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arXiv:1412.4969 · Physical Review Letters

Spectrum of Three-Body Bound States in a Finite Volume

Ulf-G. Meißner, Guillermo Rios, Akaki Rusetsky

Published Mar 4, 2015• 101 citations
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Abstract

The spectrum of a bound state of three identical particles with a mass $m$ in a finite cubic box is studied. It is shown that in the unitary limit, the energy shift of a shallow bound state is given by $\mathrm{\ensuremath{\Delta}}E=c({\ensuremath{\kappa}}^{2}/m)(\ensuremath{\kappa}L{)}^{\ensuremath{-}3/2}|A{|}^{2}\mathrm{exp}(\ensuremath{-}2\ensuremath{\kappa}L/\sqrt{3})$, where $\ensuremath{\kappa}$ is the bound-state momentum, $L$ is the box size, $|A{|}^{2}$ denotes the three-body analog of the asymptotic normalization coefficient of the bound state wave function, and $c$ is a numerical constant. The formula is valid for $\ensuremath{\kappa}L\ensuremath{\gg}1$.

DOI
10.1103/physrevlett.114.091602

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The spectrum of a bound state of three identical particles with a mass mmm in a finite cubic box is studied. It is shown that in the unitary limit, the energy shift of a shallow bound state is given by \ensuremathΔE=c(\ensuremathκ2/m)(\ensuremathκL)\ensuremath−3/2∣A∣2exp(\ensuremath−2\ensuremathκL/3)\mathrm{\ensuremath{\Delta}}E=c({\ensuremath{\kappa}}^{2}/m)(\ensuremath{\kappa}L{)}^{\ensuremath{-}3/2}|A{|}^{2}\mathrm{exp}(\ensuremath{-}2\ensuremath{\kappa}L/\sqrt{3})\ensuremathΔE=c(\ensuremathκ2/m)(\ensuremathκL)\ensuremath−3/2∣A∣2exp(\ensuremath−2\ensuremathκL/3​), where \ensuremathκ\ensuremath{\kappa}\ensuremathκ is the bound-state momentum, LLL is the box size, ∣A∣2|A{|}^{2}∣A∣2 denotes the three-body analog of the asymptotic normalization coefficient of the bound state wave function, and ccc is a numerical constant. The formula is valid for \ensuremathκL\ensuremath≫1\ensuremath{\kappa}L\ensuremath{\gg}1\ensuremathκL\ensuremath≫1.
The volume-dependence of a shallow three-particle bound state in the cubic box with a size LLL is studied. It is shown that, in the unitary limit, the energy-level shift from the infinite-volume position is given by ΔE=c(κ2/m) (κL)−3/2∣A∣2exp⁡(−2κL/3)\Delta E=c (\kappa^2/m)\,(\kappa L)^{-3/2}|A|^2 \exp(-2\kappa L/\sqrt{3})ΔE=c(κ2/m)(κL)−3/2∣A∣2exp(−2κL/3​), where κ\kappaκ is the bound-state momentum and ∣A∣2|A|^2∣A∣2 denotes the three-body analog of the asymptotic normalization constant, which encodes the information about the short-range interactions in the three-body system.
The universal properties of a three-boson system with large scattering length are well understood within the framework of Effective Field Theory. They include a geometric spectrum of shallow three-body bound states called “Efimov states” and log-periodic dependence of scattering observables on the scattering length. We investigate the modification of this spectrum in a finite cubic box using a partial wave expansion. The dependence of the binding energies on the box size is calculated for systems with positive and negative two-body scattering length. We compare the full results to results obtained using an expansion around the infinite volume binding energy. The renormalization of the Effective Field Theory in the finite volume is verified explicitly.
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.
The momentum space zero-range model is used to investigate universal properties of three interacting particles confined to two dimensions. The pertinent equations are first formulated for a system of two identical and one distinct particle and the two different two-body subsystems are characterized by two-body energies and masses. The three-body energy in units of one of the two-body energies is a universal function of the other two-body energy and the mass ratio. We derive convenient analytical formulae for calculations of the three-body energy as a function of these two independent parameters and exhibit the results as universal curves. In particular, we show that the three-body system can have any number of stable bound states. When the mass ratio of the distinct to identical particles is greater than 0.22, we find that at most two stable bound states exist, while for two heavy and one light mass an increasing number of bound states is possible. The specific number of stable bound states depends on the ratio of two-body bound state energies and on the mass ratio, and we map out an energy–mass phase diagram of the number of stable bound states. Realizable systems of both fermions and bosons are discussed in this framework.
Using non-relativistic effective Lagrangians in the particle-dimer picture, we rederive the expression for the energy shift of a loosely bound three-particle bound state of identical bosons in the unitary limit. The effective field theory formalism allows us to explicitly investigate the role of the three-particle force. Moreover, we discuss relaxing the unitary limit of infinite scattering length and demonstrate a smooth transition from the weakly bound three-particle state to a two-particle bound state of a particle and a deeply bound dimer.
We found universal laws for the spectrum of two-dimensional three-body systems, composed by two identical particles and a distinct one (AAB). These universal laws appear when the potential range (r_0) is much smaller than the size of the system. In two dimensions this condition is formulated as (E2 is the two-body energy and ? is the reduced mass). The zero range model, which is very appropriated to establish the universal laws, is introduced through the ?-Dirac potential. We derive the corresponding two-dimensional Faddeev equations for the three-body system and solve them numerically in momentum space. Our results showed that the three-body binding energy monotonically increases with the two-body binding energy, and such dependence is more pronounced than the mass variations. We found that the three-body energy depends logarithmic on the two-body energy for large values. Furthermore, been m=mB/mA the ratio between the masses of the B and A particles, the three-body energy is mass-independent for m ? ? and increase without bounds for m ?0. The limit of two non-interacting identical particles is also studied in the AAB system. We found that the two-dimensional three-body system always support at least two bound states and more bound states appear for m<0.22. Finally, we analyze the particular limit of m ?0 using the adiabatic approximation. This approximation can be used to study the three-body system in two-dimensions with an accuracy better than 10% compared to the solutions of the Faddeev equations, for m ? 0.01.