Effective Potential in Subleading Logarithmic Approximation in Arbitrary Non-renormalizable Scalar Field Theory

TL;DR

The paper develops a universal RG framework to sum leading and subleading logarithms in the effective potential for arbitrary non-renormalizable scalar theories, using the Bogoliubov-Parasiuk-Häpp-Zimmermann formalism and the locality requirement. It derives recurrence relations and differential equations for LL and NLLA contributions, introduces the $ R$-rule to map diagrams to their corresponding counterterm structures, and demonstrates that the LL results are scheme-independent while NLLA introduces scheme dependence that can be characterized. Verification against a renormalizable ($p=4$) case shows agreement with known beta-function coefficients and Landau poles, reinforcing the method’s consistency with standard RG results. The work provides a path to all-order logarithmic resummation in non-renormalizable scalar theories and highlights the central role of locality in constraining counterterms and RG flows, while acknowledging unresolved issues related to infinite arbitrariness in the UV completion of such theories.

Abstract

Following the previously developed approach to the calculation of quantum corrections to the effective potential in arbitrary scalar field theories in the leading logarithmic approximation, we extended it to the next-to-leading order. Based on Bogoliubov-Parasiuk-Hepp-Zimmerman renormalization procedure and the Bogoliubov-Parasiuk theorem, we construct recurrence relations and renormalization group equations that allow one to sum up the leading and subleading logarithms in all orders of perturbation theory. The formalism is applicable to an arbitrary scalar potential, renormalizable or not. To verify the results, we compare them with a renormalizable model treated within the standard renormalization group approach.

Figures (7)

• Figure 1: The one-, two- and three-loop vacuum graphs contributing to the effective potential
• Figure 2: Recurrence relation of the effective potential in the leading order. Grey blobs denote the ${\cal KR}'$ part of the subgraphs shrunken to a point, namely $V_k^{'A}$.
• Figure 3: The linear and nonlinear parts of the recurrence relation can be unified in the structure that reminds the one-loop diagram
• Figure 4: The one- and two-loop diagrams contributing to the recurrence relation for the next-to-leading divergences according to the $\mathcal{R}$-rule. $V_k^G$ corresponds to the part of the effective action with two derivatives (they are denoted as notches on the lines).
• Figure 5: Part of the recurrence relation contributing to the subleading order containing the one-loop survived diagram.