The three-particle quantization condition is partially diagonalized in the center-of-mass frame by using cubic symmetry on the lattice. To this end, instead of spherical harmonics, the kernel of the Bethe-Salpeter equation for particle-dimer scattering is expanded in the basis functions of different irreducible representations of the octahedral group. Such a projection is of particular importance for the three-body problem in the finite volume due to the occurrence of three-body singularities above breakup. Additionally, we study the numerical solution and properties of such a projected quantization condition in a simple model. It is shown that, for large volumes, these solutions allow for an instructive interpretation of the energy eigenvalues in terms of bound and scattering states.
It is often computationally advantageous to model space as a discrete set of points forming a lattice grid. This technique is particularly useful for computationally difficult problems such as quantum many-body systems. For reasons of simplicity and familiarity, nearly all quantum many-body calculations have been performed on simple cubic lattices. Since the removal of lattice artifacts is often an important concern, it would be useful to perform calculations using more than one lattice geometry. In this work we show how to perform quantum many-body calculations using auxiliary-field Monte Car…
We study the effects of a finite cubic volume with twisted boundary conditions on pseudoscalar mesons. We first apply chiral perturbation theory in the p-regime and calculate the corrections for masses, decay constants, pseudoscalar coupling constants and form factors at next-to-leading order. We show that the Feynman-Hellmann theorem and the relevant Ward-Takahashi identity are satisfied. We then derive asymptotic formulae a la Luscher for twisted boundary conditions. We show that chiral Ward identities for masses and decay constants are satisfied by the asymptotic formulae in finite volume a…
The physical interpretation of lattice QCD simulations, performed in a small volume, requires an extrapolation to the infinite volume. A method is proposed to perform such an extrapolation for three interacting particles at energies above threshold. For this, a recently formulated relativistic $3\to 3$ amplitude based on the isobar formulation is adapted to the finite volume. The guiding principle is two- and three-body unitarity that imposes the imaginary parts of the amplitude in the infinite volume. In turn, these imaginary parts dictate the leading power-law finite-volume effects. It is de…
Strong three-body interactions above threshold govern the dynamics of many exotics and conventional excited mesons and baryons. Three-body finite-volume energies calculated from lattice QCD promise an ab-initio understanding of these systems. We calculate the three-$π^+$ spectrum unraveling the three-body dynamics that is tightly intertwined with the $S$-matrix principle of three-body unitarity and compare it with recent lattice QCD results. For this purpose, we develop a formalism for three-body systems in moving frames and apply it numerically.
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 many-nucleon states, but also for other fields such as simulations …
We study systems of bosons and fermions in finite periodic boxes and show how the existence and properties of few-body resonances can be extracted from studying the volume dependence of the calculated energy spectra. Using a plane-wave-based discrete variable representation to conveniently implement periodic boundary conditions, we establish that avoided level crossings occur in the spectra of up to four particles and can be linked to the existence of multi-body resonances. To benchmark our method we use two-body calculations, where resonance properties can be determined with other methods, as…
We explore variational approach to the finite-volume $N$-body problem. The general formalism for N non-relativistic spinless particles interacting with periodic pair-wise potentials yields N-body secular equations. The solutions depend on the infinite-volume N-body wave functions. Given that the infinite-volume N-body dynamics may be solved by the standard Faddeev approach, the variational N-body formalism can provide a convenient numerical framework for finding discrete energy spectra in periodic lattice structures.
We perform a fit of the finite-volume QCD spectrum of three pions at maximal isospin to constrain the three-body force. We use the unitarity-based relativistic three-particle quantization condition, with the GWUQCD spectrum obtained at 315 MeV and 220 MeV pion mass in two-flavor QCD. For the heavier pion mass we find that the data is consistent with a constant contact term close to zero, whereas for the lighter mass we see a statistically significant energy dependence in tension with the prediction of leading order ChPT. Our results also suggest that with enough three-body energy levels, the t…
Under the assumption of separable interactions, we illustrate how the few-body quantization condition may be formulated in terms of phase shifts in general, which may be useful for describing and modeling of few-body resonances in finite volume.
In this work, based on consideration of periodicity and asymptotic forms of wave function, we propose a novel approach to the solution of finite volume three-body problem by mapping a three-body problem into a higher dimensional two-body problem. The idea is demonstrated by an example of two light spinless particles and one heavy particle scattering in one spatial dimension. This 1D three-body problem resembles a two-body problem in two spatial dimensions mathematically, and quantization condition of 1D three-body problem is thus derived accordingly.
In this talk, we present a framework for studying structural information of resonances and bound states coupling to two-hadron scattering states. This makes use of a recently proposed finite-volume formalism to determine a class of observables that are experimentally inaccessible but can be accessed via lattice QCD. In particular, we shown that finite-volume two-body matrix elements with one current insertion can be directly related to scattering amplitudes coupling to the external current. For two-hadron systems with resonances or bound states, one can extract the corresponding form factors o…
In present work, we study an numerical approach to one dimensional finite volume three-body interaction, the method is demonstrated by considering a toy model of three spinless particles interacting with pair-wise $δ$-function potentials. The numerical results are compared with the exact solutions of three spinless bosons interaction when strength of short-range interactions are set equal for all pairs.
Lattice QCD calculations of scattering phaseshifts and resonance parameters in the two-body sector are becoming precision studies. Early calculations employed Lüscher's formula for extracting these quantities at lowest order. As the calculations become more ambitious, higher-order relations are required. In this study we present a way to validate the higher-order quantization conditions. This is an important step given the involved derivations of these formulae. We derive and validate quantization conditions up to $\ell=5$ partial waves in both cubic and elongated geometries, and for states ze…
In this work, we use McGuire's model to describe scattering of three spinless identical particles in one spatial dimension, we first present analytic solutions of Faddeev's equation for scattering of three spinless particles in free space. The three particles interaction in finite volume is derived subsequently, and the quantization conditions by matching wave functions in free space and finite volume are presented in terms of two-body scattering phase shifts. The quantization conditions obtained in this work for short range interaction are Lüscher's formula like and consistent with Yang's res…
