  We compute the connected four point correlation function (the trispectrum in
Fourier space) of cosmological density perturbations at one-loop order in
Standard Perturbation Theory (SPT) and the Effective Field Theory of Large
Scale Structure (EFT of LSS). This paper is a companion to our earlier work on
the non-Gaussian covariance of the matter power spectrum, which corresponds to
a particular wavenumber configuration of the trispectrum. In the present
calculation, we highlight and clarify some of the subtle aspects of the EFT
framework that arise at third order in perturbation theory for general
wavenumber configurations of the trispectrum. We consistently incorporate
vorticity and non-locality in time into the EFT counterterms and lay out a
complete basis of building blocks for the stress tensor. We show predictions
for the one-loop SPT trispectrum and the EFT contributions, focusing on
configurations which have particular relevance for using LSS to constrain
primordial non-Gaussianity.
