Sensitivity of Nonrenormalizable Trajectories to the Bare Scale

TL;DR

This paper investigates how nonrenormalizable trajectories in scalar field theory depend on the bare scale, deriving equations that describe this dependence and exploring their generalizations using nonperturbative methods.

Contribution

It derives a Polchinski-like equation for the effective action's dependence on the bare scale and analyzes its applicability to various generalizations.

Findings

Effective action dependence on bare scale follows a Polchinski-like equation.

Existence of an action satisfying the equation is guaranteed for certain flow equations.

The identified action corresponds to the effective action at zero scale in specific cases.

Abstract

Working in scalar field theory, we consider RG trajectories which correspond to nonrenormalizable theories, in the Wilsonian sense. An interesting question to ask of such trajectories is, given some fixed starting point in parameter space, how the effective action at the effective scale, Lambda, changes as the bare scale (and hence the duration of the flow down to Lambda) is changed. When the effective action satisfies Polchinski's version of the Exact Renormalization Group equation, we prove, directly from the path integral, that the dependence of the effective action on the bare scale, keeping the interaction part of the bare action fixed, is given by an equation of the same form as the Polchinski equation but with a kernel of the opposite sign. We then investigate whether similar equations exist for various generalizations of the Polchinski equation. Using nonperturbative, diagrammatic arguments we find that an action can always be constructed which satisfies the Polchinski-like equation under variation of the bare scale. For the family of flow equations in which the field is renormalized, but the blocking functional is the simplest allowed, this action is essentially identified with the effective action at Lambda = 0. This does not seem to hold for more elaborate generalizations.