Bounds on Crossing Symmetry
Sebastian Mizera
TL;DR
The paper addresses whether crossing symmetry can be established on-shell within perturbative quantum field theory by deriving explicit bounds on internal masses that guarantee crossing symmetry to all loop orders. Using a worldline, Schwinger-parameter formulation and the $i\varepsilon$ prescription, it connects causality to analytic structure and provides a concrete mass bound $m_e > \frac{\sqrt{n}}{2\sqrt{2}} \sqrt{ \max_i( M_i^2, (\sum_j M_j^2 - 2 M_i^2)/(n-2) ) }$, ensuring crossing between channels for general $n$ and multiplicities (with refinements for planar cases and equal-mass examples yielding specific thresholds). The work also discusses the limitations of naive deformations, introduces a causality-preserving deformation strategy, and highlights avenues to tighten bounds or extend results to higher dimensions or special diagram classes. These results offer a perturbative, on-shell route toward crossing symmetry and clarify how the $i\varepsilon$ prescription constrains analytic continuation in scattering amplitudes. Overall, the paper provides quantitative criteria under which crossing symmetry can be expected to hold, advancing our understanding of the analytic properties of multi-particle amplitudes in quantum field theory.
Abstract
Proposed in 1954 by Gell-Mann, Goldberger, and Thirring, crossing symmetry postulates that particles are indistinguishable from anti-particles traveling back in time. Its elusive proof amounts to demonstrating that scattering matrices in different crossing channels are boundary values of the same analytic function, as a consequence of physical axioms such as causality, locality, or unitarity. In this work we report on the progress in proving crossing symmetry on-shell within the framework of perturbative quantum field theory. We derive bounds on internal masses above which scattering amplitudes are crossing-symmetric to all loop orders. They are valid for four- and five-point processes, or to all multiplicity if one allows deformations of momenta into higher dimensions at intermediate steps.