Renormalization in gravitational leptogenesis with pseudo-scalar-tensor coupling

We consider the renormalization in the pseudo-scalar inflation models with the gravitational Chern-Simons term. In this model, lepton asymmetry is generated from the chiral gravitational waves produced due to the Chern-Simons term through the gravitational chiral anomaly. However, it is known that the naive estimate of the expectation value of the gravitational Chern-Pontryagin density as well as the resultant lepton number density depend on the UV-cutoff scale, which raises a question on their validity. In this paper, we propose a way to renormalize the expectation value of the Chern-Pontryagin density to remove the UV-cutoff dependence. We also discuss the renormalized lepton number density when we adopt the minimal subtraction scheme and the viability of the gravitational leptogenesis scenario.