Hamiltonians to all Orders in Perturbation Theory and Higher Loop Corrections in Single Field Inflation with PBHs Formation

We calculate the action and the interaction Hamiltonians to all orders in perturbation theory in the model of single field inflation with a transient ultra slow-roll phase. Employing the formalism of EFT of inflation, we obtain a compact non-perturbative expression for the interaction Hamiltonian in terms of the Goldstone field $\pi$ in the decoupling limit. In addition, we also present a non-linear relation between $\pi$ and the curvature perturbations to all orders in perturbation theory. These are powerful results which enable us to calculate the cosmological correlators and loop corrections to any order in perturbation theory. As a non-trivial example, we calculate the $L$-loop corrections on long CMB scale perturbations in the USR models which are used for PBHs formation. We show that the loop corrections scale like $(\mathrm{\Delta }N{\mathcal{P}}_{e}L{)}^L$ in which ${\mathcal{P}}_e$ is the peak of the power spectrum and $\mathrm{\Delta }N$ is the duration of the USR phase. This indicates that the loop corrections grow quickly out of perturbative control for large values of $L$. In the conventional USR setup for PBHs formation with $\mathrm{\Delta }N\simeq 2.5$, this happens at $L=4$.