Renormalized one-Loop Corrections in Power Spectrum in USR Inflation

Abstract: The nature of one-loop corrections on long CMB scale modes in models of single field inflation incorporating an intermediate USR phase is under debate. In this work, we investigate the regularization and renormalization of the one-loop corrections of curvature perturbation power spectrum. Employing the UV-IR regularizations and performing the in-in analysis, we calculate the regularized one-loop corrections, including tadpole, in the power spectrum associated with cubic and quartic Hamiltonians. We show that the fully regularized and renormalized fractional loop correction in the power spectrum is controlled by the peak of the power spectrum at the end of USR phase, scaling like $ e^{6 \Delta {\cal N}}$ in which $\Delta {\cal N} $ is the duration of the USR phase. This confirms the original conclusion that the loop corrections can get out of perturbative control if the transition from the intermediate USR phase to the final SR phase is instantaneous and sharp.