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

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.

💡 Research Summary

The paper presents a field‑theoretic renormalization‑group (RG) study of the directed percolation (DP) process, focusing on the perturbative calculation of the equation of state up to three‑loop order in the ε‑expansion (ε = 4 − d). Starting from the standard Janssen‑Grassberger action written in terms of the density field ψ and the Martin‑Siggia‑Rose response field ˜ψ, the authors shift the field by its mean value m₀ (ψ = m₀ + φ) to isolate fluctuations φ and the corresponding response ˜φ. This shift generates two interaction vertices (∝ g₀) and two propagators: the retarded ⟨φ ˜φ⟩ and the symmetric ⟨φ φ⟩. A key technical step is the identity ⟨φ φ⟩ = D₀ g₀ m₀ ⟨φ ˜φ⟩⟨˜φ φ⟩, which allows the symmetric propagator to be represented as a pair of retarded lines connected by an effective vertex. This representation makes it possible to map a large class of three‑loop diagrams onto already known three‑loop results from earlier DP calculations, dramatically reducing the number of genuinely new diagrams that must be evaluated.

The combinatorial analysis of diagrams is carried out using relations between the number of loops (l), internal lines (p), and vertices (v). For a diagram with a single external ˜φ leg, the authors derive p = 3l − 2 and v = 2l − 1, and further decompose the propagators and vertices into those involving ⟨φ ˜φ⟩ and ⟨φ φ⟩. From these constraints they show that the number of ⟨φ φ⟩ lines cannot exceed the loop order, with the maximal case (p₂ = l) corresponding to diagrams built solely from the ⟨φ φ ˜φ⟩ vertex. The minimal case (p₂ = 1) avoids closed retarded loops. These structural limits guide the identification of “truly novel” three‑loop contributions that are not covered by the mapping.

Dimensional regularization and the minimal subtraction (MS) scheme are employed to handle ultraviolet divergences. Poles in ε appear in the Green functions, and the renormalization constants are extracted to obtain the β‑function for the coupling and the anomalous dimensions of fields and parameters. The equation of state follows from the condition ⟨˜φ⟩ = 0, leading to the relation
0 = D₀ h₀ − D₀ m₀ τ₀ + ½ g₀ m₀² + X‑graphs,
where X‑graphs denote the sum of all one‑particle‑irreducible diagrams with a single external ˜φ leg. The authors compute these graphs up to two loops to verify existing results, and outline a semi‑analytic procedure for the remaining three‑loop pieces: (i) use the mapping to reuse known integrals, (ii) evaluate the genuinely new diagrams analytically where possible and numerically otherwise.

The paper’s main contributions are: (1) a systematic mapping that reduces the workload of three‑loop calculations for DP, (2) a concrete semi‑analytic algorithm for evaluating the residual three‑loop diagrams, and (3) an independent confirmation of earlier two‑loop results, thereby establishing the reliability of the method. The authors emphasize that the ultimate goal is to obtain the full three‑loop scaling function for the equation of state, including non‑universal amplitude ratios, which are more sensitive to the entire RG flow than critical exponents. They argue that achieving this for the relatively simple DP model will pave the way for similar multi‑loop analyses of more complex non‑equilibrium systems such as reaction‑diffusion models, turbulent flows, and absorbing‑state phase transitions. The manuscript concludes with a roadmap for completing the three‑loop program and discusses the expected impact on high‑precision comparisons with Monte‑Carlo simulations and experimental data.