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A perturbative description of Large Scale Structure is a cornerstone of ourunderstanding of the observed distribution of matter in the universe.Renormalization is an essential and defining step to make this descriptionphysical and predictive. Here we introduce a systematic renormalizationprocedure, which neatly associates counterterms to the UV-sensitive diagramsorder by order, as it is commonly done in quantum field theory. As a concreteexample, we renormalize the one-loop power spectrum and bispectrum of bothdensity and velocity. In addition, we present a series of results that arevalid to all orders in perturbation theory. First, we show that whilesystematic renormalization requires temporally non-local counterterms, inpractice one can use an equivalent basis made of local operators. We give anexplicit prescription to generate all counterterms allowed by the symmetries.Second, we present a formal proof of the well-known general argument that thecontribution of short distance perturbations to large scale density contrast$\delta$ and momentum density $\mathbf\pi(\mathbf k)$ scale as $k^2$ and $k$,respectively. Third, we demonstrate that the common practice of introducingcounterterms only in the Euler equation when one is interested in correlatorsof $ \delta$ is indeed valid to all orders.

Systematic Renormalization of the Effective Theory of Large Scale Structure