With the physical Higgs mass the Standard Model symmetry restoration phase transition is a smooth cross-over. We study the thermodynamics of the cross-over using numerical lattice Monte Carlo simulations of an effective SU(2) X U(1) gauge + Higgs theory, significantly improving on previously published results. We measure the Higgs field expectation value, thermodynamic quantities like pressure, energy density, speed of sound and heat capacity, and screening masses associated with the Higgs and Z fields. While the cross-over is smooth, it is very well defined with a width of only approximately 5 GeV. We measure the cross-over temperature from the maximum of the susceptibility of the Higgs condensate, with the result $T_c = 159.5 pm 1.5$ GeV. Outside of the narrow cross-over region the perturbative results agree well with non-perturbative ones.
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