Unitary Nonplanar Closed Loops

We show how one can calculate the Veneziano amplitude for an arbitrary nonplanar diagram with one closed loop. In particular, we calculate explicitly the nonplanar diagram with four external particles and one "twisted vertex." If we write the amplitude M=∫d⁴Π M(Π), we find that our result for M(Π) is periodic in the loop momentum Π, so that the expression for M does not converge. We show that, if we integrate over just one period, we obtain the correct result for the imaginary part of the amplitude in the s channel. Consequently, we adopt this as our definition of the unitary nonplanar amplitude. Finally, we show that with this redefinition, M satisfies the duality requirement that it be completely symmetric in the "untwisted" momenta.