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

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$2$\mathrm{.}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.