[Paper Review] The local potential approximation in the background field formalism

I. Hamzaan Bridle, Juergen A. Dietz+1 more|arXiv (Cornell University)|Dec 10, 2013

Cosmology and Gravitation Theories31 references4 citations

TL;DR

This paper resolves pathologies in the local potential approximation (LPA) of scalar field theory caused by field-dependent cutoff operators in the background field formalism. By introducing a modified shift Ward identity (sWI), it restores universality and recovers the correct fixed point structure and critical exponents—matching results from field-independent cutoffs—demonstrating that physical predictions are robust when the sWI is consistently applied alongside the flow equation.

ABSTRACT

Working within the familiar local potential approximation, and concentrating on the example of a single scalar field in three dimensions, we show that the commonly used approximation method of identifying the total and background fields, leads to pathologies in the resulting fixed point structure and the associated spaces of eigenoperators. We then show how a consistent treatment of the background field through the corresponding modified shift Ward identity, can cure these pathologies, restoring universality of physical quantities with respect to the choice of dependence on the background field, even within the local potential approximation. Along the way we point out similarities to what has been previously found in the f(R) approximation in asymptotic safety for gravity.

Motivation & Objective

• To address severe pathologies in the local potential approximation (LPA) when using field-dependent cutoff operators in the background field formalism.
• To restore universality of physical results—such as fixed point structure and critical exponents—despite arbitrary background field dependence in the cutoff operator.
• To demonstrate that the modified shift Ward identity (sWI) is essential for consistent and reliable results in non-perturbative renormalisation group flows with field-dependent regulators.
• To provide a framework applicable to quantum gravity's asymptotic safety scenario, where similar cutoff dependence issues arise in the f(R) truncation.
• To show that the sWI complements the flow equation and enables exact equivalence to standard results even with complex cutoff dependencies.

Proposed method

• Introduces a field-dependent cutoff operator R_k(-∇², φ̄) in the background field formalism, where the cutoff depends on the background field φ̄.
• Applies the standard functional renormalisation group flow equation for the effective average action, truncated to the local potential approximation (LPA).
• Derives and enforces a modified shift Ward identity (sWI) that accounts for the field dependence of the cutoff, ensuring consistency with the underlying symmetries.
• Uses the sWI as a constraint alongside the flow equation to uniquely determine the effective average action, preventing unphysical solutions.
• Analyzes the fixed point structure and eigenoperator spectrum under various field-dependent cutoffs, comparing results with and without the sWI.
• Demonstrates that with the sWI, the flow remains equivalent to the standard field-independent cutoff case, even when the cutoff depends on the potential or background field.

Experimental results

Research questions

• RQ1 How does the inclusion of field-dependent cutoff operators in the background field formalism affect the fixed point structure and critical exponents in the LPA?
• RQ2 Why do standard LPA flows with field-dependent cutoffs lead to unphysical results such as missing vacua or collapsed eigenspaces?
• RQ3 Can a modified shift Ward identity restore universality and consistency in the presence of field-dependent regulators?
• RQ4 To what extent is the physical content of the Wilson-Fisher and Gaussian fixed points preserved when the cutoff depends on the background field?
• RQ5 Can the sWI-based approach be generalized to more complex truncations, such as the f(R) approximation in quantum gravity?

Key findings

• The modified shift Ward identity (sWI) successfully restores the correct fixed point structure and critical exponents in the LPA, even when the cutoff operator depends explicitly on the background field.
• Without the sWI, field-dependent cutoffs lead to unphysical results: the Gaussian fixed point is lost, and the eigenspace collapses to a point for certain parameter values.
• The sWI ensures that the full physical content—including the universality of the flow—remains intact across a large class of field-dependent cutoffs.
• The system of the flow equation and the sWI leads to exact equivalence with the standard LPA flow using a field-independent cutoff, confirming universality.
• The method resolves the issue of redundant operators and missing vacua that arise when the cutoff depends on the field, particularly for α < 0 in the parameterized cutoff model.
• The results strongly suggest that the sWI is essential for reliable non-perturbative calculations in asymptotic safety scenarios, where similar cutoff dependencies appear in the f(R) truncation.