Crossing symmetry of scattering amplitudes across kinematic sign regions

Establish whether scattering amplitudes can be analytically continued from a kinematic region with one sign vector to a region with a second sign vector in such a way that the amplitude has a similar form on both regions.

Background

The paper studies massless momentum-conserving kinematic regions, whose full-dimensional strata are indexed by sign vectors describing the incoming and outgoing particle configuration. In scattering theory, crossing symmetry concerns analytically relating amplitudes associated with different sign vectors. The authors note that poles and branching singularities in the complex Mandelstam variables make this continuation difficult to prove, leaving the conjectured relation unresolved in the discussion.

References

Physicists conjecture that the amplitude can be analytically continued from the region with one sign vector, $\sigma$, to a region with a second sign vector, $\sigma'$, in such a way that the function takes a similar form on both regions. This is called crossing symmetry. However, this is difficult to prove, because amplitudes have both poles and branching singularities as analytic functions of the $s_{ij}$, regarded as complex variables.

— Kinematic Stratifications  (2503.09571 - Cortes et al., 12 Mar 2025) in Section 6, “Stratifications and Scattering”