We prove that the (real or complex) chromatic roots of a series-parallel graph with maxmaxflow ? lie in the disc |q - 1| < (∧ - 1)/log2. More generally, the same bound holds for the (real or complex) roots of the multivariate Tutte polynomial when the edge weights lie in the "real antiferromagnetic regime" - 1 ≤ ve ≤ 0. For each ∧ ≥ 3, we exhibit a family of graphs, namely, the "leaf-joined trees", with maxmaxflow ? and chromatic roots accumulating densely on the circle |q - 1| = ∧ - 1, thereby showing that our result is within a factor 1/log 2 ≈ 1.442695 of being sharp.