[Paper Review] A new functional RG flow: regulator-sourced 2PI versus average 1PI

Elizabeth Alexander, Peter Millington+2 more|arXiv (Cornell University)|Aug 6, 2019

Advanced MRI Techniques and Applications24 references4 citations

TL;DR

This paper introduces a new functional renormalization group (RG) flow derived from a regulator-sourced two-particle irreducible (2PI) effective action, contrasting it with the standard average one-particle irreducible (1PI) approach. For the quartic scalar theory with spontaneous symmetry breaking, using the Litim regulator and derivative expansion, it finds that the 2PI flow leads to faster evolution of potential minima and cosmological constant, while the quartic coupling evolves more slowly—differences that may impact asymptotic safety and Higgs sector non-perturbative dynamics.

ABSTRACT

We derive the renormalization group evolution of the quartic scalar theory with spontaneous symmetry breaking from an alternative flow equation, obtained within the externally sourced two-particle irreducible framework due to Garbrecht and Millington. In order to make a straightforward comparison with the evolution from the standard Wetterich-Morris-Ellwanger equation, we employ the Litim regulator, work to lowest order in the derivative expansion and neglect anomalous scaling. By this means, we illustrate the leading differences between analytic expressions for the resulting threshold and (non-perturbative) beta functions. In four dimensions, we find that the positions of the potential minima and the cosmological constant evolve more rapidly with scale compared to the standard approach, whereas the quartic coupling evolves more slowly, albeit by a small amount. These differences may have implications for the asymptotic safety programme, as well as our understanding of the non-perturbative scale evolution of the Standard Model Higgs sector.

Motivation & Objective

• To investigate the implications of a newly derived functional RG flow based on the regulator-sourced 2PI formalism for scalar field theories with spontaneous symmetry breaking.
• To compare the new 2PI-based flow with the standard Wetterich-Morris-Ellwanger 1PI flow in terms of threshold functions and beta functions.
• To assess the impact of differing RG flow structures on the non-perturbative scale evolution of the Higgs sector and asymptotic safety scenarios.
• To establish a foundation for higher-order derivative expansion and anomalous scaling studies in the 2PI framework.

Proposed method

• Derive the regulator-sourced 2PI effective action via a double Legendre transform involving the two-point source and regulator, distinct from the standard 1PI approach.
• Employ the Litim regulator to ensure a sharp momentum cutoff and simplify threshold function analysis.
• Apply the derivative expansion to the 2PI effective action, retaining only the lowest-order terms in the kinetic and potential energy functionals.
• Use the ansatz $ U_k(\rho) = \frac{1}{2}g_k(\rho - \bar{\rho}_k)^2 + \Lambda_k $ to model the scale-dependent potential with constant parameters.
• Introduce dimensionless couplings $ \kappa_k $, $ \lambda_k $, and $ \Lambda_k/k^d $ to analyze RG evolution in $ d=2,3,4 $ spacetime dimensions.
• Compute the flow equations for $ \partial_t \Lambda_k $, $ \partial_t \kappa_k $, and $ \partial_t \lambda_k $ using supertraces and threshold functions derived from the regulator-sourced 2PI formalism.

Experimental results

Research questions

• RQ1 How do the beta functions for the cosmological constant and potential parameters differ between the regulator-sourced 2PI and standard 1PI RG flows?
• RQ2 What is the quantitative impact of the 2PI flow on the scale evolution of the Higgs potential minimum and vacuum energy?
• RQ3 How does the quartic coupling $ \lambda_k $ evolve under the 2PI flow compared to the 1PI flow, and what are the implications for asymptotic safety?
• RQ4 To what extent do the differences in threshold functions between the two flows lead to divergent non-perturbative behavior in four dimensions?

Key findings

• In four dimensions, the cosmological constant $ \Lambda_k $ and the potential minimum $ \bar{\rho}_k $ evolve more rapidly under the regulator-sourced 2PI flow than under the standard 1PI flow.
• The quartic coupling $ \lambda_k $ evolves more slowly in the 2PI framework, though the difference is small, with the 2PI beta function being reduced by a factor of approximately 1/6 compared to the 1PI case.
• The threshold functions in the 2PI flow exhibit different functional forms—specifically, $ \ell_1^d $, $ \delta\ell_1^d $, $ \ell_2^d $, and $ \delta\ell_2^d $—leading to distinct scaling behavior in the beta functions.
• For $ d=4 $, the 2PI flow yields $ \partial_t \lambda_k = (d-4)\lambda_k + \frac{432v_d}{d(d+2)} \frac{\lambda_k^2}{(1+2\kappa_k\lambda_k)^4} $, while the 1PI flow gives $ \partial_t \lambda_k = (d-4)\lambda_k + \frac{72v_d}{d} \frac{\lambda_k^2}{(1+2\kappa_k\lambda_k)^3} $, showing a significant difference in the denominator structure.
• The 2PI flow leads to faster evolution of $ \kappa_k $ in $ d=3 $ and $ d=4 $, as confirmed by numerical plots of the RG trajectories, indicating stronger scale dependence of the vacuum expectation value.
• Despite small quantitative differences in the quartic coupling, the qualitative behavior of the RG flows remains similar across $ d=2,3,4 $, suggesting that the 2PI approach may yield more significant deviations in non-perturbative regimes.