Multiloop calculations with parametric integration in critical dynamics: the four-loop analytic study of model A of $φ^4$ theory

TL;DR

This paper tackles the problem of determining the dynamic critical exponent $z$ for model A in $d=4-2\varepsilon$ by performing a four-loop analytic renormalization group calculation. The authors develop and apply a revised diagram-reduction scheme and then compute renormalized, convergent integrals using parametric integration with Goncharov polylogarithms via HyperInt, aided by a novel stream-based method to handle non-reducible integrals arising from time cuts. They provide explicit four-loop contributions to the renormalization constant $Z_1$, extract $z=2+\gamma_1^* - \eta$, and present a detailed $\varepsilon$-expansion of $z$ as a function of the number of order-parameter components $n$, including cross-checks with the $1/n$ expansion. The work demonstrates the feasibility of analytic multiloop calculations in dynamic critical phenomena, introduces techniques (streams) to manage non-linear reducibility, and offers a foundation for extending analytic methods to more complex dynamic models and higher-loop orders.

Abstract

We perform an analytical four loop calculation of exponent $z$ in model A of critical dynamics in $d=4-2\varepsilon$ dimensions. This is the first time such a large order of perturbation theory has been calculated analytically for models of critical dynamics. To do this, we apply the modern method of parametrical integration with hyperlogaritms. We discuss in detail peculiarities of application of this method to critical dynamics, e.g. the problem of linear-irreducible diagrams already present in four loop (contrary to statics where the first linear-irreducible diagram appears in six loop).

Figures (5)

• Figure 1: The six types of dynamic subgraphs appearing in the the three- and four-loop diagrams. (a)-(e) are logarithmically divergent, (f) is quadratically divergent.
• Figure 2: The set of three-loop reduced diagrams of $\Gamma^R_{ \psi' \psi'}$ from AIKV_4lSD17.
• Figure 3: All two-, three- and four-loop diagrams obtained after the additional reduction that contribute to $\Gamma^R_{\psi'\psi'}$ at four-loop order. The notation of the diagrams is chosen to coincide with that of AIKV_4lSD17.
• Figure 4: Two subgraphs in the set of the four-loop reduced diagrams, requiring separate subtractions.
• Figure 5: All spanning trees of diagram $Y_3$ and their corresponding contributions to the first Symanzik polynomial. Black solid lines represent spanning trees; black dotted lines denotes edges not included in the spanning tree; blue dotted lines correspond to dynamic cuts.