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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 3ϩ1 dimensional lattice QCD model with Wilson action, SU͑3͒ f global and SU͑3͒ c local symmetries. We work in the strong coupling regime, such that the hopping parameter Ͼ 0 is small and much larger than the plaquette coupling  Ͼ 1 / g 0 2 ജ 0 ͑ Ӷ Ӷ 1͒. 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 / 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 SU͑3͒ 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 SU͑3͒ f nonet of pseudo-scalar mesons (J =0͒ and the three nonets of vector mesons ͑J =1,J z = Ϯ 1,0͒. Within each nonet a further decomposition can be made using C 2 to obtain the singlet state ͑C 2 =0͒ and the eight members of the octet ͑C 2 =3͒. 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 m͑ , ͒ are found exactly and are given by convergent expansions in the parameters and . The masses are all of the form m͑ ,  =0͒ϵm͑͒ = −2ln −3 2 / 2+ 4 r͑͒ with r͑0͒ 0 and r͑͒ real analytic; for  Ͼ 0,m͑ , ͒ +2ln is jointly analytic in and . 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  =0, up to and including O͑ 4 ͒, the masses of the octet and the singlet are found to be equal. But there is a pseudoscalar-vector meson mass splitting given by 2 4 + O͑ 6 ͒ and the splitting persists for  Ͼ 0. For  = 0, the dispersion curves are all of the form w͑p ជ͒ =−2 ln −3 2 / 2+ ͑ 1 4 ͒ 2 ͚ P ͑0͒ ͑u,w͒⌳ ͑2͒ ͑w,z͒Q ͑0͒ ͑z,v͒.However, for w 0 ഛ p, z 0 ജ p +1, ⌳ ͑2͒ ͑w , z͒ =−͓⌳ ͑0͒ J ͑2͒ ⌳ ͑0͒ ͔͑w , z͒, which is obtained by taking the second derivative of the relation ⌳J = 1 and observing that ͓⌳ ͑0͒ J ͑1͒ ⌳ ͑1͒ ͔͑w , z͒ = ͓⌳ ͑1͒ J ͑1͒ ⌳ ͑0͒ ͔ ϫ͑w , z͒ = 0 for w 0 ഛ p, z 0 ജ p + 1. With these restrictions on sums, we get J ͑2͒ ͑x,y͒ = G M M ͑2͒ ͑x,y͒ = ͓G M M ͑0͒ ؠ G M M ͑0͒ ͔͑x,y͒, so that 072301-27 Eightf...
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 3ϩ1 dimensional lattice QCD model with Wilson action, SU͑3͒ f global and SU͑3͒ c local symmetries. We work in the strong coupling regime, such that the hopping parameter Ͼ 0 is small and much larger than the plaquette coupling  Ͼ 1 / g 0 2 ജ 0 ͑ Ӷ Ӷ 1͒. 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 / 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 SU͑3͒ 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 SU͑3͒ f nonet of pseudo-scalar mesons (J =0͒ and the three nonets of vector mesons ͑J =1,J z = Ϯ 1,0͒. Within each nonet a further decomposition can be made using C 2 to obtain the singlet state ͑C 2 =0͒ and the eight members of the octet ͑C 2 =3͒. 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 m͑ , ͒ are found exactly and are given by convergent expansions in the parameters and . The masses are all of the form m͑ ,  =0͒ϵm͑͒ = −2ln −3 2 / 2+ 4 r͑͒ with r͑0͒ 0 and r͑͒ real analytic; for  Ͼ 0,m͑ , ͒ +2ln is jointly analytic in and . 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  =0, up to and including O͑ 4 ͒, the masses of the octet and the singlet are found to be equal. But there is a pseudoscalar-vector meson mass splitting given by 2 4 + O͑ 6 ͒ and the splitting persists for  Ͼ 0. For  = 0, the dispersion curves are all of the form w͑p ជ͒ =−2 ln −3 2 / 2+ ͑ 1 4 ͒ 2 ͚ P ͑0͒ ͑u,w͒⌳ ͑2͒ ͑w,z͒Q ͑0͒ ͑z,v͒.However, for w 0 ഛ p, z 0 ജ p +1, ⌳ ͑2͒ ͑w , z͒ =−͓⌳ ͑0͒ J ͑2͒ ⌳ ͑0͒ ͔͑w , z͒, which is obtained by taking the second derivative of the relation ⌳J = 1 and observing that ͓⌳ ͑0͒ J ͑1͒ ⌳ ͑1͒ ͔͑w , z͒ = ͓⌳ ͑1͒ J ͑1͒ ⌳ ͑0͒ ͔ ϫ͑w , z͒ = 0 for w 0 ഛ p, z 0 ജ p + 1. With these restrictions on sums, we get J ͑2͒ ͑x,y͒ = G M M ͑2͒ ͑x,y͒ = ͓G M M ͑0͒ ؠ G M M ͑0͒ ͔͑x,y͒, so that 072301-27 Eightf...
We consider a 3 + 1 lattice QCD model with three quark flavors, local SU(3)c gauge symmetry, global SU(3) f isospin or flavor symmetry, in an imaginary-time formulation and with strong coupling (a small hopping parameter κ > 0 and a plaquette coupling β > 0, 0 < β κ 1). Associated with the model there is an underlying physical quantum mechanical Hilbert space H which, via a Feynman-Kac formula, enables us to introduce spectral representations for correlations and obtain the low-lying energy-momentum spectrum exactly. Using the decoupling of hyperplane method and concentrating on the subspace He ⊂ H of vectors with an even number of quarks, we obtain the one-particle spectrum showing the existence of 36 meson states from dynamical first principles, i.e. directly from the quark-gluon dynamics. Besides the SU(3) f quantum numbers (total hypercharge, quadratic Casimir C 2 , total isospin and its 3rd component), the basic excitations also carry spin labels. The total spin operator J and its z-component Jz are defined 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 eightfold way meson particles are given by linear combinations of these 36 states and can be grouped into three SU(3) f nonets associated with the vector mesons (J = 1, Jz = 0, ±1) and one nonet associated with the pseudo-scalar mesons (J = 0). Each nonet admits a further decomposition into a SU(3) f singlet (C2 = 0) and octet (C2 = 3). The particles are detected by isolated dispersion curves w(p) in the energy-momentum spectrum. They are all of the form, for β = 0, w(p) = −2 ln κ − 3κ 2 /2 + (1/4)κ 2 3 j=1 2(1 − cos p j) + κ 4 r(κ, p), with |r(κ, p)| ≤ const. For the pseudo-scalar mesons r(κ, p) is jointly analytic in κ and p j , for |κ| and |Im p j | small. The meson masses are given by m(κ) = −2 ln κ − 3κ 2 /2 + κ 4 r(κ), with r(0) = 0 and r(κ) real analytic; they are also analytic in β. For a fixed nonet, the mass of the vector mesons are independent of J z and are all equal within each octet. All singlet masses are also equal for the vector mesons. For β = 0, up to and including O(κ 4), for each nonet, the masses of the octet and the singlet are found to be equal. All members of each octet have identical dispersions. Other dispersion curves may differ. Indeed, there is a pseudo-scalar, vector meson mass splitting (between J = 0 and J = 1) given by 2κ 4 + O(κ 6); at β = 0, analytic in β and the splitting persists for β << κ. Using a correlation subtraction method, we show the 36 meson states give the only spectrum in H e up to near the two-meson threshold of ≈ −4 ln κ. Combining our present result with a similar one for baryons (of asymptotic mass −3 ln κ) shows that the model does exhibit confinement up to near the two-meson threshold.
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