Universal Constraints on Conformal Operator Dimensions (0905.2211v2)

Abstract: We continue the study of model-independent constraints on the unitary Conformal Field Theories in 4-Dimensions, initiated in arXiv:0807.0004. Our main result is an improved upper bound on the dimension \Delta of the leading scalar operator appearing in the OPE of two identical scalars of dimension d. In the interval 1<d<1.7 this universal bound takes the form \Delta<2+0.7(d-1)^{1/2}+2.1(d-1)+0.43(d-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 2-D analogue, to string theory.

• The paper presents improved universal upper bounds for the leading scalar operator dimension by increasing the derivative order from N=6 to N=18.
• It employs unitarity, crossing symmetry, and OPE analysis to set stringent, model-independent constraints in four-dimensional CFTs.
• These refined bounds impact the study of non-supersymmetric CFTs and offer potential insights for particle phenomenology and string theory.

Universal Constraints on Conformal Operator Dimensions

The paper authored by Vyacheslav S. Rychkov and Alessandro Vichi addresses a significant problem in the field of four-dimensional (4D) Conformal Field Theories (CFTs). It extends the methodologies initially explored in [arxiv:(0807.0004)] and focuses on refining model-independent constraints on operator dimensions within unitary CFTs. The research is centered on the derivation and enhancement of universal upper bounds for the dimension of scalar operators conforming with the principles of unitarity and crossing symmetry, which are central tenets of CFT.

Key Results and Methodology

The authors present substantial advancements over previous results by providing improved upper bounds on the dimension $\Delta$ of the leading scalar operator appearing in the Operator Product Expansion (OPE) of two identical scalar operators of dimension $d$. The core result is: $$\Delta \leq 2 + 0.7(d-1)^{1/2} + 2.1(d-1) + 0.43(d-1)^{3/2}$$ for the range $1 < d < 1.7$. This bound significantly refines the existing ones, achieved through an increase in the number of derivatives considered in the expansion, which improved from $N=6$ to $N=18$.

The approach to deriving these bounds involves an intricate application of the unitarity condition, crossing symmetry, and OPEs, alongside conformal block decomposition. The methodology rests on the use of inequalities derived from these principles that must be satisfied in any valid CFT, effectively setting universal constraints.

Implications and Theoretical Considerations

This work presents critical theoretical implications, particularly in the understanding of non-supersymmetric CFTs where few complete models exist. By establishing strong constraints on operator dimensions, the film indirectly maps out the permissible landscape for CFTs and pinpoints potential theories meriting further investigation.

Moreover, the results carry potential applications to particle phenomenology, and intriguingly, to string theory, where analogous bounds in two dimensions are hypothesized to help understand the constraints in higher-dimensional theories.

Future Directions

This inquiry opens several fertile avenues for further research:

• Symmetry Extensions: Incorporating global symmetries into the framework to derive bounds could hold significant implications for theories of Electroweak Symmetry Breaking.
• Generalization Across Dimensions: Extending these methods to constrain OPE coefficients and applying analogous techniques in two-dimensional CFTs could yield insights into string compactifications and the role of massive string states, especially via the connection to central charge constraints.
• Numerical and Algorithmic Enhancements: There remains potential to extend this approach through algorithmic improvements that could ultimately converge the bound faster and more accurately.

In summary, Rychkov and Vichi's paper provides a comprehensive and refined set of constraints that sharpen our understanding of CFT dynamics by leveraging the mathematical beauty of conformal symmetry and its implications across a spectrum of physical domains. This theoretical framework not only refines our grasp of known theories but also guides the search for those still undiscovered, fulfilling a need for new structuring principles in the ongoing exploration of quantum field theories.