Sphalerons, baryogenesis, and helical magnetogenesis in the electroweak transition of the minimal standard model

We start by considering the production rates of sphalerons with different sizes ρ in the symmetric phase, T>TEW. At small ρ, the distribution is cut off by the growing mass M∼1/ρ, and at large ρ by the magnetic screening mass. In the broken phase, T<TEW, the scale is set by the Higgs vacuum expectation value v(T). We introduce the concept of "sphaleron freeze-out"whereby the sphaleron production rate matches the Hubble Universe expansion rate. At freeze-out the sphalerons are out of equilibrium. Sphaleron explosions generate sound and even gravity waves, when a nonzero Weinberg angle makes them nonspherical. We evaluate the magnitude of CP violation during the sphaleron explosions. We assess its magnitude using the Standard Model Cabbibo-Kobayashi-Maskawa (CKM) quark matrix, first for nonzero and then zero Dirac eigenstates. We find that its magnitude is maximal at the sphaleron freeze-out condition with T≈130 GeV. We proceed to estimate the amount of CP violation needed to generate the observed magnitude of baryon asymmetry of the Universe, and find that it is about an order of magnitude above our CKM-based estimates. We also relate the baryon asymmetry to the generation of U(1) magnetic chirality, which is expected to be conserved and perhaps visible in polarized intergalactic magnetic fields.