Mesonic eightfold way from dynamics and confinement in strongly coupled lattice quantum chromodynamics

• Published in: Journal of mathematical physics 2008-07, Vol.49 (7), p.1
• Main Authors: Francisco Neto, Antônio, O’Carroll, Michael, Faria da Veiga, Paulo A.
• Format: Article
• Language: English
• Subjects: BARYONS; Calculus; Correlation analysis; Exact sciences and technology; FLAVOR MODEL; FOUR-DIMENSIONAL CALCULATIONS; GAUGE INVARIANCE; HILBERT SPACE; ISOSPIN; LATTICE FIELD THEORY; Lattice theory; MASS; MASS FORMULAE; Mathematical methods in physics; Mathematics; OCTET MODEL; Physics; PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; PSEUDOSCALAR MESONS; QUANTUM CHROMODYNAMICS; QUANTUM MECHANICS; QUANTUM NUMBERS; Quantum theory; QUARKS; RESONANCE PARTICLES; Sciences and techniques of general use; SINGULARITY; STRONG-COUPLING MODEL; SU-3 GROUPS; VECTOR MESONS; Vector space
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.2903751

We show the existence of all the 36 eightfold way mesons and determine their masses and dispersion curves exactly, from dynamical first principles such as directly from the quark-fluon dynamics. We also give a proof of confinement below the two-meson energy threshold. For this purpose, we consider an imaginary time functional integral representation of a 3+1 dimensional lattice QCD model with Wilson action, SU ( 3 ) f global and SU ( 3 ) c local symmetries. We work in the strong coupling regime, such that the hopping parameter κ > 0 is small and much larger than the plaquette coupling β > 1 / g 0 2 ⩾ 0 ( β ⪡ κ ⪡ 1 ). In the quantum mechanical physical Hilbert space H , a Feynman-Kac type representation for the two-meson correlation and its spectral representation are used to establish an exact rigorous connection between the complex momentum singularities of the two-meson truncated correlation and the energy-momentum spectrum of the model. The total spin operator J and its z -component J z are defined by 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 mesons admit a labelling in terms of the quantum numbers of total isospin I , the third component I 3 of total isospin, the z -component J z of total spin and quadratic Casimir C 2 for SU ( 3 ) f . With this labelling, the mesons can be organized into two sets of states, distinguished by the total spin J . These two sets are identified with the SU ( 3 ) f nonet of pseudo-scalar mesons ( J = 0 ) and the three nonets of vector mesons ( J = 1 , J z = ± 1 , 0 ) . Within each nonet a further decomposition can be made using C 2 to obtain the singlet state ( C 2 = 0 ) and the eight members of the octet ( C 2 = 3 ) . By casting the problem of determination of the meson masses and dispersion curves into the framework of the the anaytic implicit function theorem, all the masses m ( κ , β ) are found exactly and are given by convergent expansions in the parameters κ and β . The masses are all of the form m ( κ , β = 0 ) ≡ m ( κ ) = − 2 ln κ − 3 κ 2 / 2 + κ 4 r ( κ ) with r ( 0 ) ≠ 0 and r ( κ ) real analytic; for β > 0 , m ( κ , β ) + 2 ln κ is jointly analytic in κ and β . The masses of the vector mesons are independent of J z and are all equal within each octet. All isospin singlet masses are also equal for the vector mesons. For each nonet and β = 0 , up to and including O ( κ 4 ) , the masses of the oct