Unitary Nonplanar Closed Loops. II
Abstract
Using the methods developed in a previous paper, we generalize the calculation of the dual amplitude for a nonplanar diagram with a single closed loop to the case with an arbitrary number of "twisted vertices." Just as with the fourpoint function discussed in the previous paper, we find that, if we write the amplitude M=_{Σ}d^{4}ΠM(Π), M(Π) is periodic in Π, the period being given by the sum of the fourmomenta of the "twisted vertices." We then show in the general case that the prescription of choosing Σ to range over just one period yields the imaginary part required by perturbative unitarity. We verify that M so defined is dual in three different ways. We show explicitly that our result is equivalent to the KikkawaKleinSakitaVirasoro (KKSV) prescription; we also prove duality directly from the BardakciRuegglike form without reference to the KKSV structure; and finally we show that duality is manifest within the operator formalism before the trace is performed.
 Publication:

Physical Review D
 Pub Date:
 September 1970
 DOI:
 10.1103/PhysRevD.2.1071
 Bibcode:
 1970PhRvD...2.1071T