Finite-volume Hamiltonian method for coupled-channels interactions in lattice QCD

Within a multichannel formulation of $\pi \pi$ scattering, we investigate the use of the finite-volume Hamiltonian approach to resolve scattering observables from lattice QCD spectra. The asymptotic matching of the well-known Lüscher formalism encodes a unique finite-volume spectrum. Nevertheless, in many practical situations, such as coupled-channels systems, it is advantageous to interpolate isolated lattice spectra in order to extract physical scattering parameters. Here we study the use of the Hamiltonian framework as a parametrization that can be fit directly to lattice spectra. We find that, with a modest amount of lattice data, the scattering parameters can be reproduced rather well, with only a minor degree of model dependence.

Figure 1

The phase shifts of $\pi \pi$ scattering from Model 1b-1c (cf. Table 1) compared with the data.

Figure 2

The spectrum of $\pi \pi$ states in the 1b-1c model. The black curves are calculated by using the finite-volume Hamiltonian approach. The boxes denote discrete points on these curves which are used in the phase extraction shown in Fig. 3.

Figure 3

Phase shifts. The black curve is generated from directly solving scattering equations (6) and (7), and the solid squares are calculated from Lüscher's method by using the spectrum appearing in Fig. 2.

Figure 4

The phase shift ${\delta }_{\pi \pi }$ for $\pi \pi$ scattering from the 1b-2c model compared with the data.

Figure 5

The phase shift ${\delta }_{K\overline{K}}$ of $K\overline{K}$ scattering calculated in the 1b-2c model.

Figure 6

The inelasticity $\eta$ in the 1b-2c model compared with the data.

Figure 7

The computed spectrum is shown as a function of the volume. The black curves show the energy spectra generated by using the finite-box Hamiltonian approach within the the 1b-2c model. The solid and open squares are selected solutions below and above the inelastic threshold, respectively. These solutions have been inverted through the extended Lüscher formalism to determine the phase shifts and inelasticities (see Fig. 9).

Figure 8

The solutions of Eq. (25) (in the inelastic region). The solid dots represent the finite-volume spectrum as determined by the extended Lüscher formalism, computed directly from the model phase shifts and inelasticities. These are in excellent agreement with the spectra computed with the Hamiltonian approach, as shown by the continuous curves.

Figure 9

The extraction of ${\delta }_{\pi \pi }$ for a range of energies. The black curve denotes the model $\pi \pi$ phase shift. The solid and open squares denote the inversion of the solutions shown in Fig. 7 below and above the inelastic thresholds. Below the inelastic threshold, each solution uniquely determines the phase shift. Above the inelastic threshold, the unique solution requires the impractical determination of three identical energy levels at different $L$. In this region, ${\delta }_{K\overline{K}}$ and $\eta$ (Figs. 5 and 6) are equally well described by this inversion.

Figure 10

(a) The spectrum data generated from the 1b-1c model. (b) The phase shifts calculated from the one-channel model with parametrizations A (1b-1c model) and B and C specified in Eqs. (28, 29, 30, 31, 32, 33) compared with the data (from the 1b-2c model).

Figure 11

(a) The spectrum data generated from the 1b-1c model. (b) The phase shifts calculated from the one-channel model with parametrizations A (1b-1c model) and B and C specified in Eqs. (28, 29, 30, 31, 32, 33) compared with the data (from the 1b-1c model).

Figure 12

(a) The spectrum data generated from the 1b-2c model. (b)–(d) The phase shifts and the inelasticity calculated from the two-channel model with parametrizations A (1b-2c model) and B and C specified in Eqs. (28, 29, 30, 31, 32, 33) compared with the data (from the 1b-2c model).

Figure 13

(a) The spectrum data generated from the 1b-2c model. (b)–(d) The phase shifts and the inelasticity calculated from the one-channel model with parametrizations A (1b-2c model) and B and C specified in Eqs. (28, 29, 30, 31, 32, 33) compared with the data (from the 1b-2c model).

Figure 14

The phase shifts ${\delta }_{\pi \pi }$ and ${\delta }_{K\overline{K}}$ and inelasticity $\eta$ of $s$-wave $\pi \pi$ scattering in the $J^{IP}=1^{0+}$ partial wave. The solid squares are the experimental data. The red solid, blue dashed, and green dotted lines are from the NKLS model, Model B, and Model C, respectively.

Figure 15

The phase shifts ${\delta }_{\pi \pi }$ and ${\delta }_{K\overline{K}}$ and inelasticity $\eta$ of $p$-wave $\pi \pi$ scattering in the $J^{IP}=1^{1-}$ partial wave. The solid squares are the experimental data. The red solid, blue dashed, and green dotted lines are from the NKLS model, Model B, and Model C, respectively.

Figure 16

Spectra for (a) $J^{IP}=0^{0+}$ and (b) $J^{IP}=1^{1-}$ partial waves from the NKLS model, Model B, and Model C. The spectra have been displaced for clarity.