A low-energy β-function in a finite super-Yang-Mills model with multiple mass scales

We compute the one-loop contribution to the low-energy light-fermion gauge coupling in a finite supersymmetric gauge theory with two mass scales: a heavy mass that breaks an initial N=4 supersymmetry down to N=2, but respects the finiteness, and a light mass that, for simplicity, is set to zero. We find that the coupling grows with the mass of the heavy intermediate states. Hence the latter do not decouple at low energies, leading to large logarithms that invalidate low-energy perturbation theory. Consequently, further manipulations are required to obtain a meaningful perturbative expansion. Enforcing decoupling through finite renormalizations, that absorb the heavy mass effects into a redefinition of the parameters of the Lagrangian, introduces an arbitrary subtraction mass μ. The requirement that the S-matrix elements be independent of μ leads to a non-trivial renormalization-group equation for the low-energy theory, with a non-vanishing β-function. (author)