Regularizing infrared divergences in de Sitter spacetime: Loops, dimensional regularization, and cutoffs

Abstract

Correlation functions of light scalar fields in de Sitter spacetime, computed via standard perturbation theory, often exhibit secular growth characterized by time-dependent divergent terms in the form of powers of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>ln</a:mi> <a:mi>a</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>t</a:mi> <a:mo stretchy="false">)</a:mo> </a:mrow> </a:math> , where <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>a</e:mi> <e:mo stretchy="false">(</e:mo> <e:mi>t</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> is the scale factor describing cosmic expansion. It is widely believed that loop corrections further enhance this secular growth. We argue that this is not necessarily the case: Loop corrections can be systematically handled using standard perturbative techniques, such as dimensional regularization, without introducing new <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>ln</i:mi> <i:mi>a</i:mi> <i:mo stretchy="false">(</i:mo> <i:mi>t</i:mi> <i:mo stretchy="false">)</i:mo> </i:math> terms. We focus on a canonical massless scalar field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>φ</m:mi> </m:math> with self-interactions described by a potential <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi mathvariant="script">V</o:mi> <o:mo stretchy="false">(</o:mo> <o:mi>φ</o:mi> <o:mo stretchy="false">)</o:mo> </o:math> , and analyze correlation functions represented by diagrams with a single vertex and an arbitrary number of loops. In this framework, infrared divergences can be systematically eliminated with counterterms at each order in perturbation theory, leading to loop-corrected correlation functions that are indistinguishable from their tree-level forms, with no secular growth from loops. Furthermore, adopting a Wilsonian perspective, we explore the role of cutoffs in computing loop corrections within effective field theory and identify the effective potential <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:msub> <t:mi mathvariant="script">V</t:mi> <t:mi>eff</t:mi> </t:msub> <t:mo stretchy="false">(</t:mo> <t:mi>φ</t:mi> <t:mo stretchy="false">)</t:mo> </t:math> , which guarantees cutoff-independent observables. We conclude that when infrared comoving cutoffs are used to regularize loop integrals, time-dependent Wilsonian coefficients are necessary to maintain cutoff-free correlation functions. Neglecting this time dependence results in secular growth from loops. Published by the American Physical Society 2024

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