Effective theories of scattering with an attractive inverse-square potential and the three-body problem

A distorted-wave version of the renormalisation group is applied to scattering by an inverse-square potential and to three-body systems. In attractive three-body systems, the short-distance wave function satisfies a Schroedinger equation with an attractive inverse-square potential, as shown by Efimov. The resulting oscillatory behaviour controls the renormalisation of the three-body interactions, with the renormalisation-group flow tending to a limit cycle as the cut-off is lowered. The approach used here leads to single-valued potentials with discontinuities as the bound states are cut off. The perturbations around the cycle start with a marginal term whose effect is simply to change the phase of the short-distance oscillations, or the self-adjoint extension of the singular Hamiltonian. The full power counting in terms of the energy and two-body scattering length is constructed for short-range three-body forces.Comment: 19 pages (RevTeX), 2 figure

Goal Setting and Raising the Bar: A Field Experiment

We study goal setting using a randomized field experiment involving 1092 first-year undergraduate students. Students have private mentor-student meetings during the year. We instructed a random subset of mentors to encourage students to set a course-specific grade goal during one of the mentor-student meetings (goal treatment). A random subset of those mentors was further instructed to challenge students to set more ambitious goals if deemed appropriate (raise treatment). We find that students in the goal treatment perform significantly better as compared to students in the control group, and more so when they performed poorly prior to the experiment. Next, we find that students in the raise treatment do not perform significantly different from the control group. Finally, students who set a goal and are challenged to set a more ambitious goal perform significantly worse than comparable students in the goal treatment