Improved bounds for CFT's with global symmetries

Abstract

The four point function of Conformal Field Theories (CFT's) with global symmetry gives rise to multiple crossing symmetry constraints. We explicitly study the correlator of four scalar operators transforming in the fundamental representation of a global SO(N) and the correlator of chiral and anti-chiral superfields in a superconformal field theory. In both cases the constraints take the form of a triple sum rule, whose feasibility can be translated into restrictions on the CFT spectrum and interactions. In the case of SO(N) global symmetry we derive bounds for the first scalar singlet operator entering the Operator Product Expansion (OPE) of two fundamental representations for different value of N. Bounds for the first scalar traceless-symmetric representation of the global symmetry are computed as well. Results for superconformal field theories improve previous investigations due to the use of the full set of constraints. Our analysis only assumes unitarity of the CFT, crossing symmetry of the four point function and existence of an OPE for scalars.

Figures (4)

• Figure 1: Bound for the smallest dimension of a scalar operators singlet under a global $SO(N)$ symmetries. The bounds corresponds, from the strongest to the weaker, to $SO(N),\,N=2,3,4$ and have been computed with 4 derivatives. The line is an interpolation between the points where the bound has been computed exactly. We assume as usual a smooth interpolation.
• Figure 2: Bounds for the smallest dimension operators appearing in the OPE of two scalar fields transforming under the fundamental representation of a global $SO(4)$. The weaker bound (blue line) corresponds to scalar operators neutral under $SO(4)$. The strongest bound (red) refers to scalar operators transforming as a symmetric traceless tensor. Again we assumed a smooth interpolation between the points where the bound has been computed exactly.
• Figure 3: Bound for the smallest dimension of a vector superfield appearing in the OPE of a chiral field with its conjugate. On the left: bound obtained obtained using only the first sum rule of (\ref{['susyvecsumrule']}); the bound $f_{10}(d)$ reproduces the results of poland. On the right: bound obtained with 6 derivatives using the vectorial sum rule. Irregularities are due to the gap in dimension of the operators allowed by superconformal symmetry to appear in the $\Phi\times\Phi$ OPE.
• Figure 4: On the left: Lower bound on the central charge computed with $N_{der}=10$ (red) and $N_{der}=14$ (blue) using only one sum rule, the red bound reproduces the results of poland . On the right: Lower bound on the central charge computed with $N_{der}=6$ using the vectorial sum rule. The dashed line corresponds to the central charge of a supersymmetric theory with one chiral superfield.