Renormalizability of Effective Scalar Field Theory

TL;DR

This paper proves the perturbative renormalizability, unitarity, and causality of a specific effective scalar quantum field theory using Wilson's renormalization group approach, ensuring its consistency at all energy scales.

Contribution

It demonstrates the all-order perturbative renormalizability and analytic S-matrix of a $Z_2$ symmetric scalar field theory within an effective field theory framework, extending previous results.

Findings

The theory is bounded, convergent, and universal at low energies.

Redundant Lagrangian terms can be eliminated without affecting the S-matrix.

The effective theory maintains unitarity and causality at all energy scales.

Abstract

We present a comprehensive discussion of the consistency of the effective quantum field theory of a single $Z_2$ symmetric scalar field. The theory is constructed from a bare Euclidean action which at a scale much greater than the particle's mass is constrained only by the most basic requirements; stability, finiteness, analyticity, naturalness, and global symmetry. We prove to all orders in perturbation theory the boundedness, convergence, and universality of the theory at low energy scales, and thus that the theory is perturbatively renormalizable in the sense that to a certain precision over a range of such scales it depends only on a finite number of parameters. We then demonstrate that the effective theory has a well defined unitary and causal analytic S--matrix at all energy scales. We also show that redundant terms in the Lagrangian may be systematically eliminated by field redefinitions without changing the S--matrix, and discuss the extent to which effective field theory and analytic S--matrix theory are actually equivalent. All this is achieved by a systematic exploitation of Wilson's exact renormalization group flow equation, as used by Polchinski in his original proof of the renormalizability of conventional $\varphi^4$-theory.