Few-body physics in a Finite Volume

Simon Kreuzer, H.‐W. Hammer

Nikolaj Thomas Zinner

S. König

Abstract Simulating quantum systems in a finite volume is a powerful theoretical tool to extract information about them. Real-world properties of the system are encoded in how its discrete energy levels change with the size of the volume. This approach is relevant not only for nuclear physics, where lattice methods for few- and many-nucleon states complement phenomenological shell-model descriptions and ab initio calculations of atomic nuclei based on harmonic oscillator expansions, but also for other fields such as simulations of cold atomic systems. This contribution presents recent progress concerning finite-volume simulations of few-body systems. In particular, it discusses details regarding the efficient numerical implementation of separable interactions and it presents eigenvector continuation as a method for performing robust and efficient volume extrapolations.

Simon Kreuzer

In this work, three-body bound states were studied in finite volume using an Effective Field Theory framework. The finite volumes under consideration are cubic volumes with periodic boundary conditions. The Effective Field Theory framework used in this work employs only contact interactions and is particularly well suited for studies of universal properties, i.e. properties independent of the details of the interaction on short distances. A particular example in the three-body sector is the Efimov effect, the emergence of a geometrically spaced bound state spectrum. In the first part, systems of three identical bosons are investigated. As a consequence of the breakdown of the spherical symmetry to cubic symmetry, the partial waves of the bound state amplitude are coupled. An infinite set of coupled integral equations for these partial waves is derived. These equations have to be solved numerically in order to obtain the binding energies in the finite volume. The dependence of the energies on the box size is calculated and the results are explicitly verified to be renormalized. Results for positive and negative scattering lengths are shown. The effects of higher partial waves are investigated. The behavior of shallow trimers near the dimer energy as well as deeply bound trimers is studied. The shallowest state investigated crosses the dimer energy at a certain volume and behaves like a scattering state for smaller volumes. Numerical evidence for a universal scaling of the finite volume corrections is provided. Subsequently, the formalism is extended to systems of three nucleons. This case provides the main motivation for this work due to its applicability to Lattice Quantum Chromodynamics (QCD) calculations of the triton. Such calculations always take place inside a finite volume which makes control over the corresponding effects crucial for an understanding of results from the lattice. For the triton, there are two coupled channels already in the infinite volume corresponding to two different spin-isospin combinations. An infinite set of coupled integral equations for the partial waves of the bound state amplitudes is derived. The renormalization of all results is again explicitly verified. The physical triton inside a finite volume is investigated as well as the triton spectrum for unphysical pion masses. The former case qualitatively shows the same behavior as the three-boson case, and the volume dependence is calculated. The smallest volumes investigated are of the order of magnitude typical for present day Lattice calculations. The motivation for the latter part is twofold. On the one hand, Lattice QCD calculations are performed at pion masses larger than the physical one for computational reasons. On the other hand, it has been conjectured that QCD is close to the critical trajectory for an infrared renormalization group limit cycle, in which case the Efimov effect would occur for a critical pion mass. Close to this critical pion mass, the triton has excited states. The behavior of the ground state and of the excited states inside a finite volume is investigated for various pion masses around the critical one. The excited states cross the energy of the bound di-nucleon, as it was already observed for the shallowest bosonic trimer. The results for the ground state were used to provide strong numerical evidence for a universal scaling of the finite volume corrections.

Uttam Kumar Chowdhury

Jing-Lun Li, Paul S. Julienne, Johannes Hecker Denschlag +1 more

We derive an analog of the Lellouch-Lüscher (LL) relation for few-body bosonic systems, linking few-body scattering loss rates to the energies and widths of the corresponding harmonically trapped few-body states. Three-body numerical simulations show that the LL relation applies across a broad range of interaction strengths and energies and allows the determination of scattering rates within a single partial wave. Our Letter establishes a robust theoretical framework for understanding the role of the finite-volume effect in few-body observables in optical lattice and tweezer experiments, enabling precise determination of multibody scattering rates.