Abstract
We study the transition widths of $\Upsilon(10753)$ and $\Upsilon(11020)$ into standard bottomonium under the hypothesis that they correspond to the two lowest laying $1^{--}$ hybrid bottomonium states. We employ weakly coupled potential NRQCD an effective filed theory incorporating the heavy quark and multipole expansions. We consider the transitions generated by the leading order and next-to-leading order singlet-octet operators. In the multipole expansion the heavy quark matrix elements factorize from the production of light-quark mesons by gluonic operators. For the leading order operator we compute the widths with a single $\pi^0$, $\eta$ or $\eta'$ in the final state and for the next-to-leading operator for $\pi^+\pi^-$ or $K^+K^-$. The hadronization of the gluonic operators is obtained, in the first case, from the axial anomaly and a standard $\pi^0-\eta-\eta'$ mixing scheme and, in the second case, we employ a coupled-channel dispersive representation matched to chiral perturbation theory for both the $S$ and $D$ wave pieces of the gluonic operator. We compare with experimental values and semi-inclusive widths. Our results strongly suggest that $\Upsilon(11020)$ is indeed a hybrid bottomonium state.
Highlights
Hadrons have been traditionally classified according to their number of valence quarks
We have worked under the assumption that these two states are the first two lowest laying 1−− hybrid bottomonium states
It is plausible that the hybrid bottomonium states used in our approach are a good approximation of a more general isospin I 1⁄4 0 exotic quarkonium state mixing a nontrivial gluonic component with a light-quark–antiquark pair component
Summary
Hadrons have been traditionally classified according to their number of valence quarks. Often several interpretations are consistent with the observed spectrum and not enough quantum numbers of exotic quarkonium are accessible experimentally to be able to check heavy-quark spin symmetry multiplet predictions Another avenue to understand the structure of exotic quarkonium is the study of their decays, in particular transitions into standard quarkonium states with one or two light-quark mesons in the final state, since many of the known exotic quarkonium states have been discovered through these decay channels. Since the heavy quarks in exotic quarkonium are nonrelativistic, the natural starting point for their study is NRQCD [3,4] at leading order, that is in the static limit In this limit the spectrum is composed of the so-called static energies, which depend on the quantum numbers of the light quarks and gluon degrees of freedom, the heavy-quark antiquark distance, and the representation of the cylindrical symmetry group D∞h.1. In Appendix C, we collect the definitions of the Mandelstam variables and several formulas employed in the evaluation of widths from the transition amplitudes
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