Universal constraints on conformal operator dimensions
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Abstract
We continue the study of model-independent constraints on the unitary conformal field theories (CFTs) in four dimensions, initiated in [R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, J. High Energy Phys. 12 (2008) 031]. Our main result is an improved upper bound on the dimension $\ensuremath{\Delta}$ of the leading scalar operator appearing in the operator product expansion (OPE) of two identical scalars of dimension $d$: ${\ensuremath{\phi}}_{d}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\phi}}_{d}=\mathbb{1}+{O}_{\ensuremath{\Delta}}+\dots{}$. In the interval $1<d<1.7$ this universal bound takes the form $\ensuremath{\Delta}\ensuremath{\le}2+0.7(d\ensuremath{-}1{)}^{1/2}+2.1(d\ensuremath{-}1)+0.43(d\ensuremath{-}1{)}^{3/2}$. The proof is based on prime principles of CFT: unitarity, crossing symmetry, OPE, and conformal block decomposition. We also discuss possible applications to particle phenomenology and, via a 2D analogue, to string theory.
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