Abstract
We introduce a new numerical algorithm based on semidefinite programming to efficiently compute bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and N=1 superconformal field theories. Using our algorithm, we dramatically improve previous bounds on a number of CFT quantities, particularly for theories with global symmetries. In the case of SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal technicolor. In N=1 superconformal theories, we place strong bounds on dim(Phi*Phi), where Phi is a chiral operator. These bounds asymptote to the line dim(Phi*Phi) <= 2 dim(Phi) near dim(Phi) ~ 1, forbidding positive anomalous dimensions in this region. We also place novel upper and lower bounds on OPE coefficients of protected operators in the Phi x Phi OPE. Finally, we find examples of lower bounds on central charges and flavor current two-point functions that scale with the size of global symmetry representations. In the case of N=1 theories with an SU(N) flavor symmetry, our bounds on current two-point functions lie within an O(1) factor of the values realized in supersymmetric QCD in the conformal window.
Highlights
Conformal phases in four dimensions are ubiquitous and may play a crucial role in beyond the Standard Model physics
We introduce a new numerical algorithm based on semidefinite programming to efficiently compute bounds on operator dimensions, central charges, and operator product expansion (OPE) coefficients in 4D conformal and N = 1 superconformal field theories
While constraints on the form of simple correlation functions (e.g., [45, 46]) and unitarity restrictions on operator dimensions [47, 48] were worked out long ago, it was pointed out in [49] that crossing symmetry of four-point functions combined with the constraints of unitarity imply additional bounds on operator dimensions that must be satisfied in any consistent conformal field theories (CFTs)
Summary
Conformal phases in four dimensions are ubiquitous and may play a crucial role in beyond the Standard Model physics. In the case of general CFTs with SO(N ) global symmetries, we will place upper bounds on the dimension of the lowest-dimension SO(N )-singlet operator appearing in the φi × φj OPE. As a new application of these methods in superconformal theories, we place both upper and lower bounds the OPE coefficient of the chiral Φ2 operator which always appears in the Φ × Φ OPE In this case, lower bounds are possible because unitarity requires that there is a gap in the spectrum of dimensions, so no other nearby operators can mimic the effects of the Φ2 operator in the conformal block decomposition.
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