Infrared limit of left-handed string at genus one

Abstract

We extend the left-handed string formalism at one-loop level to focus on only the infrared limit, where the Green's function for the left-handed string is expanded around the cusp of the modular parameter. This expansion leads to the separating degeneration limit of a Riemann surface corresponding to a sphere and a torus connected by a long tube. The well-behaved short-distance behavior of the Green's function requires all marked points to be inserted on the sphere. Analogous to the tree-level calculations, we obtain Dirac $\delta$-functions by integrating out the anti-holomorphic variables. The constraints embedded in these $\delta$-functions, associated with the marked points on the sphere part of the Riemann surface, are the same Scattering Equations at the tree-level. After the integration over the modular parameter, we observe the expected pattern of the infrared divergence, consistent with the one-loop results from the box diagram calculations in field theory.