The Gauge Theory Bootstrap: Predicting pion dynamics from QCD

TL;DR

The paper advances the Gauge Theory Bootstrap by strengthening IR constraints on spectral densities, introducing a Watsonian iterative procedure, and incorporating refined high-energy form-factor inputs to pin down the strongly coupled pion dynamics. By iteratively saturating unitarity and aligning the UV and IR data, GTB converges to a unique pion S-matrix, form factors, and resonance spectrum, without assuming the number of resonances. The converged solution yields low-energy constants, scattering lengths, radii, and resonance parameters in qualitative and quantitative agreement with chiral EFT benchmarks and experimental inputs, while enabling a thermodynamic analysis of a pion gas from first principles. The approach highlights the interplay between UV QCD inputs and IR pion physics, and provides a publicly available numerical tool for exploring the intermediate-energy regime of QCD with controlled uncertainties.

Abstract

The Gauge Theory Bootstrap [arXiv:2309.12402, arXiv:2403.10772] computes the strongly coupled pion dynamics by considering the most general scattering matrix, form factors and spectral densities and matching them with perturbative QCD at high energy and with weakly coupled pions at low energy. In this work, we show that further constraints on the spectral densities significantly reduce the possible solutions to a small set of qualitatively similar ones. Quantitatively, the precise solution is controlled by the asymptotic value of the form factors and SVZ sum rules. We also introduce an iterative procedure that, starting from a generic feasible point, converges to a unique solution parameterized by the UV input. For the converged solution we compute masses and widths of resonances that appear, scattering lengths and effective ranges of partial waves, low energy coefficients in the effective action. Additionally, we use these results to discuss the thermodynamics of a pion gas including pair correlations of pions with same and opposite charge.

Figures (15)

• Figure 1: The matrix of state overlaps should saturate the positive semidefinite condition and have zero modes.
• Figure 2: Convergence of max-min $\lambda$ with Watsonian iterations. maximizing: , minimizing: , no functional:
• Figure 3: Partial waves of the maximum and minimum $\lambda$ after 1 Watsonian iteration. As a reference we plot the experimental data fromProtopopescu:1973shLOSTY1974185Hyams:1975mc (gray dots) as well as phenomenological fits from Pelaez:2004vs (gray lines). In the experimental data, the $S0$ wave has a bump from Kaon pair production that we do not have since we do not include the $s$ quark. Resonances are clearly visible in the $S0$, $P1$ and $D0$ channels.
• Figure 4: Convergence of max-min $a^{(I)}_{\ell}$ with Watsonian iterations. maximizing: , minimizing: , no functional:
• Figure 5: Convergence of max-min $\langle r_\pi^2\rangle^I_\ell$ with Watsonian iterations. maximizing: , minimizing: , no functional: . As conventional we denote $\langle r_\pi^2\rangle^0_0=\langle r_\pi^2\rangle_S$, $\langle r_\pi^2\rangle^1_1=\langle r_\pi^2\rangle_V$, $\langle r_\pi^2\rangle^0_2=\langle r_\pi^2\rangle_T$, the scalar vector and tensor radii.