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

Recently, Grabowska and Kaplan constructed a four-dimensional lattice formulation of chiral gauge theories on the basis of the chiral overlap operator. At least in the tree-level approximation, the left-handed fermion is coupled only to the original gauge field~$A$, while the right-handed one is coupled only to the gauge field~$A_{\star }$, a deformation of~$A$ by the gradient flow with infinite flow time. In this paper, we study the fermion one-loop effective action in their formulation. We show that the continuum limit of this effective action contains local interaction terms between $A$ and~$A_{\star }$, even if the anomaly cancellation condition is met. These non-vanishing terms would lead an undesired perturbative spectrum in the formulation.