Tree-level bispectrum in the effective field theory of large-scale structure extended to massive neutrinos

We compute the tree-level bispectrum of dark matter in the presence of massive neutrinos in the mildly non-linear regime in the context of the effective field theory of large-scale structure (EFTofLSS). For neutrinos, whose typical free streaming wavenumber (k(fs)) is longer than the non-linear scale (k(NL)), we solve a Boltzmann equation coupled to the effective fluid equation for dark matter. We solve perturbatively the coupled system by expanding in powers of the neutrino density fraction (f(nu)) and the ratio of the wavenumber of interest over the non-linear scale (k/k(NL)) and add suitable counterterms to remove the dependence from short distance physics. For equilateral configurations, we find that the total-matter tree-level bispectrum is approximately 16f(nu) times the dark matter one on short scales (k > k(fs)). The largest contribution stems from the back-reaction of massive neutrinos on the dark matter growth factor. On large scales (k < k(fs)) the contribution of neutrinos to the bispectrum is smaller by up to two orders of magnitude.