Existence of Baryons, Baryon Spectrum and Mass Splitting in Strong Coupling Lattice QCD

Abstract

We consider a functional integral formulation for one-flavor lattice Quantum Chromodynamics in d=2,3 space dimensions and imaginary time, and work in the regime of the small hopping parameter , and zero plaquette coupling. Following the standard construction, this model exhibits positivity which is used to obtain the underlying physical Hilbert space . Then, using a Feynman-Kac formalism, we write the correlation functions for the model as functional integrals over the space of Grassmannian (fermionic) fields for one quark specie and the SU(3) gauge fields. We determine the energy-momentum spectrum associated with gauge invariant local baryon (anti-baryon) fields which are composites of three quark (anti-quark) fields. With the associated correlation functions, we establish a Feynman-Kac formula, and a spectral representation for the Fourier transform of the two-point functions. This representation allows us to show that baryons and anti-baryons arise as tightly bound, bound states of three (anti-)quarks. Labelling the components of the baryon fields by s=3/2,1/2,-1/2,-3/2, we show that the baryon and anti-baryon mass spectrum only depends on |s|, and the associated masses are given by M s = −3lnκ+r s (κ), where r s (κ) is real analytic in κ, for each d=2,3. The mass splitting is M 3/2 −M 1/2 =18κ6, for d=2 and, if any, is at least of (κ7), for d=3. In the subspace o ⊂ generated by an odd number of fermions, the baryon and anti-baryon energy-momentum dispersion curves are isolated up to near the baryon-meson threshold −5lnκ (upper gap property), identical and are determined up to (κ5). The symmetries of coordinate reflections, spatial lattice rotations, parity and charge conjugation are established for the correlation functions, and are shown to be implemented on by unitary (anti-unitary, for time reversal) operators.