RENORMALIZATION-GROUP CALCULATION FOR THE REACTION KA-]CIRCLE-DIVIDE

The diffusion-controlled reaction kA --> empty set is known to be strongly dependent on fluctuations in dimensions d less-than-or-equal-to 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 approximately 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 approximately 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 --> lA, l < k, with the consequence that the density exponent is the same, but the amplitude is modified.