Recursion relations for scattering amplitudes with massive particles

Sourav Ballav,Arkajyoti Manna

doi:10.1007/jhep03(2021)295

Abstract

We use the recently developed massive spinor-helicity formalism [1] of Arkani-Hamed et al. to study a new class of recursion relations for tree-level amplitudes in gauge theories. These relations are based on a combined complex deformation of massless as well as massive external momenta. We use these relations to study tree-level amplitudes in scalar QCD as well as amplitudes involving massive vector bosons in the Higgsed phase of Yang-Mills theory. We prove the validity of our proposal by showing that in the limit of infinite momenta of two of the external particles, the amplitude once again is controlled by an enhanced Spin-Lorentz symmetry paralleling the proof of BCFW shift for massless gauge theories. Simple examples illustrate that the proposed shift may lead to an efficient computation of tree-level amplitudes.

Highlights

Have made seemingly impossible computations possible within a few pages
We proved the validity of this recursion relation for massive scalar QCD and Higgsed Yang-Mills theories by adapting the background field methods of Arkani-Hamed and Kaplan [25]
The five-particle vector boson amplitudes for different helicities of gluons that we obtained with the newly proposed recursion relation is a new result of this formalism

Summary

Review of massive spinor-helicity formalism

The scattering amplitude is defined as an inner product of the “in” and “out” states. The massive spinor helicity variables (λiαI , λiα J ) carry an extra set of indices (I, J ) which labels the little group SU(2). Where ξI± are two orthonormal 2-component vectors which serve as a basis in the SU(2) space This expansion allows one to take the high energy limit of the massive spinor helicity variables which in turn will prove to be useful while analyzing the high energy limit of scattering amplitudes involving massive particles [1]. We move on to defining polarization tensors for massive and massless particles For massless particles, this is well studied and the polarization vector is expressed in terms of spinor helicity variables in the following way [10]: eμ+(j) =.

Three-particle amplitude: basic building blocks

BCFW recursion scheme for massive particles

Proposal for a valid massive-massless shift

Scalar QCD

Scalar QCD: five-particle amplitude

Compton amplitude with massive vector bosons

Massive spin-1 with gluon: five-particle amplitude

High energy limit

Validity of massive-massless shift

Higgsed Yang-Mills theory

Structure of Aμν

Validity for transverse modes