IR finite correlation functions in de Sitter space, a smooth massless limit, and an autonomous equation

A bstract We explore two-point and four-point correlation functions of a massive scalar field on the flat de Sitter background in the long-wavelength approximation. By employing the Yang-Feldman-type equation, we compute the two-point correlation function up to the λ 3 order and the four-point correlation function up to the λ 2 one. In contrast to the standard theory of a massive scalar field based on the de Sitter-invariant vacuum, we develop the vacuum-independent reasoning that may not possess de Sitter invariance but results in a smooth massless limit of the correlation function’s infrared part. Our elaboration affords to calculate correlation functions of a free massive scalar field and to proceed with quantum corrections, relying only on the known two-point correlation function’s infrared part of a free massless one. Remarkably, the two-point correlation function of a free massive scalar field coincides with the Ornstein-Uhlenbeck stochastic process’s one and has a clear physical interpretation. We compared our results with those obtained with the Schwinger-Keldysh diagrammatic technique, Starobinsky’s stochastic approach, and the Hartree-Fock approximation. At last, we have constructed a renormalization group-inspired autonomous equation for the two-point correlation function. Integrating its approximate version, one obtains the non-analytic expression with respect to a self-interaction coupling constant λ . That solution reproduces the correct perturbative series up to the two-loop level. At the late-time limit, it almost coincides with the result of Starobinsky’s stochastic approach in the whole interval of a new dimensionless parameter $$ 0\le \frac{\pi^2{m}^4}{3\lambda {H}^4}&lt;\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>λ</mml:mi> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>&lt;</mml:mo> <mml:mo>∞</mml:mo> </mml:math> .