Bounds on 4D conformal and superconformal field theories

Abstract

We derive general bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and $\mathcal{N} = 1$ superconformal field theories. In any CFT containing a scalar primary ϕ of dimension d we show that crossing symmetry of $\left\langle {\phi \phi \phi \phi } \right\rangle$ 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 $\mathcal{N} = 1$ 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 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.