Eikonal expansion

An eikonal expansion of the potential scattering T matrix is evaluated, without approximation, through third order in the inverse momentum. Based on the results, their correspondence with the WKB approximation and a new statement of the unitarity constraint, we propose a sequence of four approximations to the exact impact parameter (Fourier-Bessel) representation of the scattering matrix. The sequence consists of the Glauber approximation and three systematic corrections to the Glauber approximation. The corrections are analytic functions of the impact parameter for Yukawa and Gaussian potentials; they vanish for a Coulomb potential.The sequence of eikonal amplitudes is convergent at high energy and is clearly established for small momentum transfer. Validity for all momentum transfer is conjectured based on systematic cancellation, explicitly verified through third order in the expansion, of momentum transfer dependence in the eikonal impact parameter representation. Such cancellation is shown to occur in the explicit construction of the eikonal expansion of the second Born amplitude for a Yukawa potential.Numerical tests of the sequence of eikonal amplitudes show systematic increase of the angular range of validity by comparison with partial wave results for continuous potentials; the theory is not convergent for discontinuous potentials.The WKB phase shift formula is shown to produce a systematic connection with eikonal expansion results. From this we deduce a generating function for the eikonal phase corrections of arbitrary order and also conjecture a sum of the eikonal expansion valid in the limit of high energy and arbitrary potential strength.     

Fresnel corrections in the relativistic scattering of particles with spin 0 and 1/2

The corrections to the eikonal scattering amplitude are investigated. The Fresnel corrections to the eikonal solution of a quasipotential equation are obtained. In the construction, use is made of the iteration-separation method generalized to the case of the scattering of particles with spin on potentials containing a spin-orbit interaction.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 75–81, October, 1976.     

Second-order eikonal corrections for

The first-order eikonal approximation is frequently adopted in interpreting the results of A(e,e^′p) measurements. Glauber calculations, for example, typically adopt the first-order eikonal approximation. We present an extension of the relativistic eikonal approach to A(e,e^′p) which accounts for second-order eikonal corrections. The numerical calculations are performed within the relativistic optical model eikonal approximation. The nuclear transparency results indicate that the effect of the second-order eikonal corrections is rather modest, even at Q²≈0.2 (GeV/c)². The same applies to polarization observables, left–right asymmetries, and differential cross sections at low missing momenta. At high missing momenta, however, the second-order eikonal corrections are significant and bring the calculations in closer agreement with the data and/or the exact results from models adopting partial-wave expansions.     

Infrared and high energy fixed angle domains of scattering and the role of eikonal approximation

We examine the infrared structure and fixed angle behaviour of generalised ladder diagrams in spinor electrodynamics. The external line rules for infrared divergence is proved within the framework of parametric representation of the amplitudes. We derive a set of rules for computing the precise form of the leading infrared behaviour of any generalised ladder diagram. The same set of rules is shown to yield the leading term in the high energy fixed angle limit. Our analysis shows that both in the infrared and in the fixed-angle high energy limits the usual propagator eikonal approximation fails even in the leading order. The deficiencies of eikonal approximation can be partially removed if one also makes the small angle approximation. Leading corrections of relative $\text{O(}\text{|t}\text{|}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{/s}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ to the small-angle eikonal amplitude are calculated.     

Eikonal approximation in the theory of energy loss by fast charged particles

Energy losses in fast charged particles as a result of collisions with atoms are considered in the eikonal approximation. It is shown that the nonperturbative contribution to effective stopping in the range of intermediate impact parameters (comparable with the characteristic sizes of the electron shells of the target atoms) may turn out to be significant as compared to shell corrections to the Bethe-Bloch formula calculated in perturbation theory. The simplifying assumptions are formulated under which the Bethe-Bloch formula can be derived in the eikonal approximation. It is shown that the allowance for nonperturbative effects may lead to considerable (up to 50%) corrections to the Bethe-Bloch formula. The applicability range for the Bethe-Bloch formula is analyzed. It is concluded that calculation of the energy loss in the eikonal approximation (in the range of impact parameters for which the Bethe-Bloch formula is normally used) is much more advantageous than analysis based on the Bethe-Bloch formula and its modifications because not only the Bloch correction is included in the former calculations, the range of intermediate impact parameters is also taken into account nonperturbatively; in addition, direct generalization to the cases of collisions of complex projectiles and targets is possible in this case.     

Long-range Coulomb forces in DIS: missed radiative corrections?

The Born approximation, one photon exchange, used for DIS (deep inelastic scattering) is subject to virtual radiative corrections which are related to the long-range Coulomb forces. They may be sizeable for heavy nuclei since Zα is not a small parameter. So far, these corrections are known only for two processes, elastic scattering and bremsstrahlung on the Coulomb field of a point-like target. While the former amplitude acquires only a phase, in the latter case also the cross-section is modified. Although the problem of Coulomb corrections for DIS on nuclei is extremely difficult, it should be challenged rather than “swept under the carpet”. The importance of these radiative corrections is questioned in the present paper. We show that, in the simplest case of a constant hadronic current, the Coulomb corrections provide a phase to the Born amplitude, therefore the cross-section remains the same. Inclusion of more realistic hadronic dynamics changes this conclusion. The example of coherent production of vector mesons off nuclei reveals large effects. So far a little progress has been made deriving lepton wave functions in the Coulomb field of an extended target. Employing available results based on the first-order approximation in Zα, we conclude that the Coulomb corrections are still important for heavy nuclei. We also consider an alternative approach for extended nuclear targets, the eikonal approximation, which we demonstrate to reproduce the known exact results for Coulomb corrections. Calculating electroproduction of vector mesons, we again arrive at a large deviation from the Born approximation. We conclude that one should accept with caution the experimental results for nuclear effects in DIS based on analyses done in the Born approximation. Received: 16 May 2001 / Accepted: 4 July 2001     

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.     

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).     

The eikonal approximation and E(2) invariance

A group theoretic interpretation is given for the eikonal approximation in potential scattering. This is based upon the approximate invariance at high energies under translations and rotations in the transverse scattering plane; that is, symmetry under the group E(2). The Lippman-Schwinger equation is formulated in a set of basis states which transform invariantly under irreducible representations of E(2) and the solution for the eikonal Hamiltonian, together with lowest order (Saxon-Schiff) corrections is obtained within this basis. A formulation of unitarity in the impact-parameter representation, based upon the E(2) invariance is given. The “geometrical” interpretation of this representation, in connection with the eikonal approximation, is made clear by our approach.     

Confinement of massless particles: Photon

Confinement of massless particles in a suitably chosen dielectric medium is considered. Light waves of selected frequencies are shown to be confined in a medium with dielectric constantε(r)=a/r−b ². A wave theoretical analysis gives equispaced frequency spectrum for the confined light, the radial dependence of its electric wave vector resembling that of hydrogen atom wave functions. In the large frequency limit an eikonal approximation of the problem gives elliptic orbits for the confined rays. Higher frequency orbits are shown to be closer to the centre of the medium than the lower frequency ones.     

On the energy losses of fast charged particles

The energy losses of fast charged particles colliding with atoms have been considered in the eikonal approximation. It has been shown that the nonperturbative contribution to the effective stopping from the region of the intermediate impact parameters (comparable with the characteristic sizes of the electron shells of the target) not only can be significant as compared to shell corrections to the Bethe-Bloch formula (usually considered in the first order of perturbation theory), but also can provide significant (up to 50%) corrections to this formula.     

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.     

Violation of the factorization theorem in large angle radiative Bhabha scattering

The lowest order QED radiative corrections to the radiative large angle Bhabha scattering process in the region where all kinematic invariants are large compared to the electron mass are considered. We show that the leading logarithmic corrections do not factorize before the Born cross section, contrary to the picture assumed in the renormalization group approach. The leading and non leading contributions for typical kinematics of the hard process at the energy of the Φ factory are estimated. Zh. éksp. Teor. Fiz. 115, 392–403 (February 1999) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Coulomb corrections for coherent electroproduction of vector mesons: Eikonal approximation

Virtual radiative corrections due to the long-range Coulomb forces of heavy nuclei with charge Z may lead to sizeable corrections to the Born cross-section usually used for lepton-nucleus scattering processes. An introduction and presentation of the most important issues of the eikonal approximation is given. We present calculations for forward electroproduction of rho-mesons in a framework suggested by the VDM (vector dominance model), using the eikonal approximation. It turns out that Coulomb corrections may become relatively large. Some minor errors in the literature are corrected.Received: 3 October 2003, Revised: 2 December 2003, Published online: 6 July 2004PACS: 25.30.-c Lepton-induced reactions - 25.30.Rw Electroproduction reactions - 13.40.-f Electromagnetic processes and properties - 25.30.Bf Elastic electron scattering     

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.     

Scaling violation and shadowing corrections at HERA

We study the value of shadowing corrections (SC) in the HERA kinematic region in the Glauber–Mueller approach. Since the Glauber–Mueller approach was proven in perturbative quantum chromodynamics (QCD) in the double-logarithmic approximation (DLA), we develop the DLA approach for the deep inelastic structure function which takes into account the SC. Our estimates show small SC for in the HERA kinematic region while they turn out to be sizable for the gluon structure function. We compare our estimates with those for gluon distribution in leading order (LO) and next to leading order (NLO) in the DGLAP evolution equations. Received: 21 September 1998 /Published online: 14 January 1999     

Off-shell corrections to the glauber theory of nuclear multiple scattering

We study the first-order off-shell corrections to Glauber's multiple scattering model. These consist of a direct and a rescattering term, and are obtained from the eikonal corrections studied in a previous paper by a simple substitution. For an independent particle nuclear model the direct term becomes identical to an expression previously derived by Moniz. We show that the rescattering term is in general not small compared to the direct term. For high incident energy a comparison is made with other corrections to the Glauber model. An optical potential is proposed which consists of the Glauber potential and its off-shell, eikonal, Fresnel and spatial correlation corrections.