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.
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Poland, D., Simmons-Duffin, D. Bounds on 4D conformal and superconformal field theories. J. High Energ. Phys. 2011, 17 (2011). https://doi.org/10.1007/JHEP05(2011)017
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DOI: https://doi.org/10.1007/JHEP05(2011)017