variable z = q~/(2m~w), where q,, is the 4-momentum transfer

and mN is the nucleon mass. The ratios exhibit an interesting feature, a plateau in the region of

I between 1.5 and 2.

These cross section ratios have previously been studied by a number of authors. Vary et al [36, 311

have performed a calculation based on a quark cluster model which involves 3- and 6-quark clusters.

The B-quark cluster contribution leads to a plateau of the cross section ratios in the region 1.5 < z 1

corresponds to the low-w tail of the quasi-elastic peak, z = 2 is the kinematic limit for scattering off

the A=2 system. In order to give a rough idea on the relation between k and z, we quote in figure 15

the values of the initial momentum k that give, in IA and for the deuteron, the largest contribution

to the inclusive cross section.

The full calculation reproduces the data quite well. In particular, it yields the behaviour discussed

above, a sort of plateau at z >1.4.

This

plateau is partly attributable to the fact that the ratio of

the nucleus and deuteron momentum distribution at large

k

is indeed close to a constant. The peak

at 2=1.4 results from the fact that near k = k F the momentum space density of iron is much bigger

than the one of the deuteron. The rise of the cross section ratios at z N 2 results from the fact, that

due to the phase space the deuteron inclusive cross section has to approach zero at 2=2.

Two additional curves shown in figure 15 allow to gauge the sensitivity of the calculation to various

ingredients.

The dot-dashed curve corresponds to the approximation used in (32) where the spread

7/25/2019 FIRAS (482)

16/21

8

2

Fiie 15: htioa of inclusive cro81 section6 of iron md deuterium at

3.6GeV and 25. Full calculation (solid line), calculation with no FSI for

both iron and the deuteron (dashed), calculation with spectral function

projected on E=k2/2mN (dashdot). The dominating initial momenta in

the deuteron are indicated in MeV/c at the top.

Figure 16: Ratios of inclusive cross sections of nuclear matter and deu-

terium at 3.6GeV and 25.

of P(k, E) M a function of

E ia

eglected. The dashed curve correspond6 to the case, where the FSI

of the recoil nucleon is neglected, BB done in refn. (37,321. Contrary to the assumption of [32], the

FSI in the nucleus and the deuteron do not cancel; the N-N distribution function6 of the deuteron

and heavier nuchzi differ, and the FSI with the (A-2)

spectator uudeonr is not negligible. For a

quan t i t a t i v e

underetanding of

the cross section ratior, it is dearly imperative to use both a realistic

P(k, E) and a realistic description of the FSI.

While, at k significantly above hi, the ratio of iron and deuteron momentum distribution p(k) indeed

7/25/2019 FIRAS (482)

17/21

Inclusive

Electron Scattering

:X3)

approaches a constcmt value of 3.8, the numerical value of the calculated cross section ratio at z >1.4

does not appear to relate to this feature in a direct way; the average calculated cross section ratio

is N 5.2 f 0.2. The effects of FSI and the spread of P(k, E)

in

E are

too large to allow for a direct

comparison between the two quantities.

We have calculated the cross section ratios for the s8me kinematics and 8ll nuclei where similar data

are available (A=4, 12, 27, 56, 197), and find similarly good agreement. In figure 16 we show ss

s.n illustration the data and calculation for infinite nuclear matter, where we have determined the

experimental nuclear matter to deuteron ratios using the nuclear matter cross sections of ref. [25]

and the deuteron cross sections of refs. [33, 34, 351,

in 8 way analogous to the one used in ref. [32].

In figure 17 we show the comparison between experiment and calculation for a higher value of the

h h-k 00.5

Figure 17: Ratios of inclusive cross sections of iron and deuterium at

3.6GeV 8nd 30. Full crrlculation (solid line), calculation with the corre-

lated part of P(k,

E

reduced by a factor of two (dafhdot).

momentum transfer. The agreement of calculation and data is similar to fig. 15.

As pointed out above, the cross section ratios at z >1.4 are not close to the ratio of nucleus to

deuteron momentum distributions at huge k, and therefore cannot be used to deduce this ratio

directly from the data 8s done in ref. [32]. Both FSI

and the spreading of the spectral function in

E significantly affect the cross section ratios. Despite these complications, the data at large z 8re

sensitive to the properties of P(k, E) at large k. This is demonstrated in figure 17, where the dsshdot

curve gives the cross section ratios for the c8se where the iron P(k,E) is artificially modified such

that the correlated part has a normalization reduced by 8 factor of 2. The reduction of the high-k

components by a factor of two, together with the corresponding change of the short-range FSI which

is included for consistency, has a significant effect at z >1.3.

From the agreement between calculation and data we conclude that the experimental ratio of the

nucleus-to-deuteron momentum distributions at large k is close to the one given by the calculation

employed here, which for iron (nuclear matter) gives 3.8 (4.9). Considering the vsrious uncertrinties

that enter the calculation of the cross section ratios, we assign an uncertainty of f0.6 to this ratio.

We however reiterate the caveat that the data are sensitive to an integral over P(k,

E ,

and do not

directly measure the ratio of momentum distributions.

7/25/2019 FIRAS (482)

18/21

340

10. Scaling

I. Sick

For the rtudy of the inclusive response, it is useful to look at the result8 in terms of the rcaling

function F(y). For brevity, we here limit the discueeion to nuclear matter.

In general, the inclusive response ie a function of two independent variables, momentum transfer q

and energy loee w. In PWIA it can be shown [40,41] that, w q tends to infinity, the cross nectiona

will scale, i.e. become a function of a single variable y, tti y being it&f a function of q and w. The

variable y may be thought of (~1 he

minimal value of the momentum k of a nucleon bound with the

minimal removal energy E. At very large q, and for negligible excitation energy of the final (A-l)

system, i.e. when the rpcctral function P(k, E) may be represented by momentum distribution

n(k), F(y) should depend on y only. It represents the momentum distribution n(k$.

It has been shown that scaling works extremely weJl for very light nuclei [40], while for heavier

nuclei [24] important deviationa have been obsuped. Ref. 141)give review of both experimental

obserpotions and theoretical studies. The occurrence of scaling and the approach to the

q = oo

limit

can give useful clues on the reaction mechauiem.

Deviation8 from scaling occur for two main reasone: 1) The strength of P(k, E) ia distributed over an

extended range of E; thiB eads to a convergence of F(y) &

m e ow

I f

or incressing momentum trauefer

q. 2) The knocked-out nucleon has a fmal state interaction which in general leads to a convergence

from &we. In the kinematical range of the data presently available

it is

the latter which has a large

elect. As a consequence of the high density of nuclear matter, the effect of FSI ir particularly large.

0.0

Fiie 18: Convergence of the scaling function F(y, q) of nuclear matter

for two values of y. The full (solid lines) and PWIA calculations (dashed)

are aLso shown.

We show in figure 18 the convergence of F(y,q) for two aelected valuea of the scaling variable y.

At 1 = -lOOMeV/c, a value below the Fermi momentum

kF

the quality of scaling is very good,

and F(u) changes little over the q-range accessible despite the large change of the croaa section.

Experiment and calculation are in good agreement. For y =

-5OOMeV/c,

on the other baud, F(u,

q)

change a factor of 3 over the q-range accessible. The rate of convergence of F(y) ir reasonably

close to the one caLdated. In this region of y, the change of F(y) with q resultr basically from the

folding of the response due to FSI. The trill of the folding function move strength from k < kF to

the region k > kF where, in IA (dashed curve), the rtrength in very low. The rather good agreement

between convergence of experiment and calculation indicates that the tails of the folding function

are properly predicted.

7/25/2019 FIRAS (482)

19/21


341

11. Conclusions

In thir paper we have discussed an attempt to quantitatively understand the inclusive response

function of nuclear matter and &ite nuclei at high q. The study of the low-w wing of the quasi-

elastic peak provides fundamental information on the short range structure of the nuclear wave

function.

Realistic analyses of this interesting kinematical region of the nuclear response could not be performed

in the past both for experimental and theoretical reasons. Data for the most interesting case -

infinite nuclear matter - can only be obtained once the inclusive cross sections for many nuclei have

been measured under the same kinematical conditions; such data became available only recently. The

major theoretical difficulties consisted in the microscopic evaluation of the spectral function, and the

quantitative treatment of the FSI of the recoiling nucleon which needs to be treated relativistically.

In this paper, we have discussed calculations of the cross section for nuclear matter and finite nuclei

based on spectral functions calculated for a realistic N-N interaction, by using correlated basis

function theory for nuclear matter. The P(k, E) of finite nuclei are derived by splitting the P(k,

E

into the correlated and uncorrelated pieces; while the uncorrelated ones are taken from mean-field

approaches, the correlated ones are derived from the nuclear matter results using the local density

approximation.

The FSI of the struck nucleon has been evaluated by generalizing Glauber theory to the case of a

relativistic nucleon propagating in the same nuclear medium to which it was bound before being

struck by the electron. This amounts to taking into account the fact, that such a nucleon, being

a part of the ground state before the interaction with the electron, experiences a nucleonic density

pg(r) instead of p, where g(r) represents the NN distribution function.

We found that the sensitivity of the cross-section to g(r) is quite pronounced. This sensitivity to

g(T) is most welcome, as in most observable8 the effects of N-N correlations are hidden and indirect.

This sensitivity provides a strong motivation to study (e, e) at large q in more detail in the future.

Corrections to the FSI due to colour transparency have been included in the correlated Glauber

treatment. We find that colour transparency is indeed necessary to obtain good agreement with the

data, and that observable effects of colour transparency are much larger than for the exclusive (e, ep

reaction at similar momentum transfers.

We have shown that the study of nucleus-to-deuteron cross section ratios at z < 1.8 offers sensitive

means to investigate the behaviour of the spectral function at large initial momenta k considered.

From our analysis of these ratios we conclude that the nuclear matter momentum distribution at

high k agrees with experiment to within ~20%.

Overall, our calculations -which are performed without free parameters - show a good agreement

with the data. At very low momentum transfer, this agreement deteriorates, a fact we assign to the

inadequacy of Glauber theory for low recoil-nucleon energies (low w). Data at higher values of q and

w would be most helpful to study further the short-range properties of the wave function of finite

nuclei and nuclear matter.

12. Acknowledgements

Much of the work presented in this paper has been carried out in collaboration with 0. Benhar, A.

Fabrocini and S. Fantoni. This work was supported by the Schweizerische Nationalfonds.

7/25/2019 FIRAS (482)

20/21

342

13. References

I

Sick

[l] D. Day, J.S. McCarthy, Z.E. Meziani, R. Mimehart, R. Sealock, S.T. Thornton, J. Jourdan,

I. Sick, B.W. Filippone, R.D. McKeown, R.G. Miier, D.H. Potterveld, and Z. Szalata.

Phys.

Rev.,

C481849, 1993.

[2] D. Day I. Sick.

Phys.Left.,

B274:16, 1992.

(31 0. Benhar, A. Fabrocini, and S. Fantoni. Nucl.

Phys.,

A505:267, 1989.

[4] 0. Benhar, A.Fabrocini, S. Fantoni, and I. Sick. Nucl.Phys.A, in print, 1994.

[5] 0. Benhar, A. Fabrocini, and S. Fantoni.

Phys. Rev., C41:24,

1990.

[6]

I. Sick, S. Fantoni, A. Fabrocini, and 0. Benhar.

Phy s. Left . B, 323:267,

1994.

[7]

I.E. Lagaris and V.R. Pandharipande. Nucl.

Phys.,

A359:331, 1981.

[8] I.E. Lagaris and V.R. Pandharipande.

NuclPhys.,

A359:349, 1981.

[9] 0. Benhar, A.Fabrocini, S. Fantoni, G.A. Miller, V.R. Pandharipande, and I. Sick.

Phy s. Rev.,

C44:2328,

1991.

[lo] L. J. Rodriguez, H.A. Gersch, and H.A. Mook.

Phys. Rev. A, 9:2085,

1974.

[ll] R. Silver.

Phy s.Reu. I ?, 38:2283,

1988.

[12] Y. Horikawa, F. Lenz, and N.C. Mukhopadhyay.

Phys. Rev., C22:1680,

1980.

[13] 0. Benhar, A. Fabrocini, S.Fantoni, V.R. Pandharipande, and I. Sick.

Phy s. Reu.Leff .,

69:881,

1992.

(141 A.V. Dobrovolsky, A.V. Khanzadeev, G.A. Korolev, E.M.Maev, V.I. Medvedev, G.L. Sokolov,

N.K. Terentyev, Y. Terrien, G.N. Vehchko, A.A. Vorobyov, and Yu.K. Z&e.

Nucl. Phys.B,

214:1,

1983.

[15] B.H. Silverman, J.C. Lugol, J. Saudinos, Y. Terrien, F.WeIlers, A.V. Dobrovolsky, A.V. Khan-

zadcev, G.A. Korolev, G.E. Petrov, and A.A. Vorobyov.

Nucl. Phy s.,

A499:763, 1989.

[16] T. Uchiyama, A.E.L. Dieperink, and 0. Scholten.

Phys.Lett.B, 233:31,

1989.

[17] AS. Rinat and M.F. Taragin.

Phy s. Left., B267:447,

1991.

[18] N.N. NiioIaev, A. Szczurek, J. Speth, J. Wambach, B.G. Zakharov, and V.R. Zoller.

Phy s. Leff.,

B317:281,

1993.

[19] 0. Benhar, A. Fabrocini, S. Fantoni, and I. Sick. publication in preparation.

[20] L.L. Frankfurt and M.I.

Strikman Phys. Rep s, 160:235,

1988.

[21]

G.R. Farrar, H. Liu, L.L. Frankfurt, and M.I. Strikman.

Phy s.Reu.Let f., 61:686,

1988.

[22]

A. Bodek, M. Breidenbach, D.L.Dubm, J.E. Elias, J.I. Friedman, H.W.KendalI, J.S.Poucher,

E.M. Riordan, M.R. Sogard, D.H. Coward, and D.J. Sherden.

Phys. Rev., D20:1471,

1979.

[23]

A. Bodek and J.L. Ritchie.

Phys. Rev., D23:1070,

1981.

[24] D. Day, J.S. McCarthy, Z.E. Meziani, R. Minehart, R. Sealock, S.T. Thornton, J. Jourdan,

I. Sick, B.W. Fihppone, R.D. McKeown, R.G.Miier, D.H. Potterveld, and Z. SzaIata.

Phys.

Rev. Leff., 59:427,

1987.

7/25/2019 FIRAS (482)

21/21


343

[25] D. Day, J.S. McCarthy, Z.E. Me&&, R. Minehart, R.M. Sealock, S. Thornton, J. Jourdau,

I. Sick, B.W. Filippone, RD. McKeown, R.G. Miier, D. Potterveld, and Z. Ssalata. Phys.Reu.C,

40:1011, 1989.

[26] S.J. Brodsky. Proc. 13th

Int. Symp. on

Mu&particle Llgnamies.

World

Scient. Singapore, page

963,1982.

[27]

A. Mueller.

Proc. 17th

Rencontre Moriond. Ed. J.Than Thanh

page 13, 1982.

[28]

B.K. Jeuings and G.A. Miller. Phys.Lett., B237:209, 1990.

[29] N. C. R. Makins

et al. Phys. Rev. Left.,

72:1986, 1994.

Van, Ed.

Z+ont.,

Gif-sur-

Yvette,

(301 SC. Pieper, R.B. Wiringa, and V.R. Pandharipaude. Phys.Reu., C46:1741, 1992.

(311 D. Day. iVucZ.Phys., A478:397c, 1988.

[32] L.L. Frankfurt, M.I. Strikmau, D.B. Day, and M. Sargeyau. Phya.

Rev. C,

48:2451, 1993.

[33] S. Rock, R.G. Arnold, P. Bosted, B.T. Chertok, B.A. Meckiug, I. Schmidt, Z.M. Szalata, R.D.

York, and R. Zdarko. Phys. Rev.

Left.,

49:1139, 1982.

[34] R.G. Arnold, D. Benton, P. Boated, L. Clogher, G. DeChambrier, A.T. Katramatou, J. Lambert,

B. Debebe, M. Frodyma, R.S. Hicks, A. Hotta, G.A. Peterson, A. Gearhart R., J. AIster,

J. Lichtenstadt, F. Dietrich, and K. van Bibber. Phys.

Rev. Left.,

61:806, 1988.

[35] W.P. Schuetz

et d

Phys.

Rev. Left.,

38:259, 1977.

136) J. Vary. Lect. Notes in Phys., 260:422, 1986.

[37] S. Liuti. Phys.

Rev. C,

47:1854, 1993.

[38] C. Ciofi degh Atti and S. Liuti. Phys.Reu., C44:1269, 1991.

[39] 0. Benhar and V.R. Paudharipaude. Phys.Rev., 47:2218, 1993.

[40] I. Sick, D. Day, and J.S. McCarthy. Phys.

Reu. Left.,

45:871, 1980.

[41] D.Day, J.S.McCarthy, T.W. Donnelly, and I. Sick. Ann.Reu.NucZ.Part.Sci., 40:357, 1990.

firas (482)

Documents