Renormalisation and matching of massless scalar correlation functions in Soft de Sitter Effective Theory

For light and massless scalar fields, cosmological correlation functions suffer from infrared divergences and secular logarithms. Soft de Sitter Effective Theory (SdSET) has been proposed by Cohen and Green as the effective description of the non-trivial dynamics of long-wavelength modes $k_{\rm phys} < H$ in de Sitter space, which is responsible for the infrared and late-time logarithms, and as a systematic extension of the stochastic approach. In this article, we construct SdSET in dimensional regularisation, including an initial-condition functional. We demonstrate by examples that renormalisation and matching works as for flat-space effective field theories. Adopting massless $\kappa \phi^4$ theory as the UV theory, we match the tree-level trispectrum and six-point function, and the one-loop power spectrum to SdSET, verifying explicitly that SdSET is the appropriate effective field theory for the quantum dynamics of superhorizon modes.