Abstract
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.
Highlights
Three-particle channels are important in spectroscopy of excited mesons
G parity forbids a two-pion decay of the a1 ð1260Þ meson so that the width of this axialvector resonance is determined by its coupling to three pions
Isobar models [54,55,56,57,58,59,60,61,62,63,64,65,66,67] have been a useful tool to describe few-body interactions, in which the fewbody interaction is treated by taking into account all possible recombinations of two-body subsystems
Summary
Three-particle channels are important in spectroscopy of excited mesons. For example, G parity forbids a two-pion decay of the a1 ð1260Þ meson so that the width of this axialvector resonance is determined by its coupling to three pions. Energy eigenvalues for the Roper [43] resonance above the ππN threshold have been calculated recently including nonlocal πN and σN operators. In a rough and incomplete classification, there are three major approaches to solving the three-body problem in finite volume in the momentum-space representation: an relativistic, all-orders perturbation theory pursued by Briceño, Hansen, and Sharpe [12,13,14,18,21], a nonrelativistic dimer formalism by Hammer et al [10,15,16,17,20], and a method based on three-body unitarity to identify on-shell configurations and, power-law finite-volume effects by Döring and Mai [8,19,22] For the latter two approaches, the partial diagonalization of the amplitude according to cubic symmetry was discussed in Ref. The first-ever prediction of excited three-body energy eigenvalues of a physical system (πþπþπþ ) from two-body scattering information and lattice threshold eigenvalues [41,42] was achieved recently [8]
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