Temperature Dependence of Standard Model $CP$ Violation

We analyze the temperature dependence of $CP$ violation effects in the standard model by determining the effective action of its bosonic fields, obtained after integrating out the fermions from the theory and performing a covariant gradient expansion. We find nonvanishing $CP$ violating terms starting at the sixth order of the expansion, albeit only in the $C$-odd–$P$-even sector, with coefficients that depend on quark masses, Cabibbo-Kobayashi-Maskawa matrix elements, temperature and the magnitude of the Higgs field. The $CP$ violating effects are observed to decrease rapidly with temperature, which has important implications for the generation of a matter-antimatter asymmetry in the early Universe. Our results suggest that the cold electroweak baryogenesis scenario may be viable within the standard model, provided the electroweak transition temperature is at most of order 1 GeV.

Figure 1

The coefficients $c_1–c_{13}$ plotted as functions of the effective temperature $T_{\mathrm{eff}}\equiv vT/\varphi$. The bold lines correspond to the smallest and largest of the $c_i$ that approach one at zero temperature, i.e., $c_1$ and $c_{10}$.

Figure 2

The dependence of the $c_i$ on the Higgs field $\varphi$, plotted for three different temperatures, $T=1/3$, 1, and 3 GeV. In each case, the gray band is spanned by $c_1$ and $c_{10}$.