  The low energy quasiparticle dispersion of various narrow gap and gapless
semiconductors are respectively described by three dimensional massive and
massless Dirac fermions. The three dimensional Dirac spinor structure admits a
time-reversal invariant, odd parity and Lorentz pseudoscalar topological
superconducting state. Here we derive the effective field theory of this
topological paired state for massless Dirac fermions in the presence of a
fluctuating Zeeman term, which appears as a chiral gauge field. The effective
theory consists of a mixed electromagnetic and chiral anomaly term in the bulk,
and a combination of pure and mixed anomalies for the surface. In this paper we
demonstrate the existence of fermion zero modes in the dilute vortex phase
under generic conditions. Guided by the existence of the zero modes and its
intimate connection with the anomaly, we propose an effective topological field
theory in the presence of Dirac mass. We briefly discuss the experimental
consequences of the effective field theory and the zero modes for the low
temperature unconventional superconducting states of
$\mathrm{Cu}_x\mathrm{Bi}_2\mathrm{Se}_3$ and
$\mathrm{Sn}_{1-x}\mathrm{In}_x\mathrm{Te}$.
