Induced Chern-Simons-like action in Lorentz-violating massless QED

In the present work, we study different aspects of Lorentz and CPT symmetry violation in extended massless QED. By following the observation that the 2+1-dimensional Maxwell-Chern-Simons theory can be originated from the 3+1-dimensional Chern-Simons-like action, we also focus on the fermion sector to relate the 3+1 dimensional extended massless QED to 2+1-dimensional massive QED. We take advantage of this to state that the Chern-Simons-like action in extended massless QED can be induced with its coefficient being well defined and finite just as its 2+1 counterpart. We make use of three different regularization schemes by inducting the Chern-Simons-like term in 3+1 dimensions to support the conjecture.