Loop Diagrams in Dual-Resonance Theories with Nonlinear Trajectories

Rules are presented for calculating loop diagrams in a dual-resonance theory with nonlinear trajectories. Explicit expressions for the self-energy graph, the vacuum bubble graph, and the vertex graph are given. In the limit of linear trajectories the integrands of these graphs diverge. This is in contrast to the behavior of the tree graphs which approach the Veneziano tree graphs in the same linear limit. It is also shown that when the trajectories are nonlinear, the self-energy graph including the $\int d^{4}k$ is finite. It should be emphasized that these rules have been developed without consideration of the factorization properties of the nonlinear $N$-point Born term. In this sense they are analogous to the original Veneziano-loop-diagram rules of Kikkawa, Sakita, and Virasoro, which were subsequently modified by the requirements of factorization. However, although our rules may likewise be modified, they present a simplified context for developing techniques which are of general value for the study of the nonlinear theory.