Status and Prospects of the χ_c1(3872) Study at BESIII

Abstract

The ${\chi }_{c1}\left(3872\right)$ plays a pivotal role in understanding hadronic structures, remaining one of the most extensively studied exotic particles among numerous observed unconventional hadronic states. Sustained experimental and theoretical investigations into the particle over the past two decades have propelled its study into a high-precision regime, marked by refined measurements of its decay dynamics and line shape, thereby offering critical insights for resolving longstanding debates between molecular, tetraquark, hybrid, and charmonium interpretations of this particle. Furthermore, the heavy-quark symmetry in the molecular picture predicts a series of ${\chi }_{c1}\left(3872\right)$ partners. The BESIII experiment has made seminal contributions to the study of the ${\chi }_{c1}\left(3872\right)$, leveraging its unique capabilities in high-statistics data acquisition and low-background condition, such as observations of the productions $e^{+}{e}^{-}\to \gamma {\chi }_{c1}\left(3872\right)$ and $\omega {\chi }_{c1}\left(3872\right)$ and investigations of its decays. This article gives a concise review and prospects of the study of the ${\chi }_{c1}\left(3872\right)$ from the BESIII experiment.

1. Introduction

The ${\chi }_{c1}\left(3872\right)$, first observed in 2003 by the Belle collaboration [1], remains the most enigmatic state among the charmonium-like particles two decades after its discovery. The LHCb experiment determined the quantum number of the particle $J^{PC}=1^{++}$ [2]. The precise determination of the ${\chi }_{c1}\left(3872\right)$ mass and width is critical for understanding its nature, given its striking proximity to the $D^{\ast 0}{\overline{D}}^0$ mass threshold at $3871.69\pm 0.07\mathrm{MeV}$ [3]. Multiple experiments have employed Breit–Wigner parametrizations to analyze the resonant parameters in the decay channels containing $J/\psi$. The Particle Data Group (PDG) reports the global averages of the mass and width to be $M=3871.64\pm 0.06\mathrm{MeV}$ and $\mathrm{\Gamma }=1.19\pm 0.21\mathrm{MeV}$ [3]. However, the measurements through the open-charm channel $B\to D^{\ast 0}{\overline{D}}^{0}K$ decays made by Belle and BaBar yield systematically higher values ($M>3873\mathrm{MeV}$, $\mathrm{\Gamma }>3\mathrm{MeV}$) [4,5], significantly exceeding those from the hidden-charm final states. This discrepancy in lineshapes between the hidden- and open-charm decay modes of the ${\chi }_{c1}\left(3872\right)$ likely arises from coupled-channel effects inducing threshold distortions near $D^{\ast 0}{\overline{D}}^0$, rendering symmetric Breit–Wigner descriptions inadequate in the $D^{\ast 0}{\overline{D}}^0$ spectrum. To address the coupled-channel distortions, Flatté-model parametrization [6] is applied to describe the line shape. Refs. [6,7] implemented Flatté-model analyses of Belle and BaBar data for ${\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi$, ${\pi }^{+}{\pi }^{-}{\pi }^{0}J/\psi$, and $D^{\ast 0}{\overline{D}}^0$, revealing significant ${\chi }_{c1}\left(2P\right)$ admixture in Belle datasets. Subsequent Flatté analyses by LHCb, Belle, and BESIII [4,8,9] remain inconclusive due to limited statistics, particularly in the ${\chi }_{c1}\left(3872\right)\to D^{\ast 0}{\overline{D}}^0$ channel where signal yields are severely constrained. Substantial increase in $D^{\ast 0}{\overline{D}}^0$ signal statistics is therefore imperative for precision line shape studies to disentangle the ${\chi }_{c1}\left(3872\right)$ internal structure.

Over the past two decades, extensive measurements have revealed multiple decay channels, including ${\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi$, $D^{\ast 0}{\overline{D}}^0$, $\omega J/\psi$, $\gamma J/\psi$, $\gamma \psi \left(2S\right)$, and ${\pi }^0{\chi }_{c1}$ [5,10,11,12,13,14,15,16]. Among its decay modes, the radiative transitions ${\chi }_{c1}\left(3872\right)\to \gamma J/\psi$ and $\gamma \psi \left(2S\right)$ are particularly interesting. These channels exhibit stark contrasts in branching fractions between conventional charmonium and exotic configurations. For example, the decay width for $\gamma \psi \left(2S\right)$ is significantly enhanced in conventional charmonium compared to the molecular scenarios. This difference provides critical leverage for elucidating the ${\chi }_{c1}\left(3872\right)$ composition. Nevertheless, consensus regarding its nature remains elusive. The theoretical interpretations include a $D^{\ast 0}{\overline{D}}^0$ molecular configuration [17], a state mixing molecular $D^{\ast 0}{\overline{D}}^0$ components with conventional ${\chi }_{c1}\left(2P\right)$ charmonium [18], a compact tetraquark state [19], and a threshold cusp at the $D^{\ast 0}{\overline{D}}^0$ mass limit [20].

Since the hadronic transition between ${\chi }_{c1}\left(3872\right)$ and $J/\psi$ is via ${\chi }_{c1}\left(3872\right)\to {\rho }^{0}J/\psi$, then it must be the charged ${\chi }_{c1}\left(3872\right)$ which can decay to $J/\psi$ via ${\chi }_{c1}{\left(3872\right)}^{\pm }\to {\rho }^{\pm }J/\psi$ if the ${\chi }_{c1}\left(3872\right)$ and its decay obey the isospin symmetry. In addition, some tetraquark models propose ${\chi }_{c1}\left(3872\right)$ is a four-quark bound state and further predict the existence of the charged ${\chi }_{c1}\left(3872\right)$ [19,21,22]. BaBar and Belle both reported the search of the charged ${\chi }_{c1}\left(3872\right)$ via the B meson decay $B\to X^{\pm }K$ [10,23]. BaBar provided the upper limits $\mathcal{B}(B^{-}\to X^{-}{\overline{K}}^0,X^{-}\to J/\psi {\pi }^{-}{\pi }^0)<22\times {10}^{-6}$ and $\mathcal{B}(B^0\to X^{-}{K}^{+},X^{-}\to J/\psi {\pi }^{-}{\pi }^0)<5.4\times {10}^{-6}$ at 90% confidence level (CL). Belle provided more strict 90% CL upper limits $\mathcal{B}(B^{+}\to X^{+}{K}^0)\times \mathcal{B}(X^{+}\to J/\psi {\rho }^{+})<6.1\times {10}^{-6}$ and $\mathcal{B}({\overline{B}}^0\to X^{+}{K}^{-})\times \mathcal{B}(X^{+}\to J/\psi {\rho }^{+})<4.2\times {10}^{-6}$. The heavy-quark spin and flavor symmetry predict the existance of a ${\chi }_{c1}$(3872) partner in the molecular picture [24]. Recently, Refs. [25,26] predicted the existence of isovector $D{\overline{D}}^{\ast }$ hadronic molecules with $J^{PC}=1^{++}$ denoted as $W_{c1}^{0,\pm }$. They are isovector partners of ${\chi }_{c1}\left(3872\right)$ in the hadronic molecular picture. These states are virtual state poles at the $D{D}^{\ast }$ threshold and appear as threshold cusps. The pole positions are predicated at $W_{c1}^0:{3881.7}_{-0.7}^{+1.0}+i\left({1.2}_{-0.7}^{+0.8}\right)$ MeV and $W_{c1}^{\pm }:{3862.5}_{-10.3}^{+6.4}-i\left(0.07\right)\pm 0.00$ MeV. The $W_{c1}^0$ pole is ${10}_{-0.7}^{+1.0}$ MeV above the $D^0{\overline{D}}^{\ast 0}$ threshold and ${1.8}_{-0.7}^{+1.0}$ above the $D^{+}{D}^{\ast -}$ threshold. The $W_{c1}^{-}$ pole is ${13.3}_{-6.4}^{+10.3}$ MeV below the ${\overline{D}}^0{D}^{\ast +}$ threshold.

The BESIII experiment has made seminal contributions to the study of the ${\chi }_{c1}\left(3872\right)$ in its decays, productions, and line shape, leveraging its unique capabilities in high-statistics data acquisition and low-background condition. This article gives a concise review and prospects of the radiative decays and line shape study of the ${\chi }_{c1}\left(3872\right)$ from the BESIII experiment.

2. Datasets at the BESIII Experiment

BESIII at the BEPCII accelerator is a major upgrade of BESII at the BEPC for the studies of hadron physics and $\tau$-charm physics with high accuracy. It is located in IHEP, Bejing, China. In the past ten years, the BESIII experiment has accumulated a substantial dataset at the center-of-mass energies above 3.8 GeV. Figure 1 shows the integrated luminosities at these energy points. These datasets are used to investigate the physics of the charmonium-like states, charm mesons, and charmed baryons. BESIII has published a series of influential research outcomes including the discoveries of the $Z_c\left(3900\right)$ and $Z_{cs}\left(3885\right)$ using these data [27,28].

BESIII made significant contributions to the investigation of the ${\chi }_{c1}\left(3872\right)$. BESIII observed the ${\chi }_{c1}\left(3872\right)$ in the processes $e^{+}{e}^{-}\to \gamma {\chi }_{c1}\left(3872\right)$ at $\sqrt{s}$= 4.18–4.3 GeV [13,29] and $e^{+}{e}^{-}\to \omega {\chi }_{c1}\left(3872\right)$ above 4.66 GeV [30]. Figure 2 shows the $\sqrt{s}$-dependent cross sections of the $e^{+}{e}^{-}\to \gamma {\chi }_{c1}\left(3872\right)\to \gamma {\pi }^{+}{\pi }^{-}J/\psi$ (left) and $e^{+}{e}^{-}\to \omega {\chi }_{c1}\left(3872\right)$ (right) by reconstructing the decay ${\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi$. To investigate the source of the observed ${\chi }_{c1}\left(3872\right)$ signal, Refs. [13,29] fit the line shape of the cross section using a single Breit–Wigner function as shown in Figure 2, which is discussed in detail in Section 3.

3. ${\mathit{\chi }}_{\mathit{c}\mathbf{1}}\left(\mathbf{3872}\right)$ and $\mathit{\psi }\left(\mathbf{4230}\right)$

As mentioned before, one of the main ways of producing the ${\chi }_{c1}\left(3872\right)$ at BESIII experiment is via the radiative process $e^{+}{e}^{-}\to \gamma {\chi }_{c1}\left(3872\right)$. Profiting from the features of the machine, BESIII constructed the connection between the ${\chi }_{c1}\left(3872\right)$ and another famous vector charmonium-like state, $\psi \left(4230\right)$, which was first observed by the BaBar experiment, via the decay of $\psi \left(4230\right)\to \gamma {\chi }_{c1}\left(3872\right)$. One popular interpretation on the $\psi \left(4230\right)$ nature is considering it a $D{D}_1$ molecule [31,32,33]. With this molecule picture, Ref. [34] suggested the radiative decay of $\psi \left(4230\right)$ to ${\chi }_{c1}\left(3872\right)$ is dominated by the mechanism in Figure 3, and calculated the decay width.

Taking advantage of the energy scan data samples collected by the detectors, BESIII measured the $\sqrt{s}$-dependent cross sections of a series of $e^{+}{e}^{-}$ annihilation processes [35,36,37,38,39,40,41,42,43,44], and further reported the masses and widths of the structures observed in the line-shape of the cross section, which are shown in Figure 4. The combined mass and width of the $\psi \left(4230\right)$ and $\psi \left(4160\right)$, provided in PDG [3] by globally analyzing the measurements from the different processes, are also shown in the Figure 4. In addition, BESIII measured the $\sqrt{s}$-dependent cross sections of the process $e^{+}{e}^{-}\to \gamma {\chi }_{c1}\left(3872\right)$ via the decays ${\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi$ and $\omega J/\psi$ [13]. The shape of the cross section supports that the produced ${\chi }_{c1}\left(3872\right)$ originates from the radiative decay of $\psi \left(4230\right)$. By comparing the mass and width of the structure measured via $e^{+}{e}^{-}\to \gamma {\chi }_{c1}\left(3872\right)$ to the global values of $\psi \left(4230\right)$ and $\psi \left(4160\right)$ as shown in Figure 4, we can see the discrepancies between these values. The mass of the single resonance determined by the process $e^{+}{e}^{-}\to \gamma {\chi }_{c1}\left(3872\right)$ is smaller than that of $\psi \left(4230\right)$ but larger than that of $\psi \left(4160\right)$, while the width is larger than those of both $\psi \left(4230\right)$ and $\psi \left(4160\right)$.

To explain this discrepancy, Ref. [45] suggested that the contributions to the $e^{+}{e}^{-}\to \gamma {\chi }_{c1}\left(3872\right)$ comes only from the $\psi \left(4230\right)$ decay but also the $\psi \left(4160\right)$. In addition, Ref. [34] calculated the partial decay width of $\psi \left(4160\right)\to \gamma {\chi }_{c1}\left(3872\right)$, and predicted the value with an accuracy within a few keV. A precise measurement of the line shape of $e^{+}{e}^{-}\to \gamma {\chi }_{c1}\left(3872\right)$ cross section at BESIII is essential for clarifying the production mechanism of $e^{+}{e}^{-}\to \gamma {\chi }_{c1}\left(3872\right)$ and improving our understanding of the vector charmonium-like states, e.g., $\psi \left(4230\right)$ and $\psi \left(4160\right)$.

4. The Radiative Decays of the ${\mathit{\chi }}_{\mathit{c}\mathbf{1}}\left(\mathbf{3872}\right)$

4.1. ${\chi }_{c1}\left(3872\right)\to \gamma J/\psi ,\gamma \psi \left(2S\right)$

The radiative decays of the ${\chi }_{c1}\left(3872\right)$ serve as crucial probes for elucidating the intrinsic properties of this particle. Theoretical predictions for the partial decay widths $\mathrm{\Gamma }({\chi }_{c1}\left(3872\right)\to \gamma \psi )$ and their ratio $\mathrm{\Gamma }({\chi }_{c1}\left(3872\right)\to \gamma \psi \left(2S\right))/\mathrm{\Gamma }({\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi )$ exhibit significant model dependence. The conventional charmonium ${\chi }_{c1}\left(2P\right)$ hypothesis of ${\chi }_{c1}\left(3872\right)$ predicts a larger value of $\mathrm{\Gamma }({\chi }_{c1}\left(3872\right)\to \gamma \psi \left(2S\right))$ and gives the ratio in a range of 1–6 [46,47,48,49,50], while the molecule model yields the ratio much smaller than one [51,52,53,54,55,56]. Recently, using the Born–Oppenheimer Effective Field Theory (BOEFT), Ref. [57] obtains a 8% ${\chi }_{c1}\left(2P\right)$ charmonium component in the ${\chi }_{c1}\left(3872\right)$ internal structure, and further predicts the ratio to be $2.99\pm 2.36$. Independent measurements of these decays have been conducted by the BESIII, BaBar, Belle, and LHCb collaborations. Belle, BaBar, and LHCb all reported the measurements of ${\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi ,\gamma J/\psi ,\gamma \psi \left(2S\right)$ channels in the $B^{\pm ,0}\to {\chi }_{c1}\left(3872\right)K^{\pm ,0}$ decays [10,15,16,58,59]. BESIII studied these channels via the $e^{+}{e}^{-}\to \gamma {\chi }_{c1}\left(3872\right)$ production process [60].

The ${\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi$ decay mode, characterized by its low-background conditions and small measurement uncertainties across experiments, is selected as the normalization channel to compute the branching fraction ratios: $\mathcal{B}({\chi }_{c1}\left(3872\right)\to \gamma J/\psi )/\mathcal{B}({\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi )$ and $\mathcal{B}({\chi }_{c1}\left(3872\right)\to \gamma \psi \left(2S\right))/\mathcal{B}({\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi )$. This approach enables us to check consistency of radiative decay measurements among experiments. Figure 5 shows the calculated branching fraction ratios from BESIII, BaBar, and Belle. For the ${\chi }_{c1}\left(3872\right)\to \gamma \psi \left(2S\right)$ channel, Belle and BESIII observed no significant signals; thus, their 90% CL upper limits are denoted by arrows. It is worth pointing out that the denominator $\mathcal{B}({\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi )$ in each ratio corresponds to the measurement from the same experiment as the numerator, ensuring independence between results from different experiments. Furthermore, LHCb reported an observation of the ${\chi }_{c1}\left(3872\right)\to \gamma \psi \left(2S\right)$ decay with a $6.7\sigma$ significance, and measured the ratio $\mathcal{B}({\chi }_{c1}\left(3872\right)\to \gamma \psi \left(2S\right))/\mathcal{B}({\chi }_{c1}\left(3872\right)\to \gamma J/\psi )$ [58], which is incorporated into Figure 5. This measurement strongly indicates a sizeable compact charmonium or tetraquark component within the ${\chi }_{c1}\left(3872\right)$ particle.

For the ratio $\mathcal{B}({\chi }_{c1}\left(3872\right)\to \gamma J/\psi )/\mathcal{B}({\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi )$, Belle provides the most precise measurement, and the ratios between Belle and BESIII are inconsistent within 1$\sigma$ with each other within one standard deviation. For the ${\chi }_{c1}\left(3872\right)\to \gamma J/\psi$ study at BESIII, the $J/\psi$ is reconstructed with the decays $J/\psi \to e^{+}{e}^{-}/{\mu }^{+}{\mu }^{-}$, the dominant backgrounds are from the $e^{+}{e}^{-}\to \left(\gamma \gamma \right)e^{+}{e}^{-}$ and $\left(\gamma \gamma \right){\mu }^{+}{\mu }^{-}$ processes. These backgrounds are very serious and difficult to be suppressed in a study employing conventional methods. Some techniques, like machine learning combined with the single charm meson tag approach, may may help improve the measurements. For the ratio $\mathcal{B}({\chi }_{c1}\left(3872\right)\to \gamma \psi \left(2S\right))/\mathcal{B}({\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi )$, both Belle and BESIII provide an upper limit which contradicts the measurement from BaBar, as shown in the Figure 5. Although LHCb does not give the absolute branching fraction of $B\to {\chi }_{c1}\left(3872\right)K\to \gamma \psi \left(2S\right)K$, it does observe this decay, and report the ratio $\mathcal{B}({\chi }_{c1}\left(3872\right)\to \gamma \psi \left(2S\right))/\mathcal{B}({\chi }_{c1}\left(3872\right)\to \gamma J/\psi )=1.67\pm 0.21\pm 0.12\pm 0.04$, where the first uncertainty is statistical, the second systematic and the third is due to the uncertainties on the branching fractions of the $\psi \left(2S\right)$ and $J/\psi$ mesons. At BESIII, the ${\chi }_{c1}\left(3872\right)\to \gamma \psi \left(2S\right)$ decay is studied with the $\psi \left(2S\right)\to {\pi }^{+}{\pi }^{-}J/\psi ,J/\psi \to e^{+}{e}^{-}/{\mu }^{+}{\mu }^{-}$ decay. The background condition is clean in the analysis, but taking into account the branching fractions of $\psi \left(2S\right)\to {\pi }^{+}{\pi }^{-}J/\psi ,J/\psi \to e^{+}{e}^{-}/{\mu }^{+}{\mu }^{-}$ [$(4.14\pm 0.05)\%$] [3], the total selection efficiency is less than 1%. The discrepancy of the ${\chi }_{c1}\left(3872\right)\to \gamma \psi \left(2S\right)$ measurements between these experiments has existed for a long time. For BESIII, larger statistical data and refined reconstruction technique are both needed to improve the sensitivity on this channel.

4.2. ${\chi }_{c1}\left(3872\right)\to \gamma {\psi }_2\left(3823\right)$

Recently, BESIII reported the search for the radiative transition ${\chi }_{c1}\left(3872\right)\to \gamma {\psi }_2\left(3823\right)$ via the process $e^{+}{e}^{-}\to \gamma {\chi }_{c1}\left(3872\right)$ at the center-of-mass energies $\sqrt{s}=4.178–4.278$ GeV [61]. Figure 6 shows the obtained distribution of the $\gamma {\psi }_2\left(3823\right)$ invariant mass. This is the first measurement on the ${\chi }_{c1}\left(3872\right)$ transition to a D-wave charmonium. E1 transition is a natural decay between the P-wave and D-wave resonances and is expected to have a large decay width. The radiative decay ${\chi }_{c1}\left(3872\right)\to \gamma {\psi }_2\left(3823\right)$ may happen via the E1 transition if the ${\chi }_{c1}\left(3872\right)$ which contains a component of the excited spin-triplet state ${\chi }_{c1}\left(2P\right)$, where the ${\psi }_2\left(3823\right)$ is considered as the $1^3{D}_2$ charmonium state. The branching fraction ratio of this decay relative to the ${\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi$ decay, ${\mathcal{R}}_{{\chi }_{c1}\left(3872\right)}\equiv \mathcal{B}({\chi }_{c1}\left(3872\right)\to \gamma {\psi }_2\left(3823\right),{\psi }_2\left(3823\right)\to \gamma {\chi }_{c1})/\mathcal{B}({\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi )$ is investigated, and the upper limit at the 90% CL is set at 0.075. Many theoretical models predict the partial widths of the radiative transitions between different conventional charmonium states, including P-wave and D-wave states. The partial widths of ${\chi }_{c1}\left(2P\right)\to \gamma \psi \left(1^3{D}_2\right)$ and $\psi \left(1^3{D}_2\right)\to \gamma {\chi }_{c1}\left(1P\right)$ are calculated with the non-relativistic (NR) potential model and the Godfrey–Isgur (GI) relativistic potential model [47]. In addition, lattice QCD (LQCD) [62] calculated the partial width of $\psi \left(1^3{D}_2\right)\to \gamma {\chi }_{c1}\left(1P\right)$. Combined with the total width of the $\psi \left(1^3{D}_2\right)$ estimated according to the BESIII measurements and some phenomenological results, the branching fraction ratio, $\mathcal{B}({\chi }_{c1}\left(2P\right)\to \gamma \psi \left(1^3{D}_2\right),\psi \left(1^3{D}_2\right)\to \gamma {\chi }_{c1}\left(1P\right))/\mathcal{B}({\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi )$, is estimated to be $0.46\pm 0.19$, $0.21\pm 0.09$, and $0.26\pm 0.11$, based on NR [47], GI [47], and LQCD [62], respectively. The determined upper limit of 0.075 contradict these estimation, thus challenging the pure charmonium interpretation of ${\chi }_{c1}\left(3872\right)$.

BESIII implemented the ${\chi }_{c1}\left(3872\right)\to \gamma {\psi }_2\left(3823\right)$ study [61] with a full reconstruction technique by reconstructing the ${\psi }_2\left(3823\right)$ from the cascade decay ${\psi }_2\left(3823\right)\to \gamma {\chi }_{c1}$, ${\chi }_{c1}\to \gamma J/\psi$, $J/\psi \to {\ell }^{+}{\ell }^{-}$ ($\ell =e,\mu$), in which there are four radiative photons in the whole decay chain. In order to determine the sequence of the photons in the cascade radiative decays, mass constraints are applied to select the photons from the ${\psi }_2\left(3823\right)$ and ${\chi }_{c1}\left(1P\right)$ decays. This method is helpful in suppressing background, particularly combinatorial ones. A disadvantage of the approach, however, is that it constrains the involved intermediate particles to the masses of ${\psi }_2\left(3823\right)$ and ${\chi }_{c1}\left(1P\right)$, which limits the physical insights of the study. In fact, the study could be enhanced by using a more ambitious algorithm to search for all possible intermediate states, e.g., ${\chi }_{c0,1,2}\left(2P\right)$ and $\psi \left(1D\right)$, in the transitions $e^{+}{e}^{-}\to \gamma {\chi }_{c0,1,2}\left(2P\right)$, ${\chi }_{c0,1,2}\left(2P\right)\to \gamma \psi \left(1D\right)$, $\psi \left(1D\right)\to \gamma {\chi }_{c0,1,2}\left(1P\right)$, ${\chi }_{c0,1,2}\left(1P\right)\to \gamma J/\psi$ by removing the mass constrains of the ${\psi }_2\left(3823\right)$ and ${\chi }_{c1}\left(1P\right)$ in the reconstruction approach. Of course, a detailed study is needed to deal with the sequence problem of the four radiative photons. So there is still space for BESIII to improve the sensitivity and provide more abundant physical outputs for this research.

5. The ${\mathit{\chi }}_{\mathit{c}\mathbf{1}}\left(\mathbf{3872}\right)$ Line Shape

The almost perfect match between its mass and the $D^{\ast 0}\overline{D^0}$ mass threshold reveals the distinctly asymmetric behavior of ${\chi }_{c1}\left(3872\right)$ line shape, which may contain essential information on the nature of this particle. A Flatté-inpired parametrization [6,7] is widely applied to study its line shape. In the model, the differential decay rate to the energy relative to the $D^{\ast 0}{\overline{D}}^0$ threshold (E) is expressed as

$${\displaystyle \frac{dBr}{dE}}\propto {\displaystyle \frac{g_1{k}_1}{{\left|D\left(E\right)\right|}^2}}$$

with

$$D\left(E\right)=E-E_f+{\displaystyle \frac{i}{2}}(g_1{k}_1+g_2{k}_2+\mathrm{\Gamma }\left(E\right))$$

and $k_1=\sqrt{2{\mu }_{1}E}$ and $k_2=\sqrt{2{\mu }_2(E-\delta )}$ are the momenta for the $D^{\ast 0}\overline{D^0}$ and $D^{\ast +}{D}^{-}$ in the rest frame of the state ${\chi }_{c1}\left(3872\right)$, respectively, where $\delta$ is the difference of the mass threshold of the charged channel to the neutral, and ${\mu }_1$ and ${\mu }_2$ are corresponding the reduced masses in the two channels; $g_1$ and $g_2$ are the coupling constants for the neutral and charged channels; $\mathrm{\Gamma }\left(E\right)$ accounts for the non-$D^{\ast }D$ modes; $E_f$ is the difference of the ${\chi }_{c1}\left(3872\right)$ mass to the $D^{\ast 0}{\overline{D}}^0$ threshold. E is approximately in a range of −10 MeV to 10 MeV, which could cover both the $D^0{\overline{D}}^0{\pi }^0$ and $D^{\ast +}{D}^{-}$ thresholds and make the above formula a truncation function for the different E region. Fitting data with this formula, we can extract the parameters, e.g. the state mass, and the coupling constants or partical decay widths. The ${\chi }_{c1}\left(3872\right)\to D^0{\overline{D}}^0{\pi }^0$ decay is a good channel to study the ${\chi }_{c1}\left(3872\right)$ line shape in experiments since the detection resolutions of ${\pi }^0$ and the D mass could be removed by applying mass constrains, so we can obtain a ${\chi }_{c1}\left(3872\right)$ line shape with good resolution.

In 2020, LHCb experiment reported a study on the spectrum of the decay ${\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi$ [8], but the channel applied in the work is not very sensitive to all the parameters in the Flatté formula due to being far away from the threshold for ${\pi }^{+}{\pi }^{-}J/\psi$ channel. Furthermore, the Belle experiment analyzed the ${\chi }_{c1}\left(3872\right)\to D^{\ast 0}\overline{D^0}+c.c.$ channel and provided the limitation on the coupling constant of ${\chi }_{c1}\left(3872\right)$ to $D^{\ast 0}\overline{D^0}$ [4], while the statistics limited the measurement precision. In addition, the ${\pi }^0{D}^0$ is constrained to the $D^{\ast 0}$ nominal mass in the event selection in order to suppress the backgrounds, which make the data lower than the $D^{\ast 0}\overline{D^0}$ threshold be unavailable. In 2024, BESIII reported the coupled-channel analysis of the line shape on the channels ${\chi }_{c1}\left(3872\right)\to D^0{\overline{D}}^0{\pi }^0$ and ${\pi }^{+}{\pi }^{-}J/\psi$ [9]. In the analysis, instead of using a mass constraint on ${\pi }^0{D}^0$ to the $D^{\ast 0}$ mass, a mass window cut around the $D^{\ast 0}$ mass is applied, which provided the information below the $D^{\ast 0}\overline{D^0}$ threshold.

The parameters of the Flatté-inspired model, e.g., the coupling constant, mass, and the decay width, are determined. Figure 7 shows the fit line shape of the ${\chi }_{c1}\left(3872\right)$. The authors estimated the scattering length $a=(-{16.5}_{-27.6-27.7}^{+7.0+5.6})$ fm and the effective range $r_e=-{4.1}_{-3.3-4.4}^{+0.9+2.8}$ fm. An improvement of the measurement precision is necessary, since these parameters could be used to evaluate the compositeness of the ${\chi }_{c1}\left(3872\right)$, e.g., the possibility of a compact tetraquark component existing in the internal structure.

In the ${\chi }_{c1}\left(3872\right)\to D^0{\overline{D}}^0{\pi }^0$ spectrum, BESIII yields $36.1\pm 7.6$ signals for the $D^{\ast 0}\to {\pi }^0{D}^0$ and $20.5\pm 7.4$ signals for $D^{\ast 0}\to \gamma D^0$ by double tagging the charm mesons [60]. The reconstruction algorithm could be tuned to improve the efficiency. Of course, a larger statistics dataset could be essential to achieve higher precise measurement.

6. Summary and Prospects

As a dedicated electron–positron collider experiment operating at the $\tau$-charm energy region, the BESIII possesses unique advantages in studying the ${\chi }_{c1}\left(3872\right)$. Its optimized detector geometry and high luminosity at center-of-mass energies near 3.8–5.0 GeV enable threshold-scanning precision for charmonium-like states. With the dataset accumulated in the past decade, BESIII provided the measurements of the ${\chi }_{c1}\left(3872\right)\to \gamma J/\psi ,\gamma \psi \left(2S\right),\gamma {\psi }_2\left(3823\right)$, ${\overline{D}}^{\ast 0}{D}^0,\omega J/\psi ,{\pi }^0{\chi }_{c1}\left(1P\right)$ decays, and discovered $\psi \left(4230\right)\to \gamma {\chi }_{c1}\left(3872\right)$ and $e^{+}{e}^{-}\to \omega {\chi }_{c1}\left(3872\right)$. The coupled-channel analysis of the ${\chi }_{c1}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi$ and ${\overline{D}}^{\ast 0}{D}^0$ decays is implemented. Due to the limited measurement precision so far, some questions remain to be answered at BESIII. With improved signal sensitivity, we may be able to determine if $\psi \left(4160\right)\to \gamma {\chi }_{c1}\left(3872\right)$ exists and gain insight into the internal structure of the particle via the study of its line shape. We could search for radiative transition of new vector resonance to ${\chi }_{c1}\left(3872\right)$, e.g., $\psi \left(4660\right)\to \gamma {\chi }_{c1}\left(3872\right)$. It will be very helpful to understand the particle nature if BESIII could provide higher sensitivity of the ${\chi }_{c1}\left(3872\right)\to \gamma \psi \left(2S\right)$ measurements, since only LHCb experiment reported a strong observation of this decay so far.

Building upon the inherent advantages of the BESIII experiment, there are opportunities to further refine its event reconstruction capabilities, which could substantially enhance precision in exotic hadron studies. An imminent opportunity arises from the upcoming upgrade of the BEPCII collider, which will deliver a threefold enhancement in instantaneous luminosity within the XYZ energy region. This luminosity gain is projected to enable the collection of data samples with statistical significance far exceeding current levels across these critical energy points. Such a leap in data volume will provide an essential foundation for high-precision investigations of the ${\chi }_{c1}\left(3872\right)$ at BESIII. In addition, global analysis of the measurements from all experiments, e.g., BESIII, Belle, Barbar, and LHCb, could provide another path to investigate the nature of the ${\chi }_{c1}\left(3872\right)$.