RG-Flow Renormalized One-Loop Corrections to the Power Spectrum in USR Inflation

Abstract

The nature of one-loop corrections to long-wavelength CMB-scale modes in single-field inflation models with an intermediate ultra-slow-roll (USR) phase remains a subject of active debate. In this work, we perform a detailed investigation into the regularization and renormalization of these one-loop corrections to the curvature perturbation power spectrum. Employing a combined UV-IR regularization scheme within the renormalization group (RG) flow formalism, we compute the renormalized one-loop contributions, including those from the tadpole diagram, arising from both the cubic and quartic interaction Hamiltonians. This allows us to study meaningfully the running of the coupling constant aiming at removing the divergences appear in our study. We demonstrate that the fully regularized and renormalized fractional loop correction to the power spectrum is controlled by its peak value at the end of the USR phase, scaling as $\mathcal{P}_\mathrm{peak} \sim e^{6 \Delta N}$, where $\Delta N$ is the duration of the USR phase. This result confirms the original conclusion that loop corrections can become non-perturbatively large if the transition from the USR phase to the final slow-roll phase is instantaneous and sharp, potentially challenging the perturbative framework of such inflationary scenarios for primordial black hole formation.