The nonplanar one-loop amplitude in Witten's string field theory

We develop methods for calculating the one-open-string-irreducible diagrams for Witten's string field theory. These provide the coefficients for an expansion of the quantum effective action in a power series in the string field Φ. Each external leg of a diagram is labelled by an element of the BRST first-quantized string state space. We choose the standard oscillator Fock states as a convenient basis for this labelling. Our method is to represent each Feynman diagram as a path integral over space-time coordinates x^μ and the bosonized world-sheet ghost field φ. The dependence of the diagram on the external strings and ghost insertions is obtained by completing the square. The coefficient of this dependence, the "measure", is then inferred by exploiting the Weyl invariance of the Polyakov path integrals. As an illustration of our methods, we analyze in detail the one loop nonplanar two string function. This is the simplest diagram containing information on closed strings. Calling p^μ the energy-momentum carried by one of the external legs, we find poles in p² at -α′p²= 1 2n - 4, with n=0,1,2,.... The pole locations with n≠8k are unphysical and such poles must disappear for physical open string states. We confirm this explicitly for n=1,2 and argue that this decoupling happens generally.