Unitarity Bound of the Wave Function Renormalization Constant

TL;DR

This paper investigates the unitarity bound of the wave function renormalization constant in quantum field theories, revealing conditions under which the traditional bound is violated and deriving a more accurate unitarity constraint.

Contribution

It derives the true unitarity bound of the wave function renormalization constant, accounting for field dependence and contributions from multi-particle states in nonlinear sigma models.

Findings

The traditional bound $0 \,\leq Z \leq 1$ can be violated in certain models.

Bubble diagram contributions to the anomalous dimension can be negative.

Multi-particle state contributions uphold the positivity of the anomalous dimension.

Abstract

The wave function renormalization constant $Z$, the probability to find the bare particle in the physical particle, usually satisfies the unitarity bound $0 \leq Z \leq 1$ in field theories without negative metric states. This unitarity bound implies the positivity of the anomalous dimension of the field in the one-loop approximation. In nonlinear sigma models, however, this bound is apparently broken because of the field dependence of the canonical momentum. The contribution of the bubble diagrams to the anomalous dimension can be negative, while the contributions from more than two particle states satisfies the positivity of the anomalous dimension as expected. We derive the genuine unitarity bound of the wave function renormalization constant.