Bootstrapping the $a$-anomaly in $4d$ QFTs

TL;DR

The paper develops a nonperturbative S-matrix bootstrap framework in four dimensions to bound the UV Weyl anomaly a using gapped QFTs connected to a UV-CFT via relevant deformations. It introduces a dilaton as a universal probe, enforces unitarity, crossing, and analyticity, and derives a dispersive sum rule linking a^{UV} to the low-energy dilaton scattering amplitude. Numerically, it finds a universal lower bound a^{UV}/a_{\text{free}} ≳ 0.3, with a robust absolute minimum around 0.316±0.015 after extrapolation, and demonstrates consistency with free-theory limits and elastic scattering in the spin-0 sector. These results establish a quantitative bridge between IR scattering data and UV conformal data, while outlining clear avenues for generalizing the setup to more complex spectra and symmetries, potentially illuminating the landscape of 4d QFTs.

Abstract

We study gapped 4d quantum field theories (QFTs) obtained from a relevant deformation of a UV conformal field theory (CFT). For simplicity, we assume the existence of a $\mathbb{Z}_2$ symmetry and a single $\mathbb{Z}_2$-odd stable particle and no $\mathbb{Z}_2$-even particles at low energies. Using unitarity, crossing and the assumption of maximal analyticity we compute numerically a lower bound on the value of the $a$-anomaly of the UV CFT as a function of various non-perturbative parameters describing the two-to-two scattering amplitude of the particle.

Figures (14)

• Figure 1: The complete system of scattering amplitudes of the $\mathbb{Z}_2$ odd particle $A$ with mass $m$ and the massless dilaton $B$.
• Figure 2: Minimum of the $a$-anomaly of the UV CFT as a function of the parameters $\lambda_0$ and $\lambda_2$ defined in \ref{['eq:lambdas']}. The red dot marks the absolute minimum. The red vertical lines indicate the boundaries of the allowed regions for $\lambda_0$ and $\lambda_2$.
• Figure 3: Minimum of the $a$-anomaly of the UV CFT as a function of the parameters $\Lambda_0$ and $\Lambda_2$ defined in \ref{['eq:Lambdas']}. The red dot marks the absolute minimum. The red vertical lines indicate the boundaries of the allowed regions for $\Lambda_0$ and $\Lambda_2$.
• Figure 4: The set of diagrams contributing to the $\mathcal{T}_{AB\rightarrow AB}$ amplitudes. Here the solid lines represent scalar particles $A$ and dashed lines represent dilatons $B$.
• Figure 5: Minimum possible value of the a-anomaly without any further assumptions as a function of $1/N_{max}$ with $L_{max}=N_{max}+10$. The numerical results are depicted by blue points. Linear extrapolation to $N_{max} \rightarrow \infty$ depicted by the red line gives $0.316\pm 0.015$ for the minimum of $a/a_\text{free}$.