Dimensional crossover in non-relativistic effective field theory

Abstract

Isotropic scattering in various spatial dimensions is considered for arbitrary finite-range potentials using non-relativistic effective field theory. With periodic boundary conditions, compactifications from a box to a plane and to a wire, and from a plane to a wire, are considered by matching S-matrix elements. The problem is greatly simplified by regulating the ultraviolet divergences using dimensional regularization with minimal subtraction. General relations among (all) effective-range parameters in the various dimensions are derived, and the dependence of bound states on changing dimensionality are considered. Generally, it is found that compactification binds the two-body system, even if the uncompactified system is unbound. For instance, compactification from a box to a plane gives rise to a bound state with binding momentum given by $\ln \left({\scriptstyle \frac{1}{2}}\left(3+\sqrt{5} \right) \right)$ in units of the inverse compactification length. This binding momentum is universal in the sense that it does not depend on the two-body interaction in the box. When the two-body system in the box is at unitarity, the S-matrices of the compactified two-body system on the plane and on the wire are given exactly as universal functions of the compactification length