On Loops in Inflation
Leonardo Senatore, Matias Zaldarriaga
TL;DR
This work resolves a long-standing puzzle about loop corrections in inflation by showing that inflaton and spectator-field loops induce a physical $\log(H/\mu)$ running, not the previously claimed $\log(k/\mu)$, and that super-horizon curvature perturbations $\zeta$ remain time-independent at one loop after proper renormalization. By analyzing two representative theories (a large $\dot\pi^3$ self-interaction and gravitational couplings to $N$ spectator scalars) with three regularization schemes, the authors establish consistency between cutoff, dimensional regularization, and dimensionless-ization methods. They also provide multiple arguments ensuring the time-independence and scale invariance of $\zeta$ outside the horizon, including tadpole-cancellation and an alternative diagrammatic approach with a correct $i\epsilon$ prescription. The results reinforce inflationary predictivity and have implications for eternal inflation, suggesting that loop corrections remain under control and do not drive dangerous time evolution after horizon crossing. The analysis lays groundwork for extending to higher loops and more general inflationary EFTs, with potential connections to renormalization group interpretations in de Sitter space.
Abstract
We study loop corrections to correlation functions of inflationary perturbations. Previous calculations have found that the two-point function can have a logarithmic running of the form log(k/mu), where k is the wavenumber of the perturbation, and mu is the renormalization scale. We highlight that this result would have profound consequences for both eternal inflation and the predictivity of standard inflation. We find a different result. We consider two sets of theories: one where the inflaton has a large cubic self-interaction and one where the inflaton interacts gravitationally with N massless spectator scalar fields. We find that there is a logarithmic running but of the form log(H/mu), where H is the Hubble constant during inflation. We find this result in three independent ways: by performing the calculation with a sharp cutoff in frequency-momentum space, in dimensional regularization and by the simple procedure of making the loop integral dimensionless. For the simplest of our theories we explicitly renormalize the correlation function proving that the divergencies can be reabsorbed and that the correlation function for super-horizon modes does not depend on time (once the tadpole terms have been properly taken into account). We prove the time-independence of the super-horizon correlation function in several additional ways: by doing the calculation of the correlation function at finite time using both the regularizations and by developing a formalism which expresses loop corrections directly in terms of renormalized quantities at each time. We find this last formalism particularly helpful to develop intuition which we then use to generalize our results to higher loops and different interactions.