Hybrid to Quarkonia transitions

TL;DR

This work addresses the identification of heavy-quark hybrids in the charmonium and bottomonium sectors using Born-Oppenheimer Effective Field Theory (BOEFT) informed by the latest lattice QCD static potentials. It updates the hybrid spectrum, recalculates both spin-conserved and spin-flip hybrid-to-quarkonium transitions, and performs a comprehensive error analysis that includes masses, higher-order couplings, multipole effects, and the weak/strong coupling transition. By confronting the predictions with PDG XYZ resonances, the study proposes hybrid or quarkonium assignments for most zero-isospin heavy mesons and predicts semi-inclusive decay widths for testable channels. The results strengthen the case for hybrid interpretations of several XYZ states and demonstrate the impact of improved potentials on spectra and decay amplitudes, while also noting limitations in bottomonium due to quenched lattice inputs.

Abstract

Hybrid quarkonia -exotic hadrons with explicit gluonic degrees of freedom- have gained increasing attention in hadron spectroscopy, particularly with the ongoing discovery of new XYZ mesons. In this work, we update the spectrum of heavy hybrid mesons in the charmonium and bottomonium sectors using the Born-Oppenheimer Effective Field Theory framework, by incorporating the latest lattice QCD results for hybrid static potentials. We refine earlier calculations and analyze allowed transitions from hybrids to conventional quarkonia, including both spin-conserved and spin-flip decays. We carry out a comprehensive error analysis and discuss the reliability of our results. We compare them to experimental data of the Particle Data Group, which allows us to identify hybrid candidates among the observed XYZ states. We provide hybrid or quarkonium interpretations for nearly all heavy isospin-zero mesons observed and incorporate new hybrid candidates.

Figures (3)

• Figure 1: Spectrum of charmonium (up) and bottomonium (down) from Table \ref{['fig:hybrid_spectrum']} put on top of the experimental XYZ mesons reported by the PDG. We show in green the states with known $J^{PC}$ and in orange the states with unknown or partially known $J^{PC}$. We show an error of $\pm 110$ MeV ($\pm 33$ MeV) for charmonium (bottomonium) according to our expected precision. The $(s/d)_1$, $p_1$, $p_0$, $(p/f)_2$ and $d_2$ states are usually named $H_1$, $H_2$, $H_3$, $H_4$ and $H_5$ respectively Juge:1999ieBerwein:2015vcaBrambilla:2022hhi. Our identifications are summarized in Tables \ref{['tab:hybrids']},\ref{['tab:charmonium0']}, and \ref{['tab:hybrids0']}.
• Figure 2: Octet field self-energy diagram in pNRQCD. Double line represents octet propagator (hybrid), while single lines represent singlet propagator (quarkonium). The curly line is a gluon of energy larger than $\Lambda_{\rm QCD}$, the vertices are chromoelectric dipole vertices Eq. \ref{['Lagrangian_0']} for spin-conserved transitions or chromomagnetic dipole ones Eq. \ref{['Lagrangian_1']} for spin-flip transitions. Gluons with energy similar to $\Lambda_{\rm QCD}$ are not displayed. They are necessary to make the initial and final state gauge invariant.
• Figure 3: Left panel: the blue and green lines show the fit of the hybrid potentials $\Pi_u$ and $\Sigma_u$, respectively. The solid red line shows the fit of the static potential $\Sigma_g^+$ in Eq. \ref{['VLambdaeta']} proposed in ref. Alasiri:2024nue on top of the lattice data ( dots) from ref. Capitani:2018roxSchlosser:2021wnr. The energy gap of $\sim1.2$ GeV corresponds to the energy difference between the minimum of the hybrid potentials and the value of the $\Sigma_g^+$ potential at the distance where the minimum occurs. Right panel: the red dash-dotted lines show the Coulomb potential and the red dotted lines the confining potential in order to illustrate the energy gap between long- and short- distance regimens, $\mu=E_0-E_{0^{++}}=-28$ MeV. We also show in black the expected long (EST) and short (perturbative) distance behavior. The red dashed line shows the corresponding Cornell approximation that we use in section \ref{['sec:Co']} to estimate the error.