$b_1$ resonance in coupled $\pi \omega$, $\pi \varphi$ scattering from lattice QCD

We present the first lattice QCD calculation of coupled $\pi \omega$ and $\pi \varphi$ scattering, incorporating coupled $S$ and $D$-wave $\pi \omega$ in $J^P=1^{+}$. Finite-volume spectra in three volumes are determined via a variational analysis of matrices of two-point correlation functions, computed using large bases of operators resembling single-meson, two-meson, and three-meson structures, with the light-quark mass corresponding to a pion mass of $m_{\pi }\approx 391\mathrm{MeV}$. Utilizing the relationship between the discrete spectrum of finite-volume energies and infinite-volume scattering amplitudes, we find a narrow axial-vector resonance ($J^{PC}=1^{+-}$), the analogue of the $b_1$ meson, with mass $m_R\approx 1380\mathrm{MeV}$ and width ${\mathrm{\Gamma }}_R\approx 91\mathrm{MeV}$. The resonance is found to couple dominantly to $S$-wave $\pi \omega$, with a much-suppressed coupling to $D$-wave $\pi \omega$, and a negligible coupling to $\pi \varphi$ consistent with the Okubo-Zweig-Iizuka rule. No resonant behavior is observed in $\pi \varphi$, indicating the absence of a putative low-mass $Z_s$ analogue of the $Z_c$ seen in $\pi J/\psi$. In order to minimally present the contents of a unitary three-channel scattering matrix, we introduce an $n$-channel generalization of the traditional two-channel Stapp parametrization.

Figure 1

Quark propagation lines (black are light quarks, green are strange quarks) from operator constructions featuring in an optimized $a_0$-like operator.

Figure 2

Wick contraction topologies for $b_1-a_0\pi$. Left meson resembles the $b_1$, upper right meson the $\pi$ and the remaining one or two mesons the $a_0$ (only a subset of the topologies in Fig. 1 are relevant here).

Figure 3

Momentum dependence of $\omega$ and $\varphi$ energies and fits to Eq. (9). Blue and red lines correspond to the $\omega$ meson with $|\lambda |=0$ and 1 respectively. Similarly, green and orange lines correspond to the $\varphi$ meson with $|\lambda |=0$ and 1. Points are shown with statistical uncertainties and grey points show the $(L/a_s)=16$ in-flight energies which are not included in the fit.

Figure 4

Left: Finite-volume spectrum in the $[000]T_1^{+}$ irrep on three lattice volumes. Black points give the energy levels, including statistical uncertainties, from a variational analysis using the operator bases in Table 4. Solid curves are two-meson noninteracting energies, $a_t{E}_{\mathrm{n}.\mathrm{i}.}^{(2)}$, short dashed horizontal lines are $a_t{E}_{\mathrm{n}.\mathrm{i}.}^{(2+1)}$, and long dashed horizontal lines show the two–, three–, and four–meson thresholds. Multiplicities (if greater than one) are shown as $\{n\}$. For each energy level on the largest volume, we show the principal correlators, plotted as ${\lambda }_{\mathfrak{n}}(t,t_0)e^{E_{\mathfrak{n}}(t-t_0)}$ for $t_0=10a_t$ so that a horizontal line is observed when a single exponential dominates. Points show ${\lambda }_{\mathfrak{n}}(t,10)$ and error bars correspond to the one-sigma statistical uncertainty. Curves show fits to the form described in the text; the curves show the fit range and gray points are not included in the fit. The histograms show the operator-state overlap factors, $Z_i^{\mathfrak{n}}=⟨\mathfrak{n}|{\mathcal{O}}_i^{†}(0)|0⟩$, for each energy level on the largest volume for the $\mathbb{M}\mathbb{M}=\pi \omega$ (dark blue), $\pi \varphi$ (green) and $\mathbb{R}\mathbb{M}=\rho \eta$ (blue-green), $K^{*}\overline{K}$ (purple) operators along with a sample set of single-meson operators subduced from $J^P=1^{+}$ (red) and $J^P=3^{+}$ (orange). The overlaps are normalized such that the largest value for any given operator across all energy levels is equal to one. Right: The spectrum extracted when $\rho \eta$ and $K^{*}\overline{K}$ operators are excluded from the basis (black) compared with the complete spectrum (gray).

Figure 5

Finite-volume energy levels in the $\mathsf{cm}$-frame for $[000]T_1^{+}$ and $\overrightarrow{P}{A}_2$ below the lowest $E_{\mathrm{n}.\mathrm{i}.}^{(2+1)}$ or $E_{\mathrm{n}.\mathrm{i}.}^{(3)}$. Black points are used in the scattering analysis in Sec. 6 while gray points are excluded from the main analysis as discussed in the text. Solid curves are two-meson noninteracting energies, $a_t{E}_{\mathrm{n}.\mathrm{i}.}^{(2)}$, short solid gray horizontal lines show the lowest $E_{\mathrm{n}.\mathrm{i}.}^{(2+1)}$ or $E_{\mathrm{n}.\mathrm{i}.}^{(3)}$, and long dashed horizontal lines show the two–, three–, and four–meson thresholds. Multiplicities (if greater than one) are shown as $\{n\}$. The horizontal axes are in units of $L/a_s$.

Figure 6

As Fig. 5 but for irreps $[000]T_2^{+}$, $[000]E^{-}$, $[001]B_1$ and $[001]B_2$ on the largest lattice volume. Dashed curves show noninteracting two-meson energies where the corresponding operator was not included in the basis.

Figure 7

$\pi \omega \{{{}^{3}S}_1\}$ elastic phase-shift assuming no ${{}^{3}D}_1$ amplitude. The blue line shows the reference amplitude given in Eq. (19) with the blue bands reflecting the statistical (inner) plus systematic (outer) uncertainty. Gray lines and bands correspond to a range of parametrizations presented in Table 14 of Appendix pp3 with only the statistical uncertainties shown. The point size (small to large) of the discrete phase-shift point encodes the lattice volume (small to large).

Figure 8

Upper: $\pi \omega \{{{}^{3}S}_1\}$ (blue) and $\pi \omega \{{{}^{3}D}_1\}$ (purple) phase-shifts for the reference amplitude in Eq. (20) with the bands reflecting the statistical (inner) plus systematic (outer) uncertainties. In gray are parametrizations given in Table 15 of Appendix pp3 with only statistical uncertainties shown. Middle: As upper but for the mixing-angle, $\overline{\epsilon }(\pi \omega \{{{}^{3}S}_1\}|\pi \omega \{{{}^{3}D}_1\})$. Lower: Black points are the finite-volume energy levels used to constrain the fit and orange points are the energy levels calculated using Eq. (11) for the reference amplitude in Eq. (20).

Figure 9

Upper: As in Fig. 8 but for the $\pi \omega \{{{}^{3}S}_1\}$ (blue), $\pi \omega \{{{}^{3}D}_1\}$ (purple) and $\pi \varphi \{{{}^{3}S}_1\}$ (green) phase-shifts for the reference amplitude in Eqs. (21) and (22), and for other parametrizations presented in Table 16 of Appendix pp3 (gray). Middle: As upper but for the mixing-angle $\overline{\epsilon }(\pi \omega \{{{}^{3}S}_1\}|\pi \omega \{{{}^{3}D}_1\})$. The other mixing angles, $\overline{\epsilon }(\pi \omega \{{{}^{3}S}_1\}|\pi \varphi \{{{}^{3}S}_1\})$ and $\overline{\epsilon }(\pi \omega \{{{}^{3}D}_1\}|\pi \varphi \{{{}^{3}S}_1\})$, are extremely small and consistent with zero for all parametrizations and are not plotted. Lower: The energy levels used to constrain the scattering amplitude (black) and their corresponding description by the amplitude in Eqs. (21) and (22) (orange).

Figure 10

As Fig. 9 but for ${\rho }_a{\rho }_b|t_{\ell Ja,{\ell }^{\prime }Jb}{|}^2$. Colored curves illustrate the reference amplitude in Eqs. (21) and (22) with bands reflecting the statistical (inner) plus systematic (outer) uncertainty. Other parametrizations presented in Table 16 of Appendix pp3 are in gray with bands reflecting only the statistical uncertainties. ${\rho }_a{\rho }_b|t_{\ell Ja,{\ell }^{\prime }Jb}{|}^2$ not plotted are significantly smaller than those shown and consistent with zero.

Figure 11

As Fig. 5 but including, as orange bands, the energy levels calculated from the reference amplitude in Eqs. (21) and (22) using Eq. (11) as a function $L/a_s$. The thickness of the bands reflect the combined statistical and systematic uncertainties. The vertical red band on the right of the figure indicates the position of the resonant pole of $m_R$ and width ${\mathrm{\Gamma }}_R$ as determined in Sec. 7. The red horizontal line at the resonant mass is shown in each irrep to guide the eye.

Figure 12

Top: Lower half-plane sheet $\mathsf{II}$ poles. Red ellipses reflect the statistical uncertainties, oriented to account for correlations between the real and imaginary parts, for poles from all the parametrizations shown in Table 16 of Appendix pp3. Black ellipses correspond to the reference amplitude in Eq. (22) reflecting the statistical (inner) plus systematic (outer) uncertainties. Bottom: As top but for the corresponding couplings, $c(\pi \omega \{{{}^{3}S}_1\}{)}_{\mathsf{II}}$ (blue), $c(\pi \omega \{{{}^{3}D}_1\}{)}_{\mathsf{II}}$ (purple) and $c(\pi \varphi \{{{}^{3}S}_1\}{)}_{\mathsf{II}}$ (green). Black ellipses again correspond to the couplings of the reference amplitude in Eq. (22) where $c(\pi \varphi \{{{}^{3}S}_1\}{)}_{\mathsf{II}}=0$.

Figure 13

Sensitivity of the finite-volume spectra to $g_{\pi \omega \{{{}^{3}D}_1\}}$. Lighter to darker red curves reflect smaller to larger values of $g_{\pi \omega \{{{}^{3}D}_1\}}$ as shown in the key. The central curves corresponds to $g_{\pi \omega {\{}^3{D}_1\}}=1.08$, i.e., the mean value in the reference amplitude in Eq. (22). The gray bands reflect the combined statistical and systematic uncertainties of Eq. (22). The horizontal axes are in units of $L/a_s$.

Figure 14

As Fig. 13 but for varying ${\gamma }_{\pi \omega \{{{}^{3}P}_0\},\pi \omega \{{{}^{3}P}_0\}}^{(0)}$. The central curves corresponds to ${\gamma }_{\pi \omega \{{{}^{3}P}_0\},\pi \omega \{{{}^{3}P}_0\}}^{(0)}=0$. The phase-shifts on the left reflect the strengths of the $\pi \omega \{{{}^{3}P}_0\}$ amplitudes.

Figure 15

As Fig. 14 but for varying ${\gamma }_{\pi \omega \{{{}^{3}P}_2\},\pi \omega \{{{}^{3}P}_2\}}^{(0)}$.

Figure 16

As Fig. 11 but for the amplitude in Eq. (27). Orange bands reflect only the statistical uncertainty on the scattering parameters. The gray bands are transcribed from Fig. 11.

Figure 17

Top: The scattering amplitudes-squared, ${\rho }_a{\rho }_b|t_{\ell Ja,{\ell }^{\prime }Jb}{|}^2$, transcribed from Fig. 10 with the energy axis converted to physical units. Below the amplitudes are the energy levels used to constrain the amplitudes (black points). Bottom: The best estimate of the resonant pole position, where uncertainties combine statistical and systematic uncertainties with variations across parametrizations. The histograms show the best estimate of the magnitude of each coupling with the lightly shaded region reflecting the combined uncertainties. The $\pi \varphi \{{{}^{3}S}_1\}$ coupling is an estimate of the upper bound.

Figure 18

The $b_1$ pole position for various pion masses. Blue shows the ground-state mass of the axial-vector octet from a lattice calculation with $m_{\pi }\approx 700\mathrm{MeV}$ [46], red shows the estimate from this work with $m_{\pi }\approx 391\mathrm{MeV}$ and black is the experimentally determined mass and width of the $b_1$ resonance [15].

Figure 19

Upper: As in Fig. 8 but for the $\pi \omega \{{{}^{3}S}_1\}$ (blue), $\pi \omega \{{{}^{3}D}_1\}$ (purple), $\pi \varphi \{{{}^{3}S}_1\}$ (green), $\rho \eta \{{{}^{3}S}_1\}$ (orange), and $K^{*}\overline{K}\{{{}^{3}S}_1\}$ (red) phase-shifts for the reference amplitude in Eq. (27). The faded error bands reflect the statistical uncertainty on the scattering parameters. The $\rho \eta$ and $K^{*}\overline{K}$ “thresholds” are calculated using the $\rho$ and $K^{*}$ masses given in Sec. 8c. Lower: As upper but for the mixing angles $\overline{\epsilon }(\pi \omega \{{{}^{3}S}_1\}|\pi \omega \{{{}^{3}D}_1\})$ (blue), $\overline{\epsilon }(\pi \omega \{{{}^{3}S}_1\}|K^{*}\overline{K}\{{{}^{3}S}_1\})$ (gray) and $\overline{\epsilon }(\pi \omega \{{{}^{3}D}_1\}|K^{*}\overline{K}\{{{}^{3}S}_1\})$ (brown). All other mixing angles are extremely small and consistent with zero as discussed in the text.