Bootstrapping gauge theories

We consider asymptotically free gauge theories with gauge group 𝑆⁢𝑈⁡(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 𝑆⁢0, 𝑃⁢1 and 𝑆⁢2 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.