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 epsilon C satisfying \a\ > c, \arg a\ < 3pi/4 - b with epsilon > 0 and 0 < b < pi/4. We restrict \a\ > c > 0 in order to use the killing rate as an infrared cutoff, which allows us to construct 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.