  In this work, we make a study of the $\tau^- \to \omega\pi^-\nu_\tau$ decay
in the framework of low-energy effective field theory. The $J^{\mathcal{P}G}$
decompositions of the quark currents and the $\omega\pi$ final state show that,
besides the Standard Model vector interaction, only the non-standard tensor
interaction can have a non-zero contribution to the decay. To discuss its
effect, a reliable calculation of the $\omega\pi$ tensor form factors is
necessary. After constructing the Lagrangian of resonance chiral theory with
external tensor sources, we calculate both the vector and tensor form factors
with the relevant resonance couplings determined by combining the QCD
short-distance constraints, the fit to the spectral function of $\tau^- \to
\omega\pi^-\nu_\tau$ decay, as well as the matching between the
$\mathcal{O}(p^4)$ odd-intrinsic-parity operators after integrating out the
vector resonances and the $\mathcal{O}(p^6)$ operators of chiral perturbation
theory. The new physics effect is then investigated in the distributions of the
spectral function and the forward-backward asymmetry of $\tau^- \to
\omega\pi^-\nu_\tau$ decay. We find that the spectral function is dominated by
the Standard Model, and the non-standard tensor contribution is negligible.
However, since the forward-backward asymmetry can be only generated with a
non-zero tensor interaction, the observable is quite sensitive to this kind of
new physics. A future measurement of the observable at the Belle II experiment
as well as at the proposed Tera-Z and STCF facilities is, therefore, strongly
called for to check the existence of such a non-standard tensor interaction.
