Bounds on 4D conformal and superconformal field theories

We derive general bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and superconformal field theories. In any CFT containing a scalar primary ϕ 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 τ IJ and flavor charges. We extend these bounds to 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 dim Φ 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.