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 20 years, a precise nonperturbative 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 $\mathcal{N}=1$ supersymmetric quantum chromodynamics are superconformal.

• Received 20 February 2011

DOI:https://doi.org/10.1103/PhysRevD.83.105011