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For the case of charged particles colliding with atoms, an analytical procedure, capable of providing quantitative estimates of the Glauber cross section, is proposed.

Ray techniques are used to construct an approximate solution for radio waves propagating in the ionosphere when the loss of energy due to collisions is not negligible. For the time-harmonic problem, the solution is in the form of an asymptotic expansion in a large parameter scaling the operating frequency and the natural frequencies characterizing the ionospheric properties. Complex rays are introduced as the characteristics of the resulting eikonal equation, analytically extended to complex space points. The rapidly varying exponential factor and a series for the wave amplitude are both determined by integrating along the rays. The propagation of narrow-band modulation is also determined by integrating along the same complex rays. It is shown that the nature of the initial disturbance plays a vital role in determining the character of the rays required for the solution of a particular problem.

The purpose of this volume is to present from a unified point of view a description of the major soluble and approximate models of relativistic quantum field theory. While the proper content of (realistic) field theories remains a matter of speculation, it seems to be clear that an understanding of the structure of matter, deeper than a phenomenological one, will rest on an understanding of the dynamics of interacting fields. For this reason alone it is worthwhile to collect, as this book does, those results that are now known about model theories. Considerations of efficiency and the anticipation of future usefulness suggest the employment of the beautiful functional methods developed by Schwinger, Symanzik, and Feynman, among others.The level of presentation is such that students familiar with conventional field theoretic arguments should make the transition to a functional description both without difficulty and with a certain sense of excitement.The first half of the book is a compact expression and derivation of functional methods that are applicable to relativistic quantum field theory, while the second deals with the models themselves--it employs these rather elegant functional techniques in describing almost all the soluble and approximate models of field theory.After introducing functional field theory, the first part takes up in turn the S-matrix and the generating functional; construction of the generating functional: the Symanzik construction, the Schwinger construction, several interacting fields, rearrangements/grouping of Feynman graphs, and fields at the same point; noncanonical (e.g., Chiral) generalizations; and special properties of quantum electrodynamics: the heavy proton limit, Green's function equations, and gauge transformations and Ward identities.The second half of the book, on soluble models, covers perturbation expansions: the Born (tree-graph) functional, lowest-order radiative corrections, the Lamb shift, and renormalization procedures; two-dimensional electrodynamics and the Thirring model; the Lee model; static and Bloch-Nordsieck models; and relativistic eikonal physics.

The high-energy properties of a simple class of unitary, crossing-symmetric eikonal models of multiproduction are discussed on the basis of the general closed expression given for the S -matrix elements in a previous publication. In particular, the high-energy behaviour of the multiplicity moments is discussed and it is shown that the KNO scaling relation emerges in a very natural fashion in this class of models.

Heuristic derivations are given for a relation linking elastic and inelastic cross sections, satisfied in the simplest cases by the sets of field-theoretic tower graphs.

The application of eikonal methods to electron-atom scattering is reviewed. After recalling the basic features of the eikonal approximation, we discuss the Glauber theory, the eikonal-Born series method, the eikonal optical model theory and the eikonal distorted wave approach. We also give a survey of recent results obtained for electron-hydrogen and electron-helium collisions at intermediate energies.

We develop a Lagrangian field-theoretic laboratory where one can rigorously investigate ideas and problems in high-energy hadronic interactions. In this paper (the first of a series) the general field-theoretic framework is outlined in the oversimplified model of a scalar-scalar Yukawa interaction. Functional methods are used to cast all Green's functions in an “operator eikonal” form. The eikonal approximations (EA's) in Lagrangian relativistic quantum mechanics are reviewed and discussed. We then derive an exact eikonal equation in quantum field theory. The perturbation theoretic solution of this equation leads to a new kind of eikonal perturbation theory (EPT) which generalizes simultaneously the EA's as well as the ordinary perturbation theory (OPT). Some salient features of Green's functions in the EPT are as follows: (i) the lowest-order EPT amplitudes correspond to a kind of semiclassical approximation; (ii) the lowest-order four-point amplitudes contain the high-energy part of the full radiatively corrected crossed ladder series, without vacuum polarization effects; (iii) for spin-one gluons, the latter amplitude develops diffractive behavior in the direct channel and, for spin-one and spin-zero gluons, Regge behavior in the crossed channel; (iv) for vanishing gluon mass, this amplitude develops poles, in the direct channel, corresponding to a positronium-like bound-state spectrum. Properties (i)–(iv) are generalized to EPT from EA's and are absent in OPT. Unlike in the case of EA's we also have that (v) the EPT is a quantum field theory, which properly includes selfinteraction effects; (vi) the EPT is an iterative perturbation theoretic scheme, which shares with OPT the properties of renormalizability.

A medium which may be slightly inhomogeneous in space and time is characterized by its current response. The author derives a system of equations describing the linear and non-linear coupling of WKB modes.

A technique to derive covariant recoil to the eikonal model in relativistic quantum field theory is developed. Recoil is defined relative to the rectilinear-path approximation which gives the eikonal model. It appears in terms of a partially factorizable correlation expansion, where the first term is noncorrelated and is the usual eikonal model. The second term displays pair correlations, etc. By correlation here we mean dynamical correlation. Kinematical correlations due to four-momentum conservation are of course superimposed everywhere. We first consider soft recoil, and then generalize to the hard re:oil, without introducing any cutoff. The technique is developed here for a scalar $'-type theory but it is general enough to be extended to, for example, quantum electrodynamics and scalar electrodynamics.

Eikonalization of dual pole amplitudes, such as the Veneziano amplitude, is shown to lead to singularities in the impact parameter plane, which, in the eikonal approximation, may be interpreted as branch points in the direct channel angular momentum plane. This result is discussed in the light of dual absorption models.

We have studied the effect of fragmentation and energy conservation in the eikonal-Regge model. A generalized eikonal representation involving multi-impact parameters is given for the elastic and inelastic amplitudes when fragmentation takes place. The generalized eikonal function which describes a many-body potential depends on more than one impact parameter. In the strong-absorption model and at high energy, however, the elastic amplitude can be approximated by a single-impact-parameter representation with an effective eikonal function. As a result of the fragmentation, we find that although ${\ensuremath{\sigma}}_{E}$ and ${\ensuremath{\sigma}}_{T}$ still increase as ${\mathrm{ln}}^{2}s$, their ratio is no longer \textonehalf{}. Instead, it is the sum of the elastic and diffractive cross sections which remains to be one half of the total cross section. To enforce the energy conservation, we propose a thermodynamic approach by introducing an impact-parameter-dependent temperature. Using well-known thermodynamics relations, we obtain various corrections to the naive eikonal-Regge model predictions due to energy conservation. Experimental consequences are discussed.

A numerical solution method is outlined for solving Maxwell's wave equation in parabolic approximation, which is of importance in nonlinear optics. The method is based on cubic splines which guarantee continuity of first and second derivatives. In practice the method has proven to be both accurate and flexible.

The collision of He2+ with H(1s) is studied in the impact parameter (eikonal) approximation by expanding the wavefunction on a set of electronic states of the molecule HeH2+. Exact eigenenergies and matrix elements between the states of this basis set are used in the calculations. The results for the total charge-exchange cross section are in good agreement with the experimental data. The structures observed experimentally in the variation of the charge-exchange probability as a function of energy for a fixed scattering angle are fairly well reproduced in the theory. However, the origin of these structures is not that proposed by Keever and Everhart (1966) on the basis of their experimental work.

The problem of quantum collisions involving several particle systems is reviewed within the framework of multiple scattering theory. The basic apparatus of collision theory for nonrelativistic potential problems is first developed, and the Born and eikonal series are introduced. A general analysis is then given of multiple scattering expansions for several particle problems. We discuss in particular the Born developments, the Faddeev-Watson expansions, the Glauber method, and various multiple scattering approaches to the determination of the optical potential. Applications to atomic collision problems and to high-energy hadron-deuteron scattering are discussed at length. © 1974 American Physical Society.

We treat massive electrodynamics as a model for fixed angle scattering. The dominant momentum flow through Feynman diagrams at high energies is examined. In spinor electrodynamics this leads to the “eikonal” answer. In scalar electrodynamics we must also consider other flows which cancel between different diagrams. The analysis is carried out in Feynman parameter space. As well as the d -line contributions we also analyse the pinch effects. Although for a given diagram these may be larger they again cancel between diagrams. To leading powers of log s we thus prove the “eikonal” answer for spinor and scalar electrodynamics.

From the Takagi-Taupin equations for two-beam cases, the exact wave fields for a spherical incident wave are obtained in the Laue case for a crystal having a constant strain gradient. Absorption is taken into account. Both direct and Bragg-reflected waves are essentially expressed in terms of confluent hypergeometric functions. Their characters depend on strain gradient, structure factor and crystal thickness. The wave fields tend to those obtained by Eikonal theory as the strain gradient decreases. For an extremely large strain gradient, the wave fields reduce to those predicted by the kinematical theory.

The imaginary part of the proton-proton elastic scattering amplitude, as measured at the ISR, is examined in impact parameter. The transformed amplitude has two important properties. First, it is very accurately Gaussian in the impact parameter from the center to two fm with very little flattening near the center. Flattening would be expected from eikonal- ization. Secondly, there is a tail beyong two fm with a much flatter slope. This tail in impact parameter is equivalent to the break of do/d t at t = - 0.15 GeV. We discuss the. physical origin of the tail. It cannot reasonably be diffraction dissociation, since diffraction should be large where absorption is large. We suggest that the tail is due to dissociation which must be distinguished from its diffractive part, and make a physical model which gives the tail and describes do/dt very well. This model predicts the Deck model part of the diffraction inelastic cross section. We discuss the interpretation of elastic scattering in terms of s-channel unitarity rather than a t-channel exchange or the structure of a single hadron.

The solution of the reduced scalar wave equation for an almost stratified medium is written in the form of an asymptotic power series. The vertical structure of the solution is expressed as a linear combination of the normal‐mode eigenfunctions whose coefficients satisfy two‐dimensional eikonal and transport equations. Although smooth caustics may be treated for the two‐dimensional scalar wave equation, the problem of a uniform approximation near a point source has not yet been resolved. Finally, the theory is applied to acoustic propagation in a realistic model ocean and the results are compared with measurements.

The application of the Glauber-type eikonal approximation to multi-centre atomic collisions, such as atom-atom and electron-molecule collisions, is considered. The eikonal phases for such systems can be evaluated in closed forms and are independent of the z-component of the internal coordinates.

The basic equation for the scattering amplitude as determined from a multistate eikonal description of electron-atom collisions is presented. The relationship with other semiclassical treatments is examined. Four-state eikonal calculations of the cross sections for elastic and the 2s and 2p excitations of H(1s) by electrons with incident energy E in the range 13.6 eV

An extension of Fermat's principle to the stochastic case is presented in order to treat ray propagation in random media. The concept of dynamic programming permits one to derive a sequence of stochastic eikonal equations from which a sequence of rays can be traced using Hamilton's equations. The extension is motivated by analogous stochastic control problems in which the concepts and methods are adapted to the present problem. To facilitate understanding of the main ideas the presentation is given in a simple and somewhat naive manner. Two simple examples are presented to demonstrate the ideas and techniques.

A comparison is made between a number of recently proposed approximations of the eikonal type for potential scattering. The simple amplitudes proposed by Glauber, and later by Abarbanel and Itzykson, are found to be as good as any of the approximations studied. (auth)

Eikonal formulae for the production process NN-7NN+mc are studied on the basis of a relativistic field-theoretical model in which the potential between nucleons, pion emission and absorption are all induced by a scalar field A. The functional integral method is used to derive formulae. Within the " straight path" approximation for the initial particles, it is shown that an exact impact parameter representation for production amplitude can be. obtained, including all internal interactions, in particular, nucleon loops. The amplitude is properly represented with the aid of three variables; i.e., transverse vector and two longitudinal coordinates. The latter two correspond to two moving intermediate nucleons respectively. These variables compose a vector in a four-dimensional space called "eikonal frame", and a modified rapidity is introduced. Multiple scattering effects by the A field are investigated and a cluster expansion with respect to correlated A-lines is performed for eikonal phase of production amplitude.

Thus, the study of the class of ladder diagrams in the scalar model shows that the eikonal formula corresponds to our allowing in the asymptotic behavior for the\(\overline t \)-paths that coincide with nucleon lines. In this case, the leading particle, which carries the large momentum, is a nucleon and it does not change its species in the virtual process. The noneikonal contributions to the scattering amplitude are due to processes in which the species of the leading particle changes, i.e., to a transfer of momentum from nucleons to mesons and vice versa. There then arises the important question of the role of “twisted” graphs corresponding to the original ladder graph with replacement of the final momenta q1↔q2 (compare Fig. 1 and formula (1.2)). The possibility of transferring a large momentum to a meson means that the contribution to the asymptotic behavior of the scattering amplitude may dominate over the eikonal contribution in the same order in the coupling constant. For example, in the fourth order, the “twisted” graph (see Fig. 16) has the asymptotic behavior 1n s/s.

As is well known, the s-channel crossed ladder amplitudes calculated in the eikonal approximation (1) present a number of desirable features: i) in the limit of vanishing exchanged (( gluon )) mass /~, these on-shell amplitudes develop poles in the centre-ofmass energy squared s, which, in the nonrelativistic limit of a Coulomb potential, reproduce the Balmer formula; ii) as the invariant momentum transfer t becomes very large, these amplitudes develop a Regge behaviour t ~(*~ which, in the ~3-model, agrees with fhe known behaviour of the ordinary ladder for s>> #2; iii) as s-~ co with t fixed, for vector-gluon exchange these amplitudes show a purely diffractive behaviour; furthermore, iv) the eikonal approximation corresponds to a kind of semi-classical picture (2). Given that many of the presently successful models are based on semiclassical intuit ion (parton models, droplet and optical models), in view of i)-iv) one can reasonably hope that an eikonal formalism might be the appropriate tool for constructing a field theory of hadrons. Unfortunately, available eikonal approximations suffer from some drawbacks, which become particularly serious when radiative corrections and other renormalization effects are included. In order to appreciate these difficulties, consider the simple, superrenorrealizable interaction g~p~p~/2 between a scalar (( quark )) + of mass m and a scalar (~ gluon )) (I) of mass/~. The ~-propagator in the presence of an external field A(x) satisfies the equation

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 is considered for propagation in an inhomogeneous, time-dependent, anisotropic environment. The usual eikonal analysis is generalized to take into account constitutive relations which contain space and time derivatives of the field quantities. The field equations are obtained for all orders of eikonal approximation. These consist of homogeneous equations for the zeroth-order, and recursion relations which connect arbitrary consecutive higher orders of approximation. An apparent inconsistency between the zeroth-order equations and the higher order ones is avoided by the development of a set of subsidiary conditions on the field approximations. The lowest order condition in this set is considered in detail, and a general relation is derived between a certain vector quantity appearing in this condition, and the tangent vector to a ray path. For an environment which is stationary and free of spatial dispersion, this relation is shown to have, as one of its consequences, a new normalization condition for the zeroth-order fields in anisotropic propagation. The relation is applied to propagation in a cold magnetoplasma, and a number of results in ionospheric propagation are recovered and extended.

The Glauber version of the eikonal expansion in potential theory is shown to be the unique result of calculating the high-energy, fixed-momentum-transfer limit of the Born series. Correction terms are systematically calculated and agree with the form proposed by Wallace. These results are used to understand the fact that the Glauber eikonal is an accurate approximation to the exact scattering amplitude at all energies.

The effect of including the 3s and 3p pseudostates is examined for the 2s and 2p excitation of atomic hydrogen by incident electrons with energy E (eV) in the range 16.5

Dispersion corrections to elastic electron scattering are calculated using an eikonal approximation for the electron. Additional phase shifts Λ ( b ) with b = j + 1 2 /k can be defined which have to be added to the static phase shifts χ j obtained from a partial-wave calculation. Equivalent is a description by an energy-independent optical potential. Summation over the excited nuclear states is done by a special closure approximation. The only free parameterthe mean excitation energy — can be determined by the photo sum rule σ −2 . Total inelastic cross sections agree well with experimental estimates; corrections in the elastic cross section are usually negative and small except in diffraction minima.

We analyze elastic electron-atom scattering at intermediate and high energies by using a second-order optical-model potential together with the eikonal approximation. We devote particular attention to the role of long-range forces and formulate the theory in a way suitable for generalizations to complex target atoms. We also study the relationship between the optical eikonal model and the Born series, and include exchange effects to leading order in the inverse power of the energy. Our optical-model theory is then illustrated by a detailed analysis of elastic electron-helium scattering for incident-electron energies ranging from 100 to 500 eV. We find excellent agreement with experimental data without using any phenomenological parameter. © 1974 The American Physical Society.

We have fitted the experimental data on the high-energy processes γ p → π 0 p, γ n → π 0 n and γ p → η p with just the ϱ and ω Regge poles, and strong absorptive cuts generated by the P+P′ exchange amplitude from πN elastic scattering. We obtain excellent agreement with all the data, including target and photon asymmetries, without invoking secondary reggeons such as the B or H trajectories. This result, together with earlier work on the πN charge exchange scattering, leads us to conclude that the addition of P′ ⊗ reggeon cuts can solve many of the phase problems of the traditional SCRAM model.

The Regge-pole eikonal model is used to calculate the scattering amplitude for the double-charge-exchange process nπ+→Δ++π−. To restrict the number of parameters needed to connect this reaction to other similar reactions, with a member of the 3/2 decuplet in this final state, the quark model and exactSU6 have been used. It is indicated how these connections may be made, and predictions of cross-sections for nπ+→Δ++π− are discussed.