Dynamical eightfold way mesons in strongly coupled lattice QCD

Abstract

We consider a $3+1$ lattice QCD model with three quark flavors, local $\mathrm{SU}(3{)}_{c}$ gauge symmetry, global $\mathrm{SU}(3{)}_{f}$ flavor symmetry, in an imaginary-time formulation and with strong coupling (a small hopping parameter $\ensuremath{\kappa}>0$ and a plaquette coupling $\ensuremath{\beta}>0$, $0<\ensuremath{\beta}\ensuremath{\ll}\ensuremath{\kappa}\ensuremath{\ll}1$). Associated with the model there is an underlying physical quantum mechanical Hilbert space $\mathcal{H}$ which, via a Feynman-Kac formula, enables us to derive spectral representations for correlations and obtain the low-lying energy-momentum spectrum exactly. Using the decoupling of hyperplane method and concentrating on the subspace ${\mathcal{H}}_{e}\ensuremath{\subset}\mathcal{H}$ of vectors with an even number of quarks, we obtain the one-particle spectrum showing the existence of 36 meson states from dynamical first principles, i.e. directly from the quark-gluon dynamics. The particles are detected by isolated dispersion curves $w(\stackrel{\ensuremath{\rightarrow}}{p})$ in the energy-momentum spectrum. Besides the $\mathrm{SU}(3{)}_{f}$ quantum numbers (total hypercharge, quadratic Casimir ${C}_{2}$, total isospin and its 3rd component), the basic excitations also carry spin labels. The total spin operator $J$ and its $z$-component ${J}_{z}$ are defined using $\ensuremath{\pi}/2$ rotations about the spatial coordinate axes and agree with the infinitesimal generators of the continuum for improper zero-momentum meson states. The eightfold way meson particles are given by linear combinations of these 36 states and can be grouped into three $\mathrm{SU}(3{)}_{f}$ nonets associated with the vector mesons ($J=1$, ${J}_{z}=0$, $\ifmmode\pm\else\textpm\fi{}1$) and one nonet associated with the pseudoscalar mesons ($J=0$). Each nonet admits a further decomposition into a $\mathrm{SU}(3{)}_{f}$ singlet (${C}_{2}=0$) and an octet (${C}_{2}=3$). For $\ensuremath{\beta}=0$, the particle dispersion curves are all of the form $w(\stackrel{\ensuremath{\rightarrow}}{p})=\ensuremath{-}2\mathrm{ln}\ensuremath{\kappa}\ensuremath{-}3{\ensuremath{\kappa}}^{2}/2+(1/4){\ensuremath{\kappa}}^{2}\ensuremath{\sum}_{j=1}^{3}2(1\ensuremath{-}\mathrm{cos}{p}^{j})+{\ensuremath{\kappa}}^{4}r(\ensuremath{\kappa},\stackrel{\ensuremath{\rightarrow}}{p})$, with $\stackrel{\ensuremath{\rightarrow}}{p}\ensuremath{\in}(\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi}{]}^{3}$ and $|r(\ensuremath{\kappa},\stackrel{\ensuremath{\rightarrow}}{p})|\ensuremath{\le}\mathrm{const}$. For the pseudoscalar mesons, $r(\ensuremath{\kappa},\stackrel{\ensuremath{\rightarrow}}{p})$ is jointly analytic in $\ensuremath{\kappa}$ and ${p}^{j}$, for $|\ensuremath{\kappa}|$ and $|\mathcal{I}\mathrm{m}{p}^{j}|$ small. At $\ensuremath{\beta}=0$ the meson masses are given by $m(\ensuremath{\kappa})=\ensuremath{-}2\mathrm{ln}\ensuremath{\kappa}\ensuremath{-}3{\ensuremath{\kappa}}^{2}/2+{\ensuremath{\kappa}}^{4}r(\ensuremath{\kappa})$, with $r(0)\ensuremath{\ne}0$ and $r(\ensuremath{\kappa})$ real analytic; for $\ensuremath{\beta}\ensuremath{\ne}0$ the nonsingular part of the mass, $m(\ensuremath{\kappa},\ensuremath{\beta})+2\mathrm{ln}\ensuremath{\kappa}$, is jointly analytic in $\ensuremath{\kappa}$ and $\ensuremath{\beta}$. For a fixed nonet, the mass of the vector mesons are independent of ${J}_{z}$ and are all equal within each octet. All singlet masses are also equal for the vector mesons. For $\ensuremath{\beta}=0$, up to and including $\mathcal{O}({\ensuremath{\kappa}}^{4})$, for each nonet, the masses of the octet and the singlet are found to be equal. All members of each octet have identical dispersions. Other dispersion curves may differ. Indeed, there is a pseudoscalar, vector meson mass splitting (between $J=0$ and $J=1$) given by $2{\ensuremath{\kappa}}^{4}+\mathcal{O}({\ensuremath{\kappa}}^{6})$ at $\ensuremath{\beta}=0$, analytic in $\ensuremath{\beta}$ and the splitting persists for $\ensuremath{\beta}\ensuremath{\ll}\ensuremath{\kappa}$. Using a correlation subtraction method, we show the 36 meson states give the only spectrum in ${\mathcal{H}}_{e}$ up to near the two-meson threshold of $\ensuremath{\approx}\ensuremath{-}4\mathrm{ln}\ensuremath{\kappa}$. Combining our present result with a similar one for baryons (of asymptotic mass $\ensuremath{-}3\mathrm{ln}\ensuremath{\kappa}$) shows that the model does exhibit confinement up to near the two-meson threshold.