  The Maxwell equations imply that, under the background of non-zero
$\boldsymbol{B}$, varying $\theta$ term produces $\boldsymbol{E} \cdot
\boldsymbol{B}$. An interesting example is the Witten effect where a magnetic
monopole becomes a dyon which, however, should disappear in the exact massless
limit of the fermion. Underlying mechanism of this phenomenon has been
understood by Callan by the presence of an effective axion-like degree of
freedom around the monopole, which is roughly the phase of the fermions. The
configuration of this axion cancels the effect of the $\theta$ term. Now, the
chiral anomaly implies that non-vanishing $\boldsymbol{E} \cdot \boldsymbol{B}$
induces the chiral charge in the system. The question is whether the chiral
charge is generated in the massless limit when we take into account the
axion-like degree of freedom in the discussion. The discussion is relevant for
the mechanism of baryogenesis under the background of time-dependent $\theta$.
We solve the system of the massless QED with time dependent $\theta$ by
reducing it to the two-dimensional QED. We demonstrate the occurrence of chiral
charge generation in the background of static magnetic field for two cases: a
magnetic monopole and a uniform magnetic flux. For the monopole case, the
chiral charge comes out from the monopole while canceling the Witten effect.
For the case of the uniform flux, on the other hand, the effect of the
backreaction cannot be ignored, giving a more non-trivial time dependence. We
also discuss their implications on baryogenesis.
