Melting of the Higgs Vacuum: Conserved Numbers at High Temperature

Abstract

We discuss the computation of the grand canonical partition sum describing hot matter in systems with the Higgs mechanism in the presence of non-zero conserved global charges. We formulate a set of simple rules for that computation in the high-temperature approximation in the limit of small chemical potentials. As an illustration of the use of these rules, we calculate the leading term in the free energy of the standard model as a function of baryon number B. We show that this quantity depends continuously on the Higgs expectation value $φ$, with a crossover at $φ\sim T$ where Debye screening overtakes the Higgs mechanism---the Higgs vacuum ``melts". A number of confusions that exist in the literature regarding the B dependence of the free energy is clarified.

Figures (3)

• Figure 1: Tadpole diagram producing terms linear in chemical potential.
• Figure 2: Dependence of the equilibrium value of the baryon number (in units of $B-L$) on the expectation value of the Higgs field (in units of temperature) for $n_f=3,~n_s=1$.
• Figure 3: Dependence of the coefficient $\kappa$ on the expectation value of the Higgs field (in units of temperature) for $n_f=3,~n_s=1$.