temperature of electroweak phase transition

We calculate the free energy (density) for the Higgs field $\phi$ at finite temperature. In the Standard Model, this looks like

$\mathcal{F}_{SM}(\phi,T) = -\frac{\pi^2}{90}g_* T^4+V_{SM}(\phi, T) \ ,$

where $g_*$ is the number of degrees of freedom in the SM ($g_*=106.75$).

The potential has the form

$V_{SM}(\phi,T) = D(T^2-T_0^2)\phi^2 - ET\phi^3+\frac{\lambda_T}{4}\phi^4\ ,$

with $D,E,T_0^2,\lambda_T$ some factors depending on particle masses, coupling constants, the Higgs v.e.v. and temperature.

At the phase transition (PT), there are two degenerate minima of the potential. One sits at $\phi=0$, where we are in the symmetric phase, the other is at $\phi=\phi_0$, where we are in the broken phase. If my quick calculation is correct, this leads to a critical temperature $T_c\approx 163 GeV$.

Note that in this case, the order parameter of the PT $\phi_c/T_c$ is very small. This means for one thing, that the PT is only very weakly first order and else, that perturbation theory is no longer reliable and we need to do non-perturbative calculations.

(Though the procedure is standard, I took this paper from Carena, Megevand, Quirós and Wagner as reference, just because it was the closest at hand, not because I particularly like it, which I don't btw.)

Let's define $T_{EW}$ the temperature where the coefficient $m^2_H(T)$ of the operator $H^2$ in the SM lagrangian vanishes: $$ m_H^2(T=T_{EW})=0\,. $$ For $T>T_{EW}$ the Higgs vev is vanishing, the EW symmetry in unbroken, and the elementary particles are massless. For $T<T_{EW}$ the the vev is non-vanishing, $v_T\propto -m_H^2/\lambda\neq 0$, the EW is broken spontaneously, and the various elementary particles acquire masses $m_i$.

Let's now determine $T_{EW}$ quantitatively. Since $m_i\simeq 0$ for $T\simeq T_{EW}$ is OK to make an expansion in small $m/T$. In this regime the 1-loop corrections to the Higgs potential from the thermal propagators are, expanded at leading order in $m/T$, given by \begin{equation} m^2(\tilde{T}_{EW})=m^2_{T=0}+\tilde{T}_{EW}^2\left[\frac{y_t^2}{4}+\frac{\lambda}{2}+\frac{g^{\prime\,2}}{16}+\frac{3g_2^2}{16}\right] \end{equation} where $m^2_{T=0}=m_H(T=0)=-\lambda v_{T=0}^2$, with $v_{T=0}=246$ GeV and $2\lambda v_{T=0}=m_h^2=(125\mathrm{GeV})^2$. The negative mass-squared receive positive thermal mass contributions form the loops of the particles the Higgs couples to. The leading contribution to $m_H^2(T)$ is, non-sorprisingly, coming from the largest couploing, top quark yukawa $y_t=\sqrt{2}m_t/v_{T=0}$. Neglecting the gauge couplings and the higg-self coupling, and solving $m^2(\tilde{T}_{EW})=0$ for $T_{EW}$ we get \begin{equation} \tilde{T}^2_{EW}\simeq m_h^2 v^2/m_t^2\simeq (178\mathrm{GeV})^2\,. \end{equation}