Crossing Symmetry in the Planar Limit

TL;DR

This paper demonstrates that crossing symmetry holds for planar scattering amplitudes in perturbative quantum field theory by analyzing analytic continuations of external momenta and showing the absence of singularities.

Contribution

It provides a five-step sequence of deformations of external momenta that proves crossing symmetry for planar amplitudes in perturbation theory.

Findings

Crossing symmetry can be established through analytic continuation of momenta.

Singularities are absent in planar diagrams during the deformation process.

The method offers a pathway to generalize crossing symmetry beyond planar cases.

Abstract

Crossing symmetry asserts that particles are indistinguishable from anti-particles traveling back in time. In quantum field theory, this statement translates to the long-standing conjecture that probabilities for observing the two scenarios in a scattering experiment are described by one and the same function. Why could we expect it to be true? In this work we examine this question in a simplified setup and take steps towards illuminating a possible physical interpretation of crossing symmetry. To be more concrete, we consider planar scattering amplitudes involving any number of particles with arbitrary spins and masses to all loop orders in perturbation theory. We show that by deformations of the external momenta one can smoothly interpolate between pairs of crossing channels without encountering singularities or violating mass-shell conditions and momentum conservation. The analytic continuation can be realized using two types of moves. The first one makes use of an $i\varepsilon$ prescription for avoiding singularities near the physical kinematics and allows us to adjust the momenta of the external particles relative to one another within their lightcones. The second, more violent, step involves a rotation of subsets of particle momenta via their complexified lightcones from the future to the past and vice versa. We show that any singularity along such a deformation would have to correspond to two beams of particles scattering off each other. For planar Feynman diagrams, these kinds of singularities are absent because of the particular flow of energies through their propagators. We prescribe a five-step sequence of such moves that combined together proves crossing symmetry for planar scattering amplitudes in perturbation theory, paving a way towards settling this question for more general scattering processes in quantum field theories.