Bounding scalar operator dimensions in 4D CFT (0807.0004v2)

Abstract: In an arbitrary unitary 4D CFT we consider a scalar operator \phi, and the operator \phi^2 defined as the lowest dimension scalar which appears in the OPE \phi\times\phi with a nonzero coefficient. Using general considerations of OPE, conformal block decomposition, and crossing symmetry, we derive a theory-independent inequality [\phi^2] \leq f([\phi]) for the dimensions of these two operators. The function f(d) entering this bound is computed numerically. For d->1 we have f(d)=2+O(\sqrt{d-1}), which shows that the free theory limit is approached continuously. We perform some checks of our bound. We find that the bound is satisfied by all weakly coupled 4D conformal fixed points that we are able to construct. The Wilson-Fischer fixed points violate the bound by a constant O(1) factor, which must be due to the subtleties of extrapolating to 4-\epsilon dimensions. We use our method to derive an analogous bound in 2D, and check that the Minimal Models satisfy the bound, with the Ising model nearly-saturating it. Derivation of an analogous bound in 3D is currently not feasible because the explicit conformal blocks are not known in odd dimensions. We also discuss the main phenomenological motivation for studying this set of questions: constructing models of dynamical ElectroWeak Symmetry Breaking without flavor problems.

• The paper establishes a theory-independent bound on the φ² operator in the φ×φ OPE using crossing symmetry and conformal block techniques.
• It employs non-perturbative analysis to show that as the scaling dimension d approaches 1, the bound f(d) converges to 2, aligning with the free field limit.
• The results provide critical model validation and phenomenological constraints for 4D CFTs, while suggesting extensions to other dimensions and symmetry classes.

Bounding Scalar Operator Dimensions in 4D CFT

In the paper of 4D Conformal Field Theories (CFTs), understanding the constraints on operator dimensions provides essential insights into the theoretical underpinnings of these models. The paper in question investigates a specific scenario where a scalar operator, denoted as $\phi$, exists in a four-dimensional CFT. The objective is to determine a theory-independent bound on the dimension of the first scalar operator $\phi^2$ occurring in the Operator Product Expansion (OPE) of $\phi\times\phi$.

Main Results and Methodology

The authors set forth a boundary condition stipulating that $\Delta_{\min}$, the dimension of the $\phi^2$ operator, is constrained by a function $f([\phi])$. This function $f(d)$ is systematically calculated using a blend of operator product formalism, conformal block decompositions, and crossing symmetry principles. For low values of the operator dimension $d$ (where $d \to 1$), it is observed that the function $f(d)$ approaches $2$, which corroborates the intuitive notion that the free field theory limit is smoothly approached as $d$ trends towards unity.

The calculations within this paper are robustly non-perturbative and classical constructs, such as central charge, are not utilized, reinforcing the broad applicability of these findings across a range of theories. Notably, the constraints derived are independent of specific model features and rely only on general conformal symmetry properties.

The analysis also extends to 2D CFTs, where corresponding bounds yield insights aligned with known results from Minimal Models, with cases like the Ising model near the saturation of these bounds.

Implications

This research has several theoretical and phenomenological implications:

1. Model Validation: The derived bounds serve as validation checks for various 4D CFT models, ensuring their internal consistency regarding operator dimensions. This is particularly relevant for weakly and strongly coupled fixed points within 4D theories.
2. Phenomenological Constraints: The findings provide insights into constructing dynamical models, especially those addressing Electroweak Symmetry Breaking without leading to flavor-related problems. They suggest constraints on parameters such as $\phi$'s anomalous dimensions, critically influencing model stability and predictions.
3. Future Directions: The paper hints at potential future research in extending similar bounds to CFTs in other dimensions, like 3D, despite the current unavailability of simple conformal block expressions in these contexts. Additionally, it underscores the need for understanding global symmetries in CFT, which can lead to new constraints not addressed by the present analysis.

Conclusion

Overall, the paper enriches the understanding of dimensional constraints within 4D CFTs, providing a rigorous framework for assessing operator dimensions and guiding theoretical and phenomenological pursuits. The exploration of whether current findings might transfer to CFTs with differing global symmetry or other dimensions marks an exciting avenue for future research.