Benchmarking regulator-sourced 2PI and average 1PI flow equations in zero dimensions

Peter Millington,Paul M Saffin

doi:10.1088/1751-8121/ac2d00

Peter Millington, Paul M Saffin

Open Access

https://doi.org/10.1088/1751-8121/ac2d00

Copy DOI

Abstract

We elucidate the regulator-sourced 2PI and average 1PI approaches for deriving exact flow equations in the case of a zero dimensional quantum field theory, wherein the scale dependence of the usual renormalisation group evolution is replaced by a simple parametric dependence. We show that both approaches are self-consistent, while highlighting key differences in their behaviour and the structure of the would-be loop expansion.

Highlights

Quantum effective actions provide a framework within which to study the quantum dynamics of field-theoretic systems, both perturbatively and non-perturbatively, and, for instance, out of thermodynamic equilibrium
We elucidate the regulator-sourced 2PI and average 1PI approaches for deriving exact flow equations in the case of a zero dimensional quantum field theory, wherein the scale dependence of the usual renormalisation group evolution is replaced by a simple parametric dependence
We show that both approaches are self-consistent, while highlighting key differences in their behaviour and the structure of the would-be loop expansion

Summary

Introduction

Quantum effective actions provide a framework within which to study the quantum dynamics of field-theoretic systems, both perturbatively and non-perturbatively, and, for instance, out of thermodynamic equilibrium. Note that the expression for Δ−1 in equation (8c) contains would-be loop corrections built self-consistently from Δ While it has been truncated at second order λ2, the solution for Δ obtained from equation (8c) resums an infinite series of loop insertions to the two-point function. This is the power of the 2PI approach. Note that by restricting the source K, we can constrain the two-point function and thereby the effective action (see references [27, 28]), as we will do later in order to obtain analogues of the exact flow equations of the functional renormalisation group (as was done in references [8, 9]). The infinite shift is the zero-dimensional analogue of the vacuum energy, which diverges logarithmically in zero spacetime dimensions

Inverse ‘two-point function’ from convexity

First order in λ

Second order in λ

Average 1PI

Concluding remarks

Data availability statement