New types of instability and CP violation in electroweak theory

It is known that the Schwinger mechanism of vector-like QED theory is afflicted by a logarithmic singularity under background electromagnetic field due to a hypothetical massless charged fermion. We extend singularity analysis to a more realistic case of the chiral electroweak theory, to show that the effective lagrangian under background gauge field at zero temperature exhibits a similar instability proportional to $\ln (1/m_{\nu}^2)$ with $m_{\nu}$ a small neutrino mass. Moreover, the effective lagrangian of chiral fermion loop contains CP violating pieces proportional to background gauge fields in odd powers of $\vec{E}_Z\cdot\vec{B}_Z$ or $(\vec{E}_{W^+}\cdot\vec{B}_{W^-}+ \vec{E}_{W^-}\cdot\vec{B}_{W^+})/2$. This brings in a new source of CP violation and time-reversal symmetry violation in the standard particle theory independent of the Kobayashi-Maskawa phase of quark mass mixing matrix. The effective action in thermal equilibrium at finite temperature $T$ is then calculated under background SU(2)$\times$U(1) gauge fields in the spontaneously broken phase. An even more singular power-law behavior $\propto (m_{\nu} T)^{-5/2}$ is found and it contains CP violating term as well. The case of Majorana neutrino satisfies almost all necessary conditions to generate a large lepton number asymmetry, though not necessarily convertible to a baryon asymmetry due to lower cosmic temperatures at which this may occur.