Radiatively induced CPT-odd Chern-Simons term in massless QED

Dynamical Lorentz symmetry breaking in a 4D massless four-fermion model

“…Following this approach, for the rationalization of the propagator iG b (p) = i( / p − / bγ 5 ) −1 , we use the one presented in Ref. [22], given by…”

“…whose nontrivial solution (b µ = 0) is b 2 = 3π 2 G , with G > 0 (G < 0) for timelike (spacelike) b µ . The above expression (22) can be integrated, yielding the potential…”

“…From recent works [22][23][24], it is known that in Lorentz-violating theory, thermal effects imply the suppression of temporal component of b µ , leading to the restoration of parity symmetry. In our case, we expect to arise a well-defined transition from a parity-breaking phase to a paritysymmetric one.…”

“…One can immediately see that the divergent part corresponds to that one arising in the standard QED, since it is independent of the Lorentz-breaking axial vector b µ . In the context of Weyl semimetals (similar to massless QED), the CFJ contribution was calculated also with use of the 't Hooft-Veltmann prescription in [22], with the coefficient turns out to be three times greater than in Eq. ( 42).…”

“…As obviously A µ = 0, because of the unique nontrivial solution (22), the above Lagrangian assumes the form…”