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.
Highlights
Part of those that enter in the matching relation between the Higgs quartic coupling and the pole Higgs mass [1]
In this paper we show that these IR divergences are spurious, we perform a simple resummation of all IR-problematic terms known and explain how to extend the resummation to cure all such divergences to any order
The perturbative calculation of the Standard Model effective potential in Landau gauge encounters problematic terms that lead to infrared divergences in the limit G → 0 [15], where G denotes the Goldstone mass squared
Summary
The SM effective potential V (φ) is calculated order by order in perturbation theory by summing 1PI vacuum diagrams with Feynman rules derived (using Landau gauge in what follows) in a classical background value φ for the Higgs field. This is problematic as finding the minimum of the potential requires solving the minimization equation V (φ) = 0, which should be well defined for any φ value This problem is often ignored, since higher loop contributions induce a small mass to the Goldstone bosons. Less troublesome are contributions of order G2LmG in V (φ), which are IR safe but cause a divergence in V ≡ d2V /dφ, as V would contain terms of order (G )2LmG , divergent for G → 0 This is potentially harmful for calculations of the Higgs mass, which is related precisely to V evaluated at the minimum of the potential. The possible impact of such IR divergences at large field values (those relevant for stability considerations) is indirect: they could affect the connection between potential parameters like the Higgs quartic coupling and observables like the Higgs pole mass
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