Unquenched Radially Excited $P$-wave Charmonia

Abstract: The ground-state positive-parity charmonia $χ<em>{c0}(1P)$, $χ</em>{c1}(1P)$, $h_c(1P)$, and $χ<em>{c2}(1P)$ are generally well described in static (``quenched'') quark models, in which dynamical effects of actual or virtual strong decay are neglected. In contrast, the five PDG candidates for $P$-wave charmonia in the energy region 3.85-3.95 GeV, probably including the first radial excitations of the above ones, display a totally different and quite disparate mass pattern. Moreover, two scalar states are listed, viz. $χ</em>{c0}(3860)$ and $χ_{c0}(3915)$, the former one apparently being very broad. Preliminary results will be presented here for the first radial excitations of the lowest $P$-wave $c\bar{c}$ states, obtained with the Resonance-Spectrum Expansion while including in the calculation all OZI-allowed decay channels of the most relevant charm-meson pairs. Employing a generalised scheme of computing coupling constants for decays based on the ${}<sup>{3!}P_0$ model ensures that no distortion of the spectra will occur due to the different classes of allowed decay channels for the various positive-parity charmonia.

• The paper demonstrates that explicit open-charm decay channels induce two scalar poles in the χc0 sector, explaining the broad and narrow resonances observed in experiments.
• The paper employs the unitarised Resonance-Spectrum Expansion framework with the ³P₀ transition operator to compute complex pole positions that closely match experimental mass and width values.
• The paper illustrates that coupled-channel effects significantly alter the mass splittings in 2P charmonium states compared to static quark models, emphasizing the importance of unitarisation in heavy quark spectroscopy.

Unitarised Description of Radially Excited $P$-wave Charmonia

Introduction

This paper focuses on the theoretical investigation of the first radially excited $P$-wave charmonia above open-charm thresholds, deploying the Resonance-Spectrum Expansion (RSE) framework. The study systematically examines the consequences of unquenching—explicitly including open-flavor strong decay channels—on the mass spectra and widths of positive-parity charmonium states, with particular attention to interpreting the perplexing experimental landscape for the $\chi_{c0}$, $\chi_{c1}$, $h_c$, and $\chi_{c2}$ sectors. Contrasts are drawn with the regular and well-understood bottomonium spectra, highlighting the crucial role of open-charm thresholds in distorting the expected quark-model patterns in charmonium.

Motivation and Experimental Puzzles

While static (quenched) quark models accounting for scalar linear-plus-Coulomb confining potentials and perturbative spin-dependent forces successfully describe the lowest $P$-wave charmonia, the experimental data in the energy region 3.85–3.95 GeV reveal a distinctly irregular structure. The Particle Data Group lists five putative $2P$ charmonium states—$\chi_{c0}(3860)$, $\chi_{c1}(3872)$, $P$0, $P$1, and $P$2—with significant anomalies:

• Two scalar ($P$3) states instead of the expected single state, with $P$4 being very broad and $P$5 much narrower.
• Mass inversion: $P$6 is lighter than $P$7, at odds with typical quark-model predictions.
• Unexpected ordering: $P$8, interpreted as a possible $P$9, is heavier than $\chi_{c0}$0, defying simple expectations.

These empirical irregularities are absent in the bottomonium system, where the $\chi_{c0}$1 states all reside below open-bottom thresholds and exhibit regular mass splitting ratios, attributed to nodal structure in the wavefunctions.

Theoretical Framework and Methodology

The RSE model facilitates the unitarisation of quarkonium states by encompassing all OZI-allowed decay channels within a generic coupled-channel formalism. The model incorporates precise decay coupling computations derived from the $\chi_{c0}$2 transition operator, ensuring a consistent treatment across different quantum numbers and decay channels and minimizing artificial spectral distortions.

Previous analyses of $\chi_{c0}$3 and $\chi_{c0}$4 within similar unitarised models revealed the sensitivity of pole positions and widths to quark mass and effective coupling variations. These studies also illuminated the dual nature of the $\chi_{c0}$5 as a state with a substantial $\chi_{c0}$6 component at small distances, becoming $\chi_{c0}$7-dominated at large distances.

Results

A comprehensive RSE calculation for all $\chi_{c0}$8 positive-parity charmonia (with $\chi_{c0}$9, $\chi_{c1}$0, $\chi_{c1}$1, $\chi_{c1}$2) was carried out, considering coupling to all relevant open-charm meson pairs. The resulting complex pole positions (in MeV) for dominant $\chi_{c1}$3 components are reported:

• $\chi_{c1}$4 sector: $\chi_{c1}$5 and $\chi_{c1}$6 (two scalar resonances)
• $\chi_{c1}$7: $\chi_{c1}$8
• $\chi_{c1}$9: $h_c$0
• $h_c$1: $h_c$2

These results encapsulate several critical points:

• Two scalar poles emerge in the $h_c$3 sector, consistent with the experimental observation of both $h_c$4 (broad) and $h_c$5 (narrow).
• The calculated $h_c$6 pole is in precise agreement with experiment, exhibiting a small width, compatible with its proximity to the $h_c$7 threshold.
• The $h_c$8 and $h_c$9 positions indicate the necessity of incorporating spin-orbit and tensor interactions, as well as potential mixing effects, to explain the full $\chi_{c2}$0 spectrum and the presence of $\chi_{c2}$1.

A key theoretical output is the demonstration that due to strong coupled-channel effects above threshold, the $\chi_{c2}$2 mass splittings and resonance pattern in charmonium are radically altered compared to both the bottomonium system and static models.

Practical and Theoretical Implications

The findings reinforce the indispensable role of unitarisation and explicit open-flavor channel coupling for understanding charmonium states above open-charm thresholds. For phenomenology:

• The existence of multiple scalar $\chi_{c2}$3 poles substantiates the experimental listing of both $\chi_{c2}$4 and $\chi_{c2}$5 as predominantly $\chi_{c2}$6 but strongly admixed/dynamical states.
• Pole mass and width determinations elucidate the broad/narrow dichotomy in the scalar sector, directly tied to the underlying coupled-channel dynamics and spectral density near corresponding thresholds.
• The successful prediction of the $\chi_{c2}$7 mass supports its primarily $\chi_{c2}$8 nature with crucial open-charm channel admixtures, consistent with its unique decay properties and production in $\chi_{c2}$9 decays.

The theoretical implication is that the static quark model paradigm is insufficient in this regime; open-channel-induced threshold effects dominate and can invert or drastically modify expected mass orderings and splittings. The results suggest that identifying higher quarkonium states above open-flavor thresholds must always rely on unitarised treatments that permit dynamical pole generation and coupling-induced spectral shifts.

Future Directions

The immediate extension involves systematic inclusion of spin-orbit and tensor interactions and state mixing, to achieve a quantitatively accurate description of all $P$0 charmonium states and potentially disentangle intrinsic from dynamically generated resonances. Tracking pole evolution in the complex energy and momentum plane will further clarify the nature of controversial states such as $P$1 and potential hybrid admixtures. These developments are integral for robust QCD-based modelling of the heavy quarkonium spectrum in regions dominated by strong decay dynamics.

Conclusion

This study provides a technical and predictive description of radially excited $P$2-wave charmonia in the regime above open-charm thresholds, employing a consistently unitarised coupled-channel formalism. The analysis reproduces key experimental features, notably the presence of two scalar charmonium candidates and the anomalous placement of the $P$3. The research asserts that only by integrating all OZI-allowed decay modes and their couplings can the complex experimental charmonium spectrum in this sector be appropriately rationalised, with foundational implications for both hadron spectroscopy and the precise identification of non-perturbative QCD effects in heavy quark systems.