We use large-scale lattice simulations to compute the rate of baryon number violating processes (the sphaleron rate), the Higgs field expectation value, and the critical temperature in the Standard Model across the electroweak phase transition temperature. While there is no true phase transition between the high-temperature symmetric phase and the low-temperature broken phase, the crossover is sharply defined at Tc = (159 ± 1) GeV. The sphaleron rate in the symmetric phase (T > Tc) is Γ/T 4 = (18 ± 3)α 5 W , and in the broken phase in the physically interesting temperature range 130 GeV < T < Tc it can be parametrized as log(Γ/T 4 ) = (0.83 ± 0.01)T /GeV − (147.7 ± 1.9). The freeze-out temperature in the early Universe, where the Hubble rate wins over the baryon number violation rate, is T * = (131.7 ± 2.3) GeV. These values, beyond being intrinsic properties of the Standard Model, are relevant for e.g. low-scale leptogenesis scenarios. Introduction:The current results from the LHC are in complete agreement with the Standard Model of particle physics: a Higgs boson with the mass of 125 -126 GeV has been discovered [1], and no evidence of exotic physics has been observed. If the Standard Model is indeed the complete description of the physics at the electroweak scale, the electroweak symmetry breaking transition in the early Universe was a smooth cross-over from the symmetric phase at T > T c , where the (expectation value of the) Higgs field was approximately zero, to the broken phase at T < T c GeV where it is finite, reaching the experimentally determined value |φ| ≃ 246/ √ 2 GeV at zero temperature. The cross-over temperature T c is somewhat larger than the Higgs mass. The nature of the transition was settled already in 1995-98 using lattice simulations [2][3][4][5], which indicate a first-order phase transition at Higgs masses < ∼ 72 GeV, and a cross-over otherwise.A smooth cross-over means that the standard electroweak baryogenesis scenarios [6,7] are ineffective. These scenarios produce the matter-antimatter asymmetry of the Universe through electroweak physics only, and they require a strong first-order phase transition, with supercooling and associated out-of-equilibrium dynamics. Thus, the origin of the baryon asymmetry must rely on physics beyond the Standard Model.Baryogenesis at the electroweak scale is possible in the first place through the existence of the chiral anomaly relating the baryon number of fermions to the topological Chern-Simons number N cs of the electroweak SU(2) gauge fields

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)×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 ∼ 5 GeV. We measure the cross-over temperature from the maximum of the susceptibility of the Higgs condensate, with the result Tc = 159.5 ± 1.5 GeV. Outside of the narrow cross-over region the perturbative results agree well with non-perturbative ones.