  An important unanswered question in quantum field theory is to understand
precisely under which conditions scale invariance implies invariance under the
full conformal group. While the general answer in two dimensions has been known
for over twenty years, a precise non-perturbative relation between scale and
conformal invariance in higher dimensions has been lacking. In this note, we
specialize to four dimensions and give a full quantum mechanical proof that
certain unitary R-symmetric fixed points are necessarily superconformal. Among
other consequences, this result implies that the infrared fixed points of N=1
supersymmetric quantum chromodynamics are superconformal.
