Large-$N$ $π^+π^+$ scattering from twisted Eguchi--Kawai reduction

We present the first computation of the $π^+π^+$ scattering length in large-$N$ QCD in the 't Hooft limit, $N_{\scriptscriptstyle{\rm f}}/N\to 0$, using the twisted Eguchi--Kawai formulation of the lattice theory. We employ values of $N$ up to 1156 and consider several quark masses and lattice spacings to achieve controlled large-$N$, continuum, and chiral limits. The chiral behavior of the scattering length is compared with next-to-leading-order Chiral Perturbation Theory predictions. Our result for the low-energy coupling describing the sub-leading quark-mass dependence is consistent with existing S-matrix bootstrap bounds. Combining TEK results with previous lattice QCD data obtained using standard methods for $3\le N\le 6$, we are also able to extract sub-leading $\mathcal{O}(1/N)$ corrections to the large-$N$ scattering length.