Existence of mesons and mass splitting in strong coupling lattice quantum chromodynamics

We consider one flavor lattice quantum chromodynamics in the imaginary time functional integral formulation for space dimensions d=2, 3 with 4×4 Dirac spin matrices, small hopping parameter κ, 0<κ≪1, and zero plaquette coupling. We determine the energy-momentum spectrum associated with four-component gauge invariant local meson fields which are composites of a quark and an antiquark field. For the associated correlation functions, we establish a Feynman–Kac formula and a spectral representation. Using this representation, we show that the mass spectrum consists of two distinct masses ma and mb, given by mc=−2 ln κ+rc(κ), c=a,b, where rc is real analytic. For d=2, ma and mb have multiplicity two and the mass splitting is κ4+O(κ6); for d=3, one mass has multiplicity one and the other three, with mass splitting 2κ4+O(κ6). In the subspace of the Hilbert space generated by an even number of fermion fields the dispersion curves are isolated (upper gap property) up to near the two-meson threshold of asymptotic m...