We investigate the selection of certain dominant topological classes of Feynman diagrams in the large-n limit in the context of a non-Abelian gauge field theory, where the nontrivial couplings of the regular representation of the group are involved. We find that nonplanar diagrams are suppressed relative to locally planar diagrams by factors of l/n in the case of SU(n) and by factors of I/n in the case of SO(n) or Sp(n). We consider the concrete example of the meson bound states in a theory of colored quarks bound by colored gauge vector gluons, where the topological selection corresponds to that required for hadrons in a dual theory.
We show how duality diagrams may be rigorously formulated in terms of recoupling diagrams. This allows us to write down in closed form general solutions to the constraints imposed by duality and the absence of exotic resonances, and to give for the first time in closed form general expressions for the coupling constants in terms of the quark-hadron recoupling coefficients which are given by the 6j-like symbols of the appropriate group. We note the connection with the internal-symmetry factors given by the trace of a product of matrices, and we further note that SU(n) is not determined by the above constraints.
For the ranges of values of a, p, and to given in the text, (A4) represents a 5% increase over Weinberg's value of ao. Expression (A5) represents an increase in magnitude of a2 of 40%, due partly to the smallness of Weinberg's original value. For the combination 2ao-5a2 we find an increase of 20% over the value 6 4 in line with other estimates of this c~rrection.'~ 13 H . D. Dahmen and K. D. Rothe, Nucl. Phys. B15, 45 (1970).In the context of duality and the absence of exotic resonances, we discuss the question of the T-7 degeneracy. This is crucial for an understanding of the internal symmetry incorporated in the meson spectrum. We examine the situation where the ?r is exchange degenerate with A l and not B, which is in accord with experiment. The 7 should not then be degenerate with the T. Nevertheless, an inconsistency is demonstrable when factorization constraints are applied to the leading trajectory couplings in PV -+ PV meson reactions. This follows from an assumption of SU(2) only, not SU(3). The inconsistency is removed by assuming degenerate doubling of the particles on the A1 and D trajectories.
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