Locality and Unitarity of Scattering Amplitudes from Singularities and Gauge Invariance.

Nima Arkani-Hamed,Laurentiu Rodina,Jaroslav Trnka

doi:10.1103/physrevlett.120.231602

Abstract

We conjecture that the leading two-derivative tree-level amplitudes for gluons and gravitons can be derived from gauge invariance together with mild assumptions on their singularity structure. Assuming locality (that the singularities are associated with the poles of cubic graphs), we prove that gauge invariance in just n-1 particles together with minimal power counting uniquely fixes the amplitude. Unitarity in the form of factorization then follows from locality and gauge invariance. We also give evidence for a stronger conjecture: assuming only that singularities occur when the sum of a subset of external momenta go on shell, we show in nontrivial examples that gauge invariance and power counting demand a graph structure for singularities. Thus, both locality and unitarity emerge from singularities and gauge invariance. Similar statements hold for theories of Goldstone bosons like the nonlinear sigma model and Dirac-Born-Infeld by replacing the condition of gauge invariance with an appropriate degree of vanishing in soft limits.

Highlights

Gauge redundancy.—The importance of gauge invariance in our description of physics can hardly be overstated, but the fundamental status of “gauge symmetry” has evolved considerably over the decades
While many older textbooks rhapsodize about the beauty of gauge symmetry and wax eloquent on how “it fully determines interactions from symmetry principles”, from a modern point of view gauge invariance can be thought of as by itself an empty statement
We speak of gauge “redundancy” as a convenient but not necessarily fundamental way of describing the local physics of Yang-Mills and gravity theories

Summary

Introduction

Gauge redundancy.—The importance of gauge invariance in our description of physics can hardly be overstated, but the fundamental status of “gauge symmetry” has evolved considerably over the decades. We find that with mild restrictions on the form of functions we consider the requirement of on-shell gauge invariance uniquely fixes the functions to match the tree amplitudes of Yang-Mills theory for spin one and gravity for spin two. Our central claim is that while locality and unitarity must be imposed to determine amplitudes for garden-variety scalar theories like φ3, much less is needed to uniquely fix the function to be “the amplitude” for gauge theories and gravity.

Results

Conclusion