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Register a new userSpin partners of the $Z_b(10610)$ and $Z_b(10650)$ revisited
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We study the implications of the heavy-quark spin symmetry for the possible spin partners of the exotic states $Z_b(10610)$ and $Z_b(10650)$ in the spectrum of bottomonium. We formulate and solve numerically the coupled-channel equations for the $Z_b$ states that allow for a dynamical generation of these states as hadronic molecules. The force includes short-range contact terms and the one-pion exchange potential, both treated fully nonperturbatively. The strength of the potential at leading order is fixed completely by the pole positions of the $Z_b$ states such that the mass and the most prominent contributions to the width of the isovector heavy-quark spin partner states $W_{bJ}$ with the quantum numbers $J^{++}$ ($J=0,1,2$) come out as predictions. Since the accuracy of the present experimental data does not allow one to fix the pole positions of the $Z_b$s reliably enough, we also study the pole trajectories of their spin partner states as functions of the $Z_b$ binding energies. It is shown that, once the heavy-quark spin symmetry is broken by means of the physical $B$ and $B^*$ masses, especially the pion tensor force has a significant impact on the location of the partner states clearly demonstrating the need of a coupled-channel treatment of pion dynamics to understand the spin multiplet pattern of hadronic molecules.
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The most recent experimental data for all measured production and decay channels of the bottomonium-like states $Z_b(10610)$ and $Z_b(10650)$ are analysed simultaneously using solutions of the Lippmann-Schwinger equations which respect constraints from unitarity and analyticity. The interaction potential in the open-bottom channels $B^{(*)}bar{B}^{*}+mbox{c.c.}$ contains short-range interactions as well as one-pion exchange. It is found that the long-range interaction does not affect the line shapes as long as only $S$ waves are considered. Meanwhile, the line shapes can be visibly modified once $D$ waves, mediated by the strong tensor forces from the pion exchange potentials, are included. However, in the fit they get balanced largely by a momentum dependent contact term that appears to be needed also to render the results for the line shapes independent of the cut-off. The resulting line shapes are found to be insensitive to various higher-order interactions included to verify stability of the results. Both $Z_b$ states are found to be described by the poles located on the unphysical Riemann sheets in the vicinity of the corresponding thresholds. In particular, the $Z_b(10610)$ state is associated with a virtual state residing just below the $Bbar{B}^{*}/bar B{B}^{*}$ threshold while the $Z_b(10650)$ state most likely is a shallow state located just above the $B^*bar{B}^{*}$ threshold.
The dipion transitions $Upsilon(10860)topi^+pi^-Upsilon(nS)$ ($n=1,2,3$) are studied in the framework of a unitary and analytic coupled-channel formalism previously developed for analysing experimental data on the bottomoniumlike states $Z_b(10610)$ and $Z_b(10650)$ [Phys. Rev. D 98, 074023 (2018)] and predicting the properties of their spin partners [Phys. Rev. D 99, 094013 (2019)]. In this work we use a relatively simple but realistic version of this approach, where the scattering and production amplitudes are constructed employing only short-ranged interactions between the open- and hidden-flavour channels consistent with the constraints from heavy quark spin symmetry, for an extended analysis of the experimental line shapes. In particular, the transitions from the $Upsilon(10860)$ to the final states $pi pi h_b(mP)$ ($m=1,2$) and $pi B^{(*)}bar B^* $ already studied before, are now augmented by the $Upsilon(10860)topi^+pi^-Upsilon(nS)$ final states ($n=1,2,3$). This is achieved by employing dispersion theory to account for the final state interaction of the $pipi$ subsystem including its coupling to the $Kbar K$ channel. Fits to the two-dimensional Dalitz plots for the $pi^+pi^-Upsilon$ final states were performed. Two real subtraction constants are adjusted to achieve the best description of the Dalitz plot for each $Upsilon(nS)$ $(n=1,2,3)$ while all the parameters related to the properties of the $Z_b$s are kept fixed from the previous study. A good overall description of the data for all $Upsilon(10860)topi^+pi^-Upsilon(nS)$ channels achieved in this work provides additional strong support for the molecular interpretation of the $Z_b$ states.
Within the framework of dispersion theory, we analyze the dipion transitions between the lightest $Upsilon$ states, $Upsilon(nS) rightarrow Upsilon(mS) pipi$ with $m < n leq 3$. In particular, we consider the possible effects of two intermediate bottomoniumlike exotic states $Z_b(10610)$ and $Z_b(10650)$. The $pipi$ rescattering effects are taken into account in a model-independent way using dispersion theory. We confirm that matching the dispersive representation to the leading chiral amplitude alone cannot reproduce the peculiar two-peak $pipi$ mass spectrum of the decay $Upsilon(3S) rightarrow Upsilon(1S) pipi$. The existence of the bottomoniumlike $Z_b$ states can naturally explain this anomaly. We also point out the necessity of a proper extraction of the coupling strengths for the $Z_b$ states to $Upsilon(nS)pi$, which is only possible if a Flatte-like parametrization is used in the data analysis for the $Z_b$ states.
We study the dipion transitions $Upsilon(4S) rightarrow Upsilon(nS) pi^+pi^-$ $(n=1,2)$. In particular, we consider the effects of the two intermediate bottomoniumlike exotic states $Z_b(10610)$ and $Z_b(10650)$ as well as bottom meson loops. The strong pion-pion final-state interactions, especially including channel coupling to $Kbar{K}$ in the $S$-wave, are taken into account model-independently by using dispersion theory. Based on a nonrelativistic effective field theory we find that the contribution from the bottom meson loops is comparable to those from the chiral contact terms and the $Z_b$-exchange terms. For the $Upsilon(4S) rightarrow Upsilon(2S) pi^+pi^-$ decay, the result shows that including the effects of the $Z_b$-exchange and the bottom meson loops can naturally reproduce the two-hump behavior of the $pipi$ mass spectra. Future angular distribution data are decisive for the identification of different production mechanisms. For the $Upsilon(4S) rightarrow Upsilon(1S) pi^+pi^-$ decay, we show that there is a narrow dip around 1 GeV in the $pipi$ invariant mass distribution, caused by the final-state interactions. The distribution is clearly different from that in similar transitions from lower $Upsilon$ states, and needs to be verified by future data with high statistics. Also we predict the decay width and the dikaon mass distribution of the $Upsilon(4S) rightarrow Upsilon(1S) K^+ K^-$ process.
Employing the relativized quark model and the quark-interchange model, we investigate the decay of the charged heavy quarkonium-like states $Z_c(3900)$, $Z_c(4020)$, $Z_c(4430)$, $Z_b(10610)$ and $Z_b(10650)$ into the ground and radially excited heavy quarkonia via emitting a pion meson. The $Z_c$ and $Z_b$ states are assumed to be hadronic molecules composed of open-flavor heavy mesons. The calculated decay ratios can be compared with the experimental data, which are useful in judging whether the molecule state assignment for the corresponding $Z_c$ or $Z_b$ state is reasonable or not. The theoretical framework constructed in this work will be helpful in revealing the underlying structures of some exotic hadrons.
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