Scattering Matrix of the SU(n) Gauge Theory with Explicit Gauge Mass Term

TL;DR

This paper develops a method to construct the scattering matrix for SU(n) gauge theories with explicit gauge mass terms, ensuring unitarity and expressing matrix elements via renormalized Green functions.

Contribution

It introduces an operator-based approach to derive the scattering matrix in massive SU(n) gauge theories, emphasizing unitarity and the relation to Green functions.

Findings

Constructed a scattering matrix with explicit gauge mass terms.

Expressed scattering matrix elements in terms of renormalized Green functions.

Ensured unitarity of the scattering matrix in the massive gauge theory context.

Abstract

Based on the renormalisability of the SU(n) theory with massive gauge bosons, we start with the path integral of the generating functional for the renormalized Green functions and develop a method to construct the scattering matrix so that the unitarity is evident. By using as basical variables the renormalized field functions and defining the unperturbed Hamiltonian operator $H_0$ that, under the Lorentz condition, describes the free particles of the initial and final states in scattering processes, we form an operator description with which the renormalized Green functions can be expressed as the vacuum expectations of the time ordered products of the Heisenberg operators of the renormalized field functions, that satisfy the usual equal time commutation or anticommutation rules. From such an operator description we find a total Hamiltonian $\widetilde{H}$ that determine the time evolution of the Heisenberg operators of the renormalized field functions. The scattering matrix is nothing but the matrix of the operator $U(\infty, -\infty)$, which describes the time evolution from $-\infty$ to $\infty$ in the interaction picture specified by $\widetilde{H}$ and $H_0$, respect to a base formed by the physical eigen states of $H_0$. We also explain the asymptotic field viewpoint of constructing the scattering matrix within our operator description. Moreover, we find a formular to express the scattering matrix elements in terms of the truncated renormalized Green functions.