Bootstrapping gauge theories

TL;DR

The paper tackles the problem of extracting the low-energy S-matrix of pseudo-Goldstone pions in asymptotically free gauge theories with confinement and chiral symmetry breaking. It develops a bootstrap approach that unites S-matrix constraints with UV information from form-factor bootstrap and finite-energy SVZ sum rules, enabling a link from high-energy QCD data to IR pion dynamics. In the Nc=3, Nf=2 case, the method yields S0, S2 phase shifts consistent with experimental trends and a rho-resonance in P1 that improves with UV inputs. The framework offers a general route to derive IR observables for gauge theories from UV data and can be extended to other color/flavor configurations and lattice inputs.

Abstract

We consider asymptotically free gauge theories with gauge group $SU(N_c)$ and $N_f$ quarks with mass $m_q\ll Λ_{\text{QCD}}$ 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 ($N_c$, $N_f$, $m_q$, $Λ_{\text{QCD}}$) 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 $m_π$ and the quark and gluon condensates.

Figures (11)

• Figure 1: In the $s$ complex plane we have analytic functions with possible cuts on the real axis: the partial waves $f^I_\ell(s)$, the form factors $F^I_\ell(s)$ and the two current correlators $\Pi^I_\ell(s)$. The gauge theory bootstrap matches those functions across different scales described by chiral perturbation theory ($\chi_{\hbox{PT}}$) at low energy, the S-matrix/form factor bootstrap at intermediate energies where pion scattering is expected to saturate unitarity and perturbative QCD (pQCD) at large energies. We take $s_0$ such that $\alpha_s\simeq 0.4$.
• Figure 2: Contour of integration for the Finite Energy Sum Rule (FESR) of an analytic function with a cut along the blue line. The integral around the red contour is zero. This relates the integral of the jump across the cut with the asymptotic behavior on the circle for large $s_0$.
• Figure 3: The space of amplitudes projected onto a plane parameterized by $f^0_0(s=3)$ and $f^1_1(s=3)$ as a result of pure S-matrix bootstrap with the constraints of analyticity, crossing and unitarity.
• Figure 4: We plot the allowed space (only the relevant $f_0^0(3)>0$ side) restricted by the chiral constraints \ref{['chiralconstraints']} with tolerances $\epsilon^{\chi}=6\times10^{-3},4\times10^{-3},2\times10^{-3},1\times10^{-3},6\times10^{-4},2\times10^{-4}$ (from the outer shape inward). The black line are the values given by the linear Weinberg model \ref{['h5']} with varying values of $f_\pi$ and the black dot the one with $f_\pi\simeq\, 92\hbox{MeV}$. We want to impose the constraints without excluding that point. A few highlighted points are chosen to explore the partial waves in the unphysical region as plotted in fig \ref{['linearity']}.
• Figure 5: Partial waves in the unphysical region $0<s<4$ for the points highlighted in fig.\ref{['chiral']}. We compare them with the linear approximation (dashed line) and find that the green points match better the linear prediction without excluding the physical value of $f_\pi$ in fig.\ref{['chiral']}.