2 7 A pr 2 01 1 Bounds on 4 D Conformal and Superconformal Field

We derive general bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal andN = 1 superconformal field theories. In any CFT containing a scalar primary φ of dimension d we show that crossing symmetry of 〈φφφφ〉 implies a completely general lower bound on the central charge c ≥ fc(d). Similarly, in CFTs containing a complex scalar charged under global symmetries, we bound a combination of symmetry current two-point function coefficients τ 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 Φ and its conjugate. In this case we derive bounds on the OPE coefficients of scalar operators appearing in the Φ × Φ† OPE, and show that there is an upper bound on the dimension of Φ†Φ when dimΦ is close to 1. We also present even more stringent bounds on c and τ 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.