Bootstrapping gauge theories

TL;DR

This paper introduces a bootstrap approach to compute the low-energy S-matrix of pions in gauge theories, achieving results consistent with experiments for QCD-like theories.

Contribution

It develops a novel bootstrap method incorporating gauge theory data and chiral symmetry constraints to calculate pion scattering phase shifts.

Findings

Phase shifts for S0, P1, S2 waves match experimental data.

Method effectively integrates gauge theory parameters into low-energy predictions.

Numerical results validate the bootstrap approach for N_c=3, N_f=2 QCD-like theories.

Abstract

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.