Induced Lorentz- and $\mathrm{CPT}$-violating Chern-Simons term in QED: Fock-Schwinger proper time method

Using the Fock-Schwinger proper time method, we calculate the induced Chern-Simons term arising from the Lorentz- and $\mathrm{CPT}$-violating sector of quantum electrodynamics with a $b_{\mu }\overline{\psi }{\gamma }^{\mu }{\gamma }_5\psi$ term. Our result to all orders in b coincides with a recent linear-in-$b$ calculation by M. Chaichian, W. F. Chen, and A. González Felipe, hep-th/0010129v2. The coincidence was pointed out by J.-M. Chung, Phys. Lett. B 461, 138 (1999) and M. Pérez-Victoria, Phys. Rev. Lett. 83, 2518 (1999), in the standard Feynman diagram calculation with the nonperturbative-in-$b$ propagator.