Efficient solution of the multi-channel Lüscher determinant condition through eigenvalue decomposition

Abstract

We present a method for efficiently finding solutions of Lüscher's quantisation condition, the equation which relates two-particle scattering amplitudes to the discrete spectrum of states in a periodic spatial volume of finite extent such as that present in lattice QCD. The approach proposed is based on an eigenvalue decomposition in the space of coupled-channels and partial-waves, which proves to have several desirable and simplifying features that are of great benefit when considering problems beyond simple elastic scattering of spinless particles. We illustrate the method with a toy model of vector-vector scattering featuring a high density of solutions, and with an application to explicit lattice QCD energy level data describing $J^P=1^{-}$ and $1^{+}$ scattering in several coupled channels.

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• Isospin-1/2 $D\pi$ scattering and the lightest $D_0^{\ast }$ resonance from lattice QCD
• Constraining $1+\mathcal{J}\to 2$ coupled-channel amplitudes in finite-volume
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• Scalar and tensor charmonium resonances in coupled-channel scattering from QCD
• Near-threshold states in coupled $D{D}^{\ast }-D^{\ast }{D}^{\ast }$ scattering from lattice QCD
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• Charmonium ${\chi }_{c0}$ and ${\chi }_{c2}$ resonances in coupled-channel scattering from lattice QCD
• $DK/D\pi$ scattering and an exotic virtual bound state at the $SU(3)$ flavour symmetric point from lattice QCD
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• $D_1$ and $D_2$ resonances in coupled-channel scattering amplitudes from lattice QCD