Anomalous Dimension in a Two-Species Reaction-Diffusion System

TL;DR

This paper investigates the anomalous scaling behavior of B particles in a two-species reaction-diffusion system, revealing a new anomalous dimension through renormalization group analysis and numerical methods.

Contribution

It introduces a detailed renormalization group calculation of the B particle anomalous dimension, including first-order epsilon expansion and numerical verification in one dimension.

Findings

The B particle correlation function exhibits a distinct anomalous dimension.

The exponent  is computed to first order in =2-d.

Logarithmic corrections are determined at the upper critical dimension d=2.

Abstract

We study a two-species reaction-diffusion system with the reactions $A+A\to (0, A)$ and $A+B\to A$, with general diffusion constants $D_A$ and $D_B$. Previous studies showed that for dimensions $d\leq 2$ the $B$ particle density decays with a nontrivial, universal exponent that includes an anomalous dimension resulting from field renormalization. We demonstrate via renormalization group methods that the $B$ particle correlation function has a distinct anomalous dimension resulting in the asymptotic scaling $C_{BB}(r,t) \sim t^{\phi}f(r/\sqrt{t})$, where the exponent $\phi$ results from the renormalization of the square of the field associated with the $B$ particles. We compute this exponent to first order in $\epsilon=2-d$, a calculation that involves 61 Feynman diagrams, and also determine the logarithmic corrections at the upper critical dimension $d=2$. Finally, we determine the exponent $\phi$ numerically utilizing a mapping to a four-walker problem for the special case of $A$ particle coalescence in one spatial dimension.