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The low energy quasiparticle dispersion of various narrow gap and gaplesssemiconductors are respectively described by three dimensional massive andmassless Dirac fermions. The three dimensional Dirac spinor structure admits atime-reversal invariant, odd parity and Lorentz pseudoscalar topologicalsuperconducting state. Here we derive the effective field theory of thistopological paired state for massless Dirac fermions in the presence of afluctuating Zeeman term, which appears as a chiral gauge field. The effectivetheory 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 wedemonstrate the existence of fermion zero modes in the dilute vortex phaseunder generic conditions. Guided by the existence of the zero modes and itsintimate connection with the anomaly, we propose an effective topological fieldtheory in the presence of Dirac mass. We briefly discuss the experimentalconsequences of the effective field theory and the zero modes for the lowtemperature unconventional superconducting states of$\mathrm{Cu}_x\mathrm{Bi}_2\mathrm{Se}_3$ and$\mathrm{Sn}_{1-x}\mathrm{In}_x\mathrm{Te}$.

Effective field theory, chiral anomaly and vortex zero modes for odd parity topological superconducting state of three dimensional Dirac materials