Scattering of GeV electrons in the framework of the relativistic hartree approximation

Scattering of GeV electrons in the framework of the relativistic hartree approximation

A. Mariano^I; C. Barbero^I; F. Krmpotic^II

^IDepartamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de la Plata, cc. 67, 1900 La Plata, Argentina

^IIInstituto de Física, Universidade de São Paulo, C. P. 66318, 05315-970 São Paulo, SP, Brazil

ABSTRACT

The high momentum transfer electron-nucleus scattering cross section is evaluated within the plane wave impulse approximation (PWIA) supplemented by the relativistic Hartree approximation (RHA). Binding effects on the struck nucleon are introduced through the scalar and vector meson-exchange fields within the framework of quantum hadrodynamics. This model naturally satisfies the current conservation, with the off-shell nucleons behaving as being on the mass shell but having an effective mass. The nucleon inelastic response is included via different parameterizations of the structure function measured at SLAC, while the smearing of the Fermi surface is introduced through a momentum distribution obtained from a perturbative nuclear matter calculation. Recent CEBAF data on inclusive scattering of 4.05 GeV electrons on ⁵⁶Fe are well reproduced for all measured geometries by the first time. Scaling effects are analyzed as well, and the scaling properties of the nuclear response in terms of the y variable associated to the PWIA within the RHA framework are discussed. The theoretical scaling function obtained in our approach also significantly improves previous PWIA calculations within the multi-GeV electron scattering regime, and describes properly the CEBAF scaling data.

The inclusive electron-nucleus scattering is a powerful tool for investigating the effective constituents of hadronic matter and their dynamics. Experiments performed in different regions of the square four-momentum q² (q º (w, q)) and energy w transfers, provide information on the nuclei constituents and diverse excitation mechanisms. They are: (i) the quasielastic scattering region, w < Q²/2M, with M being the nucleon mass and Q²º –q² > 0, where experimental data can be analyzed in terms of scaling variables providing information on nuclear dynamics and the nucleon momentum distribution; (ii) the inelastic scattering region, w > Q²/2M + M_p, with M_p being the pion mass, where nucleon resonances are excited and medium induced modifications of their properties can be studied; (iii) the deep inelastic scattering region, W º > 2 GeV, Q²> 1 (GeV/c)², being p the initial nucleon four-momentum, where possible modifications of quarks and gluon distributions in the nucleon induced by the medium can be investigated. In the later case the response function also reflects the presence of 6-quark bags in the nuclear wave function.

In the Born approximation the differential cross section reads

where L^mn(k, k¢) is the lepton tensor describing incoming and outgoing plane-wave electron states with four-momentum k = (e º ,k) and k¢ = (e¢ º, k'), respectively, where k º |k|, k¢ º |k'|, and a = is the fine structure constant and W¢ º (q, f) the scattering angle. The most general form for W_mn satisfying simultaneously Lorentz invariance, gauge invariance (q^mW_mn = 0) and the parity conservation is

where W_1,2 are Lorentz scalars. For a nuclear target at rest (M º M_A) we have p º (M_A, 0) and p · q º M_Aw, which when combined with the above two equations lead to

where A º (N, Z), ds_M/dW¢ is the Mott cross section, and where we have used the well known longitudinal () and transverse () response functions

which must be evaluated over the entire range 0 < w < q.

The PWIA [1], is inspired on the fact that the electron probes only a small region of dimensions 1/q and is based on the following assumptions:

i) the nuclear current operator can be written as the sum of one-body nucleon currents;

ii) the target decays virtually into an on-shell (A-1) nucleus (spectator) and an off-shell (p²¹ M²) struck nucleon; and

iii) the nucleon that absorbs the photon is the same that leaves the target without interaction with the spectator.

To deal with the off-shell effects, we adopt in this work an approximation that keeps simultaneously the gauge invariance and the covariant kinematics of the struck nucleon. The nucleon will be bound trough the interaction with the scalar f and vector V_m mesons fields, within the framework of quantum hadrodynamics (QHDI)[2]. The nucleon single particle spectrum reads

with , being

M^* < M is the effective mass acquired by the nucleon by the action of the attractive scalar field and is determined self-consistently through the scalar self-energy , within the Hartree approximation. The vector self-energy accounts for the action of the repulsive vector field, and the two free parameters C_S and C_V depend on the meson coupling constants and masses being fixed to reproduce the experimental binding energy per nucleon at the baryon density r_B for the normal nuclear matter [3].

In the PWIA + RHA approximation, the response tensor can be expressed in the laboratory system as

being

where m_t º p (n) for protons (neutrons). This equation has the same form as the previously defined for an on-shell nucleon but with an effective mass M^*, which indicates that nuclear electromagnetic current is still conserved and the gauge invariance preserved [4]. Also we have

where q º |q|. This result exhibits our prescription: , where n º p · q/M ® n^*º p · q/M^*, as a consequence of using the RHA to describe the struck nucleon. As said before, the probability for exciting the nucleon becomes important for Q² > 1 (GeV/c)². Thus, we cast in the form

where and are the elastic and inelastic nucleon responses, which will be both affected for the on-shell to off-shell mentioned prescription. For the functions we assume two different parametric fits obtained from the SLAC data on p(e, e')p' and d (e,e')d' reactions. One of the parameterizations was found by Bodek et al. [5] in the kinematical range 1 < Q² < 20 (GeV/c)² and 0.1 < x º Q²/(2Mw) < 0.77. The other one was reported by Whitlow [6], and corresponds to the range 0.6 < Q² < 30 (GeV/c)² and 0.06 < x < 0.9. Finally, the inclusive cross section reads

where the contributions coming from the elastic and inelastic nucleon responses are written separately.

The nucleon momentum distribution (p) has been evaluated as follows. First, the surface effects are supposed to be of minor importance and we adopt a non-relativistic nuclear matter model for the structure of the A-target ground state

where is the residual interaction. This ''minimum'' perturbative scheme allows to include the norm correction = á0_A|0_Añ^–1, avoiding in this way contributions from unbalanced disconnected diagrams. We get

where p º p| is measured in units of the Fermi momentum p_F and . q(1 – p) is the usual 0p0h Fermi step function, while

dn(p) = dn⁽²⁾(p) + dn^(4C)(p),

contains the 2p2h and 4p4h contributions that deplete the Fermi surface, as illustrated in Fig. 1. The superscript C indicates that only the ''connected'' 4p4h diagrams have to be included. For the residual interaction we use

where the quantum numbers I = T, S, J stand, respectively, for the isospin , the spin and the total angular momentum. The operators O^I() are defined as

and a Landau-Migdal parameterization plus a the static one pion exchange potential is adopted for the strengths V^I(q) [7].

The scaling function is defined as

where p_min = |y|, being y the scaling variable

obtained from the energy-conservation relation

for a fixed (w, q) pair.

At high momentum transfers (Q² > 1 (GeV/c)²) and y < 0, we get

which indicates that, for Q² ® ¥, F scales in y, i.e., it depends only on y and not on (w, q) or (w, q) separately. Thus it should be approximately constant for a fixed value of y. In Figs. 2, 3 and 4 are confronted our theoretical results for the differential cross section in ⁵⁶Fe with the CEBAF experimental data [8] when the scattered electron are detected at angles q of 15^o, 23^o, 30^o, 37^o, 45^o, 55^o and 74^o. We evaluate as well the scaling function F(w, q) as a function of the scaling variable y, that naturally appears in the model.

In summary, to treat the scattering of GeV electrons by nuclei we have implemented a new version of the PWIA approach. The FSI has also be included to some extent. This is achieved by taking into account the binding effects of the struck nucleon in both the initial and final states. The relativistic kinematics is treated within the relativistic Hartree approximation that leads to better results than the plain relativistic mean field approach. In this picture, the binding effects are included through the effective nucleon mass M^* = 0.74 M, and current conservation is naturally preserved without ad-hoc modifications in the structure functions. More precisely, in order to treat the scattering of GeV electrons we have pursued in the PWIA approach by introducing in a very simple way the relativistic effects the FSI and a new momentum distribution for the nucleons. The model does not pretend to substitute more evolved theoretical treatments; it merely yields a consistent and simple implementation of the PWIA which is able to reproduce satisfactory the full set of presently available data.

References

[1] C. Ciofi degli Atti and S. Simula, Phys. Rev. C43, 1155 (1991). C. Ciofi degli Atti, D.B. Day and S. Liuti, Phys. Rev. C46, 1045 (1991).

[2] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 (1986).

[3] J. D. Walecka, Theoretical Nuclear and Subnuclear Physics, ch. 39, Oxford University Press, New York, 1995.

[4] T. C. Ferrée and D. S. Koltun, Phys. Rev. C55, 253 (1997).

[5] A. Bodek and J. L. Ritchie, Phys. Rev. D 23, 1070 (1981).

[6] L. W. Whitlow, Ph.D. thesis, SLAC Report No. 357, 1990.

[7] A. Mariano, F. Krmpotic, and AFR. de Toledo Piza, Phys. Rev. C53, 1664 (1996).

[8] J. R. Arrington et al., Phys. Rev. Lett. 82, 2056 (1999).

Received on 10 March, 2004

Publication Dates

• Publication in this collection
  26 Oct 2004
• Date of issue
  Sept 2004

History

• Received
  10 Mar 2004