Finite <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math> matrix
Holmfridur S. Hannesdottir, Matthew D. Schwartz
Abstract
When massless particles are involved, the traditional scattering matrix ($S$ matrix) does not exist: It has no rigorous nonperturbative definition and has infrared divergences in its perturbative expansion. The problem can be traced to the impossibility of isolating single-particle states at asymptotic times. On the other hand, the troublesome nonseparable interactions are often universal: In gauge theories, they factorize so that the asymptotic evolution is independent of the hard scattering. Exploiting this factorization property, we show how a finite ``hard'' $S$ matrix, ${S}_{H}$, can be defined by replacing the free Hamiltonian with a soft-collinear asymptotic Hamiltonian. The elements of ${S}_{H}$ are gauge invariant and infrared finite and exist even in conformal field theories. One can interpret elements of ${S}_{H}$ alternatively 1) as elements of the traditional $S$ matrix between dressed states, 2) as Wilson coefficients, or 3) as remainder functions. These multiple interpretations provide different insights into the rich structure of ${S}_{H}$.