Dynamical eightfold way mesons in strongly coupled lattice QCD

Abstract

We consider a 3 + 1 lattice QCD model with three quark flavors, local SU(3)c gauge symmetry, global SU(3) f isospin or flavor symmetry, in an imaginary-time formulation and with strong coupling (a small hopping parameter κ > 0 and a plaquette coupling β > 0, 0 < β κ 1). Associated with the model there is an underlying physical quantum mechanical Hilbert space H which, via a Feynman-Kac formula, enables us to introduce spectral representations for correlations and obtain the low-lying energy-momentum spectrum exactly. Using the decoupling of hyperplane method and concentrating on the subspace He ⊂ 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. Besides the 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 Jz are defined using π/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 SU(3) f nonets associated with the vector mesons (J = 1, Jz = 0, ±1) and one nonet associated with the pseudo-scalar mesons (J = 0). Each nonet admits a further decomposition into a SU(3) f singlet (C2 = 0) and octet (C2 = 3). The particles are detected by isolated dispersion curves w(p) in the energy-momentum spectrum. They are all of the form, for β = 0, w(p) = −2 ln κ − 3κ 2 /2 + (1/4)κ 2 3 j=1 2(1 − cos p j) + κ 4 r(κ, p), with |r(κ, p)| ≤ const. For the pseudo-scalar mesons r(κ, p) is jointly analytic in κ and p j , for |κ| and |Im p j | small. The meson masses are given by m(κ) = −2 ln κ − 3κ 2 /2 + κ 4 r(κ), with r(0) = 0 and r(κ) real analytic; they are also analytic in β. 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 β = 0, up to and including O(κ 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 pseudo-scalar, vector meson mass splitting (between J = 0 and J = 1) given by 2κ 4 + O(κ 6); at β = 0, analytic in β and the splitting persists for β << κ. Using a correlation subtraction method, we show the 36 meson states give the only spectrum in H e up to near the two-meson threshold of ≈ −4 ln κ. Combining our present result with a similar one for baryons (of asymptotic mass −3 ln κ) shows that the model does exhibit confinement up to near the two-meson threshold.

Exact Mesonic Eightfold Way From Dynamics and Confinement in Strongly Coupled Lattice QCD

“…Following Refs. [3,4], a correlation subtraction method is used to show that the eightfold way meson spectrum is the only spectrum in all H e , up to near the two-meson threshold of ≈ −4 ln κ. For this, we introduce the subtracted correlation…”

“…We use the same model presented in Refs. [3,4]. The model is the SU(3) f lattice QCD model with the partition function given formally by Z = e −S(ψ, ψ,g) dψd ψdμ(g), and for a function F (ψ, ψ, g), the normalized correlations are…”

“…As in Refs. [3,4], by taking the Fourier transform of the F-K formula, a connection is rigorously established between the spectrum of the self-adjoint E-M operators H ≥ 0 and P i=1,2,3 , and singularities of the two-meson correlation. Here, we have Ť 2 0 = e −2H and Ťj = e iP j .…”

“…In this way, the total spin operator J and its z−component J z are defined using π/2 rotations about the spatial coordinate axis and agree with the infinitesimal generators for the continuum (see Refs. [3,4]).…”

“…More recently, we showed that the the low-lying E-M spectrum consists solely of the eightfold way particles [2], i.e. the eightfold way mesons [3] and baryons [4], and no glueballs. If 0 < κ β then the low-lying spectrum consists only of glueballs.…”