Bounds on 4D Conformal and Superconformal Field Theories (1009.2087v2)

Abstract: 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 <phi phi phi phi> 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.

• The paper derives rigorous bounds on operator dimensions, central charges, and OPE coefficients using crossing symmetry and superconformal block expansions.
• It employs anomaly matching and precision tests in 4D CFTs and N=1 SCFTs to validate theoretical models and constrain supersymmetric gauge theories.
• The derived bounds enhance the precision of conformal analysis and offer actionable insights for refining theories in high-energy physics.

Overview of Bounds on 4D Conformal and Superconformal Field Theories

The paper discussed here presents a comprehensive paper on deriving general bounds within the context of four-dimensional conformal field theories (CFTs) and $\mathcal{N}=1$ superconformal field theories (SCFTs). Specifically, the paper focuses on imposing limits on operator dimensions, central charges, and operator product expansion (OPE) coefficients, with significant implications for theories exhibiting conformal symmetry.

Key Contributions

1. Conformal Bounds: The paper establishes lower bounds on central charges and other critical parameters using the properties of crossing symmetry for CFTs. For any scalar primary operator $\phi$ with a given dimension, crossing symmetry leads directly to a general lower bound on the central charge. These results extend traditional understanding and allow for more precise predictions about the spectrum of operators in a conformal field theory.
2. Superconformal Bounds: In analyzing $\mathcal N=1$ superconformal field theories, the authors derive the superconformal block expansions necessary for testing four-point functions of chiral superfields. This includes upper bounds on dimensions and OPE coefficients, particularly focusing on the $\Phi \times \Phi^\dagger$ OPE, where $\Phi$ denotes a chiral superfield. The work also scrutinizes scenarios in which the dimension of $\Phi^\dagger \Phi$ is close to the boundary value of 1, revealing tighter constraints on superconformal theories.
3. Application and Verification: The bounds are applicable to a wide array of theoretical models, including classes of supersymmetric gauge theories that flow to superconformal fixed points. Using anomaly matching, the paper checks the conformance of these theories with derived bounds, thereby cross-verifying theoretical predictions with existing models.

Implications and Future Directions

• Precision in CFT Analysis: The findings push forward the boundary of precision in analyzing 4D CFTs, especially pertinent in particle physics and understanding fundamental interactions at high energy scales.
• Theoretical Consistency and Verification: New bounds facilitate testing the consistency of proposed models in high energy physics and provide a mechanism to eliminate certain theoretical constructs that cannot physically realize any known CFT or SCFT structure.
• Potential Developments: This framework paves the way for extensions and adaptations in broader contexts—potentially exploring similar bounds in higher-dimensional models or scaling dimensions pertinent to various supersymmetric theories.

The paper effectively outlines a robust approach for utilizing mathematical properties of CFTs to decode meaningful physical information, proving invaluable for future research in theoretical and high energy physics. By establishing stringent checks on operator dimensions, these insights not only aid in the fundamental understanding of theoretical models but also serve as a guiding post for new discoveries in conformal and superconformal field theory landscapes.