RG-stable parameter relations of a scalar field theory in absence of a symmetry

TL;DR

This paper investigates a toy scalar field model where RG-stable parameter relations occur without an explicit symmetry, explaining stability through complexification and hidden symmetries in the extended theory.

Contribution

It demonstrates how RG stability of parameter relations can be understood via complexification, revealing hidden symmetries that do not exist in the original real scalar theory.

Findings

RG-stable relations can be explained by complexifying the scalar field theory.

Hidden symmetries in the complexified theory guarantee the stability of parameter relations.

Identifies algebraic equations in the complexified theory that mirror stability conditions in the original theory.

Abstract

The stability of tree-level relations among the parameters of a quantum field theory with respect to renormalization group (RG) running is typically explained by the existence of a symmetry. We examine a toy model of a quantum field theory of two real scalars in which a tree-level relation among the squared-mass parameters of the scalar potential appears to be RG-stable without the presence of an appropriate underlying symmetry. The stability of this relation with respect to renormalization group running can be explained by complexifying the original scalar field theory. It is then possible to exhibit a symmetry that guarantees the relations of relevant beta functions of squared-mass parameters of the complexified theory. Among these relations, we can identify equations that are algebraically identical to the corresponding equations that guarantee the stability of the relations among the squared-mass parameters of the original real scalar field theory where the symmetry of the complexified theory is no longer present.