Few-Body Bound States and Resonances in Finite Volume

Abstract

Since the pioneering work of Lüscher in the 1980s it is well known that considering quantum systems in finite volume, specifically, finite periodic boxes, can be used as a powerful computational tool to extract physical observables. While this formalism has been worked out in great detail in the two-body sector, much effort is currently being invested into deriving analogous relations for systems with more constituents. This work is relevant not only for nuclear physics, where lattice methods are now able to calculate few-and manynucleon states, but also for other fields such as simulations of cold atoms. This article discusses recent progress regarding the extraction of few-body bound-state and resonance properties from finite-volume calculations of systems with an arbitrary number of constituents. This contribution has been presented during the ceremony of the Few-Body Systems Award for young professionals.

The 5α condensate state in 20Ne

“…The radius eigenfunctions whose eigenvalues are smaller than the cutoff parameter R cut can be our basis wave functions in GCM calculations. This method is essentially similar with the finite-volume method 49 in other models. Moreover, the stabilization method in the theory of resonances has the consequence that, except for special broad cases, the obtained eigen energies of resonance states hardly change due to the slow increase of the R cut , which is the bounded volume or barrier, while the continuum states change dramatically.…”

Three-particle scattering amplitudes from lattice QCD

“…Since lattice simulations are necessarily performed in a finite box, the measured spectrum corresponds to that of an interacting quantum field theory in a finite volume. In a few cases, the interpretation of these energy levels is simple: if an energy level corresponds to a one-particle state (or stable bound state), these energies are exponentially close to its infinite-volume value [71][72][73]. However, if a state corresponds to a multi-particle state, the connection to infinitevolume is harder to establish.…”

Three-body resonances in pionless effective field theory

“…Fore more details regarding the DVR setup we refer to Refs. [59,79] and further references cited therein, and merely recall here that the overall strategy with this method is to represent the N -body finite-volume Hamiltonian H = H 0 + V , where H 0 = K is the relative kinetic energy operator and V denotes the sum of all interactions among the particles, as a matrix in the space spanned by the DVR states. Energy levels in the box are then obtained by calculating the spectrum (or more specifically the lowest lying states in the spectrum) via Lanczos/Arnoldi iteration.…”

“…The factor N = 2 d (L/N ) d/2 arises as normalization from the integral over spectator coordinates. Since the plane-wave DVR states we consider are closely related to a DFT [78,79], this is naturally the tool to use for calculating matrix elements s|V ij |ψ when the V ij are given in momentum space. To that end, one considers two-body momentum modes…”