Abstract
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 $B\bar{B}^{*}/\bar B{B}^{*}$ threshold while the $Z_b(10650)$ state most likely is a shallow state located just above the $B^*\bar{B}^{*}$ threshold.
Highlights
In the last decade, numerous new hadrons have been observed in the charmonium and bottomonium energy region
This work continues a series of papers aimed at a systematic description of the line shapes of near-threshold resonances in general and, in particular, at understanding the nature and properties of the ZÆb ð10610Þ and ZÆb ð10650Þ states in the spectrum of bottomonium
The results of this work can be summarized as follows: (i) We find that the distributions obtained from the direct numerical solution of the Lippmann-Schwinger equations (LSEs) with just
Summary
Numerous new hadrons have been observed in the charmonium and bottomonium energy region. The fact that the full OPE has to be incorporated into a coupled-channel approach to the BðÃÞBðÃÞ system is natural already from the momentum scales involved: For energies near the BÃB à threshold, the on shell relative momentum in the BÃBchannel is as large as pffiffiffiffiffiffiffiffi ptyp 1⁄4 mBδ ≃ 500 MeV; ð1Þ where δ 1⁄4 mBà − mB 1⁄4 45 MeV denotes the BÃ-B mass difference, with mB and mBà being the B and Bà meson mass, respectively. In this work we analyze the line shapes at leading order within an EFT approach, which is formulated based on an effective Lagrangian consistent with both chiral and heavyquark spin symmetries of QCD. Since all the other interactions are of a shorter range they are integrated out to the order we are working and included in the EFT Lagrangian via a series of contact interactions with the low-energy constants (LECs) adjusted to the data This procedure is in full analogy to the treatment for fewnucleon systems, see e.g., Ref.
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