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Renormalization group analysis of directed percolation process: Towards multiloop calculation of scaling functions

Michal Hnatič, Matej Kecer, Tomáš Lučivjanský, Lukáš Mižišin

TL;DR

The paper targets universal scaling in directed percolation by applying a field-theoretic renormalization group approach and an epsilon expansion around the upper critical dimension $d_c=4$. It develops a semi-analytic three-loop strategy that maps many diagrams to existing results and numerically evaluates the remaining truly novel ones using Sector Decomposition and the Vegas algorithm, with two-loop results used as stringent benchmarks. A key outcome is the substantial reduction in independent three-loop diagrams from 65 to 16 that require new calculations, enabling the derivation of the Widom-Griffiths scaling form for the equation of state and the associated amplitude ratios, together with a rigorous RG-based framework for universal scaling functions. The work advances high-precision predictions for DP scaling behavior and furnishes a methodological blueprint for extending multiloop analyses to other nonequilibrium critical phenomena.

Abstract

In this work, we employ a field-theoretic renormalization group approach to study a paradigmatic model of directed percolation. We focus on the perturbative calculation of the equation of state, extending the analysis to the three-loop order in the expansion parameter $\varepsilon = 4-d$. We show that a large group of the necessary three-loop Feynman diagrams can be mapped onto already existing three-loop results, and develop a technique for the calculation of the remaining -- truly novel -- ones. The described semi-analytic procedure is further used to verify existing two-loop results. The main aim of this study is to provide an update on this ongoing work, as full three-loop calculations utilizing the described procedure are in progress.

Renormalization group analysis of directed percolation process: Towards multiloop calculation of scaling functions

TL;DR

The paper targets universal scaling in directed percolation by applying a field-theoretic renormalization group approach and an epsilon expansion around the upper critical dimension . It develops a semi-analytic three-loop strategy that maps many diagrams to existing results and numerically evaluates the remaining truly novel ones using Sector Decomposition and the Vegas algorithm, with two-loop results used as stringent benchmarks. A key outcome is the substantial reduction in independent three-loop diagrams from 65 to 16 that require new calculations, enabling the derivation of the Widom-Griffiths scaling form for the equation of state and the associated amplitude ratios, together with a rigorous RG-based framework for universal scaling functions. The work advances high-precision predictions for DP scaling behavior and furnishes a methodological blueprint for extending multiloop analyses to other nonequilibrium critical phenomena.

Abstract

In this work, we employ a field-theoretic renormalization group approach to study a paradigmatic model of directed percolation. We focus on the perturbative calculation of the equation of state, extending the analysis to the three-loop order in the expansion parameter . We show that a large group of the necessary three-loop Feynman diagrams can be mapped onto already existing three-loop results, and develop a technique for the calculation of the remaining -- truly novel -- ones. The described semi-analytic procedure is further used to verify existing two-loop results. The main aim of this study is to provide an update on this ongoing work, as full three-loop calculations utilizing the described procedure are in progress.
Paper Structure (5 sections, 35 equations, 3 figures, 1 table)

This paper contains 5 sections, 35 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Field-theoretic approach
  3. Diagrammatic analysis
  4. Scaling formulation of the equation of state
  5. Conclusion

Figures (3)

  • Figure 1: The graphical representation of Feynman rules for the DP model obtained from the shifted action functional \ref{['eq:s_bare_DP_shift']}.
  • Figure 2: The graphical representation of the propagator $\langle {\varphi}_0 {\varphi}_0 \rangle$ and the newly generated interaction vertex.
  • Figure 3: The graphical representation of the two-loop diagrams with the symmetry coefficient: (a) $s_\text{shift} = 1$ (b) $s_\text{shift} = 1/2$ (c) $s_\text{shift} = 1$ (d) $s_\text{shift} = 1/4$.