Hyperfine splittings of heavy quarkonium hybrids

TL;DR

This work computes the hyperfine splittings of heavy quarkonium hybrids at leading order in the BOEFT expansion by combining short-distance pNRQCD constraints with long-distance EST predictions, interpolated through a scale $r_0$. Two spin-dependent potentials $V_{hf}$ and $V_{hf2}$ control the LO hyperfine structure, with short-distance constants $A$ and $B$ fixed from charmonium lattice data and then extended to bottomonium via NRQCD matching. The inclusion of EST-based long-distance contributions significantly improves agreement with lattice results for charmonium hybrids and yields precise predictions for bottomonium hybrids, along with quantified uncertainties. The approach demonstrates the value of blending BOEFT with EST in hadronic hybrids and provides concrete predictions for experimental search of hybrid states.

Abstract

In the framework of the Born-Oppenheimer Effective Field Theory, the hyperfine structure of heavy quarkonium hybrids at leading order in the 1/m Q expansion is determined by two potentials. We estimate those potentials by interpolating between the known short distance behavior and the long distance behavior calculated in the QCD Effective String Theory. The long distance behavior depends, at leading order, on two parameters which can be obtained from the long distance behavior of the heavy quarkonium potentials (up to sign ambiguities). The short distance behavior depends, at leading order, on two extra paramentes, which are obtained from a lattice calculation of the lower lying charmonium hybrid multiplets. This allows us to predict the hyperfine splitting both of bottomonium hybrids and of higher multiplets of charmonium hybrids. We carry out a careful error analysis and compare with other approaches.

Figures (4)

• Figure 1: The two spin-dependent potentials (\ref{['Vhfdef']}) at short (\ref{['Vhfsd']}) [blue line] and long (\ref{['Vhfl']}) [green line] distances. The orange line shows the interpolation we use (\ref{['Vhf']}). The short distance parameters $A$ and $B$ correspond to those in Table \ref{['ABerror']} and $r_0 = 3.96$ GeV$^{-1}$.
• Figure 2: The spectrum of the lower-lying $n\,(s/d)_1$ ($H_1$), $n\,p_1$ ($H_2$), $n\,(p/f)_2$ ($H_4$) and $n\,p_0$ ($H_3$) charmonium hybrids computed by adding the LO spin-dependent potentials to the static potentials used in ref. Oncala:2017hop is shown in green boxes. The average mass for each multiplet is shown as a red dashed line. Blue and magenta boxes show the results of the BOEFT with NLO spin-dependent short distance potentials only Brambilla:2018pynBrambilla:2019jfi and lattice QCD Cheung:2016bym respectively, adjusted to our spin average. The height of the boxes indicates the uncertainty.
• Figure 3: The spectrum of the lower-lying $n\,(s/d)_1$ ($H_1$), $n\,p_1$ ($H_2$), $n\,(p/f)_2$ ($H_4$) and $n\,p_0$ ($H_3$) bottommonium hybrids computed by adding the LO spin-dependent potentials to the static potentials used in ref. Oncala:2017hop is shown in green boxes. The average mass for each multiplet is shown as a red dashed line. Blue and magenta boxes show the results of the BOEFT with NLO spin-dependent short distance potentials only Brambilla:2018pynBrambilla:2019jfi and lattice QCD Ryan:2020iog respectively, adjusted to our spin average. The height of the boxes indicates the uncertainty.
• Figure 4: Comparison of the $V_{1^+11}^{sa}(r)/c_F$ (left) and $V_{1^+10}^{sb}(r)/c_F$ (right) potentials in Ref. Soto:2020xpm with our model (solid green), with lattice results from Schlosser:2025tca (red dots), and with results from Table II of Brambilla:2019jfi (dashed blue).