Taming Infrared Divergences in the Effective Potential

TL;DR

The paper identifies infrared divergences in the Standard Model Higgs effective potential arising from Goldstone loops as spurious artifacts. It develops a practical resummation by shifting the Goldstone mass to $\tilde G=G+\kappa\Pi_g$ and, when needed, further reorganizes around this shifted mass to control subleading divergences; it also offers a Wilsonian (method of regions) and a 2PI effective-action formulation to achieve resummation to arbitrary order. The authors extend the discussion to Higgs-related IR issues, analyze potential non-decoupling logarithms from broken scale invariance, and argue that, after proper matching and renormalization, the low-energy theory remains SM-like. The work provides a general, gauge-compatible framework for IR-safe effective potentials applicable beyond the SM, with practical recipes for high-precision vacuum stability studies.

Abstract

The Higgs effective potential in the Standard Model (SM), calculated perturbatively, generically suffers from infrared (IR) divergences when the (field-dependent) tree-level mass of the Goldstone bosons goes to zero. Such divergences can affect both the potential and its first derivative and become worse with increasing loop order. In this paper we show that these IR divergences are spurious, we perform a simple resummation of all IR-problematic terms known (up to three loops) and explain how to extend the resummation to cure all such divergences to any order. The method is of general applicability and would work in scenarios other than the SM. Our discussion has some bearing on a scenario recently proposed as a mechanism for gauge mediation of scale breaking in the ultraviolet, in which it is claimed that the low-energy Higgs potential is non-standard. We argue that all non-decoupling effects from the heavy sector can be absorbed in the renormalization of low-energy parameters leading to a SM-like effective theory.

Figures (5)

• Figure 1: Generic multiloop diagram with a Goldstone ring, dressed by Goldstone self-energies, that leads to IR problematic contributions to the Higgs potential.
• Figure 2: Two-loop diagram with two Goldstones and a $Z$ that naively would lead to the wrong combinatorics for resummation of IR divergences.
• Figure 3: $L$-loop diagram with a neutral Goldstone ring adorned by $(L-1)$-iterated top-quark loops.
• Figure 4: Diagrammatic representation of the decomposition of the two-loop momentum integral $I(G,G,H)$ from (\ref{['IGGH']}) (left diagram) in contributions with two soft Goldstones (first diagram on the right), one soft Goldstone (second diagram on the right) plus contributions with hard Goldstones only (ellipsis).
• Figure 5: Resummed Higgs contributions to the effective potential at one and two loops to cure $H\rightarrow 0$ IR problems. The square in the Higgs propagator indicates a corrected Higgs propagator, as explained in the text.