Bootstrapping gauge theories

Abstract

We consider asymptotically free gauge theories with gauge group $SU(N_c)$ and $N_f$ quarks with mass $m_q\ll \hat{I}{›}_{\text{QCD}}$ 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 ($N_c$, $N_f$, $m_q$, $\hat{I}{›}_{\text{QCD}}$) 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 $m_{\ddot{I}}€$ and the quark and gluon condensates.

v3: 43 pages, 11 figures, appendix added testing numerical parameters; Joint publications in Phys. Rev. Lett. (short version) and Phys. Rev. D (this version)