We consider asymptotically free gauge theories with gauge group SU(Nc) and Nf quarks with mass mq≪ΛQCD that undergo chiral symmetry breaking and confinement. We propose a bootstrap method to compute the S matrix of the pseudo-Goldstone bosons (pions) that dominate the low-energy physics. For the important case of Nc=3, Nf=2, a numerical implementation of the method gives the phase shifts of the S0, P1 and S2 waves in good agreement with experimental results. The method incorporates gauge theory information (Nc, Nf, mq, ΛQCD) by using the form-factor bootstrap recently proposed by Karateev, Kuhn and Penedones together with a finite energy version of the Shifman-Vainshtein-Zakharov (SVZ) sum rules. At low energy we impose constraints from chiral symmetry breaking. The only low-energy numerical inputs are the pion mass mπ and the quark and gluon condensates. Published by the American Physical Society 2024