Renormalization Group Calculation for the Reaction $kA\rightarrow\emptyset$

The diffusion-controlled reaction $kA\rightarrow\emptyset$ is known to be strongly dependent on fluctuations in dimensions $d\le d_c=2/(k-1)$. We develop a field theoretic renormalization group approach to this system which allows explicit calculation of the observables as expansions in $\epsilon^{1/(k-1)}$, where $\epsilon=d_c-d$. For the density it is found that, asymptotically, $n\sim A_k t^{-d/2}$. The decay exponent is exact to all orders in $\epsilon$, and the amplitude $A_k$ is universal, and is calculated to second order in $\epsilon^{1/(k-1)}$ for $k=2,3$. The correlation function is calculated to first order, along with a long wavelength expansion for the second order term. For $d=d_c$ we find $n \sim A_k (\ln t/t)^{1/(k-1)}$ with an exact expression for $A_k$. The formalism can be immediately generalized to the reaction $kA\rightarrow\ell A$, $\ell < k$, with the consequence that the density exponent is the same, but the amplitude is modified.