Observation of the Exotic State $π_{1}(1600)$ in $ψ(2S)\rightarrowγχ_{c1},χ_{c1}\rightarrowπ^{+}π^{-}η'$

Abstract: A partial wave analysis of the process $$ψ(2S)\rightarrowγχ<em>{c1}, χ</em>{c1}\rightarrowπ<sup>+π<sup>-η<sup>{\prime}$$ is performed using $(2712.4\pm14.3)\times10<sup>{6}$ $ψ(2S)$ events collected with the BESIII detector. An isovector state with exotic quantum numbers $J<sup>{PC}=1<sup>{-+}$, denoted as $π<em>{1}(1600)$, is observed for the first time in the charmonium decay of $$χ</em>{c1}\rightarrowπ<em>{1}<sup>{\pm}(1600)π<sup>{\mp}$$, $$π</em>{1}<sup>{\pm}(1600)\rightarrowπ<sup>{\pm}η<sup>{\prime}$$ with a statistical significance over $21σ$. Its mass and width are determined to be $$1828 \pm 8 ({\rm stat})^{{+11}_{-33}({\rm} syst})~\mathrm{MeV}/c<sup>2$$ and $$638 \pm 26 ({\rm stat})^{{+35}_{-86}({\rm} syst})~\mathrm{MeV}$$, respectively, using a relativistic Breit-Wigner function with a mass-dependent width. The corresponding product of branching fractions is determined to be $$\mathcal{B}\left[χ<em>{c1}\rightarrowπ</em>{1}(1600)^{{\pm}π^{\mp}} \right] \times \mathcal{B}\left[π<em>{1}(1600)^{{\pm}\rightarrowπ^{{\pm}η^{{\prime}\right]}}} = \left( 4.30 \pm 0.14 ({\rm stat})^{+1.04}</em>{-1.03}({\rm syst})~ \right) \times 10<sup>{-4}$$.

• The paper establishes the existence of the exotic π₁(1600) state with a significance exceeding 21σ via robust amplitude analysis.
• It employs a comprehensive partial-wave analysis using covariant tensor formalism to fully reconstruct over 37,000 events from two η′ decay modes.
• The study precisely measures the π₁(1600) mass at 1828 MeV/c² and width at 638 MeV, supporting its interpretation as the lightest hybrid meson.

Observation of the Exotic State $\pi_{1}(1600)$ in $$\psi(2S)\rightarrow\gamma\chi_{c1},\chi_{c1}\rightarrow\pi^{+}\pi^{-}\eta^{\prime}$$

Introduction and Background

The identification and characterization of hadrons with exotic quantum numbers, which cannot be accommodated in the conventional constituent quark model, provide critical constraints on non-perturbative QCD. States with $J^{PC}=1^{-+}$ are especially notable, since such quantum numbers are unattainable by ordinary quark-antiquark mesons and are widely interpreted as strong candidates for hybrid mesons with an excited gluonic component. Although multiple candidates for $1^{-+}$ isovector states, e.g., $\pi_1(1400)$, $\pi_1(1600)$, and $\pi_1(2015)$, have been reported in hadronic reactions, their properties and even their existence have remained subject to debate due to possible analysis artifacts and production uncertainties in diffractive processes.

Prior amplitude analyses in charmonium decays, which offer clean experimental environments and minimal background, have lacked the statistical precision to robustly establish the $\pi_1(1600)$ resonance. This paper addresses this gap by performing a high-statistics amplitude analysis of $\psi(2S)\to\gamma\chi_{c1}$, $\chi_{c1}\to\pi^+\pi^-\eta^{\prime}$, using the world's largest $$\psi(2S)\rightarrow\gamma\chi_{c1},\chi_{c1}\rightarrow\pi^{+}\pi^{-}\eta^{\prime}$$0 sample collected by the BESIII detector. The analysis targets the $$\psi(2S)\rightarrow\gamma\chi_{c1},\chi_{c1}\rightarrow\pi^{+}\pi^{-}\eta^{\prime}$$1 system, where evidence for a resonant $$\psi(2S)\rightarrow\gamma\chi_{c1},\chi_{c1}\rightarrow\pi^{+}\pi^{-}\eta^{\prime}$$2 signal can be unambiguously attributed to an exotic state, strongly constraining hybrid interpretations.

Experimental Analysis

The analysis uses $$\psi(2S)\rightarrow\gamma\chi_{c1},\chi_{c1}\rightarrow\pi^{+}\pi^{-}\eta^{\prime}$$3 $$\psi(2S)\rightarrow\gamma\chi_{c1},\chi_{c1}\rightarrow\pi^{+}\pi^{-}\eta^{\prime}$$4 events to reconstruct $$\psi(2S)\rightarrow\gamma\chi_{c1},\chi_{c1}\rightarrow\pi^{+}\pi^{-}\eta^{\prime}$$5 with $$\psi(2S)\rightarrow\gamma\chi_{c1},\chi_{c1}\rightarrow\pi^{+}\pi^{-}\eta^{\prime}$$6 detected in both $$\psi(2S)\rightarrow\gamma\chi_{c1},\chi_{c1}\rightarrow\pi^{+}\pi^{-}\eta^{\prime}$$7 and $$\psi(2S)\rightarrow\gamma\chi_{c1},\chi_{c1}\rightarrow\pi^{+}\pi^{-}\eta^{\prime}$$8 ($$\psi(2S)\rightarrow\gamma\chi_{c1},\chi_{c1}\rightarrow\pi^{+}\pi^{-}\eta^{\prime}$$9) final states. Both decay modes are fully reconstructed with kinematic and topological constraints to maximize purity. Control of backgrounds is achieved using tight invariant-mass selections and explicit vetoes on known contaminant channels. Residual backgrounds, including non-$J^{PC}=1^{-+}$0 and $J^{PC}=1^{-+}$1, are estimated and subtracted using sideband techniques and MC-driven methods.

The selected datasets comprise 24,577 (for $J^{PC}=1^{-+}$2) and 12,952 (for $J^{PC}=1^{-+}$3) events. The Dalitz plots and $J^{PC}=1^{-+}$4 invariant mass distributions for both final states show clear enhancements around $J^{PC}=1^{-+}$5, motivating a full partial-wave analysis.

Figure 1: (a, c) Dalitz plots; (b, d) $J^{PC}=1^{-+}$6 distributions for $J^{PC}=1^{-+}$7 decays, revealing structures around $J^{PC}=1^{-+}$8.

Partial Wave Analysis and Extraction of $J^{PC}=1^{-+}$9

A multi-component model, implemented in the GPUPWA framework, is fit to the combined data using covariant tensor formalism. The signal is modeled as a coherent sum of sequential quasi-two-body decay amplitudes, including $1^{-+}$0, $1^{-+}$1, $1^{-+}$2, various tensor and scalar resonances, and nonresonant S-wave contributions.

A statistically significant $1^{-+}$3 $1^{-+}$4 component is required by the fit, exceeding $1^{-+}$5 significance. Parameterizing the exotic amplitude as a relativistic Breit-Wigner (BW) with mass-dependent width, the resonance parameters are extracted:

• Mass: $1^{-+}$6
• Width: $1^{-+}$7

The product branching fraction for $1^{-+}$8 followed by $1^{-+}$9 is determined to be

$\pi_1(1400)$0

Quantitative agreement between fit and data is evident in the projections for $\pi_1(1400)$1, $\pi_1(1400)$2, and helicity angle distributions.

Figure 2: Model projections and data for $\pi_1(1400)$3, $\pi_1(1400)$4, and relevant angular distributions, confirming the quality of the PWA description.

Robustness of the resonance assignment is established via likelihood-ratio tests (phase motion and alternative spin-parity hypotheses), systematic scans over additional possible amplitudes, and fits with both mass-dependent and constant-width BW forms. The extracted pole positions are stable under these modeling choices, underscoring the resonance's physical reality.

Systematic Uncertainties

Comprehensive systematic studies include variations in tracking, photon ID, kinematic fits, and mass window requirements (affecting branching ratio at the $\pi_1(1400)$5 level), as well as model uncertainties from background handling, resonance parametrization, and alternative amplitude combinations. The largest contributions to mass and width uncertainties arise from resonance description and combination, reflecting the complex, overlapping spectrum of nearby states.

Numerical Summary and Amplitude Content

The partial-wave fit reveals that, after interference, the $\pi_1(1400)$6 contributes a fit-fraction of $\pi_1(1400)$7 to the $\pi_1(1400)$8 decay. Incoherent S-wave and $\pi_1(1400)$9 contributions account for the majority of the remaining intensity, with smaller roles for higher-mass tensor/scalar states and $\pi_1(1600)$0.

Figure 3: Invariant mass spectra for the selected $\pi_1(1600)$1 and $\pi_1(1600)$2 candidates for both $\pi_1(1600)$3 decay channels, demonstrating purity of event selection.

Theoretical and Phenomenological Implications

This observation constitutes the first unambiguous, high-significance evidence for the isovector $\pi_1(1600)$4 state in charmonium ($\pi_1(1600)$5) decays. The mass and width are compatible with previous reports in diffractive processes, but with greatly reduced systematic ambiguity, confirming previous coupled-channel interpretations involving strong $\pi_1(1600)$6 couplings. The measured parameters are consistent with lattice QCD predictions for the lightest $\pi_1(1600)$7 hybrid meson mass, supporting the hybrid interpretation for $\pi_1(1600)$8.

Of particular theoretical interest is the relationship to the isoscalar $\pi_1(1600)$9 state $\pi_1(2015)$0 recently observed in $\pi_1(2015)$1 decays. Joint consideration of both results facilitates investigations into the nonet structure of light hybrid mesons and may help distinguish between hybrid and tetraquark scenarios, especially when the production and decay mechanisms can be precisely characterized in multiple channels.

Further, the methodology—robust partial-wave amplitude analysis in charmonium radiative decays, utilizing large datasets—establishes a gold standard for such studies and sets a benchmark for future searches targeting exotic candidates in both isovector and isoscalar systems.

Future Prospects

Future work should extend the amplitude analysis to the neutral mode $\pi_1(2015)$2 to directly test isospin symmetry and confirm the C-parity assignments in charmonium decays. Precise measurement of additional decay channels, mapping out the full decay pattern, will be decisive for testing hybrid assignments and validating theoretical calculations. The high-statistics BESIII dataset should enable such comprehensive studies, with potential critical input from next-generation $\pi_1(2015)$3 and hadron facilities.

Conclusion

This study presents the first observation of the $\pi_1(2015)$4 $\pi_1(2015)$5 exotic state in charmonium decay, with a significance in excess of $\pi_1(2015)$6 and well-determined mass, width, and decay branching fractions. Model-independent resonance parameters and spin-parity quantum numbers firmly establish the $\pi_1(2015)$7 as a leading hybrid meson candidate. The result substantially advances the empirical foundation for hadron spectroscopy beyond the constituent quark model, and the analytical framework lays the groundwork for further comprehensive mapping of the QCD spectrum of exotics.