  Among so-called three 3-sigma anomalies in high energy physics, the excess of
the branching ratio $W^- \to \tau \bar \nu_\tau$ with respect to the electrons
and muons is especially interesting because (1) in the standard model (SM),
$W^-\ell\bar \nu_\ell$ is the pure left-handed charge-current which has been
tested precisely already, at least for the first two generation fermions, and
(2) the $W^\pm$ two-body leptonic decay is the cleanest one among three
anomalies due to its simpler kinematics and less hadronic uncertainties. In
this paper, we explore the possibilities to account for the anomaly in type II
two-Higgs-doublet model (2HDM) and minimal supersymmetric model (MSSM), as well
as effective lagrangian approach by introducing anomalous left- and
right-handed $W^-\tau \bar\nu_\tau$ couplings. Our results show that 2HDM and
MSSM can hardly accommodate $W^- \to \tau \nu_\tau$ anomaly, and the anomaly is
only marginally consistent to the measurements of $\tau \to \nu_\tau \ell \bar
\nu_\ell$ at 95% confidence level with the presence of anomalous couplings. In
the allowed regions, the right-handed coupling of $W^-\tau \nu_\tau$ shifts
from 0 in SM to $\sim 0.12$ while the left-handed one from 1 to $\sim 1.005$.
