Universal Constraints on Conformal Operator Dimensions

TL;DR

The work proves that unitarity and crossing symmetry impose a universal upper bound $f(d)$ on the dimension $Δ_{ m min}$ of the leading scalar in the OPE $φ_d×φ_d$ for unitary 4D CFTs, computable via a positivity-based conformal bootstrap. By increasing the derivative order $N$ to 18, the bound in the interval $1<d<1.7$ improves significantly (approximately 30–50% versus earlier results) and shows signs of convergence toward an optimal value, with $Δ_{ m min}-2$ following an approximate $f_N(d) o f_ ext{∞}(d)$ behavior. The analysis also yields a 2D analogue with faster convergence and concrete connections to minimal models and the Ising model, providing a cross-check and insight into potential string-theoretic applications. These results offer model-independent constraints on operator spectra and OPE coefficients, with implications for conformal windows, phenomenology, and the lightest massive states in string compactifications.

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 Δof 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 Δ<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.

Figures (9)

• Figure 1: The conformal bootstrap equation. The thick red line denotes a conformal block, summing up exchanges of a primary operator $O$ and all its descendants.
• Figure 2: The bound $f(d)$ is the smallest $\Delta_{\min}$ for which a positivity property exists.
• Figure 3: The bound $f_{6}(d)\simeq2+1.79\sqrt{d-1}+2.9(d-1)$, $1\leq d\leq1.35$, corresponding to $N=6$, reproduced from R.
• Figure 4: The operator product expansion of $\phi(x)\phi(0)$ converges for this configuration.
• Figure 5: The auxiliary $z$ coordinate. The conformal blocks are regular outside the cut denoted by the zigzag line.