  We consider asymptotically free gauge theories with gauge group $SU(N_c)$ and
$N_f$ quarks with mass $m_q\ll \Lambda_{\text{QCD}}$ 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 $N_c=3$, $N_f=2$, 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 ($N_c$, $N_f$, $m_q$, $\Lambda_{\text{QCD}}$) by using the
form-factor bootstrap recently proposed by Karateev, Kuhn and Penedones
together with a finite energy version of the SVZ sum rules. At low energy we
impose constraints from chiral symmetry breaking. The only low energy numerical
inputs are the pion mass $m_\pi$ and the quark and gluon condensates.
