Efficiently evaluating loop integrals in the EFTofLSS using QFT integrals with massive propagators

Abstract

We develop a new way to analytically calculate loop integrals in the Effective Field Theory of Large Scale-Structure. Previous available methods show severe limitations beyond the one-loop power spectrum due to analytical challenges and computational and memory costs. Our new method is based on fitting the linear power spectrum with cosmology-independent functions that resemble integer powers of quantum field theory massive propagators with complex masses. A remarkably small number of them is sufficient to reach enough accuracy. Similarly to former approaches, the cosmology dependence is encoded in the coordinate vector of the expansion of the linear power spectrum in our basis. We first produce cosmology-independent tensors where each entry is the loop integral evaluated on a given combination of basis vectors. For each cosmology, the evaluation of a loop integral amounts to contracting this tensor with the coordinate vector of the linear power spectrum. The 3-dimensional loop integrals for our basis functions can be evaluated using techniques familiar to particle physics, such as recursion relations and Feynman parametrization. We apply our formalism to evaluate the one-loop bispectrum of galaxies in redshift space. The final analytical expressions are quite simple and can be evaluated with little computational and memory cost. We show that the same expressions resolve the integration of all one-loop N-point function in the EFTofLSS. This method, which is originally presented here, has already been applied in the first one-loop bispectrum analysis of the BOSS data to constraint ΛCDM parameters and primordial non-Gaussianities [1, 2].

The BOSS bispectrum analysis at one loop from the Effective Field Theory of Large-Scale Structure

“…The development of a pipeline that allows us to analyze the one-loop bispectrum predicted by the EFTofLSS has required much theoretical work, and the explanation of such techniques will be presented in two upcoming papers [90,91]. While the application to constrain primordial non-Gaussianities was already presented in [24], here, instead, we will just give the essential details and focus on constraints of the ΛCDM parameters.…”

“…where we have grouped perturbation theory contributions with the counterterm contributions that renormalize them in parentheses. The loop integrals are evaluated using the techniques described in [91]. We note that the arguments of the above functions are the same as those given in eq.…”

“…We thank Babis Anastasiou, Diogo Bracanca and Henry Zheng for letting us use the code to evaluate the loop integrals in [91]. We thank Matteo Biagetti, Hector Gil-Marín, Emiliano Sefusatti, and Cheng Zhao for useful discussions.…”