On the Energy-Momentum Spectrum and One-Meson Dispersion Curves in (3 + 1)-Dimensional Strongly Coupled Lattice QCD with three Flavors

Abstract

We use an imaginary-time formulation of quantum field theory to analyze a lattice QCD model with the improved Wilson action on a Euclidean (3 + 1)-dimensional unit hypercubic lattice, with three quark flavors, local SU(3)c gauge symmetry and global SU(3)f flavor symmetry. We work in the strong coupling regime, with a small hopping parameter κ > 0 and a much smaller plaquette coupling β > 0. This work extends our previous results on the pointwisely determined one-meson dispersion curves in the low-lying one-particle energy-momentum spectrum of strongly coupled lattice QCD models. Here, we give a smoothness characterization of the dispersion curves and provide bounds on the more than one-particle contributions to the spectral measure for the two-meson correlations. Denoting by H the underlying quantum mechanical physical Hilbert space, constructed according to the Osterwalder-Seiler prescription, our spectral results are derived from spectral representations for the positive self-adjoint energy and self-adjoint momentum operators in H. After establishing a lattice Feynman-Kac formula, these representations are used to rigorously identify complex momentum singularities of the lattice Fourier transform of special two-particle correlations with energy-momentum spectral points and identify the meson particles with isolated dispersion curves. To show isolated dispersion curves we must show both lower and upper spectral gaps. To determine the eightfold-way mesons, the usual spacetime and internal symmetries are heavily used to block decompose the lattice Fourier transform of the two-meson correlation matrix. Then, to solve the implicit equations determining the dispersion curves, we apply the classical analytic implicit function theorem to the blocks associated with total spin zero mesons and the Weierstrass preparation theorem to the blocks corresponding to total spin one. The Weierstrass preparation theorem is the natural tool to deal with the determination of the spectrum when degeneracies are present and the hypotheses for the analytic implicit function theorem are not fulfilled. All the 36 meson dispersion curves are of the form, for β = 0, w(p→)=−2lnκ+r(κ,p→), with r(κ,p→) of order κ2, giving an asymptotic mass -2 ln κ. A mass splitting of 2κ4 between the pseudoscalar and vector mesons occurs at order κ4. We determine smoothness properties of the functions r(κ,p→). The results for β = 0 are extended to small β > 0 using β-analyticity properties of correlations in the infinite-volume limit.