Renormalization and Unitarity in the Dual-Resonance Model

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Abstract
All the one-loop graphs of the dual-resonance model are explicity calculated. These graphs fall into three categories:planar, nonorientable, and orientable nonplanar. Using the properties of various elliptic functions, we are able to generalize the renormalization procedure, obtained previously for the planar diagrams, to the other two categories. The orientable nonplanar diagrams turn out to be particularly interesting. First, their integration regions have to be reduced from the ones naively obtained in order to avoid multiple counting. Secondly, they give rise to new singularities (branch points) in channels that are naturally identified as having vacuum quantum numbers. These singularities are probably related to the Pomeranchukon. The question of unitarity is explored at the one-loop level, i.e., to the first nontrivial order in the perturbation series. Although the counting of diagrams is somewhat subtle, a rather simple result emerges: All inequivalent diagrams (with respect to duality transformations) should be counted with equal weight. Finally, it is indicated that three of the four primitive renormalized loop operators of the theory can be obtained from the formulas of this paper.