Exact RG invariance and symmetry improved 2PI effective potential

Apostolos Pilaftsis,Daniele Teresi

doi:10.1016/j.nuclphysb.2017.04.015

Apostolos Pilaftsis, Daniele Teresi

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https://doi.org/10.1016/j.nuclphysb.2017.04.015

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Abstract

The Symmetry Improved Two-Particle-Irreducible (SI2PI) formalism is a powerful tool to calculate the effective potential beyond perturbation theory, whereby infinite sets of selective loop-graph topologies can be resummed in a systematic and consistent manner. In this paper we study the Renormalization-Group (RG) properties of this formalism, by proving for the first time a number of new field-theoretic results. First, the RG runnings of all proper 2PI couplings are found to be UV finite, in the Hartree–Fock and sunset approximations of the 2PI effective action. Second, the SI2PI effective potential is exactly RG invariant, in contrast to what happens in the ordinary One-Particle-Irreducible (1PI) perturbation theory, where the effective potential is RG invariant only up to higher orders. Finally, we show how the effective potential of an O(2) theory evaluated in the SI2PI framework, appropriately RG improved, can reach a higher level of accuracy, even up to one order of magnitude, with respect to the corresponding one obtained in the 1PI formalism.

Highlights

The effective potential constitutes a fundamental tool in Quantum Field Theory, widely used to study a multitude of physical phenomena, such as spontaneous symmetry breaking, tunnelling rates due to vacuum instability, and thermal phase transitions
We have studied the RG properties of the Symmetry Improved Two-Particle-Irreducible effective action truncated to two-loop order, within the context of a simple O(2) scalar theory
Unlike the perturbative 1PI effective action, the loop-truncated 2PI effective action of a simple φ4-theory requires different renormalizations for all mass and coupling parameters, in order to obtain UV finite results for all possible operators consisting of the renormalized field φ and its dressed propagator ∆

Summary

Introduction

The effective potential constitutes a fundamental tool in Quantum Field Theory, widely used to study a multitude of physical phenomena, such as spontaneous symmetry breaking, tunnelling rates due to vacuum instability, and thermal phase transitions. This is a remarkable result, as it helps us to clarify the relation between the 2PI resummation and the frequently considered RG-improved approach to the 1PI effective potential. The former takes into account precisely the higher-order contributions needed to restore exact RG invariance when the effective potential is evaluated at a given loop level, whereas the latter attempts to minimize these higher-order terms by plausibly guessing the value of the RG scale μ, so as to avoid large logarithmic contributions.

Symmetry Improved 2PI Effective Potential

Finiteness of the RG Running

Finiteness of the 1PI RG Running

Exact RG Invariance

Renormalization Group and Symmetry Improved 2PI Effective Potential

Conclusions