On the universal content of the proper time flow in scalar and Yang-Mills theories

TL;DR

The paper addresses whether the perturbative content of the Proper Time (PT) renormalization group flow can reproduce universal renormalization properties, focusing on the ${O}(N)$ scalar theory and ${\rm SU}(N)$ Yang-Mills theory. Using the PT flow, the authors obtain the one- and two-loop $\beta$-function coefficients by treating the running action as a Wilsonian local action and expanding in the loop order, with explicit results $\beta^{1L}_{\lambda_R}= \frac{\lambda^2}{(16\pi^2)} \frac{N+8}{3}$, $\beta^{2L}_{\lambda_R}= -\frac{\lambda^3}{(16\pi^2)^2} \frac{3N+14}{3}$ for the scalar case, and $\beta_1= \frac{11}{3}$, $\beta_2= \frac{34}{3}$ for Yang-Mills. The YM analysis employs the background-field method, extracting $Z_A$ and showing that the PT flow does not generate gauge-violating $A^2$ terms, while three contributions to $Z_A$ at two loops sum to reproduce $\beta_2$. Overall, the work demonstrates that the PT flow preserves the universal content of renormalization and remains reliable for extracting perturbative coefficients, despite not capturing the full effective action. This supports the PT framework as a useful, gauge-consistent tool in diverse contexts, including gravity-inspired settings, while acknowledging its limitations in reconstructing the complete diagrammatic structure.

Abstract

We investigate the perturbative structure of the proper time renormalization group flow in scalar and Yang-Mills theories. Although the PT flow does not belong to the class of exact functional renormalization group equations, we show that it correctly reproduces the universal coefficients of the $β$-functions at one and two loops. For the ${\rm O(N)}$ scalar theory, we derive the one- and two-loop contributions to the running quartic coupling and also confirm the expected anomalous dimension. For the ${\rm SU(N)}$ Yang-Mills theory, using the background field method, we compute the gauge coupling renormalization recovering the correct two-loop $β$-function without generating any gauge-symmetry-violating term. These results highlight that, despite its limitations for reconstructing the full effective action, the PT flow retains the essential universal content of renormalization, accounting for its reliability in diverse applications ranging from statistical models to quantum gravity.

Figures (1)

• Figure 1: Two-loop diagrams contributing to the effective action. Diagram (a) is order $\lambda$ and diagram (b) is order $\lambda^2$.