Non-perturbative determination of meson masses and low-energy constants in large-$N$ QCD

TL;DR

This work computes non-perturbative, first-principles results for the low-lying meson spectrum and chiral low-energy constants in the large-$N$ limit of QCD using twisted volume reduction (TEK). It leverages TEK to reach $N$ up to 841 with multiple lattice spacings and quark masses, enabling controlled chiral and continuum extrapolations and enabling access to excited meson states and Regge trajectories. The authors determine $\Sigma$, $F_π$, $B=\Sigma/F_π^2$, and $\bar{\ell}_4$ (and their $1/N$ corrections) by combining Banks–Casher, GMOR relations, and mode-number methods, with non-perturbative renormalization; they also extract $Z_P/Z_S$ from Dirac spectra. Key findings include $B_R/\sqrt{\sigma}=5.58(26)$, $\Sigma_R/(N\sqrt{σ^3})=0.0889(23)$, $F_π/(\sqrt{σ}\sqrt{N})=0.1262(34)$, and $\bar{\ell}_4/N=0.446(55)$, along with parallel radial Regge slopes $μ_r/\sqrt{σ}=3.65(21)$ (π) and $3.95(24)$ (ρ), supporting universality and providing a quantified picture of sub-leading $1/N$ effects when combined with finite-$N$ data.

Abstract

We provide first-principles non-perturbative determinations of the low-lying meson mass spectrum of large-$N$ QCD in the 't Hooft limit $N_{\scriptscriptstyle{\rm f}}/N\to 0$, as well as of three low-energy constants appearing in the QCD chiral expansion: the quark condensate $Σ$, the pion decay constant $F_π$, and the next-to-leading-order coupling $\bar{\ell}_4$. Using the excited state masses in the $π$ and $ρ$ channels, we are able to investigate the behavior of their radial Regge trajectories. Concerning QCD low-energy constants, we are able to assess the magnitude of sub-leading corrections in $1/N$ by combining our $N=\infty$ results with previous finite-$N$ determinations. Our calculation exploits large-$N$ twisted volume reduction to efficiently perform numerical Monte Carlo simulations of the large-$N$ lattice discretized theory. We employ several values of $N$ up to $N=841$, 5 values of the lattice spacing, and several values of the quark mass, to achieve controlled continuum and chiral extrapolations.

Figures (18)

• Figure 1: Continuum limit of ratios of quantities used for scale setting assuming standard $\mathcal{O}(a^2)$ lattice artifacts for gluonic quantities.
• Figure 2: Chiral behavior of the pion mass (top panel) and of the PCAC quark mass (bottom panel) as a function of the Wilson hopping parameter $\kappa$.
• Figure 3: Examples of exponential decays of the $\rho$ and $a_0$ optimal correlators obtained from the resolution of the GEVP, and related plateaus in the effective masses, cf. Eq. \ref{['eq:fit_opt_tcorr']} and Eq. \ref{['eq:effmass_def']}. All plots in this figure refer to $N=841$, $b=0.370$, $\kappa=0.1540$.
• Figure 4: Chiral-continuum extrapolations of the $\rho$, $a_0$, $a_1$, $b_1$, $\pi^{*}$ and $\rho^{*}$ masses.. In all cases it is sufficient to assume a linear dependence in $m_\pi^2/\sigma$ to describe the pion mass dependence of these masses, except for the $a_0$ meson, for which it is necessary to also include a further $m_\pi^4/\sigma^2$ correction. The displayed points represent the lattice determinations for each choice of $(b,\kappa)$ after the subtraction of the lattice artifact term $k_{{{\rm A}}} a\sqrt{\sigma}$. The dashed lines and shaded bands represent our continuum results for $m_{{\rm A}}(m_\pi)/\sqrt{\sigma}$. The starred points represent the chiral limit in the continuum, while the crossed points represent the previous large-$N$ TEK chiral determinations of Ref. Perez:2020vbn.
• Figure 5: In these figures we display our best determination of the final continuum results for the mass of the $\pi^{**}$ (left panel) and $\rho^{**}$ (right panel) mesons in the chiral limit, see the text for more details.