An $S$-Matrix for Massless Particles

TL;DR

This paper constructs a divergence-free $S$-matrix for massless particles in gauge theories by employing dressed states and factorization, enabling consistent perturbative calculations in theories like QED, QCD, and super Yang-Mills.

Contribution

It introduces a novel framework for the perturbative calculation of an infrared-finite $S$-matrix using dressed states and factorization principles.

Findings

Constructed an $S$-matrix free of infrared divergences.

Demonstrated the framework in QED, QCD, and super Yang-Mills.

Showed divergence cancellation explicitly with cutoffs and in dimensional regularization.

Abstract

The traditional $S$-matrix does not exist for theories with massless particles, such as quantum electrodynamics. The difficulty in isolating asymptotic states manifests itself as infrared divergences at each order in perturbation theory. Building on insights from the literature on coherent states and factorization, we construct an $S$-matrix that is free of singularities order-by-order in perturbation theory. Factorization guarantees that the asymptotic evolution in gauge theories is universal, i.e. independent of the hard process. Although the hard $S$-matrix element is computed between well-defined few particle Fock states, dressed/coherent states can be seen to form as intermediate states in the calculation of hard $S$-matrix elements. We present a framework for the perturbative calculation of hard $S$-matrix elements combining Lorentz-covariant Feynman rules for the dressed-state scattering with time-ordered perturbation theory for the asymptotic evolution. With hard cutoffs on the asymptotic Hamiltonian, the cancellation of divergences can be seen explicitly. In dimensional regularization, where the hard cutoffs are replaced by a renormalization scale, the contribution from the asymptotic evolution produces scaleless integrals that vanish. A number of illustrative examples are given in QED, QCD, and $\mathcal{N}=4$ super Yang-Mills theory.