# The Effective Field Theory of Large Scale Structure at Three Loops

## TL;DR

This paper advances the Effective Field Theory of Large Scale Structures to three loops, showing improved accuracy in modeling dark matter power spectra up to certain scales, but also highlighting the asymptotic nature of the perturbative series.

## Contribution

It demonstrates that three-loop calculations in EFTofLSS achieve near-percent accuracy with few counter-terms and discusses the limitations due to the asymptotic series behavior.

## Key findings

- Three-loop EFTofLSS matches simulations with percent accuracy up to k≈0.4 h/Mpc at z=0.
- Adding higher loops does not significantly improve accuracy and may worsen it due to asymptotic series behavior.
- Large contributions from mildly non-linear scales limit the predictive power of higher-order loops.

## Abstract

We study the power spectrum of dark matter density fluctuations in the framework of the Effective Field Theory of Large Scale Structures (EFTofLSS) up to three loop orders. In principle, several counter-terms may be needed to handle the short-distance sensitivity in perturbation theory. However, we show that a small number of extra coefficients are sufficient to match numerical simulations with percent accuracy when a generic renormalization prescription is implemented (allowing for running of the individual counter-terms). We show that the level of accuracy increases with respect to the two loop results, up to $k \simeq 0.4\, h$Mpc$^{-1}$ at redshift $z=0$, although the overall improvement is somewhat marginal. At the same time, we argue there is evidence that the behavior of the loop expansion in the EFTofLSS is typical of an asymptotic series, already on the brink of its maximum predictive power (at $z=0$). Hence, the inclusion of higher orders will likely deteriorate the matching to data, even at moderate values of $k$. Part of the reason for this behavior is due to large contributions to the (renormalized) power spectrum at three loop order from mildly non-linear scales, even after the UV counter-terms are included. In conclusion, the EFTofLSS to three loop orders provides the best approximation to the (deterministic part of the) power spectrum in the weakly non-linear regime at $z=0$, and higher loops are not expected to improve our level of accuracy.