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...
