Interplay of quark and meson degrees of freedom in near-threshold states: A practical parametrization for line shapes

We propose a practical parametrization for the line shapes of near-threshold states compatible with all requirements of unitarity and analyticity. The coupled-channel system underlying the proposed parametrization includes bare poles and an arbitrary number of elastic and inelastic channels treated fully nonperturbatively. The resulting formulas are general enough to be used for a simultaneous analysis of the data in all available production and decay channels of the (system of) state(s) under consideration for a quite wide class of reactions. As an example, we fit the experimental data currently available for several decay channels for the charged $Z_b^{(\prime )}$ states in the spectrum of bottomonia and find a good overall description of the data. We find the present data to be consistent with the $Z_b(10610)$ as a virtual state and with the $Z_b(10650)$ as a resonance, both residing very close to the $B{\overline{B}}^{*}$ and $B^{*}{\overline{B}}^{*}$ threshold, respectively.

Figure 1

Graphical representation of the Lippmann-Schwinger equation for the elastic scattering $t$ matrix $t_{AB}$—see Eq. (17).

Figure 2

Graphical representation of the expression for the $t$ matrix component $t_{iA}$ given in Eq. (21). Representation for the component $t_{Ai}$ takes a similar form and it is not shown. The solid lines are for the heavy-light $(\overline{q}Q)$ and $(\overline{Q}q)$ mesons with open flavor, the fat solid line is for the hidden-flavor heavy meson ($\overline{Q}Q$), and the dashed line is for the light meson ($\overline{q}q$).

Figure 3

Graphical representation of the expression for the $t$ matrix component $t_{ij}$ given in Eq. (21). For the lines’ identification see the caption of Fig. 2.

Figure 4

The full effective interaction potential in the elastic channels—see Eq. (27). The double solid line is for the pole term propagator $G_0$ [see Eq. (24)] while for the other lines’ identification see the caption of Fig. 2. Potentials $V_{\alpha a}$ and $V_{b\beta }$ are defined in Eqs. (14) and (15).

Figure 5

Graphical representation of the contributions to the production amplitude for the channel $x$ from a pointlike source: Born term (first diagram) and rescattering term (second diagram).

Figure 6

Fitted line shapes of the $Z_b(10610)$ and $Z_b(10650)$ in the $B^{(*)}{\overline{B}}^{*}$ channels [plots (a) and (b)] and in the $\pi h_b(mP)$ ($m=1$, 2) channels [plots (c) and (d)]. Parameters of fits A, B, and C are used for the plots in the upper, middle, and lower rows, respectively.

Figure 7

Plots (a), (b), and (c) are for the predicted line shapes of the $Z_b(10610)$ and $Z_b(10650)$ in the $\pi \mathrm{\Upsilon }(2S)$ channel for fits A, B, and C, respectively. To guide the eye, as plot (d), we also show the corresponding experimental figure adapted from Ref. [46]. Notice that the behavior of the line shape below the left shoulder of the lower peak is influenced by the effects which lie beyond the scope of the present paper and will be addressed in future publications. Notice also that the presence of the nonresonant background in the experimental figure does not allow its direct comparison with the predicted line shapes.

Figure 8

First plot: Four Riemann sheets mapped into the $\omega$ plane. The thick solid line corresponds to the real axis in the first Riemann sheet of the complex energy plane. Second plot: The poles of the full $t$ matrix which correspond to the quantum numbers $1^{+-}$ and to the set of parameters from fit B—see Table 1. The red crosses mark the central values and the green dots show the upper and the lower bounds due to the uncertainties in the fitted parameters. Third plot: The same as in the second plot but for the set of parameters from fit C.

Figure 9

The path of the RS-III pole to RS-I. The complex conjugated pole is also shown but not its path.

Figure 10

The space of parameters (${\gamma }_t$ versus ${\gamma }_s$) for the poles of $t^v$ in the channel $1^{+-}$ close to the $B{\overline{B}}^{*}$ threshold [$Z_b(10610)$ state]. The blue and green curves correspond to different Riemann sheets (RS-I versus RS-II, respectively) and the red line gives the boundary between the two regions. The energy of the state relative to the threshold is quoted, in MeV, at every curve. The black dots with the error bars show the actual values of the parameters ${\gamma }_s$ and ${\gamma }_t$ for fits B and C taken from Table 1.