Abstract
We develop a method for treating a series of secularly growing terms obtained from quantum perturbative calculations: autonomous first-order differential equations are constructed such that they reproduce this series to the given order. The exact solutions of these equations are free of secular terms and approach a finite limit at late times. This technique is illustrated for the well-known problem of secular growth of correlation functions of a massless scalar field with a quartic self-interaction in de Sitter space. For the expectation value of the product of two fields at coinciding space-time points we obtain a finite late-time result that is very close to the one following from Starobinsky's stochastic approach.
Highlights
The renormalization group (RG), born in the framework of quantum field theory, has become one of its most efficient tools. The origin of this concept is connected with the fact that removal of ultraviolet divergences leads to some arbitrariness in defining the renormalized parameters of the theory
Physics should not be affected by this arbitrariness: observable quantities must be independent of the renormalization scale
Using this requirement, combined with the information obtained from perturbation theory, we can derive differential equations whose solutions are equivalent to partial resummation of the perturbative series
Summary
The renormalization group (RG), born in the framework of quantum field theory, has become one of its most efficient tools (see, e.g., the reviews [1,2] and references therein). The authors developed the so-called dynamical renormalization group method by considering differential equations that involve a small parameter and whose zeroth order solutions are bounded functions, while the first iteration reveals a presence of secularly growing terms. These terms spoil the validity of the perturbative expansion past a certain point in time; in order to deal with them, an arbitrary intermediate timescale is introduced and the initial conditions are renormalized. In the present paper we develop a semi-heuristic method for taking the late-time limit of a series of secularly growing terms obtained from quantum perturbative calculations. In the appendixes we present perturbative calculations of the leading secular terms in the twoand four-point functions
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