Divergence of the Two-Loop Planar Graph in the Dual-Resonance Model

The two-loop planar tadpole is constructed by inserting two projected planar tadpoles on a symmetric three-Reggeon vertex. The determinant, which is the generalization of the partition function appearing in the one-loop diagram, is studied as a function of the integration variables of the loops and propagators. It is found that although the determinant contains an infinite number of possibly exponentially divergent factors, only three of them actually diverge. The first two divergent factors are those associated with each tadpole. The additional divergence is associated with the closed path going around the two loops which does not cross itself. The calculation is also performed with the Koba-Nielsen variables of the original N-loop graph, and the same result is derived.