Harmonic-Oscillator-Based Effective Theory (HOBET): Effective Interactions without a Potential

HOBET is a treatment of the few-body nuclear problem in which an expansion around an intermediate momentum scale -- defined by the oscillator parameter and the number of shells in the P space -- provides the separation of scales necessary for a successful effective theory. This leads to a bound-state theory with both infrared and ultraviolet corrections: the former depends sensitively on binding energy and can be summed to all orders by a Green's function technique, while the latter can be replaced by a rapidly converging contact-gradient expansion with energy-independent strong-interaction coefficients. Here we demonstrate that the scattering (Lippmann-Schwinger) equation can be reorganized in precisely the same way, so that these coefficients (the effective interaction) can be determined directly from phase shifts, eliminating the need for a Q-space potential.