Abstract

In a recent paper Phys.Rev. D98, 074023 (2018), the most up-to-date experimental data for all measured production and decay channels of the bottomonium-like states $Z_b(10610)$ and $Z_b(10650)$ were analysed in a field-theoretical coupled-channel approach which respects analyticity and unitarity and incorporates both the pion exchange as well as a short-ranged potential nonperturbatively. All parameters of the interaction were fixed directly from data, and pole positions for both $Z_b$ states were determined. In this work we employ the same approach to predict in a parameter-free way the pole positions and the line shapes in the elastic and inelastic channels of the (still to be discovered) spin partners of the $Z_b$ states. They are conventionally referred to as $W_{bJ}$'s with the quantum numbers $J^{PC}=J^{++}$ ($J=0,1,2$). It is demonstrated that the results of our most advanced pionful fit, which gives the best $\chi^2/{\rm d.o.f.}$ for the data in the $Z_b$ channels, are consistent with all $W_{bJ}$ states being above-threshold resonances which manifest themselves as well pronounced hump structures in the line shapes. On the contrary, in the pionless approach, all $W_{bJ}$'s are virtual states which can be seen as enhanced threshold cusps in the inelastic line shapes. Since the two above scenarios provide different imprints on the observables, the role of the one-pion exchange in the $B^{(*)}\bar{B}^{(*)}$ systems can be inferred from the once available experimental data directly.

Highlights

  • Heavy-quark spin symmetry (HQSS) is an approximate symmetry of QCD

  • In this paper we address the properties of the spin partners WbJ with the quantum numbers Jþþ (J 1⁄4 0, 1, 2) of the bottomoniumlike states Zbð10610Þ and Zbð10650Þ

  • The same effective field theory (EFT) approach consistent with requirements from unitarity, analyticity and HQSS is employed to predict in a parameter-free way the line shapes of the positive C-parity spin partner states WbJ in the corresponding elastic [BðÃÞBðÃÞ] and inelastic [ηbðnSÞπ and χbJðmPÞπ] channels

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Summary

INTRODUCTION

Heavy-quark spin symmetry (HQSS) is an approximate symmetry of QCD. It states that in the limit of an infinite mass of the heavy quark its spin is conserved by the strong interactions. [15] (and from the discussion below) that the existing experimental data are well described without the necessity of including compact quark components explicitly This could be anticipated since the inclusion of the S-wave-to-S-wave contact terms with two-derivatives (which enter our EFT at next-to-leading (NLO) order) results in a perturbative effect, in line with our power counting, and, indicates convergence of the EFT series. We exploit the fact that all the parameters of the elastic and inelastic potentials extracted from the experimental line shapes in the JPC 1⁄4 1þ− channel are the same in the partner channels up to spin symmetry violating corrections in the contact interactions that are expected to be small. Appendix A contains the details of the NLO Lagrangian OðQ2Þ used to build the suitable EFT, while Appendix B provides the details of the partial wave projection operators applied to the effective potential

Some generalities and definitions
Contact interactions
Inelastic channels
ÁÁÁ ÁÁÁ ÁÁÁ
Pion exchange
Production vertex
Coupled-channel system
Production rates
Renormalizability of the heavy-hadron EFT with pions
Line shapes in the spin partner channels
Extracting the poles in a multichannel scattering problem
Rα g2α 2MRα
Uncertainty estimate
SUMMARY AND CONCLUSION
Full Text
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