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Title: Eightfold way from dynamical first principles in strongly coupled lattice quantum chromodynamics

Abstract

We obtain from first principles, i.e., from the quark-gluon dynamics, the Gell'Mann-Ne'eman baryonic eightfold way energy momentum spectrum exactly in an imaginary-time functional integral formulation of strongly coupled lattice quantum chromodynamics in 3+1 dimensions, with local SU(3){sub c} gauge and global SU(3){sub f} flavor symmetries. We take the hopping parameter {kappa} and the pure gauge coupling {beta} satisfying the strong coupling regime condition 0{<=}{beta}<<{kappa}<<1. The form of the 56 baryon fields emerges naturally from the dynamics and is unveiled using the hyperplane decoupling method. There is no a priori guesswork. In the associated physical quantum mechanical Hilbert space H, spectral representations are derived for the two-baryon functions, which are used to rigorously detect the particles in the energy-momentum spectrum. Using the SU(3){sub f} symmetry, the 56 baryon states admit a decomposition into 8x2 states associated with a spin 1/2 octet and 10x4 states associated with a spin 3/2 decuplet. The states are labeled by the quantum numbers of total hypercharge Y, total isospin I, its third component I{sub 3}, and the value of the quadratic Casimir of SU(3){sub f}; they also carry a label of total spin J and its z component J{sub z}. The total spin operators are definedmore » using {pi}/2 rotations about the spatial coordinate axes and for improper zero momentum baryon states agree with the infinitesimal generators of the continuum. We show there is a partial restoration of continuous rotational symmetry which implies that all the octet (decuplet) masses are the same. For {beta}=0, the masses of the 56 baryon states have the form M=-3 ln {kappa}-3{kappa}{sup 3}/4+{kappa}{sup 6}r({kappa}), with r({kappa}) analytic. There is no mass splitting within the octet (decuplet). However, we find an octet-decuplet mass splitting given by 3{kappa}{sup 6}/4+O({kappa}{sup 7}). For {beta}=0, [M({kappa},{beta})-(-3 ln{kappa})], the non-singular part of the masses, is analytic in {kappa} and {beta} and the mass splitting persists for {beta}{ne}0. For spatial momentum p{ne}0, p=(p{sup 1},p{sup 2},p{sup 3})(set-membership sign)(-{pi},{pi}]{sup 3}, the 56 baryon dispersion curves have the form w({kappa},p)=-3 ln {kappa}-3{kappa}{sup 3}/4+{kappa}{sup 3}{sigma}{sub j=1,2,3}(1-cos p{sup j})/4+r({kappa},p), where r({kappa},p) is of O({kappa}{sup 6}). For the octet, r({kappa},p) is jointly analytic in {kappa} and in each p{sup j} for small |Im p{sup j}|. For each baryon, there is an antibaryon related to it by charge conjugation and with identical spectral properties. It is shown that the spectrum associated with baryons and antibaryons is the only spectrum in the subspace of H with an odd number of quarks, up to near the meson-baryon energy threshold of {approx_equal}-5 ln {kappa}. A new time reflection is found which is used to define a local spin flip symmetry. The spin flip symmetry, together with the usual parity, time reversal, and spatial {pi}/2 rotation symmetries and analytic implicit function arguments, are used to obtain these results. Our method extends to the SU(N){sub f} case to uncover (2N+2){exclamation_point}/[3{exclamation_point}(2N-1){exclamation_point}] baryon states and also to treat mesons. Coupling our baryon results with our similar results for the eightfold mesons (of asymptotic mass -2 ln{kappa}) shows that the model exhibits confinement up to near the two-meson threshold.« less

Authors:
;  [1]
  1. Departamento de Matematica Aplicada e Estatistica, ICMC-USP, C.P. 668, 13560-970 Sao Carlos SP (Brazil)
Publication Date:
OSTI Identifier:
21100258
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 49; Journal Issue: 4; Other Information: DOI: 10.1063/1.2804858; (c) 2008 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ANTIBARYONS; ASYMPTOTIC SOLUTIONS; C INVARIANCE; CASIMIR EFFECT; FLAVOR MODEL; GLUONS; HILBERT SPACE; HYPERCHARGE; ISOSPIN; LATTICE FIELD THEORY; MASS FORMULAE; MESONS; OCTET MODEL; QUANTUM CHROMODYNAMICS; QUANTUM MECHANICS; QUARKS; SPIN FLIP; STRONG-COUPLING MODEL; SU-3 GROUPS; SYMMETRY

Citation Formats

Faria da Veiga, Paulo A, and O'Carroll, Michael. Eightfold way from dynamical first principles in strongly coupled lattice quantum chromodynamics. United States: N. p., 2008. Web. doi:10.1063/1.2804858.
Faria da Veiga, Paulo A, & O'Carroll, Michael. Eightfold way from dynamical first principles in strongly coupled lattice quantum chromodynamics. United States. https://doi.org/10.1063/1.2804858
Faria da Veiga, Paulo A, and O'Carroll, Michael. Tue . "Eightfold way from dynamical first principles in strongly coupled lattice quantum chromodynamics". United States. https://doi.org/10.1063/1.2804858.
@article{osti_21100258,
title = {Eightfold way from dynamical first principles in strongly coupled lattice quantum chromodynamics},
author = {Faria da Veiga, Paulo A and O'Carroll, Michael},
abstractNote = {We obtain from first principles, i.e., from the quark-gluon dynamics, the Gell'Mann-Ne'eman baryonic eightfold way energy momentum spectrum exactly in an imaginary-time functional integral formulation of strongly coupled lattice quantum chromodynamics in 3+1 dimensions, with local SU(3){sub c} gauge and global SU(3){sub f} flavor symmetries. We take the hopping parameter {kappa} and the pure gauge coupling {beta} satisfying the strong coupling regime condition 0{<=}{beta}<<{kappa}<<1. The form of the 56 baryon fields emerges naturally from the dynamics and is unveiled using the hyperplane decoupling method. There is no a priori guesswork. In the associated physical quantum mechanical Hilbert space H, spectral representations are derived for the two-baryon functions, which are used to rigorously detect the particles in the energy-momentum spectrum. Using the SU(3){sub f} symmetry, the 56 baryon states admit a decomposition into 8x2 states associated with a spin 1/2 octet and 10x4 states associated with a spin 3/2 decuplet. The states are labeled by the quantum numbers of total hypercharge Y, total isospin I, its third component I{sub 3}, and the value of the quadratic Casimir of SU(3){sub f}; they also carry a label of total spin J and its z component J{sub z}. The total spin operators are defined using {pi}/2 rotations about the spatial coordinate axes and for improper zero momentum baryon states agree with the infinitesimal generators of the continuum. We show there is a partial restoration of continuous rotational symmetry which implies that all the octet (decuplet) masses are the same. For {beta}=0, the masses of the 56 baryon states have the form M=-3 ln {kappa}-3{kappa}{sup 3}/4+{kappa}{sup 6}r({kappa}), with r({kappa}) analytic. There is no mass splitting within the octet (decuplet). However, we find an octet-decuplet mass splitting given by 3{kappa}{sup 6}/4+O({kappa}{sup 7}). For {beta}=0, [M({kappa},{beta})-(-3 ln{kappa})], the non-singular part of the masses, is analytic in {kappa} and {beta} and the mass splitting persists for {beta}{ne}0. For spatial momentum p{ne}0, p=(p{sup 1},p{sup 2},p{sup 3})(set-membership sign)(-{pi},{pi}]{sup 3}, the 56 baryon dispersion curves have the form w({kappa},p)=-3 ln {kappa}-3{kappa}{sup 3}/4+{kappa}{sup 3}{sigma}{sub j=1,2,3}(1-cos p{sup j})/4+r({kappa},p), where r({kappa},p) is of O({kappa}{sup 6}). For the octet, r({kappa},p) is jointly analytic in {kappa} and in each p{sup j} for small |Im p{sup j}|. For each baryon, there is an antibaryon related to it by charge conjugation and with identical spectral properties. It is shown that the spectrum associated with baryons and antibaryons is the only spectrum in the subspace of H with an odd number of quarks, up to near the meson-baryon energy threshold of {approx_equal}-5 ln {kappa}. A new time reflection is found which is used to define a local spin flip symmetry. The spin flip symmetry, together with the usual parity, time reversal, and spatial {pi}/2 rotation symmetries and analytic implicit function arguments, are used to obtain these results. Our method extends to the SU(N){sub f} case to uncover (2N+2){exclamation_point}/[3{exclamation_point}(2N-1){exclamation_point}] baryon states and also to treat mesons. Coupling our baryon results with our similar results for the eightfold mesons (of asymptotic mass -2 ln{kappa}) shows that the model exhibits confinement up to near the two-meson threshold.},
doi = {10.1063/1.2804858},
url = {https://www.osti.gov/biblio/21100258}, journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 4,
volume = 49,
place = {United States},
year = {2008},
month = {4}
}