Crossing Symmetry in the Planar Limit

Abstract

Crossing symmetry asserts that particles are indistinguishable from anti-particles traveling back in time. In quantum field theory, this statement translates to the longstanding 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 iε 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.

Positivity of the Veneziano Amplitude in $D=4$

“…(2.2) This is manifest for scattering of identical particles in the description in terms of Feynman diagrams, see [11] for recent developments on an interesting physical interpretation behind it.…”

AdS Bulk Locality from Sharp CFT Bounds

“…Such test functions suffice to prove all our bounds (in D > 4) and these properties explain why R AdS does not appear in the bounds. This could help address concerns raised in [81].…”

Kinematic singularities of Feynman integrals and principal A-determinants

“…The equations (4.14), (4.15) are called Landau equations and were independently derived in 1959 by Bjorken [8], Landau [55] and Nakanishi [60]. We recommend [58] for a comprehensive summary of the known research results in Landau's analysis of over 60 years. Further we will refer to [26] for a classical and [18,59] for modern derivations of Landau equations.…”

“…The structural investigation of the analytic properties of Feynman integrals was started in 1959 independently by Bjorken [8], Nakanishi [60] and Landau [55] and is also known as Landau analysis. Since a comprehensive historical review on Landau analysis can be found in [58], we restrict ourselves to a very short historical overview. For a summary of the first steps of this subject from the 1960s we refer to the monograph of Eden et al [26].…”