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 λ3 order and the four-point correlation function up to the λ2 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 <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mn>0</mn> <mo>≤</mo> <mfrac> <mrow> <msup> <mi>π</mi> <mn>2</mn> </msup> <msup> <mi>m</mi> <mn>4</mn> </msup> </mrow> <mrow> <mn>3</mn> <mi>λ</mi> <msup> <mi>H</mi> <mn>4</mn> </msup> </mrow> </mfrac> <mo><</mo> <mo>∞</mo> </math></EquationSource> <EquationSource Format="TEX">$ 0\le \frac{\pi^2{m}^4}{3\lambda {H}^4}<\infty $</EquationSource> </InlineEquation>.

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

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2025年4月16日

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Alexander Kamenshchik,
Polina Petriakova

摘要

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 λ³ order and the four-point correlation function up to the λ² 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 $0 \leq \frac{π^{2} m^{4}}{3 λ H^{4}} < \infty$ $ 0\le \frac{\pi^2{m}^4}{3\lambda {H}^4}<\infty $ .