We study the polarization tensor of a Dirac field in ($3+1$) dimensions confined to a half-space—a problem motivated by applications to the condensed matter physics, and to topological insulators in particular. Although the Pauli-Villars regularization scheme has a number of advantages, like explicit gauge invariance and decoupling of heavy modes, it is not applicable on manifolds with boundaries. Here, we modify this scheme by giving an axial mass to the regulators and to the physical field. We compute the renormalized polarization tensor in coordinate representation. We discuss then the induced Chern-Simons type action on the boundary and compare it to the effective action of a ($2+1$)-dimensional Dirac fermion.

• Received 2 July 2019
• Corrected 4 February 2021

DOI:https://doi.org/10.1103/PhysRevD.100.045026

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Published by the American Physical Society