Non-Perturbative Four-Point Scattering from First-Quantized Relativistic JWKB

TL;DR

This paper applies a first-quantized JWKB approximation to relativistic two-body scattering, deriving non-perturbative results in various dimensions and revealing Regge trajectories, infrared phases, and singularities in scattering amplitudes.

Contribution

It introduces a non-perturbative, first-quantized JWKB approach to relativistic scattering, providing explicit results and new insights into Regge trajectories and infrared effects.

Findings

Regge trajectory function matches quantum mechanical spectrum in D=4

Infrared divergence becomes a pure phase factor in physical region

Multi-mass branch points appear as Regge poles in D=3

Abstract

We apply the quantum mechanical (first-quantized) JWKB approximation to a two-body path integral describing the near-forward scattering of two relativistic, heavy, non-identical, scalar particles in $D$ spacetime dimensions. In contrast to the loop expansion, in $D = 4$ this gives a strong-coupling expansion, and in $D = 3$ a non-perturbative weak-coupling expansion. When the interaction is mediated by massless quanta with spin $N$, we obtain explicit, relativistic results for the scattering amplitude when $N = 0$, $1$ and $2$. In $D = 4$ we find a Regge trajectory function that agrees with the usual quantum mechanical spectrum. We also find an exponentiated infrared divergence that becomes a pure phase factor when the Mandelstam invariants $s$ and $t$ are inside of the physical scattering region. In $D = 3$ we find a singularity whose position along the $s$ axis is dependent on $t$. When the interaction is mediated by a heavy scalar with mass $M$, in $D = 3$ we find an all-order scattering amplitude where the multi-mass branch points $t = (L + 1)^{2}M^{2}$ appear as Regge poles.