Scaling in the quasielastic response of atomic nuclei

Nuclear Physics A572 (1994) 57-68 North-Holland

NUCLEAR PHYSICS A

Scaling in the quasielastic response of atomic nuclei * E. Eich, F. Beck, A. Richter htirut

fiir Kernphysik, Technische Hochschule Darmsfadt, D-64289 Darmstadt, Germany

Received 1 July 1993

Abstract The separated quasielastic response functions of electron-nucleon scattering in ‘*C and 4oCa are studied in the context of y-scaling within the uop-model. A different scaling behaviour of the longitudinal and transverse channeis is observed, i.e. the longitudinal channel is strongly quenched compared to the transverse one. This effect, which is in con~adiction to the impulse approximation, can be traced to the interactions of the nucleons inside the nucleus. The y-scaling, observed experimentally, is well reproduced in this model.

1. Introduction The interaction of photons with atomic nuclei establishes detailed and precise information on the structure of the nuclear many-body system. The reason is on one hand the weakness of the electroma~etic interaction which allows in most cases the perturbative restriction to one-photon exchange. On the other hand, the electromagnetic interaction is well known, so one obtains direct information on the nuclear charge and current distributions. Virtual photons, i.e. elastic and inelastic electron scattering, have the additional advantage that energy and momentum transfer can be varied independently. In this way information on the momentum dependence, respectively the local structure, of the currents is obtained at various positions of the excitation spectrum, While for low momentum transfer, 9, the nucleus reacts as a whole, showing the long-range behaviour of the nuclear interaction, increasing 4 brings about more and more the “granularity” of the nuclear system. At energy and momentum transfers of several hundreds of MeV the individual structure of the nucleus under investigation is lost, the virtual photon interacts with single nucleons, and the spectrum is characterized by a broad structureless distribution, called the quasielastic peak. In recent years quasielastic electron-nucleus scattering has found considerable interest experimentally as well as from the theoretical side (see refs. [1,21 and

* Dedicated to Klaus Dietrich on the occasion of his 60th birthday. 0375-9474/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0375-9474(93)E0407-Y

58

E. Eich et al. ,/ Quasielastic response

references therein). While the total inclusive cross section can be understood qualitatively already in the Fermi gas model employing the impulse approximation, the separation into longitudinal and transverse components shows several interesting features which provide a challenge to intermediate-energy hadron physics. One of the observations is the “quenching”, or “missing charge”, phenomenon [3,4] which is present in the longitudinal, but absent in the transverse channels. Many attempts to relate this behaviour to a change in the nucleon structure within the nuclear medium (“swollen nucleon”) have the difficulty that this leads to quenching in both channels [5&f. The transverse channel is, in addition, characterized by a second peak at larger energy loss which can be attributed to A-isobar excitation in the photo-nuclear absorption process [7,8]. While this provides interesting information about the behaviour of A-isobars in the nuclear medium [93, the experimentally observed [7,8] filling of the “dip-region” between the two peaks is not yet well understood theoretically. All these phenomena show, however, that the Fermi gas model is too poor an approximation, and that the scattering nucleons are at best quasiparticles carrying information about the interactions in the medium. Finaily, a systematic analysis of experimental results for different regions of momentum transfer shows a remarkable scaling behaviour of the cross sections [lO,ll]. While Bjorken scaling in deep-ineIastic electron-nucleon scattering has revealed important insights into the parton structure of nucleons, the so-called y-scaling in quasielastic electron-nucleus scattering has only recently attracted interest as a tool to study the in-medium behaviour of hadrons (see ref. [l] and references therein). Any theoretical attempt to study quasielastic electron-nucleus scattering beyond the Fermi gas model has to make a prediction about the momentum dependence of nuclear interactions. On the basis of conventional non-relati~stic nuclear many-body theory this is done by making an ansatz for the momentum dependence of the effective residual interactions and fixing the parameters from experiment 1121.Since the quasielastic peak is one of the important inputs at q of several hundreds of MeV, such approaches do not possess very much predictive power. A completely different approach is established by treating the nuclear many-body problem as a system of interacting baryons and mesons, whereby the in-medium coupling constants are fixed by nuclear-matter properties. Dynamic processes like photon absorption have then to follow without introducing new parameters [2]. They are essentially determined by meson masses, coupling constants and retardation effects. Fu~hermore, this approach allows one to make contact with low-energy theorems [13] of quantum chromodynamics (QCD). In this paper we study y-scaling in the Walecka owp-model [141 of quantum hadrodynamics (QHD). This quantum field theory of interacting nucleons and mesons has been successfully applied to quasielastic electron scattering in recent years [2], and it accounts fairly well for the observed longitudinal quenching, as well as for A-isobar excitation in the transverse channel [9]. As has been demon-

E. Eich et al. / Quasielmtic re.ponse

59

strated [4,15], quenching is produced by successive particle-hole and nucleon-antinucleon excitations which can be treated parameter free and on equal footing in QHD since it is a renormalizable field theory. Consequently we base our present analysis on the relativistic Hartree and random-phase approximations which include particle-hole excitations and vacuum polarization effects. This paper is organized as follows: sect. 2 presents an outline of y-scaling, while sect. 3 is devoted to the calculation of electromagnetic response in QHD. Sect. 4 contains our results in comparison with a compiIation of experimenta cross sections [8,16,171, and sect. 5 gives our conclusions and the summary.

2. Formalism The basic assumption leading to y-scaling is that at sufficiently high momentum, 4=/q], and e nergy, w, transfers of the exchanged photon the scattering of electrons from nuclei takes place in a quasifree manner. This means that the nucleons are sampled incoherently by the photon and the process is rather insensitive to final-state interactions of the recoiling nucleon propagating through the nucleus. Furthermore, interactions of the photon with constituents other than nucleons are neglected [ll,lS-201. Fig. 1 shows a diagrammatic description of the process. We closely follow the treatments of refs. [ll,lS] which present excellent

Fig. 1. Inclusive (e, e’N) scattering in plane-wave impulse approximation. The virtual photon is exchanged between the electron and an off-shell nucleon within the nucleus carrying energy E and momentum 8. The ejected nucleon has momentum k’. The excited residual nucleus recoils with momentum -k.

60

E. Eich et al. / Quasielastic response

reviews on y-scaling. This approach, called plane-wave impulse approximation (PWIA), is commonly used as a starting point in treatments of (e, e’N) reactions at high q and w. Under these assumptions the inclusive cross section can be written as an energy-momentum integral over a cross section a, of an electron scattered from an off-shell nucleon carrying momentum k, = (E, k), and the product of a spectral function Sick, E) with a S-function ensuring energy conservation. The spectral function describes the energy and momentum distribution of the nucleon, where A and N denote the number of protons P and neutrons N, respectively, &

= /d3k/dE +Nu,S,(k,

(Aa,S,(

E))+

k, E) +E -

:mZ).

(1)

The energy E of the nucleon before scattering is given by (2)

Here MA denotes the mass of the initial nucleus, MjY1 is the mass of the residual (A - 1) system, whereas k’ = k + q is the momentum of the scattered nucleon. To proceed further, one transforms the integration variable E to the excitation energy 8 of the residual nucleus using the relation E(k,;R)=M,-,/+-8’.

(3)

Performing the angular integrations over the azimuthal angle 4 and the angle B between k and q, using the restrictions imposed by the S-function, one obtains -$&

= 2ajT:tk’

dkig’(‘)

dB (Au&(

k, 8)

(4) where A=A(q;k,costI,8)=/~+/~-b&+8.

(5)

The last factor I &i/d cos 8, I - ’ is due to the change of the integration variable from cos 0 to A. The integration boundaries are obtained by the condition -1 ,(cos 0~ 1, with

cos 8 = [(M,+w-,/iM~-I)2-(M2+kifq2)]/2kq.

(6)

E. Eich et al. / Quasielastic response

61

The boundaries of the k-integration are taken at 8 = 0. One finds that the minimal value of k corresponds to a photon moving antiparallel to the nucleon, i.e. at a negative value of y for w Q w0 = q2/2M, where the quasielastic peak occurs. In a non-relativistic treatment, energy conservation would require k *q to be constant [21]. Therefore, y would simply denote the component of k parallel to q. The variables Y and y can be obtained by solving the equations

One also assumes isospin independence of the spectral function. Furthermore, the assumption is made that the cross section of a single nucleon varies only slightly for different values of k and 8 over the region where the spectral function contributes significantly. In this case one is allowed to move it out of the integral together with the weakly k-dependent kinematic factor I( = I k I * I&-I,0 cos 8, I-‘, with k and 8 fixed. By this procedure, with G.(q, y, k, 8:) = fWc+(p(q, y, k, 8) + NuJq, y, k, 8)), one obtains d2a

Of course, not much is known about the off-shell behaviour of a single-nucleon cross section. Since different ~~rapolations are plausible, we cannot make a general statement about the validity of this assumption. Rather it has to be checked individually for the model used. It turns out that for the popular off-shell extrapolation by de Forest [22] - the so-called ccl prescription - this approximation is valid [181 at about a 5% level if k and 8 are fixed at k = I y I and 8 = 0. Hence the inclusive cross section becomes

&

-+I,

Y;

k=lyl, a=O) *F(q,

Y),

(10)

where

F(q, Y) =

2$k

dkf(s,

Y;

k),

(11)

with

(12)

One further assumes that at sufficiently high values of q with y fixed, the upper limit, Y, of the k-integration lies far outside the region where the spectral function S contributes. This can be justified since Y grows rapidly with q for fixed values of y. Also the upper limit of the ~-integration gM(cI, Y; k) = jiM2+(q

+ d11/1,2_,+yZ

-&gJTTq~

(13)

is replaced by cY,&q=~,

y; k)=/m-/m+k-ty.

(14)

After these manipulations the cross section factorizes into a part which still depends on all kinematic variables (i.e. q and y(q, w>>separately and a unction F(y) of a single variable y

2

=(Aa(q,

Y)

ltNa(q,

Y))

*F(y)*

(15)

This behaviour, derived by starting from the initial assumption of PWIA, is called y-scaling. If it would be justified to extend gM to infinity, a relation between the scaling function and the probability f(y) to find a nucleon with momentum y in the nucleus can be obtained

1 WY)

(16)

f(Y)=ydy.

In this limit the scaling function contains interesting the Rosenbluth separation formula,

physical information.

Using

(17) i

one defines the longitudinal, standard way I193,

FL, and transverse,

SL(47Y> Ad(q, Y) +wQ

FL(Y)

=

G(Y)

= A&+L

F,,

scaling functions

in the

K-l

(18)

K-l.

(19)

Y)

and

&x47 Y) Y> +N4+(4,

Y>

E. Eich et al. / Quasielastic response

63

Here S, and S, denote the longitudinal and transverse response functions, respectively. Under the assumption of the PWIA the longitudinal and transverse channels should scale identically, i.e. there should be a single scaling function F(Y)

=FL(Y)

=FT(Y).

In Eqs. (18), (19) (+,_ und ar are the separated single-nucleon response functions.

longitudinal

and transverse

3. Electromagnetic response functions of the nucleus in QHD Theoretical scaling functions have been analyzed in refs. [23-251 using non-relativistic models which include final-state interactions. To obtain these functions in a relativistic framework which also includes nucleon interactions in the final state we calculate the response functions in a relativistic field theory [2] whose fundamental degrees of freedom are the hadrons (quantum hadrodynamics, QHD). The interactions are due to the a-meson, the w-meson and the p-meson isotriplet with quantum numbers (J”, T) = (O+, 01, (I, 0) and cl-, 11, respectively. The lagrangian of this so-established awp-model [14,261 takes the form

(21) Here Gpy = a,VV- a,,V,

and

Hpy = a,b, - a,,b,

(22)

denote the field strength tensors of the w- and the p- (b,) mesons of mass m, and m,, respectively. The scalar a-meson field of mass ms is called @, whereas rC, stands for the nucleon spinor field with mass M. Due to the large strength of the coupling constants a perturbative solution of the field equations is not possible. We therefore start from the relativistic Hartree approximation treating the meson fields in one-loop (tadpole) order [28]. This corresponds, in the non-relativistic mean-field treatment, to the Hartree-Fock, or Brueckner-Hartree-Fock approximations which have been successfully used in nuclear-structure calculations for spherical as well as for deformed nuclei [29]. The meson mass m, as well as the coupling constants are determined such that the bulk properties of nuclear matter and the charge radius of 40Ca are reproduced [14,30]. The nucleon propagator in

E. Eick et ai. / Quasielastic response

64

Hartree approximation is calculated in a self-consistent manner. It receives an effective mass M* compared to the free propagator. We also include the vacuum contributions to the self-energy terms, which are renormalizable. For this part of the calculation we consider the case of nuclear matter. The response functions per volume I/ are proportional to the imaginary part of the polarization propagator

“(“” ml=- iIm{ll(j,, V WqL

4 v

In Hartree

ifl(A,

- I-Im{fl(&, =7r

approximation

B; 4) = /$$

j,; /q/, w)},

(23)

j,;

(24)

141, w)}.

the polarization

propagator

Tr(G&Mq)GnV

is given by

+ q)N

-4)-

(25)

The Heisenberg operators A and B represent the longitudinal and transverse parts, j, and jr, of the phenomenological electromagnetic current [27] jiff(s)

=Fir(q:)E(q)t(I

+Q$)(

+ r3)yp4q)

E(q)&

$‘.(l+r&&J4q)) (26)

Here the free-nucleon Sachs form factors have been used, with Fir, F,, for the Dirac current and Fzp, FzN for the anomalous current of the nucleon. The resulting Hartree polarizations can be found in ref. [31]. This propagator does not satisfy the Ward-Takahashi identity [32] G-‘( k’) - G-‘(k)

= (k - k’),P‘_

(27)

To obtain a consistent theory we therefore evaluate the polarization propagator in the relativistic random-phase approximation (RPA), thus including particle-hole correlations. To make our results comparable to experimental data, we finally evaluate the response functions in local-density approximation (LDA) where the nucleus is treated locally as nuclear matter. We sum over volume elements with local proton and neutron densities (i.e. Fermi momenta) and their effective masses. These values are obtained by solving the field equations for spherically

E. Eich et al. / Quasielastic response

65

symmetric nuclei. The results differ only slightly from those obtained by using the exact one-particle density [331. In LDA one obtains

4. Results We evaluated the longitudinal and transverse scaling functions in both, Hartree approximation and RPA, for the two double-magic nuclei ‘*C and 40Ca. We calculated the theoretical scaling functions using Eqs. (18) and (19) with the LDA response functions for S, and S,, and the single-nucleon responses according to the ccl prescription of de Forest. Additionally we took u: = 0. One notes, that in essence the Hartree approximation can be viewed as a PWIA since no final-state interaction is considered, while in RPA parts of it are included through particle-

-400

-200

0

200

y(MeV/c) Fig. 2. Separated scaling functions F,(y) and F,(y) for “C and 40Ca calculated in the aup-model Hartree approximation. The scaling functions for the q-range of 300 to 600 MeV/c lie within shaded area.

in the

E. Eich et al. / Quasielastic

response m 300 MeV/c m 350 MeV/c

400 A 500 0600 + 300 0 350 0 400 A 500 0 600

MeV/c MeV/c MeV/c MeV/c MeV/c MeV/c MeV/c MeV/c

* 300 n 370 + 410 A 500 l 550 + 300 0 370 0 410 A 500 0 550

MeV/c MeV/c MeV/c MeV/c MeV/c MeV/c MeV/c MeV/c MeV/c MeV/c

l

-400

-200

0

200

-400

-200

0

200

400

y(MeV/c) Fig. 3. Experimental scaling functions [8,16,17] for ‘*C and 40Ca for q between 300 and 600 MeV/c (1.h.s.) compared to theoretical predictions (r.h.s.1. These are obtained in the same way as in Fig. 2, however under inclusion of nucleon-hole and nucleon-antinucleon correlations (RPA).

hole correlations. The Hartree approximation is basically given by an integration over the response of a single nucleon with an effective mass M* - as an in-medium effect - weighted by a distribution function which in case of zero temperature takes on a very simple form. We would like to point out that the free mass was replaced by M* in the propagator only; the form factors and coupling terms to the magnetic moments were left unchanged - a procedure which is consistent to reproduce the nucleon magnetic moments [34]. Fig. 2 shows the results of the Hartree approximation in the momentum range I q I = 300-600 MeV/c which has been analyzed experimentally also. We observe an excellent scaling behaviour of F;(y). The scaling functions in this momentum range vary within the shaded bands. We find, however, a different form of F(y) in the longitudinal - charge dependent - and the transverse - current dependent channel. The quantity F, is suppressed compared to FT due to medium effects. In RPA (Fig. 3 r.h.s.1 this effect becomes even stronger: the longitudinal scaling function is additionally depleated by 6-8%, while FT is not affected significantly. This effect is known as quenching or “missing charge” [4,34] as it violates the Coulomb sum rule. The corrections are mainly due to the u- and w-mesons, since the square of the coupling constant of the p-meson isotriplet is only about half of

E. Eich et al. / Quasielastic response

67

the one for the w-meson. In the longitudinal channel the a-meson causes a strong increase of the response due to its attractive interaction. The latter is slightly overcompensated by a strong depletion due to the repulsive forces of the w- and, to a lesser extent, p-mesons. On the other hand, in the transverse channel the a-meson as a scalar particle does not contribute at all. The contributions of the isoscalar w-meson cancel to a large extend, since the magnetic moments of the proton and neutron are comparable in magnitude but carry opposite signs. Due to the smaller coupling constant, the p-meson contribution is not significant. The calculated responses are in good agreement with the experimental data (Fig. 3 1.h.s.). The position of the maximum is shifted to a somewhat smaller y for the lighter nuclei. Below the quasielastic peak, where only nucleonic degrees of freedom play a significant role, both calculated scaling functions coincide with the data. Above the maximum the experimental transverse scaling functions show a dip region - where meson exchange currents and 2p2h terms should be important [12] - before the lower end of the A-peak begins. Due to the different masses of the nucleon and the A, scaling is violated in this region. Our theoretical calculations do not take these effects into account.

5. Conclusions and summary We calculated the longitudinal and transverse scaling functions in Hartree and random-phase approximations within the awp-model of QHD, and including the reno~alized nucleon-antinucleon contributions. We find that the e~erimentally observed scaling behaviour is well reproduced by the RPA response. The magnitude of the quenching in the longitudinal channel is in good agreement with the data. Medium effects such as particle-hole correlations only modify the bulk properties of the response functions, and in this way do not destroy y-scaling. In other words, the (+- and w-meson interactions just cause a net reduction of the response in the longitudinal channel, not affecting the transverse response significantly. It has been discussed already, that RPA corrections to the Hartree approximation can be viewed as a medium modification of the free-nucleon form factor [4]. We find no need for a universal scaling of the masses in the medium, as introduced in ref. 1131, where all parameters with mass dimensions - except the pion mass - scale, to obtain the correct quenching of the longitudinal channel. This fact therefore deserves further attention [35]. It is a pleasure for us to devote this article to our friend Klaus Dietrich who, in many years of fruitful communications, impressed us by his clarity of thought and his unmistakable feeling for soundness in physical reasoning. We are very grateful to J. Morgenstern for providing us with the experimental data as displayed in Fig. 3.

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E. Eich et al. / Quasielastic

response

This work has been supported in part by the German Federal Minister for Research and Technology (BMFT) under contract number 06DA6411, and the Deutsche Forschungsgemeinschaft.

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