Bounds on 4D conformal and superconformal field theories

We derive general bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and N = 1 superconformal field theories. In any CFT containing a scalar primary phi of dimension d we show that crossing symmetry of implies a completely general lower bound on the central charge c >= f(c)(d). Similarly, in CFTs containing a complex scalar charged under global symmetries, we bound a combination of symmetry current two-point function coefficients tau(IJ) and flavor charges. We extend these bounds to N = 1 superconformal theories by deriving the superconformal block expansions for four-point functions of a chiral superfield Phi and its conjugate. In this case we derive bounds on the OPE coefficients of scalar operators appearing in the Phi x Phi(dagger) OPE, and show that there is an upper bound on the dimension of Phi(dagger)Phi when dim Phi is close to 1. We also present even more stringent bounds on c and tau(IJ). In supersymmetric gauge theories believed to flow to superconformal fixed points one can use anomaly matching to explicitly check whether these bounds are satisfied.