Anomalous Dimension in a Two-Species Reaction-Diffusion System

We study a two-species reaction-diffusion system with the reactions $A+A\to (0, A)$and$A+B\to A$,withgeneraldiffusionconstants$D_A$and$D_B$. Previous studies showed that for dimensions $d\le 2$ 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 $C_{BB}(r,t) \sim t^{\phi}f(r/\sqrt{t})$,wheretheexponent$\phi$ results from the renormalization of the square of the field associated with the $B$ particles. We compute this exponent to first order in $ϵ=2-d$, 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 $\varphi$ numerically utilizing a mapping to a four-walker problem for the special case of $A$ particle coalescence in one spatial dimension.