Anomaly and Anomaly-Free Treatment of QFTs Based on Symmetry-preserving Loop Regularization

The triangle anomaly in massless and massive QED is investigated by adopting the symmetry-preserving loop regularization method proposed recently in \cite{LR}. The method is realized in the initial dimension of theory without modifying the original lagrangian, it preserves symmetries under non-Abelian gauge and Poincare transformations in spite of the existence of two intrinsic mass scales $M_c$ and $\mu_s$ which actually play the roles of UV- and IR-cut off respectively. The axialvector-vector-vector (AVV) triangle diagrams in massless and massive QED are evaluated explicitly by using the loop regularization. It is shown that when the momentum $k$ of external state is soft with $k^2 \ll \mu_s^2, m^2 $ ($m$ is the mass of loop fermions) and $ M_c \to \infty$, both massless and massive QED become anomaly free. The triangle anomaly is found to appear as quantum corrections in the case that $ m^2, \mu_s^2 \ll k^2 $ and $M_c \to \infty$. Especially, it is justified that in the massless QED with $\mu_s =0$ and $M_c\to \infty$, the triangle anomaly naturally exists as quantum effects in the axial-vector current when the ambiguity caused by the trace of gamma matrices with $\gamma_5$ is eliminated by simply using the definition of $\gamma_5$. It is explicitly demonstrated how the Ward identity anomaly of currents depends on the treatment for the trace of gamma matrices, which enables us to make a clarification whether the ambiguity of triangle anomaly is caused by the regularization scheme in the perturbation calculations or by the trace of gamma matrices with $\gamma_5$.