Decays of an exotic $1^{-+}$ hybrid meson resonance in QCD

We present the first determination of the hadronic decays of the lightest exotic $J^{PC}=1^{-+}$ resonance in lattice QCD. Working with SU(3) flavor symmetry, where the up, down and strange-quark masses approximately match the physical strange-quark mass giving $m_{\pi }\sim 700\mathrm{MeV}$, we compute finite-volume spectra on six lattice volumes which constrain a scattering system featuring eight coupled channels. Analytically continuing the scattering amplitudes into the complex-energy plane, we find a pole singularity corresponding to a narrow resonance which shows relatively weak coupling to the open pseudoscalar–pseudoscalar, vector–pseudoscalar and vector–vector decay channels, but large couplings to at least one kinematically closed axial-vector–pseudoscalar channel. Attempting a simple extrapolation of the couplings to physical light-quark mass suggests a broad ${\pi }_1$ resonance decaying dominantly through the $b_1\pi$ mode with much smaller decays into $f_1\pi$, $\rho \pi$, ${\eta }^{\prime }\pi$ and $\eta \pi$. A large total width is potentially in agreement with the experimental ${\pi }_1(1564)$ candidate state observed in $\eta \pi$, ${\eta }^{\prime }\pi$, which we suggest may be heavily suppressed decay channels.

Figure 1

Spectrum of low-lying octet (red) and singlet (cyan) mesons by $J^{P(C)}$ obtained using only single-meson operators. Solid boxes show mesons which lie below relevant meson-meson thresholds and are thus stable, while hatched boxes show mesons which lie above threshold and which will require a full finite-volume analysis to resolve their resonant nature. Dashed lines show the lowest relevant meson-meson thresholds.

Figure 2

Masses of $C=+$ octet mesons obtained using only single-meson operators (taken from Ref. [26]). Thresholds relevant for $J^{P(C)}=1^{-(+)}$ are shown.

Figure 3

Finite-volume spectrum in the $T_1^{-(+)}$ irrep on six lattice volumes. Points show the extracted energy levels, including uncertainties, from a variational analysis using the operator bases in Table 5; black points are included in the subsequent scattering analysis and gray points are not. Some points are slightly displaced horizontally for clarity when near-degenerate energies appear. Curves show meson-meson noninteracting energies, with multiplicities greater than 1 labeled by $\{n\}$ and shown as slightly split curves. Dashed curves correspond to meson-meson operators not included in the basis. Relevant thresholds transcribed from Table 4 are shown on the vertical axis. Accompanying each energy level is a histogram of the operator-state overlap factors, $Z_i^{\mathfrak{n}}=⟨\mathfrak{n}|{\mathcal{O}}_i^{†}(0)|0⟩$, for ${\eta }^{\mathbf{1}}{\eta }^{\mathbf{8}}$ (dark blue), ${\omega }^{\mathbf{8}}{\eta }^{\mathbf{8}}$ (red), ${\omega }^{\mathbf{8}}{\omega }^{\mathbf{8}}$ (sand), ${\omega }^{\mathbf{1}}{\omega }^{\mathbf{8}}$ (green), $f_1^{\mathbf{8}}{\eta }^{\mathbf{8}}$ (cyan), $h_1^{\mathbf{8}}{\eta }^{\mathbf{8}}$ (purple) and $f_1^{\mathbf{1}}{\eta }^{\mathbf{8}}$ (brown) meson-meson operators and a sample of $\overline{\psi }\mathbf{\Gamma }\psi$ (orange) fermion-bilinear operators. The overlaps are normalized such that the largest value for any given operator across all energy levels is equal to 1. For clarity, the histograms accompanying the cluster of levels on the $L/a_s=18$, 20, 24 volumes are displayed at the top of the figure.

Figure 4

Analogous to Fig. 3 but for the $A_2^{-(+)}$ irrep (operator lists shown in Table 6). Note the vertical axis is broken to emphasize the relatively low-lying ${\eta }^{\mathbf{1}}{\eta }^{\mathbf{8}}$ and ${\omega }^{\mathbf{8}}{\eta }^{\mathbf{8}}$ thresholds.

Figure 5

As Fig. 4 but including, as orange bands, the energy levels calculated from the amplitude in Eq. (6). The thickness of the bands reflects the statistical uncertainty. The dashed curves show the noninteracting energy levels for ${\omega }^{\mathbf{8}}{\eta }^{\mathbf{8}}$ (red), ${\eta }^{\mathbf{1}}{\eta }^{\mathbf{8}}$ (blue) and ${\omega }^{\mathbf{1}}{\omega }^{\mathbf{8}}$ (green).

Figure 6

As Fig. 5 but for the $T_1^{-(+)}$ irrep using the illustrative amplitude described in Eq. (8).

Figure 7

(a) Diagonal $t$-matrix elements plotted as ${\rho }_a{\rho }_b|t_{ab}{|}^2$, for the illustrative amplitude presented in Eq. (8) for non-vector-vector channels: ${\eta }^{\mathbf{1}}{\eta }^{\mathbf{8}}\{{{}^{1}P}_1\}$, ${\omega }^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}P}_1\}$, $f_1^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}S}_1\}$ and $h_1^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}S}_1\}$. Shaded bands reflect statistical uncertainties on the scattering parameters. (b)–(d) As above, but for the off-diagonal amplitudes between the four channels displayed above. (e),(f) Diagonal vector-vector amplitudes: ${\omega }^{\mathbf{8}}{\omega }^{\mathbf{8}}\{{{}^{3}P}_1\}$, ${\omega }^{\mathbf{1}}{\omega }^{\mathbf{8}}\{{{}^{1}P}_1,{{}^{3}P}_1,{{}^{5}P}_1\}$. As discussed in the text, the ${\omega }^{\mathbf{1}}{\omega }^{\mathbf{8}}$ partial waves are indistinguishable and are combined in a single plot labeled ${\omega }^{\mathbf{1}}{\omega }^{\mathbf{8}}\{{}^X{P}_1\}$, the shaded band reflecting an envelope over the statistical uncertainties for each partial wave.

Figure 8

Diagonal $t$-matrix elements plotted as ${\rho }_a{\rho }_b|t_{ab}{|}^2$, for each parametrization successfully describing the finite-volume spectra as discussed in the text, for non-vector-vector channels: ${\eta }^{\mathbf{1}}{\eta }^{\mathbf{8}}\{{{}^{1}P}_1\}$, ${\omega }^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}P}_1\}$, $f_1^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}S}_1\}$ and $h_1^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}S}_1\}$. Shaded bands reflect statistical uncertainties on the illustrative amplitude shown in Fig. 7.

Figure 9

As Fig. 8 but for off-diagonal ${\eta }^{\mathbf{1}}{\eta }^{\mathbf{8}}\{{{}^{1}P}_1\}\to {\omega }^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}P}_1\}$, $f_1^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}S}_1\}$, $h_1^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}S}_1\}$ amplitudes.

Figure 10

As Fig. 9 but for ${\omega }^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}P}_1\}\to f_1^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}S}_1\}$, $h_1^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}S}_1\}$ amplitudes.

Figure 11

As Fig. 9 but for the $f_1^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}S}_1\}\to h_1^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}S}_1\}$ amplitude.

Figure 12

As Fig. 8 but for vector-vector channels: ${\omega }^{\mathbf{8}}{\omega }^{\mathbf{8}}\{{{}^{3}P}_1\}$, ${\omega }^{\mathbf{1}}{\omega }^{\mathbf{8}}\{{}^X{P}_1\}$ amplitudes.

Figure 13

Pole singularities on the proximal sheet for all successful parametrizations as described in the text. Error bars reflect the statistical uncertainties on the pole position for each parametrization.

Figure 14

Couplings corresponding to the pole singularities shown in Fig. 13 as described in the text. Error bars reflect the statistical uncertainties on each coupling for each parametrization. Shaded bars show ranges and upper limits on the couplings described in the text. Top: couplings to non-vector-vector channels for parametrizations where the $f_1^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}S}_1\}$ coupling was found to be significantly nonzero. Middle: as top but for parametrizations where the $f_1^{\mathbf{8}}{\eta }^{\mathbf{8}}\{{{}^{3}S}_1\}$ coupling was found to be zero or fixed to be identically zero. Bottom: couplings to vector-vector channels, ${\omega }^{\mathbf{8}}{\omega }^{\mathbf{8}},{\omega }^{\mathbf{1}}{\omega }^{\mathbf{8}}$.

Figure 15

Wick contraction topologies for $\mathbf{8}\to \mathbf{8}\otimes \mathbf{8}$ (left) and $\mathbf{8}\to \mathbf{8}\otimes \mathbf{1}$ (right).

Figure 16

Partial widths as a function of the ${\pi }_1$ pole mass. The bands reflect the coupling ranges given in Table 8. The total width obtained by summing the partial widths is shown by the gray band.

Figure 17

Energy dependence of $-k_1(s){\mathcal{M}}_1(s)$ and $-k_2(s){\mathcal{M}}_2(s)$ for a lattice of spatial extent $L/a_s=24$. Both functions are purely real across this energy region. Dashed vertical lines indicate the location of noninteracting energies.

Figure 18

Top: the function $-g_1^2{k}_1(s){\mathcal{M}}_1(s)-g_2^2{k}_2(s){\mathcal{M}}_2(s)$, for $L/a_s=24$ and $g_1=g_2=1$, plotted in black, and the function $\frac{1}{2}\sqrt{s}(s-m^2)$ with $m=0.461057$ plotted in gray. The points of intersection (circles) correspond to the finite-volume energy levels. Dashed vertical lines indicate the location of noninteracting energies. Bottom: same as the top plot but with an enlarged vertical scale.

Figure 19

Finite-volume spectra for the toy model as described in the text for the $T_1^{-}$ irrep at rest and the $A_1$ irreps in flight. $a^2\equiv g_1^2+g_2^2=1$ is kept fixed, while the ratio $b\equiv g_1/g_2$ is varied.

Figure 20

As Fig. 19 but for a fixed ratio $b=1$ and varying magnitude $a$.