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

Abstract

We start by considering the production rates of sphalerons with different size $\rho$ in the symmetric phase, $T>T_{EW}$. At small $\rho$, the distribution is cut off by the growing mass $M\sim 1/\rho$, and at large $\rho$ by the magnetic screening mass. In the broken phase, $T<T_{EW}$ the scale is set by the Higgs VEV $v(T)$. We introduce the concept of "Sphaleron freezeout" whereby the sphaleron production rate matches the Hubble Universe expansion rate. At freezeout the sphalerons are out of equilibrium. Sphaleron explosions generate sound and even gravity waves, when nonzero Weinberg angle make them non-spherical. We revisit CP violation during the sphaleron explosions. We assess its magnitude using the Standard Model CKM quark matrix, first for nonzero and then zero Dirac eigenstates. We find that its magnitude is maximal at the sphaleron freezeout condition with $T\approx 130\, GeV$. We proceed to estimate the amount of CP violation needed to generate the observed magnitude of baryon asymmetry of Universe. The result is about an order of magnitude below 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.