Discrete Basis Parameterization for the Gauge Theory Bootstrap

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 $\rho(770)$ and $f_2(1270)$ resonances identified in arXiv:2403.10772, in the extraction of the $\sigma$ meson pole located far from the real axis.