Heavy quark diffusion in QCD and N=4 SYM at next-to-leading order

We present the full details of a calculation at next-to-leading order of the momentum diffusion coefficient of a heavy quark in a hot, weakly coupled, QCD plasma. Corrections arise at O(g(s)); physically they represent interference between overlapping scatterings, as well as soft, electric scale ( p similar to gT) gauge field physics, which we treat using the hard thermal loop ( HTL) effective theory. In 3-color, 3-flavor QCD, the momentum diffusion constant of a fundamental representation heavy quark at NLO is. k = 16 pi/3 alpha T-2(S)3( ln 1/g(s) + 0.07428 + 1.9026gs). We extend the computation to a heavy fundamental representation " probe" quark in large N-c, N = 4 Super Yang-Mills theory, where the result is.( SYM) k = lambda T-2(3)/6 pi ln 1/root lambda + 0.4304 + 0.8010 root lambda ( where lambda = g(2)N(c) is the t'Hooft coupling). In the absence of some resummation technique, the convergence of perturbation theory is poor.