  In this paper, we consider the Cauchy problem of local well-posedness of the
Chern-Simons-Dirac system in the Lorenz gauge for $B^{\frac14}_{2,1}$ initial
data. We improve the low regularity well-posedness, compared to Huh-Oh
\cite{huhoh} and Okamoto \cite{oka}, by using the localization of space-time
Fourier side and bilinear estimates given by Selberg \cite{selb}, whereas the
authors of \cite{huhoh, oka} used global estimates of \cite{danfoselb}. Then we
show the Dirac spinor flow of Chern-Simons-Dirac system is not $C^2$ at the
origin in $H^s$ if $ s < \frac14$. From this point of view, the space
$B_{2,1}^\frac14$ can be regarded as a critical space for the local
well-posedness. We apply the argument for failure of smoothness to the Dirac
equation decoupled from Chern-Simons-Dirac system and show the flow is not
$C^3$ in $H^s, s < 0$.
