We derive general bounds on operator dimensions, central charges, and OPE
coefficients in 4D conformal and N=1 superconformal field theories. In any CFT
containing a scalar primary phi 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 tau^{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 Phi and its conjugate. In this case
we derive bounds on the OPE coefficients of scalar operators appearing in the
Phi 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 on
c and tau^{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.Comment: 47 pages, 9 figures; V2: small corrections and clarification
