Taming the Goldstone contributions to the effective potential

TL;DR

The paper tackles two principal problems in the perturbative effective potential: spurious imaginary parts arising when the Goldstone mass squared $G$ is negative, and divergences as $G \to 0$ at higher loops. It resolves these by resumming the leading Goldstone contributions, yielding a resummed potential $\widehat{V}_{\rm eff}$ whose minimum and minimization condition do not involve $G$ explicitly; at the minimum, $G+\Delta=0$ so the resummed terms vanish. The resummed form, $\widehat{V}_{\rm eff} = V_{\rm eff} + \frac{3}{16\pi^2}[f(G+\Delta) - \sum_{n=0}^{\ell-1} \frac{\Delta^n}{n!} \left(\frac{d}{dG}\right)^n f(G)] - \frac{1}{(16\pi^2)^2}\frac{3\Delta_1}{2} A(G) - \frac{1}{(16\pi^2)^3} 27 y_t^4 A(t)^2 \overline{\ln}(G)$, removes the problematic terms and stabilizes the minimization. The resulting $G$-independent minimization conditions are shown to be RG-consistent and practically stable, with only small numerical shifts in typical scale choices and a clear elimination of instability near $G=0$, making this approach valuable for precise SM Higgs mass calculations and adaptable to other symmetry-breaking theories.

Abstract

The standard perturbative effective potential suffers from two related problems of principle involving the field-dependent Goldstone boson squared mass, G. First, in general G can be negative, and it actually is negative in the Standard Model; this leads to imaginary contributions to the effective potential that are not associated with a physical instability, and therefore spurious. Second, in the limit that G approaches zero, the effective potential minimization condition is logarithmically divergent already at two-loop order, and has increasingly severe power-law singularities at higher loop orders. I resolve both issues by resumming the Goldstone boson contributions to the effective potential. For the resulting resummed effective potential, the minimum value and the minimization condition that gives the vacuum expectation value are obtained in forms that do not involve G at all.

Figures (4)

• Figure 1.1: The running of the Landau gauge Standard Model Goldstone boson squared mass $G$, evaluated at the minimum of the effective potential, as a function of the renormalization scale $Q$, for the choice of input parameters specified in the text.
• Figure 2.1: These Feynman diagrams give the leading non-zero contribution to $V_{\rm eff}$ as $G \rightarrow 0$, at the leading order in $y_t$, for 1-loop, 2-loop, and 3-loop orders.
• Figure 2.2: Chains of Goldstone boson propagators interspersed with top and top/bottom loops. Rings of these (and similar diagrams involving loops with $Z$, $W$, and $H$) give rise to the most singular contributions as $G \rightarrow 0$, at any given loop order.
• Figure 5.1: Dependence of the VEV $v$ (left panel) and the Higgs Lagrangian mass parameter $\sqrt{-m^2}$ (right panel), as a function of the renormalization scale, as computed from the effective potential minimization condition at 2-loop order from ref. Ford:1992pn, at partial 3-loop order including also Martin:2013gka, and 3-loop order after resummation using eqs. (\ref{['eq:resummedmincon']})-(\ref{['eq:Deltahat3']}). The input parameters are specified in eqs. (\ref{['eq:inputlambda']})-(\ref{['eq:inputgp']}) at the input scale $Q=173.35$ GeV. In the left panel, the parameters including $m^2$ are run from the input scale to $Q$, and $v$ is solved for. In the right panel, the parameters (including $v=246.954$ GeV at $Q=173.35$ GeV) are run to $Q$, and $m^2$ is solved for. The 3-loop case without resummation has a numerical instability associated with a failure of the iterative solution process to converge, represented by the vertical line of arbitrary height, for a narrow range near $Q=100.4$ GeV, due to the $1/G$ term in the minimization condition.