Renormalization Group Calculation for the Reaction kA → ∅

The diffusion-controlled reaction kA → ∅ is known to be strongly dependent on fluctuations in dimensions d ≤ dc = 2/(k − 1). We develop a field theoretic renormalization group approach to this system which allows explicit calculation of the observables as expansions in ǫ, where ǫ = dc − d. For the density it is found that, asymptotically, n ∼ Akt. The decay exponent is exact to all orders in ǫ, and the amplitude Ak is universal, and is calculated to second order in ǫ 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 = dc we find n ∼ Ak(ln t/t) with an exact expression for Ak. The formalism can be immediately generalized to the reaction kA → lA, l < k, with the consequence that the density exponent is the same, but the amplitude is modified.