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We derive general bounds on operator dimensions, central charges, and OPEcoefficients in 4D conformal and N=1 superconformal field theories. In any CFTcontaining a scalar primary phi of dimension d we show that crossing symmetryof <phi phi phi phi> implies a completely general lower bound on the centralcharge c >= f_c(d). Similarly, in CFTs containing a complex scalar chargedunder global symmetries, we bound a combination of symmetry current two-pointfunction coefficients tau^{IJ} and flavor charges. We extend these bounds toN=1 superconformal theories by deriving the superconformal block expansions forfour-point functions of a chiral superfield Phi and its conjugate. In this casewe derive bounds on the OPE coefficients of scalar operators appearing in thePhi x Phi* OPE, and show that there is an upper bound on the dimension of Phi*Phi when dim(Phi) is close to 1. We also present even more stringent bounds onc and tau^{IJ}. In supersymmetric gauge theories believed to flow tosuperconformal fixed points one can use anomaly matching to explicitly checkwhether these bounds are satisfied.

Bounds on 4D Conformal and Superconformal Field Theories