Systematic Renormalization of the Effective Theory of Large Scale Structure

TL;DR

The paper tackles the challenge of making perturbative predictions for large-scale structure physical by introducing a systematic renormalization framework that maps UV-sensitive loop contributions to counterterms ordered by perturbation theory. It shows how to construct all symmetry-allowed counterterms, demonstrates that short-distance perturbations contribute to large-scale δ as $k^2$ and to momentum density π as $k$, and proves that Euler-equation counterterms suffice for δ correlators to all orders, provided nonlocal-in-time effects are accounted for (with a practical local reformulation). The work provides explicit illustrations on the one-loop power spectrum and bispectrum, clarifies the roles of 1PR vs 1PI diagrams, and derives a comprehensive, symmetry-constrained basis of counterterms, including a local, Lagrangian-derived operator basis that preserves double softness and momentum conservation. These results establish a robust EFT-like approach to LSS that is systematically improvable and directly tied to physical, observable quantities, enabling accurate predictions and interpretation of large-scale clustering data.

Abstract

A perturbative description of Large Scale Structure is a cornerstone of our understanding of the observed distribution of matter in the universe. Renormalization is an essential and defining step to make this description physical and predictive. Here we introduce a systematic renormalization procedure, which neatly associates counterterms to the UV-sensitive diagrams order by order, as it is commonly done in quantum field theory. As a concrete example, we renormalize the one-loop power spectrum and bispectrum of both density and velocity. In addition, we present a series of results that are valid to all orders in perturbation theory. First, we show that while systematic renormalization requires temporally non-local counterterms, in practice one can use an equivalent basis made of local operators. We give an explicit prescription to generate all counterterms allowed by the symmetries. Second, we present a formal proof of the well-known general argument that the contribution of short distance perturbations to large scale density contrast $δ$ and momentum density $\mathbfπ(\mathbf k)$ scale as $k^2$ and $k$, respectively. Third, we demonstrate that the common practice of introducing counterterms only in the Euler equation when one is interested in correlators of $ δ$ is indeed valid to all orders.

Figures (3)

• Figure 1: 1-loop corrections to the cubic vertex function
• Figure 2: Hard ingoing lines combining into a soft line.
• Figure 3: Hard and soft ingoing lines combining into a soft line.