Conformal Transformations of S-Matrix in Scalar Field Theory

TL;DR

This paper develops three methods to describe conformal transformations of the S-matrix in scalar quantum field theory, demonstrating scheme independence and deriving results to all loop orders.

Contribution

It introduces three novel approaches to analyze conformal transformations of the S-matrix, including algebraic renormalization, functional formalism, and local coupling techniques.

Findings

Derived scheme-independent conformal Ward identities.

Calculated conformal transformations of the S-matrix in various formalisms.

Extended analysis to massive scalar fields with local coupling.

Abstract

In this paper, three methods for describing the conformal transformations of the S-matrix in quantum field theory are proposed. They are illustrated by applying the algebraic renormalization procedure to the quantum scalar field theory, defined by the LSZ reduction mechanism in the BPHZ renormalization scheme. Central results are shown to be independent of scheme choices and derived to all orders in loop expansions. Firstly, the local Callan-Symanzik equation is constructed, in which the insertion of the trace of the energy-momentum tensor is related to the beta function and the anomalous dimension. With this result, the Ward identities for the conformal transformations of the Green functions are derived. Then the conformal transformations of the S-matrix defined by the LSZ reduction procedure are calculated. Secondly, the conformal transformations of the S-matrix in the functional formalism are related to charge constructions. The commutators between the charges and the S-matrix operator are written in a compact way to represent the conformal transformations of the S-matrix. Lastly, the massive scalar field theory with local coupling is introduced in order to control breaking of the conformal invariance further. The conformal transformations of the S-matrix with local coupling are calculated.