Using the method of renormalization group, we improve the two-loop effective potential of the massive φ 4 theory to obtain the next-next-to-leading logarithm correction in the MS scheme. Our result well reproduces the nextnext-to-leading logarithm parts of the ordinary loop expansion result known up to the four-loop order.

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Using the Fock-Schwinger proper time method, we calculate the induced Chern-Simons term arising from the Lorentz-and CPT-violating sector of quantum electrodynamics with a b µψ γ µ γ 5 ψ term. Our result to all orders in b coincides with a recent linear-in-b calculation by Chaichian et al. [hepth/0010129 v2]. The coincidence was pointed out by Chung [Phys. Lett. B461 (1999) 138] and Pérez-Victoria [Phys. Rev. Lett. 83 (1999) 2518] in the standard Feynman diagram calculation with the nonperturbative-in-b propagator.