Theory of hadron-deuteron elastic scattering in the GeV region

A multiple scattering formalism for the medium energy hadron-deuteron elastic scattering is developed. This approach provides an extension of the Glauber theory incorporating the complete spin structure and the corrections to the eikonal and fixed scatterer approximations. The systematics of the spin observables and their relation to the hadron-deuteron as well as to hadron-nucleon amplitudes are discussed. An approximate cancellation between few corrections to the double scattering term is found which leaves the non-eikonal correction as the dominant one, and often, very important numerically, e.g., in the deuteron tensor polarization. Comparison of the parameter-free predictions of the theory based on the recent NN phase shift amplitudes with the data on polarization observables in pd elastic scattering is made.     

Corrections to the Glauber multiple scattering series

The asymptotic accuracy of the Glauber multiple scattering formula is examined in the fixed scatterer approximation in one dimension. If the potentials overlap, the off-shell corrections that dominate are asymptotically of the same order as the corrections to the eikonal approximation results. In the case of nonoverlapping potentials, the corrections to the Glauber formula arise from multiple scattering. The asymptotic behavior of these latter corrections depend essentially on the smoothness of the potentials.     

Eikonal theory of electron- and positron-atom collisions

We review recent developments in the application of eikonal methods to the field of electron and positron collisions with atoms. The foundations of the eikonal approximation are first analyzed within the framework of potential scattering, with particular attention to those aspects of the theory which can be generalized to atomic collision processes. We next discuss various many-body applications of the eikonal method, namely: the Glauber approximation, the eikonal-Born series method, optical model theories, the eikonal distorted wave method and the multistate eikonal approximation. We also analyze eikonal exchange amplitudes. Applications of these methods are then considered, first for the case of elastic scattering and then for various inelastic processes.     

Aspects of Planckian scattering beyond the eikonal

We discuss an approach to compute two-particle scattering amplitudes for spinless light particles colliding at Planckian centre-of-mass energies, with increasing momentum transfer away from the eikonal limit. The leading corrections to the eikonal amplitude, in our ‘external metric’ approach, are shown to be vanishingly small in the limit of the source particle mass going to zero. For massless charged particles, the electromagnetic and gravitational interactions decouple in the eikonal limit, but mix non-trivially for the leading order corrections.     

Eikonal and Fresnel corrections to the Glauber theory of nuclear multiple scattering

Starting from a Watson-type multiple scattering series we study first-order corrections to the Glauber high-energy collision model. These are obtained by replacing the free-space Green function between successive scatterings by the second-order eikonal propagator. We distinguish between Fresnel corrections and eikonal-type corrections. Due to large cancellation effects the eikonal contribution is very small compared to the Fresnel contribution. First-order corrections to Glauber theory are calculated explicitly for high-energy elastic scattering of protons from light nuclei (⁴He, ¹²C, ¹⁶O).     

Multiple scattering in dilute systems

Multiple scattering in the limit in which the constituent scatterers are far removed from each other is studied. It is seen how this situation leads, in the Watson multiple-scattering series, to the dominance of on-shell propagation between the constituents. It also yields a version of the Glauber multiple-scattering theory in which no assumption is made to the effect that the individual projectile-constituent scattering amplitudes are eikonal in form. This approach allows for the immediate extraction of Fresnel corrections to the Glauber result and explains several apparent contradictions as to the relationship between the Watson multiple-scattering series approximated with on-shell propagation only and the Glauber theory.     

Multiple scattering with a linearized propagator

Scattering by a many-body system is studied within the framework of the “fixed scatterer” approximation and the eikonal approximation formulated in terms of a linearized propagator. If properly treated, the “fixed scatterer” approximation is able to take into account the center-of-mass motion. We specifically study the linearized propagator proposed by Abarbanel and Itzykson. Although for potential scattering the above approximation is essentially equivalent to the Glauber eikonal approximation, its physical implications are quite different when applied to scattering by a composite system. The multiple-scattering series can generally no longer be simply expressed in terms of the individual on-shell scattering amplitudes, and the additivity of phase shifts is shown to break down for overlapping potentials. The implications for phenomenological calculations are discussed. Finally, the above approximation is explicitly applied to high-energy elastic nucleon-deuteron scattering and the results are compared with several variants of the Glauber multiple-scattering formalism.     

A RELATIVISTIC CORRECTION TO THE GLAUBER THEORY AND THE HIGH ENERGY BEHAVIOR OF THE SCATTERING AMPLITUDE

In this paper, the basis of the Glauber theory is changed from the schrodinger equation to a relativistic equation, which is then solved using eikonal approximation to investigate the high energy behavior of the scattering amplitude and its relationship with phenomenological nuclear forces. A correction to the Glauber theory including the modification of transverse and longitudinal momentum transfer is also proposed.     

Noneikonal effects in spin-dependent proton-nucleus interactions

Simple expressions are derived for the non-eikonal corrections to the Glauber diffraction approximation for the proton-nucleus scattering amplitudes, with the spin dependence of the proton-nucleon amplitudes taken into account. As an example, we study the numerical importance of these corrections for elastic p ⁵⁸Ni scattering at 800 MeV.     

Non-eikonal corrections in potential scattering on two centres

Scattering of a spinless projectile from a system of two (over-lapping or non-overlapping) local potentials is studied in the first-order eikonal expansion. Corrections to the Glauber formula are found to come mainly from the non-eikonal propagation in the “direct” double scattering, and not from multiple scattering (“reflections”).A comparison is made between the exact (to the leading order) and two approximate results: (i) based on the theory of scattering from non-overlapping potentials, and (ii) neglecting the reflection terms and adopting the local “Born-like” off-shell extrapolation of the two-body amplitudes. The second approximation is found to be more accurate and technically simpler in calculation of multiple scattering from many-body systems.     

Eikonal analysis of Coulomb distortion in quasi-elastic electron scattering

An eikonal expansion is used to provide systematic corrections to the eikonal approximation through order 1/k ², where k is the wave number. Electron wave functions are obtained for the Dirac equation with a Coulomb potential. They are used to investigate distorted-wave matrix elements for quasi-elastic electron scattering from a nucleus. A form of effective-momentum approximation is obtained using trajectory-dependent eikonal phases and focusing factors. Fixing the Coulomb distortion effects at the center of the nucleus, the often-used ema approximation is recovered. Comparisons of these approximations are made with full calculations using the electron eikonal wave functions. The ema results are found to agree well with the full calculations.     

Corrections to eikonal models for hadron-nucleus scattering at high and intermediate energies

Corrections to eikonal amplitudes for hadron-nucleus scattering are calculated for realistic optical potentials at medium and high energies. The corrections, involving only one-dimensional integrals, are found to provide rapid convergence towards the exact results. In contrast to the lowest-order eikonal approximation, the corrections lead to a sensitivity of the calculated cross section to the sign of the real part of the optical potential.     

Multiple scattering series in the forward and/or backward direction

A useful formulation of potential scattering is presented in terms of multiple scattering series. This formulation starts from an approximation of the exact Green function in momentum space by the sum of two propagators (forward and backward propagators). This replacement leads naturally to both the forward and backward approximation in the same way. The Fresnel diffraction is incorporated with the to-and-fro motion (reflection) in a potential. High-energy corrections are also presented to the eikonal approximation. This formulation permits a straightforward application to scattering off a composite system, and has a close relationship to the Glauber approximation.     

Glauber amplitude for ionization of helium by electron impact

The high-energy Glauber approximation is used to derive the formula for scattering amplitude for ionization of helium atom by electron impact. The scattering amplitude is expressed as a one dimensional integral involving MeijerG-functions. This may further be expressed in a closed form as simple sums of Meijer functions by writing down the series expansion for Bessel and Meijer functions. Further, the asymptotic behaviour of these amplitudes is examined for both large and small momentum transfers.     

The fixed scatterer approximation and nucleon deuteron scattering

A method of solving the Schroedinger Equation for the scattering from two fixed local potentials is presented. The solutions are used within the framework of the fixed scatterer approximation to perform model calculations of N-D scattering using both effective range theory potentials and a “semi-realistic” potential with a strong repulsive core. For smooth potentials approximate solutions to the fixed scatterer problem are proposed and found to be quite accurate.Other procedures for calculating elastic scattering were compared with the exact fixed scatterer approximation. The results show that the neglect of longitudinal momentum transfer in the Glauber multiple diffraction theory is a severe effect except in the forward direction. For small angle scattering the Glauber prediction for the double scattering amplitude is quite accurate, and does not depend strongly upon either the extent of potential overlap, or the ratio $\text{V}\text{E}$. Comparisons with the Agassi and Gal results for nonoverlapping potentials indicate that the effects of potential overlap are important, at least for the lowest partial waves. Conclusions about the overall importance of off-energy shell effects in nucleon-deuteron scattering, and about the interference of these effects with the determination of the correlation structure of nuclei are not free from ambiguities.     

QED radiative corrections to impact factors

We consider radiative corrections to the electron and photon impact factors. The generalized eikonal representation for the e ⁺ e ^? scattering amplitude at high energies and fixed momentum transfers is violated by nonplanar diagrams. An additional contribution to the two-loop approximation appears from the Bethe-Heitler mechanism of fermion pair production with the identity of the fermions in the final state taken into account. The violation of the generalized eikonal representation is also related to the charge parity conservation in QED. A one-loop correction to the photon impact factor for small virtualities of the exchanged photon is obtained using the known results for the cross section of the e ⁺ e ^? production during photon-nuclei interactions.     

An impact parameter representation for the off-energy-shell scattering amplitude

The unitarized Born approximation for the scattering amplitude, suggested previously, is considered in the short wavelength limit. It is found that its on-energy-shell component is equivalent to the well-known impact parameter representation developed by Glauber. An approximate expression is then derived for the off-energy-shell scattering amplitude on the basis of the unitarized Born approximation which is expected to be as accurate as the Glauber approximation.     

Eikonal description of quantal scattering

Elastic and inelastic quantal scattering is described by a theory in which the contribution of a range of impact parameters to the scattering amplitude is determined by a phase integral (“eikonal”) which is integrated along a real curved “quantal” trajectory. This amplitude reduces to the Glauber expression in the high-energy, forward-angle limit, and to the usual semiclassical amplitude in the classical limit. The formulation can be applied to the study of heavy-ion scattering. The quantal trajectories are investigated analytically for the case of Coulomb scattering. A numerical analysis of elastic ¹⁶O¹⁶O scattering is carried out. The results show appreciable improvement as compared with the Glauber approximation.     

Focusing of high-energy particles in the electrostatic field of a homogeneously charged sphere and the effective momentum approximation

The impact of the strongly attractive electromagnetic field of heavy nuclei on electrons in quasi-elastic (e, e') scattering is often accounted for by the effective momentum approximation. This method is a plane wave Born approximation which takes the twofold effect of the attractive nucleus on initial- and final-state electrons into account, namely the modification of the electron momentum in the vicinity of the nucleus, and the focusing of electrons towards the nuclear region leading to an enhancement of the corresponding wave function amplitudes. The focusing effect due to the attractive Coulomb field of a homogeneously charged sphere on a classical ensemble of charged particles incident on the field is calculated in the highly relativistic limit and compared to results obtained from exact solutions of the Dirac equation. The result is relevant for the theoretical foundation of the effective momentum approximation and describes the high-energy behavior of the amplitude of continuum Dirac waves in the potential of a homogeneously charged sphere. Our findings indicate that the effective momentum approximation is a useful approximation for the calculation of Coulomb corrections in (e, e') scattering off heavy nuclei for sufficiently high electron energies and momentum transfer.