Discrete Basis Parameterization for the Gauge Theory Bootstrap

TL;DR

The paper advances the Gauge Theory Bootstrap by introducing a discrete basis parametrization for the $S$-matrix, spectral densities, and form factors in 2-to-2 pion scattering. This approach yields a substantial computational speedup while preserving the ACU constraints and UV/IR inputs that anchor GTB, enabling robust exploration of the complex-sheet amplitude and resonance content. The authors demonstrate convergence and stability with respect to discretization parameters and show that the bootstrapped amplitudes reproduce known phenomenology, including the $\sigma$, $\rho(770)$, and $f_2(1270)$ poles. The work strengthens GTB's practical viability and paves the way for more comprehensive analyses across additional channels and higher precision pole determinations.

Abstract

We implement the Gauge Theory Bootstrap (GTB) framework, initiated by He and Kruczenski in arXiv:2309.12402 and arXiv:2403.10772, using a discrete basis parametrization of the 2-to-2 pion scattering S-matrix, the spectral densities and the form factors. This approach enables a refined analysis of the convergence of the GTB and drastically reduced computational time --from approximately half an hour in arXiv:2403.10772 to under one minute in ours. The discrete basis also facilitates the evaluation of the S-matrix across the complex sheet resulting, in additional of the dominant $ρ(770)$ and $f_2(1270)$ resonances identified in arXiv:2403.10772, in the extraction of the $σ$ meson pole located far from the real axis.

Figures (6)

• Figure 1: Gauge Theory Bootstrap: The region in green represents the space of allowed amplitudes under the constraints of ACU. Zooming-in, in blue we observe the space of allowed amplitudes constrained by both ACU and CSB constraints (IR information). Incorporating the full GTB constraints yields the GTB-allowed space shown in orange. Within this region, the relation between $f_0^0(3)$ and $f_1^1(3)$, predicted by tree-level chiral perturbation theory, is shown by a red dashed line.
• Figure 2: Dependence of the allowed amplitude space (ACU) on the parameters $M$, $\ell_{\text{max}}$ and $n_{max}$.
• Figure 3: Convergence of phase shifts $\delta_\ell^I$ and elasticities $\eta_\ell^I$ for fixed $M = 49$ and $\ell_{\text{max}} = 11$. The gray curve shows the QCD phenomenological fit from Pelez, where Kaon production has been removed. The green dots represent the experimental results of PhysRevD.7.1279. We also show the points (dots) at which unitarity was imposed (energies $s_i$ of Eq. \ref{['energies']}).
• Figure 4: Gauge Theory Bootstrap $\varepsilon_{CSB}$ tolerance. The relation between $f_0^0(3)$ and $f_1^1(3)$, predicted by tree-level chiral perturbation theory, is taken as the $x$-axis. The black dot corresponds to the physical point (i.e. $f_{\pi}=92MeV$) where tree-level chiral perturbation theory predicts the coordinates of $f_0^0(3)$ and $f_1^1(3)$. We highlight on cyan a point in the boundary of the allowed region for $\varepsilon_{CSB}=2.0\times 10^{-3}$.
• Figure 5: Bootstrapped phase shifts $\delta_\ell^I$ and elasticities $\eta_\ell^I$ for the first six partial waves. The gray curve shows the two-quark QCD phenomenological fit from Pelez and the green dots represent the experimental results of PhysRevD.7.1279, where Kaon production has been removed. The continuous blue line represents our GTB result. For reference, we also have added the points (blue dots) at which unitarity was imposed (energies $s_i$ of Eq. \ref{['energies']}).