On the relationship between gauge dependence and IR divergences in the $\hbar$-expansion of the effective potential

Perturbative calculations of the effective potential evaluated at a broken minimum, $ V_{\text{min}} $, are plagued by difficulties. It is hard to get a finite and gauge invariant result for $ V_{\text{min}} $. In fact, the methods proposed to deal with gauge dependence and IR divergences are orthogonal in their approaches. Gauge dependence is dealt with through the $ \hbar $-expansion, which establishes and maintains a strict loop-order separation of terms. On the other hand, IR divergences seem to require a resummation that mixes the different loop orders. In this paper we test these methods on Fermi gauge Abelian Higgs at two loops. We find that the resummation procedure is not capable of removing all divergences. Surprisingly, the $ \hbar $-expansion seems to be able to deal with both the divergences and the gauge dependence. In order to isolate the physical part of $ V_{\text{min}} $, we are guided by the separation of scales that motivated the resummation procedure; the key result is that only hard momentum modes contribute to $ V_{\text{min}} $.