Effective quantum theories with short- and long-range forces

Abstract

At low energies, nonrelativistic quantum systems are essentially governed by their wave functions at large distances. For this reason, it is possible to describe a wide range of phenomena with short- or even finite-range interactions. In this thesis, we discuss several topics in connection with such an effective description and consider, in particular, modifications introduced by the presence of additional long-range potentials.<br /> In the first part we derive general results for the mass (binding energy) shift of bound states with angular momentum L ≥ 1 in a periodic cubic box in two and three spatial dimensions. Our results have applications to lattice simulations of hadronic molecules, halo nuclei, and Feshbach molecules. The sign of the mass shift can be related to the symmetry properties of the state under consideration. We verify our analytical results with explicit numerical calculations. Moreover, we discuss the case of twisted boundary conditions that arise when one considers moving bound states in finite boxes. The corresponding finite-volume shifts in the binding energies play an important role in the study of composite-particle scattering on the lattice, where they give rise to topological correction factors. <br /> While the above results are derived under the assumption of a pure finite-range interaction—and are still true up to exponentially small correction in the short-range case—in the second part we consider primarily systems of charged particles, where the Coulomb force determines the long-range part of the potential. <br /> In quantum systems with short-range interactions, causality imposes nontrivial constraints on low-energy scattering parameters. We investigate these causality constraints for systems where a long-range Coulomb potential is present in addition to a short-range interaction. The main result is an upper bound for the Coulomb-modified effective range parameter. We discuss the implications of this bound to the effective feld theory (EFT) for nuclear halo systems. In particular, we consider several examples of proton-nucleus and nucleus-nucleus scattering. For the bound-state regime, we find relations for the asymptotic normalization coefficients (ANCs) of nuclear halo states. Moreover, we also consider the case of other singular inverse-power-law potentials and in particular discuss the case of an asymptotic van der Waals tail, which plays an important role in atomic physics.<br /> Finally, we consider the low-energy proton{deuteron system in pionless effective feld theory. Amending our previous work, we focus on the doublet-channel spin configuration and the ³He bound state. In particular, we study the situation at next-to-leading order in the EFT power counting and provide numerical evidence that a charge-dependent counterterm is necessary for correct renormalization of the theory at this order. We furthermore argue that the previously employed power counting for the inclusion of Coulomb contributions should be given up in favor of a scheme that is consistent throughout the bound-state and the scattering regime. In order to probe the importance of Coulomb effects directly at the zero-energy threshold, we also present a first calculation of proton-deuteron scattering lengths in pionless effective feld theory.