Small-Recoil Approximation

TL;DR

This paper reviews a technique for summing a class of Feynman diagrams involving minimal recoil, generalizing the eikonal formula to non-abelian interactions, and explores its applications in high-energy scattering and amplitude calculations.

Contribution

It introduces a decomposition formula for non-recoil diagrams that generalizes the eikonal approximation to non-abelian gauge theories, enabling summation of amplitudes and calculation of subleading effects.

Findings

The decomposition formula reduces to the eikonal formula in abelian cases.

It explains gluon reggeization and photon non-reggeization.

Allows extrapolation of the BFKL-Pomeron amplitude to high energies.

Abstract

In this review we discuss a technique to compute and to sum a class of Feynman diagrams, and some of its applications. These are diagrams containing one or more energetic particles that suffer very little recoil in their interactions. When recoil is completely neglected, a decomposition formula can be proven. This formula is a generalization of the well-known eikonal formula, to non-abelian interactions. It expresses the amplitude as a sum of products of irreducible amplitudes, with each irreducible amplitude being the amplitude to emit one, or several mutually interacting, quasi-particles. For abelian interaction a quasi-particle is nothing but the original boson, so this decomposition formula reduces to the eikonal formula. In non-abelian situations each quasi-particle can be made up of many bosons, though always with a total quantum number identical to that of a single boson. This decomposition enables certain amplitudes of all orders to be summed up into an exponential form, and it allows subleading contributions of a certain kind, which is difficult to reach in the usual way, to be computed. For bosonic emissions from a heavy source with many constituents, a quasi-particle amplitude turns out to be an amplitude in which all bosons are emitted from the same constituent. For high-energy parton-parton scattering in the near-forward direction, the quasi-particle turns out to be the Reggeon, and this formalism shows clearly why gluons reggeize but photons do not. The ablility to compute subleading terms in this formalism allows the BFKL-Pomeron amplitude to be extrapolated to asymptotic energies, in a unitary way preserving the Froissart bound. We also consider recoil corrections for abelian interactions in order to accommodate the Landau-Pomeranchuk-Migdal effect.