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

Abstract

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 $κφ^4$ 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.

Figures (14)

• Figure 1: Pictorial representation of the three $\varphi_+\varphi_+$, $\varphi_+\varphi_-$, $\varphi_-\varphi_-$ two-point functions used to connect the SdSET fields. The corresponding mathematical expressions for $\nu=\frac{3}{2}$ are given in \ref{['eq::nu32prop1']}-\ref{['eq::nu32prop2']}.
• Figure 2: Sample tree-level $\varphi_+$ eight-point function diagram with three insertions of the $\varphi^3_+\varphi_-$ vertex of both Schwinger-Keldysh types containing the various propagators shown in Fig. \ref{['fig:props']}.
• Figure 3: Examples of the three vertex objects: Lagrangian interaction vertex (a), IC counterterm (b), and renormalised non-Gaussian IC function (c). Each diagram appears twice, once with a full and once with an empty vertex, due to the doubling of each vertex type in the Schwinger-Keldysh formalism. We omitted these copies here.
• Figure 4: Upper line: The two diagrams for the insertion of the quartic vertex with coupling $c_{3,1}$ into the trispectrum. The remaining permutations where the $\varphi_+\varphi_-$ line attaches to the other external legs also contribute but are not shown. Lower line: The corresponding diagrams from the insertion of the IC counterterm $\xi_{3,1}$.
• Figure 5: The two diagrams with insertion of the renormalised IC $\Xi_{3,1}$.