The Nearest-Neighbor Self-Avoiding Walk with Complex Killing Rates

The Green's function for the nearest-neighbor self-avoiding walk on a hypercubic lattice in d>2 dimensions is constructed and shown to be analytic for values of the killing rate a in C satisfying |a|>epsilon, arg a 3pi/4 - b with epsilon>0 and 0
epsilon>0 in order to use the killing rate as an infrared cutoff, which allows us to construct the Green's function using a single scale cluster expansion. The presence of non-real killing introduces complications that we resolve through the use of an appropriate choice of decoupling scheme and a subsidiary expansion. Our methods can be used to control a single momemtum slice in a phase-space expansion.