Unitary and Analytic Renormalisation of Cosmological Correlators

Abstract

Loop contributions to cosmological correlators and to the associated wavefunction are of key theoretical and phenomenological interest. Here, we investigate and compare different renormalisation schemes proposed in the literature to handle ultraviolet divergences and develop new schemes adapting $\eta$ regulators to de Sitter spacetime. We focus on one-loop contributions to the quadratic wavefunction coefficient of a shift-symmetric massless scalar in de Sitter spacetime, which is a good toy model of primordial curvature perturbations. We show that different implementations of dimensional regularisation agree with each other and with unitarity and scale invariance in the final renormalised result. Imposing unitarity in the form of the cosmological optical theorem, we define a class of unitary and analytic $\eta$ regulators that agree with dim reg but feature considerable technical and conceptual simplifications. We show that the imaginary part of all one-loop wavefunction coefficients is universally fixed in terms of the logarithmic running of the real part, under the assumptions of scale invariance, Bunch-Davies vacuum and light bulk fields. Our work resolves discrepancies in the literature, establishes regulator-independent predictions for the imaginary part at one loop, and provides a practical framework for computing quantum contributions to cosmological correlators.