Eikonal Approximation in Quantum Field Theory

Abstract

The eikonal approximation for high-energy collisions, long familiar in the theory of potential scattering, is considered from the viewpoint of relativistic quantum field theory. We study, in particular, the Feynman amplitude $M(s, t)$ describing the scattering of two spin-0 particles, $a$ and $b$, interacting by the exchange of spin-0 mesons. We show that if ${M}_{n}(s, t)$, the contribution to $M(s, t)$ arising from all $n\mathrm{th}$-order Feynman diagrams in which exactly $n$ mesons are exchanged between $a$ and $b$, is written in an appropriately symmetrized way, and if the terms in any $a$ or $b$ particle propagator which are quadratic in the internal momenta are then dropped, the resulting expression, ${{M}_{n}}^{\mathrm{eik}}(s, t)$, may be evaluated in closed form, and the sum over $n$, which defines ${M}^{\mathrm{eik}}(s, t)$, may be carried out. The representation of ${M}^{\mathrm{eik}}(s, t)$ found in this way involves the exponential of a function $\ensuremath{\chi}$ of a relative space-time variable $x=({x}^{0}, \mathrm{x})$ and the external momenta; $\ensuremath{\chi}$ is a relativistic generalization of the eikonal ${\ensuremath{\chi}}_{\mathrm{pot}}$ familiar from the theory of high-energy potential scattering. ${M}^{\mathrm{eik}}(s, t)$ is both crossing-symmetric and time-reversal-invariant. In the static limit (${m}_{b}\ensuremath{\rightarrow}\ensuremath{\infty}$), $\ensuremath{\chi}$ tends to ${\ensuremath{\chi}}_{\mathrm{pot}}$ for the appropriate Yukawa potential and ${f}^{\mathrm{eik}}=\ensuremath{-}\frac{{M}^{\mathrm{eik}}}{8\ensuremath{\pi}\ensuremath{\surd}s}$ has a limiting form ${{f}_{\mathrm{pot}}}^{\mathrm{eik}}$, which we also derive directly from the theory of potential scattering; for small scattering angles, ${{f}_{\mathrm{pot}}}^{\mathrm{eik}}$ coincides with the standard result. The amplitude for particle-antiparticle scattering is studied in the same model. It is shown that the eikonal ${\ensuremath{\chi}}_{2}(x)$ associated with the contribution of all annihilation-type diagrams has a logarithmic singularity at $x=0$ whose coefficient is proportional to $\ensuremath{\alpha}(t)+1$, where $\ensuremath{\alpha}(t)$ is the Regge-trajectory function obtained from the asymptotic behavior of the ladder-type diagrams alone. Another connection with Regge behavior is made by showing that the summation of a certain infinite class of radiative corrections to the lowest-order $\ensuremath{\gamma}\ensuremath{-}e$ Compton amplitude gives rise, in our eikonal approximation, to an eikonal $\ensuremath{\chi}(x)$ which has a similar logarithmic singularity with strength $1+\ensuremath{\beta}(t)$; here $\ensuremath{\beta}(t)$ is the trajectory function, introduced less directly in earlier work, which reproduces the major part of the spectrum of positronium on setting $\ensuremath{\beta}(t)=l=n\ensuremath{-}1$. A generalization of a simple algebraic identity used in the derivation of the above results, in the form of an integral representation, permits their extension to the case where one or more particles are off the mass shell. This is illustrated by a computation of an eikonal-type approximation to the Green's function for a relativistic particle moving in an external scalar field and by the summation of an infinite class of contributions to the vertex function in the model referred to above. The possibility of applying an off-shell eikonal approximation to the analysis of production processes is emphasized.