One-loop perturbative coupling of $A$ and $A_{\star }$ through the chiral overlap operator

We study the one-loop effective action defined by the chiral overlap operator in the four-dimensional lattice formulation of chiral gauge theories by Grabowska and Kaplan. In the tree-level continuum limit, the left-handed component of the fermion is coupled only to the original gauge field~$A$, while the right-handed one is coupled only to~$A_{\star }$, which is given by the gradient flow of~$A$ with infinite flow time. In this paper, we show that the continuum limit of the one-loop effective action contains local interaction terms between $A$ and~$A_{\star }$, which do not generally vanish even if the gauge representation of the fermion is anomaly free. We argue that the presence of such interaction terms can be regarded as undesired gauge symmetry-breaking effects in the formulation.