Charmonium-like states

X(3872) was discovered by Belle in $B\to K\pi \pi J/\psi$ decays as a narrow peak in the invariant mass distribution of the $\pi \pi J/\psi$ final state [16]. Ten years after its discovery, LHCb performed a full five-dimensional amplitude analysis of $B^{+}\to K^{+}X(3872)\to K^{+}{\pi }^{+}{\pi }^{-}J/\psi$ decays and unambiguously yielded the $J^{PC}{=1}^{++}$ assignment [21].

The most salient feature of X(3872) is that its mass almost exactly coincides with the threshold of $D^0{\overline{D}}^{*0}$. The mass and width of X(3872), as given in the latest version of the Review of Particle Physics (RPP) [22], are $(3871.64\pm 0.06)$ MeV and $(1.19\pm 0.21)$ MeV, respectively. These values were extracted using Breit–Wigner (BW) parameterization. BW parameterization works well only for isolated resonances far from the thresholds of the channels to which the resonance can strongly couple. However, X(3872) is different; thus, the BW width reported in the RPP is not a good approximation of twice the imaginary part of the X(3872) pole position in the complex energy plane.

Using the Flatté parameterization that considers the nearby $D{\overline{D}}^{*}$ thresholds [23], the LHCb Collaboration reported the mass and width of X(3872) as ${3871.69}_{-0.04-0.13}^{+0.00+0.05}$ MeV and ${0.22}_{-0.06-0.13}^{+0.07+0.11}$ MeV, respectively, using events from b-hadron decays (the pole on the second Riemann sheet is located at $0.06-i0.13$ MeV relative to the $D^0{\overline{D}}^{*0}$ threshold) [24]. BESIII reported the mass parameter and imaginary part of the X(3872) pole as $\left(3871.63\pm {0.13}_{-0.05}^{+0.06}\right)$ MeV and $\left(-0.19\pm {0.08}_{-0.19}^{+0.14}\right)$ MeV, respectively, from the processes $e^{+}{e}^{-}\to \gamma X(3872)$, with X(3872) reconstructed from the $D^0{\overline{D}}^0{\pi }^0$ and ${\pi }^{+}{\pi }^{-}J/\psi$ final states [25]. The width from the pole position is significantly smaller than that from BW parameterization.

Although X(3872) was discovered more than 20 years ago, its internal structure remains unclear. The measurement of the absolute branching fraction $\mathcal{B}\left(X(3872)\to {\pi }^{+}{\pi }^{-}J/\psi \right)$ provides useful information regarding its complex nature, in particular, leading to insights into the isospin symmetry-breaking dynamics and possible isospin-1 partners of X(3872) [26]. In 2019, BaBar measured branching fraction $\mathcal{B}\left(B^{+}\to K^{+}X(3872)\right)$ for the first time with a signal significance of $3\sigma$ [27], which made the determination of $\mathcal{B}\left(X(3872)\to {\pi }^{+}{\pi }^{-}J/\psi \right)$ possible by combining known branching fractions as a product, $\mathcal{B}\left(B^{+}\to K^{+}X(3872)\right)$ $\mathcal{B}\left(X(3872)\to {\pi }^{+}{\pi }^{-}J/\psi \right)$, the result of which was $\mathcal{B}\left(X(3872)\to {\pi }^{+}{\pi }^{-}J/\psi \right)=(4.1\pm 1.3)\%$. The authors in Ref. [28] obtained the absolute branching fractions of six X(3872) decay modes by globally analyzing the measurements from available experiments. In particular, $\mathcal{B}\left(X(3872)\to {\pi }^{+}{\pi }^{-}J/\psi \right)$ was determined to be $\left({4.1}_{-1.1}^{+1.9}\right)\%$, and the dominant decay mode was given by $X(3872)\to D^{*0}{\overline{D}}^0+\text{c}\mathrm{.}\text{c}\mathrm{.}$¹ with a branching fraction $\left({52.4}_{-14.3}^{+25.3}\right)\%$, implying a strong coupling between X(3872) and $D{\overline{D}}^{*}$. In addition, the fraction of unknown decays of X(3872) was reported to be $\left({31.9}_{-31.5}^{+18.1}\right)\%$. Additional X(3872) decays should be sought in the future.

Because the X(3872) mass is so close to the $D^0{\overline{D}}^{*0}$ threshold at $(3871.69\pm 0.07)$ MeV [22] and owing to its strong coupling to the $D^0{\overline{D}}^{*0}$ channel, it was immediately proposed that X(3872) is a hadronic molecule of $D{\overline{D}}^{*}$ [29] predicted by Törnqvist in Ref. [30]. However, other models, particularly the compact tetraquark model, for which X(3872) is a bound state of the cq diquark and $\overline{c}\overline{q}$ antidiquark [31], are currently being discussed (for reviews, please refer to Refs. [6, 7, 10, 13]).

Another physical quantity under active discussion is the ratio of the partial radiative decay widths to the ψ(2S)γ and $J/\psi \gamma$ final states, denoted by ${\mathcal{R}}_{\psi \gamma }=\frac{{\text{Γ}}_{X(3872)\to \psi (2S)\gamma }}{{\text{Γ}}_{X(3872)\to J/\psi \gamma }}\mathrm{.}$ (1) A recent measurement of $R_{\psi \gamma }$ from LHCb was 1.67 ± 0.21 ± 0.12 ± 0.04, where the first uncertainty was statistical, the second was systematic, and the third was due to uncertainties in the branching fractions of the ψ(2S) and $J/\psi$ mesons [32]. This result is consistent with previous measurements from LHCb in 2014 [33], Belle in 2011 [34], and BaBar in 2008 [35]. However, it differs from the BESIII result, where the upper limit on the ratio Rψγ<0.59 was set to a 90% confidence level [36]. This discrepancy must be understood in other experiments, particularly Belle II.

The first vector charmonium-like state Y(4260) was observed by BaBar through the initial state radiation (ISR) process $e^{+}{e}^{-}\to {\gamma }_{\text{ISR}}{\pi }^{+}{\pi }^{-}J/\psi$ [19]. Subsequently, several Y states, including Y(4360) and Y(4660), were observed at B factories. To study these vector charmonium-like states, BESIII collected large data samples above 4 GeV and reported a precise measurement of $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}J/\psi$ cross-sections from 3.77 to 4.60 GeV [37]. In the high-statistics analysis by BESIII, the cross-sectional distribution around 4.26 GeV exhibited asymmetry and resulted in a shift in the peak position of Y(4260) to a lower mass. To describe the right shoulder of the Y(4260) line shape, a new BW resonance Y(4320) was introduced; however, this may also be due to the opening of the $D_1\overline{D}$ threshold (see, for example, Ref. [38]). Recently, using data samples with an integrated luminosity of 5.85 fb^-1, BESIII measured cross-sections for the process $e^{+}{e}^{-}\to K^{+}{K}^{-}J/\psi$ from 4.61 to 4.95 GeV, where a new vector charmonium-like state Y(4710) was observed [39]. Y(4710) was also confirmed using BESIII in $e^{+}{e}^{-}\to K_S^0{K}_S^{0}J/\psi$ with a statistical significance of 4.2σ [40]. Thus far, Y(4710) is the known vector charmonium-like state with the highest mass.

$Z_c{\left(4430\right)}^{\pm }$ reported by Belle in $B\to K{\pi }^{\pm }\psi (2S)$ is the first good candidate for an exotic state with a non-zero electric charge [17]. Although it was not confirmed by BaBar by analyzing the same B decays [41], LHCb performed a four-dimensional fit of the decay amplitude for $B^0\to K^{+}{\pi }^{-}\psi (2S)$ using pp collision data corresponding to 3 fb^-1 and found that a highly significant $Z_c{\left(4430\right)}^{-}$ component was required to describe the data [42]. More charged charmonium-like states, including Zc(3900) [43, 44], Zc(4020) [45], Zc(4050), Zc(4025) [46], Zcs(3985) [47], Zcs(4000), and Zcs(4220) [48] with a strange quark, have now been observed experimentally. For a precise determination of the Zc(3900) pole using modern dispersion theory techniques and discussions of this structure, refer to Refs. [49, 50].

Although signals of Zc(3900) have been reported in $J/\psi \pi$ invariant mass distributions, with the $J/\psi {\pi }^{+}{\pi }^{-}$ invariant mass in the energy region around Y(4260), by D0 in $p\overline{p}$ collisions through b-flavored hadron decays [51], no statistically significant signals of the Y(4260) and Zc(3900) states were observed in the prompt production [52].

Hidden-charm pentaquark and double open-charm tetraquark

The search for pentaquark states has a long history. The first strong experimental evidence of a pentaquark state, referred to as $\text{Θ}{\left(1540\right)}^{+}$, was reported in the LEPS experiment in $\gamma n\to n{K}^{+}{K}^{-}$ [53]. However, this finding has not yet been confirmed for the same reaction using larger statistical data samples.

In 2015, LHCb reported the discovery of hidden-charm pentaquark candidates in ${\text{Λ}}_b^0\to K^{-}pJ/\psi$ decay, where the data sample used corresponded to an integrated luminosity of 3 fb^-1 acquired from 7- and 8- TeV pp collisions [20]. A prominent narrow peak was observed in the $pJ/\psi$ invariant mass spectrum, which was named Pc(4450)⁺. In the amplitude analysis, a second BW resonance, Pc(4380)⁺, was introduced to describe the broad bump under the Pc(4450) peak. The significance of these structures was greater than 9σ. Subsequently, LHCb updated this process using Run 2 data with statistics greater by almost one order of magnitude for ${\text{Λ}}_b^0$ events. They found that in the $pJ/\psi$ mass spectrum, there were three obvious narrow structures [54], one for the pentaquark state Pc(4312)⁺ and two for Pc(4440)⁺ and Pc(4457)⁺, split from the previously observed Pc(4450)⁺.

As the valence structure of $pJ/\psi$ is $c\overline{c}uud$, the newly discovered particles must be composed of at least five (anti)quarks. Analysis of the LHCb data in Ref. [55] suggests that there is also a $1.7\sigma$ significance for a narrow Pc(4380), which differs from the considerably broader value reported by LHCb in 2015 [20]. This analysis is based on the ${\displaystyle {\sum }_c^{(*)}{\overline{D}}^{(*)}}$ molecular model, and the existence of such a narrow Pc(4380) is a consequence of the heavy quark spin symmetry. By replacing $J/\psi$ with an ηc meson, LHCb searched for Pc(4312)⁺ in ${\text{Λ}}_b^0$ decays, but no evidence was observed [56]. Because the masses of the observed pentaquark states are close to the threshold of a charm baryon and charm meson, LHCb recently scanned 32 final states, including ${\text{Λ}}_c^{+}\overline{D}$, ${\text{Λ}}_c^{+}{\overline{D}}^{*}$, ${\text{Λ}}_c^{+}\pi \overline{D}$, ${\displaystyle {\sum }_c^{(*)}{\overline{D}}^{(*)}}$, ${\text{Λ}}_c^{+}D$, ${\text{Λ}}_c^{+}{D}^{*}$, ${\text{Λ}}_c^{+}\pi D$, and ${\displaystyle {\sum }_c^{(*)}{D}^{(*)}}$; however, no significant narrow peak was found for any of the modes.

Two pentaquark candidates with strangeness, Pcs(4459)⁰ and Pcs(4338)⁰, were also observed in the $J/\psi \text{Λ}$ system in the ${\text{Ξ}}_b^{-}\to K^{-}J/\psi \text{Λ}$ and $B^{-}\to \overline{p}J/\psi \text{Λ}$ decays with significance values of less than and greater than 5σ, respectively [57, 58].

Among exotic hadrons, there is a class of hadrons that is particularly interesting: double-charm tetraquarks. Such hadrons have been discussed since the 1980s (see Ref. [59] for early treatment using the Born–Oppenheimer approximation). The first open-charm tetraquark, Tcc(3875)⁺, comprising $cc\overline{u}\overline{d}$, was discovered in the $D^0{D}^0{\pi }^{+}$ invariant mass spectrum by LHCb in 2021 [60, 61]. Its mass was just below the $D^0{D}^{*+}$ mass threshold, whereas its width was only a few tens of keV.

Existing and future experiments

The high-energy collision of heavy ions is a powerful method for generating extremely hot and dense nuclear matter, often referred to as quark–gluon plasma (QGP) [62-64], which exhibits an energy density comparable to that of the Universe a few microseconds after the Big Bang, with roughly equal numbers of quarks and antiquarks. The extreme energy density of the QGP phase leads to the creation of many strange–antistrange quark pairs from quantum vacuum. As the QGP cools, it transits into hadron gas, resulting in the formation of various baryons, mesons, and their antiparticles. Therefore, these collisions offer a unique opportunity to explore exotic particles such as antimatter, hypernuclei [65-72], and exotic hadrons [73], thereby uncovering important fundamental interactions.

While nuclei are abundant across the universe, antinuclei heavier than the antiproton have only been observed as products of relativistic HICs at facilities such as the BNL Relativistic Heavy-Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC). In these experiments, the time projection chamber (TPC) positioned within a solenoidal magnetic field plays a crucial role in identifying the particles of interest by measuring the mean energy loss per unit track length, $\langle \text{d}E/\text{d}x\rangle$, as illustrated in Fig. 1. As shown in the left panel, STAR reported the observation of ${}^4\overline{\text{He}}$, which is the heaviest stable antinucleus detected to date [66]. A total of 18 counts were detected in Au+ Au collisions at center-of-mass (c.m.) energies of 200 GeV and 62 GeV per nucleon-nucleon pair. This discovery came a century after Rutherford first observed alpha particles. The yield also aligns with the predictions from both thermodynamic and coalescent nucleosynthesis models [66].

Fig. 1Full-screen View

(Color online) TPC mean energy loss per unit track length, $\langle \text{d}E/\text{d}x\rangle$ as a function of rigidity, $p/\left|Z\right|$, where p and Z are the momentum and charge, respectively. From Ref. [66]

Hypernuclei are the bound states of nucleons and hyperons and offer valuable insights into hyperon-nucleon interactions. While nucleons (protons and neutrons) are composed solely of up and down quarks, hyperons are light-flavored baryons that contain at least one strange quark. A hypernucleus is defined as a nucleus that contains at least one hyperon in addition to nucleons. Despite being bound within hypernuclei, all hyperons are inherently unstable because they decay via weak interactions. The simplest bound hypernucleus is the hypertriton (${}_{\text{Λ}}^3\text{H}$) [74], which comprises a $\text{Λ}$ hyperon, proton, and neutron. The first hypernucleus was observed in 1952 using a nuclear emulsion cosmic-ray detector [75]. In contrast, the first observation of antihypertritons, which care composed of an antiproton, antineutron, and $\overline{\text{Λ}}$, was reported by STAR in Au+Au collisions at $\sqrt{s_{\text{NN}}}$ = 200 GeV in 2010 [65] through the decay channel ${}_{\overline{\text{Λ}}}^3\overline{\text{H}}{\to }^3\overline{\text{He}}+{\pi }^{+}$. The measured yields for ${}_{\overline{\text{Λ}}}^3\overline{\text{H}}$ and ${}^3\overline{\text{He}}$ are comparable, indicating a balance in the populations of up, down, and strange quarks and antiquarks in both coordinate and momentum space [65, 71]. This is a consequence of the approximate SU(3) flavor symmetry among the up, down, and strange quarks of QCD.

The most precise measurements of the ${}_{\text{Λ}}^3\text{H}$ lifetime ($\tau$) and $\text{Λ}$ separation energy ($B_{\text{Λ}}$) were obtained in Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=5.02$ TeV at ALICE [76] (c.f. Fig. 2). These measurements are consistent with the predictions from effective field theories and support the characterization of ${}_{\text{Λ}}^3\text{H}$ as a weakly bound system [74]. Although some discrepancies, referred to as the “hypertriton puzzle”, have been noted in the literature regarding the lifetime and $B_{\text{Λ}}$, a global average of all available measurements revealed no significant global tension, with a 23% probability for the lifetime and 57% for $B_{\text{Λ}}$, as determined by a Pearson test [76]. The upcoming Run 3 of the LHC is expected to provide these measurements with unprecedented precision.

Fig. 2Full-screen View

(Color online) $\text{Λ}$ separation energy $B_{\text{Λ}}$ of hypertriton (left) and the lifetime of various hypernuclei (right) measured in different experiments. From Refs. [76] and [71]

Recently, STAR reported the observation of an antimatter hypernucleus, ${}_{\overline{\text{Λ}}}^4\overline{\text{H}}$, which consists of an $\overline{\text{Λ}}$, antiproton, and two antineutrons [71]. This discovery was achieved through two-body decay. A total of 15.6 candidate ${}_{\overline{\text{Λ}}}^4\overline{\text{H}}$ antimatter hypernuclei were identified, with an estimated background count of 6.4 [71]. This is the heaviest unstable antinucleus reported to date. The lifetimes of the antihypernuclei ${}_{\overline{\text{Λ}}}^3\overline{\text{H}}$ and ${}_{\overline{\text{Λ}}}^4\overline{\text{H}}$ were measured and compared with those of their corresponding hypernuclei, thereby providing a test for the symmetry between matter and antimatter. The lifetimes of the (anti)hypernuclei ${}_{\overline{\text{Λ}}}^3\overline{\text{H}}$, ${}_{\overline{\text{Λ}}}^4\overline{\text{H}}$, ${}_{\text{Λ}}^3\text{H}$, and ${}_{\text{Λ}}^4\text{H}$ are depicted in Fig. 2. The differences in the lifetimes of hypernuclei and their corresponding antihypernuclei were $\tau \left({}_{\overline{\text{Λ}}}^3\overline{\text{H}}\right)-\tau \left({}_{\text{Λ}}^3\text{H}\right)=16\pm 43(\text{stat}\mathrm{.})\pm 20(\text{sys}\mathrm{.})$ ps and $\tau \left({}_{\overline{\text{Λ}}}^4\overline{\text{H}}\right)-\tau \left({}_{\text{Λ}}^4\text{H}\right)=18\pm 115(\text{stat}\mathrm{.})\pm 46(\text{sys}\mathrm{.})$ ps. Both differences are approximately zero within the uncertainties, indicating no significant disparity between the properties of matter and antimatter particles, thus providing a new test of CPT symmetry.

Various production yield ratios among (anti)hypernuclei (including hypernuclei and/or antihypernuclei) and (anti)nuclei (including nuclei and/or antinuclei) were assessed and compared with the theoretical models, offering insights into their production mechanisms. As shown in Fig. 3, the antimatter-over-matter particle yield ratios are below unity because the colliding heavy ions carry positive baryon numbers; consequently, the collision system has a positive baryon chemical potential. The data are consistent with most existing measurements within their uncertainties and with the expectation of the coalescence picture of the (anti)nucleus and (anti)hypernucleus production and statistical thermal model [71]. Relationships between the production yield ratios ${}^4\overline{\text{He}}{/}^4\text{He}{\approx }^3\overline{\text{He}}{/}^3\text{He}\times \overline{p}/p$. ${}_{\overline{\text{Λ}}}^4\overline{\text{H}}{/}_{\text{Λ}}^4\text{H}{\approx }_{\overline{\text{Λ}}}^3\overline{\text{H}}{/}_{\text{Λ}}^3\text{H}\times \overline{p}/p$, ${}_{\text{Λ}}^4\text{H}{/}^4\text{He}\approx 4{\times }_{\text{Λ}}^3\text{H}{/}^3\text{He}$, and ${}_{\overline{\text{Λ}}}^4\overline{\text{H}}{/}^4\overline{\text{He}}\approx 4{\times }_{\overline{\text{Λ}}}^3\overline{\text{H}}{/}^3\overline{\text{He}}$ are consistent with the coalescence model of (anti)nucleus and (anti)hypernucleus production. Here, a factor of four in the last two relations arises because both the spin-0 and spin-1 states of ${}_{\text{Λ}}^4\text{H}$ possess sufficiently large binding energies, leading to no energetically allowed strong decay channels. Consequently, the spin-1 state, with a spin degeneracy of three, decays electromagnetically to the spin-0 ground state, enhancing the total production yields of ${}_{\text{Λ}}^4\text{H}$ and ${}_{\overline{\text{Λ}}}^4\overline{\text{H}}$ by a factor of four compared to those of ${}^4\text{He}$ and ${}^4\overline{\text{He}}$, which have only spin-0 states. Considering this spin-degeneracy effect, the statistical–thermal model predictions align well with the experimental measurements, although the ${}_{\text{Λ}}^4\text{H}{/}^4\text{He}$ ratio is slightly lower than the predicted value. This discrepancy, although not statistically significant, may be attributed to the low binding energy of ${}_{\text{Λ}}^3\text{H}$, which suggests that the spatial extent of its wave function is comparable to that of the entire collision system [77].

Fig. 3Full-screen View

(Color online) Production yield ratios among the various (anti)nuclei and (anti)hypernuclei with the same number of (anti)baryons measured in heavy-ion collisions. From Ref. [71]

High-energy pp collisions and HICs provide excellent laboratories for exploring multistrange dibaryons [78-82]. In the search for a possible ΛΛ bound state, known as the H-dibaryon, femtoscopic correlations of ΛΛ pairs have been studied in pp, Au+Au, and pPb collisions [83, 84]. By comparing the measured data with model calculations, the scattering parameter space, which is characterized by the inverse scattering length and effective range, is constrained. The data revealed a shallow attractive interaction, which is consistent with the findings of hypernuclei studies and lattice computations.

In addition to antimatter and hypernuclei, HICs also serve as a laboratory for studying hidden-charm XYZ particles. The first evidence of $X(3872)$ production in relativistic HICs was previously reported [85]. $X(3872)$ production in Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=5.02$ TeV was investigated, utilizing the decay chain $X(3872)\to J/\psi {\pi }^{+}{\pi }^{-}\to {\mu }^{+}{\mu }^{-}{\pi }^{+}{\pi }^{-}$. The significance of the inclusive $X(3872)$ signal was measured at 4.2 standard deviations [85]. The prompt $X(3872)$ to $\psi (2S)$ yield ratio was found to be ${\rho }_{\text{Pb}-\text{Pb}}{=1.08}_{-0.52}^{+0.49}$, which is significantly higher than the typical values of 0.1 observed in pp collisions [85]. For a further discussion, see Sect. 3.3.

Another common observable in the interaction of quarkonium states with the medium created in HICs is the nuclear modification factor $R_{AA}$, which is defined as the ratio of the yield in central HICs to that in pp collisions, normalized by the number of binary collisions in the reaction. As shown in Refs. [86-89], the suppression of the production yields of $\psi (nS)$ and $\Upsilon (nS)$ mesons in Pb-Pb collisions relative to those in pp collisions was studied using data from the CMS experiment at the LHC, which are summarized in Fig. 4. For the $\psi (nS)$ states, integrated over collision centralities, the nuclear modification factors $R_{AA}$ were $0.56\pm 0.08$ (stat.) ± 0.07 (syst.) for $J/\psi$, $0.12\pm 0.04$ (stat.) ± 0.02 (syst.) for $\psi (2S)$, and less than 0.10 (at 95% confidence level) for $\psi (3S)$, revealing sequential suppression [88, 89]. The $\Upsilon (2S)$ and $\Upsilon (3S)$ mesons were also studied [88, 89], with $\Upsilon (3S)$ observed for the first time in Pb-Pb collisions with a significance above five standard deviations. The suppression of $\Upsilon$ yields increased from peripheral to central collisions and was more pronounced for $\Upsilon (3S)$ than for $\Upsilon (2S)$ (see Fig. 4), extending the sequential suppression pattern previously reported for other quarkonium states.

Fig. 4Full-screen View

(Color online) Double ratios among the different $\Upsilon (nS)$ states and nuclear modification factor RAA as functions of $\langle N_{\text{part}}\rangle$. From Refs. [86, 87]

As reported in Ref. [90], LHCb measured the production of an exotic hadron $X(3872)$ in pPb collisions at $\sqrt{s_{\text{NN}}}=8.16$ TeV. A comparison with the charmonium state $\psi (2S)$ revealed that the dynamics of $X(3872)$ in a nuclear medium differed from those of conventional hadrons. Additionally, compared with proton–proton collision data, the presence of the nucleus may affect the production rates of $X(3872)$ (see Fig. 5). This marked the first determination of a nuclear modification factor for an exotic hadron.

Fig. 5Full-screen View

(Color online) Left: Ratio of the production cross-sections for $X(3872)$ to $\psi (2S)$ in the $J/\psi {\pi }^{+}{\pi }^{-}$ decay channel, measured in various collision systems. Right: Nuclear modification factor $R_{p\text{Pb}}$ for $X(3872)$ and $\psi (2S)$ hadrons. From Ref. [90]

LHCb also measured the feed-down fractions of ${\chi }_{c1}$ and ${\chi }_{c2}$ decays, contributing to the prompt $J/\psi$ yield in pPb collisions at $\sqrt{s_{\text{NN}}}=8.16$ TeV. The results, presented as a function of the $J/\psi$ transverse momentum $p_{\text{T}\mathrm{,}J/\psi }$ in the range 1 < $p_{\text{T}\mathrm{,}J/\psi }$ < 20 GeV, revealed that the fraction at forward speed was consistent with the measurement in pp collisions at $\sqrt{s}=7$ TeV [92] (see Fig. 6). However, the fraction at backward rapidity was 2.4σ higher than that at forward rapidity for $1<p_{\text{T}\mathrm{,}J/\psi }<3$ GeV. This increase at low $p_{\text{T}\mathrm{,}J/\psi }$ in the backward region aligns with the observed suppression of $\psi (2S)$ contributions to the prompt $J/\psi$ yield. Additionally, the study placed an upper limit of 180 MeV on the free energy available in these pPb collisions, which could dissociate or inhibit the formation of charmonium states, indicating no significant in-medium dissociation of the ${\chi }_{cJ}\left(J=\mathrm{1,}2\right)$ states [91].

Fig. 6Full-screen View

(Color online) Double ratio of the quarkonium state yield relative to its ground state or open heavy quark meson in pPb compared to pp collisions as a function of the binding energy of the quarkonium state. From Ref. [91]

The LHC detectors have also observed $B_c^{+}$ mesons in HICs [93]. CMS examined the production of the $B_c^{+}$ meson in Pb-Pb and pp collisions at $\sqrt{s_{\text{NN}}}=5.02$ TeV through its decay channel $B_c^{+}\to \left(J/\psi \to {\mu }^{+}{\mu }^{-}\right){\mu }^{+}{\nu }_{\mu }$ [93]. The left panel of Fig. 7 presents the measured $B_c^{+}$ cross-section. The two bins of the trimuon p_T correspond to different rapidity ranges (see Ref. [93]].). The ratio between the low p_T and high p_T regions was ${18.2}_{-2.1}^{+1.3}$ in pp data and 24.1 in the simulation, suggesting that the simulation overestimated the spectrum steepness of the spectrum [93]. The right panel of Fig. 7 shows the nuclear modification factor for the $B_c^{+}$ meson, obtained by comparing the production cross-sections in Pb-Pb to pp collisions measured across different transverse momentum and collision centrality bins. The $B_c^{+}$ meson showed less suppression than the quarkonia and most open heavy-flavor mesons, indicating that hot and dense nuclear matter created in HICs may enhance its production [93]. This finding provides a valuable new insight into the complex dynamics of the suppression and enhancement mechanisms of heavy quarkonia in hot dense matter. This is another connection to the study of the structure of heavy quarkonium(-like) states and HICs.

Fig. 7Full-screen View

(Color online) $B_c^{+}$ meson production in pp and Pb-Pb collisions at LHC energies. From Ref. [93]

Production in $pp/p\overline{p}$ pp/p pbar collisions

3.2.1

Exotic hadrons with heavy quarks

Exotic hadrons can be produced in $pp/p\overline{p}$ collisions mainly in two ways: in weak decays of b-flavored hadrons, and in prompt processes. The former, through the $b\to c\overline{c}s$ process at the quark level, is a prominent source of exotic hadrons, with examples such as the X(3872), $Z_c\left(4430\right)$ and several Pc states discussed in Sect. 2. The latter directly produces exotic hadrons through strong interactions and is the focus of this section.

Shortly after the discovery of X(3872) in B decays by the Belle Collaboration [16], the CDF and D0 Collaborations reported the observation of X(3872) in $p\overline{p}$ collisions in semi-inclusive processes [94, 95]. Subsequently, semi-inclusive production of X(3872) in pp collisions was observed by the CMS and LHCb Collaborations [96-98]. All of these observations were made in the $J/\psi {\pi }^{+}{\pi }^{-}$ final state.

Using CDF measurements of the yields of X(3872) [99] and ψ(2S) [100], the prompt production rate of X(3872) in $p\overline{p}$ collisions at 1.96 TeV was estimated to be [101, 102] $\begin{array}{l}\sigma \left[p\overline{p}\to X\left(3872\right)+\text{all}\right]\text{Br}\left[X\to J/\psi {\pi }^{+}{\pi }^{-}\right]\\ \approx \left(3.1\pm 0.7\right)\text{nb}\mathrm{,}\end{array}$ (2) where the undetected particles produced in association with the X(3872) are denoted by “all”. Taking the branching faction of $X(3872)\to J/\psi {\pi }^{+}{\pi }^{-}$ from the RPP [22], $(3.5\pm 0.9)\%$, we obtained the X(3872) production cross-section at the CDF II detector as follows: $\sigma [p\overline{p}\to X(3872)+\text{all}]=(89\pm 30)\text{nb}\mathrm{.}$ (3) It was proposed in Ref. [101] that such a large cross-section conflicts with the hadronic molecular picture of X(3872) based on the following inequality² $\begin{array}{l}\sigma \left(\overline{p}p\to X+\text{all}\right)\\ \sim {\left|{\displaystyle \int }{\text{d}}^{3}k\langle X|D^0{\overline{D}}^{*0}\left(k\right)\rangle \langle D^0{\overline{D}}^{*0}\left(k\right)|\overline{p}p\rangle \right|}^2\\ \simeq {\left|{\displaystyle {\int }_{\mathcal{R}}}{\text{d}}^{3}k\langle X|D^0{\overline{D}}^{*0}\left(k\right)\rangle \langle D^0{\overline{D}}^{*0}\left(k\right)|\overline{p}p\rangle \right|}^2\\ ⩽{\displaystyle {\int }_{\mathcal{R}}}{\text{d}}^{3}k|\Psi \left(k\right){|}^2{\displaystyle {\int }_{\mathcal{R}}}{\text{d}}^{3}k{\left|\langle D^0{\overline{D}}^{*0}\left(k\right)|\overline{p}p\rangle \right|}^2\\ ⩽{\displaystyle {\int }_{\mathcal{R}}}{\text{d}}^{3}k{\left|\langle D^0{\overline{D}}^{*0}\left(k\right)|\overline{p}p\rangle \right|}^2\mathrm{,}\end{array}$ (4) where the undetected particles are assumed to be spectators, $\mathcal{R}$ is the region of the phase space where X(3872) is produced, and $\Psi \left(k\right)$ is the wave function of X(3872). It was argued in Ref. [101] that $\mathcal{R}$ should be of the order of the binding momentum of X(3872), ${\gamma }_X\sim 35$ MeV, such that the approximate upper bound in Eq. (4) is approximately 0.1 nb, which is three orders of magnitude smaller than the measured cross-section using Eq. (3).

The estimate of $\mathcal{R}$ as the binding momentum scale has been criticized in Refs. [102-104] (see also the discussion in Ref. [7]). In Ref. [102], the authors argued that $\mathcal{R}$ should be of the order of the inverse of the force range. As an approximate estimate, the inverse of the force range is one order of magnitude larger than γX (see below), and the upper bound in Eq. (4) is then enlarged by a factor of $\mathcal{O}\left({10}^3\right)$ compared with the estimate in Ref. [101] and agrees with the measured value from Eq. (3).

We now argue that the inverse of the force range for the $D{\overline{D}}^{*}$ S-wave interaction should be of the order of a few hundred MeV. However, because the u-channel one-pion exchange (OPE) between D and D^* has a scale of ${\mu }_{\pi }=\sqrt{{\left(M_{D^{*0}}-M_{D^0}\right)}^2-M_{{\pi }^0}^2}\approx 44$ MeV, the main binding force for X(3872) is probably not the OPE assumed in Ref. [30]. First, the renormalization of the OPE potential requires short-distance counter terms (see, for example, Ref. [105]). Second, phenomenological studies suggest the importance of light vector-meson exchanges (see, for example, Refs. [106-109]). Third, the binding energies of the hidden-charm hadronic molecules predicted in Ref. [110] change only marginally when the OPE is included, compared to the case with only constant contact terms. Fourth, by analyzing the contributions from different Wick contractions in the lattice QCD calculation of the isoscalar DD^* interaction directly related to Tcc(3875)⁺, Ref. [111] reported that the ρ exchange may be crucial for inducing the attraction between DD^*, in line with phenomenological studies [112, 113]. It is reasonable to assume that the $D{\overline{D}}^{*}$ system experiences a similar situation, which may be tested using lattice QCD calculations.

To demonstrate that the choice of $\mathcal{R}\simeq {\gamma }_X$ is too small, in Ref. [104], the authors considered deuterons as an example. The binding energy of the deuteron is approximately 2.2 MeV, and the binding momentum is ${\gamma }_d\simeq 45$ MeV. Fig. 8 shows the averaged wave functions of the deuteron in the $\mathcal{R}$ region [104], ${\overline{\Psi }}_{\lambda }\left(\mathcal{R}\right)\equiv {\displaystyle {\int }_{\mathcal{R}}}{\text{d}}^{3}k{\Psi }_{\lambda }\left(k\right)$, where λ is the regulator introduced to render the wave function well defined (for details, see Ref. [114]). It is clear that the wave function is far from saturated in the region $\underset{˜}{<}{\gamma }_X$, and saturation is achieved only when $\mathcal{R}$ is a few hundred MeV, in line with the argument in Ref. [102].

Fig. 8Full-screen View

(Color online) Averaged wave functions of the deuteron in the $\mathcal{R}$ region. The solid and dashed lines are the results with and without OPE, respectively, and the red, blue, and black lines correspond to the values of the cutoff λ=0.8, 1.5, and 4.0 GeV, respectively (the blue and black curves almost overlap each other). Adapted from Ref. [104]

This can be understood by considering the diagram corresponding to the second line of Eq. (4), as shown in Fig. 9 (a).³ Because the charmed meson pair is produced during high-energy collisions with all the particles produced in association with X(3872) assumed to be spectators, the production of $D{\overline{D}}^{*}$ occurs over short distances as a point-like vertex. The loop integral shown in Fig. 9 (a) is ultraviolet (UV) divergent and can be regularized using a cutoff. A cutoff-dependent counter term is required to absorb the divergence, as shown in Fig. 9 (b), to render the full-amplitude cutoff independent. For the counter term to be of natural size, that is, of the same order as the loop, the cutoff should also take a natural value for a hard scale, which is again of the order of the inverse of the force range, that is, a few hundred MeV.

Fig. 9Full-screen View

(Color online) Illustration of the semi-inclusive production of X(3872) in high-energy collisions. The circled cross presents the source for X(3872) and its associated particles in the production, denoted as “all.” (a) presents the $D{\overline{D}}^{*}$ loop contribution, and (b) represents a counter term required to absorb the UV divergence from the loop

The static properties of X(3872), such as the mass and JPC quantum numbers, can be understood through the $D{\overline{D}}^{*}$ hadronic molecular component. However, this does not mean that all properties are predominantly determined by the long-distance $D{\overline{D}}^{*}$ component. The production of X(3872) in high-energy collisions is a dynamic process, and the mechanism involved is generally different from that responsible for the internal structure. It involves the production of a pair of charm and anti-charm quarks at short distances, the hadronization of these quarks into charm and anti-charm mesons at intermediate distances ($\sim 1/{\text{Λ}}_{\text{QCD}}$), and the coalescence of these charm mesons into X(3872) at long distances ($\sim 1/{\gamma }_X$) (for a discussion on the factorization of long- and short-distance factors for the production and decay of X(3872), see Ref. [115]). The counter term in Fig. 9 (b) refers to the short-distance production of X(3872) from sources other than the $D{\overline{D}}^{*}$ intermediate state with low relative momenta and includes contributions from, for example, the production of X(3872) from a $c\overline{c}$ pair through hadronization. The prompt production of X(3872) from $c\overline{c}$ operators in high-energy hadronic reactions can be studied using a nonrelativistic QCD framework, as discussed in Refs. [102, 116-118].

Another method to make the entire production amplitude cutoff independent is to consider the factorization formula, as discussed in Ref. [115]. The production of X(3872), or more generally, hadronic molecules, contains both long-distance and short-distance parts. The long-distance part has a typical momentum scale of the X(3872) binding momentum and is given by the coupling of X(3872) to $D{\overline{D}}^{*}$, which is fixed from the binding momentum for a pure composite system [119, 7] or equivalently expressed in terms of the universal wave function for S-wave loosely bound states [120, 102]. The momentum scales in the short-distance part are of the order of the inverse of the force range or larger. The loop integral shown in Fig. 9 (a) regularized using a hard three-momentum cutoff is $G_{\text{Λ}}\propto 2\text{Λ}/\pi +ik$, where k is the magnitude of D or ${\overline{D}}^{*}$ in the $D{\overline{D}}^{*}$ c.m. frame. For Λ of the order of the inverse of the force range, the first terms are dominant. The Λ-dependence of the loop can be absorbed by the production vertex of $D{\overline{D}}^{*}$ from high-energy collisions, $P_{\text{Λ}}\propto 1/\text{Λ}$, and the short-distance part of the production amplitude is given by the product $P_{\text{Λ}}{G}_{\text{Λ}}$.

In Refs. [121, 104], the authors performed order-of-magnitude estimates of the cross-sections for prompt production in $pp/p\overline{p}$ collisions of X(3872) as a $D{\overline{D}}^{*}$ hadronic molecule. The $D{\overline{D}}^{*}$ cross-sections were estimated using the Monte Carlo event generators Pythia and Herwig following Refs. [101-103]. The $D{\overline{D}}^{*}$ loop shown in Fig. 9 (a) was regularized using a Gaussian form factor with a cutoff $\in [\mathrm{0.5,}1.0]$ GeV, corresponding to $\mathcal{R}\in [\mathrm{0.3,}0.6]$ GeV [104]. Correspondingly, the estimated cross-section for the production of X(3872) in $p\overline{p}$ collisions with CDF kinematics was $\sim [7(5)\mathrm{,}29(20)]$ nb [104], where the values outside (inside) the parentheses were obtained using Herwig (Pythia). This estimate agrees with the measured value from Eq. (3) at the order-of-magnitude level. The estimate for production in pp collisions at 7 TeV is approximately $\sim [13(4)\mathrm{,}55(15)]$ nb [104], which is in agreement with $(30\pm 9)$ nb⁴ measured by the CMS Collaboration [97].

X(3872) is a hidden-flavor exotic hadron, The dominant component of its wave function for describing its static properties is not necessarily its lowest Fock space components ($q\overline{q}$ with $q=u\mathrm{,}d\mathrm{,}s\mathrm{,}c$). In this case, a small $c\overline{c}$ component may be the driving force to produce X(3872) in high-energy collisions, even if X(3872) is predominantly molecular. Consequently, for a high-energy collision at a squared c.m. energy of s, the scaling (in powers of s) of the differential cross-section should follow the constituent counting rule [122] for the $c\overline{c}$ component, despite its main component still having four constituent (anti)quarks (see also Ref. [123]). This contrasts with the deuterons and other light (hyper)nuclei discussed in Sect. 3.1, whose lowest Fock space component has 3A quarks, where A is the number of baryons inside the (hyper)nuclei. Therefore, it is natural that the X(3872) production cross-section at large p_T is orders of magnitude larger than those of the deuteron and other light (hyper)nuclei measured by ALICE [68, 124]. Moreover, the critique in Ref. [125] on the molecular picture of X(3872) based on a large difference does not hold true.

With the same reasoning, because f₀(980) is also a hidden-flavor meson, the lack of observation of an enhancement of the p_T-differential $f_0\left(980\right)/K^{*}{\left(892\right)}^0$ ratio in pPb collisions by ALICE [126] should not be regarded as evidence for the f₀(980) being a normal $q\overline{q}$ meson or against the $K\overline{K}$ hadronic molecular picture [127, 128]. Similarly, the scaling of the elliptic anisotropic flow parameter v₂ for the production of f₀(980) in pPb collisions at the CMS experiment [129] likely agrees with a two-constituent hypothesis and should not be regarded as evidence for f₀(980) being a normal $q\overline{q}$ meson.

As discussed in Sect. 3.1, the LHCb measured the production of X(3872) and ψ(2S) as functions of the charged-particle multiplicity in pp collisions at 8 TeV [98]. The measured ratio of the yields of X(3872) to ψ(2S) in the $J/\psi {\pi }^{+}{\pi }^{-}$ decay channel decreased as the charged-particle multiplicity increased. This behavior is in line with the result obtained in the comover interaction model, assuming the diameter of X(3872) (as a compact tetraquark) to be 1.3 fm, and differs drastically from the hadronic molecular picture of X(3872) obtained with a coalescence model [130]. This conclusion was challenged in Ref. [131]. In Ref. [130], the cross-section for breaking up X(3872) as a hadronic molecule by scattering with comoving particles was assumed to be inversely proportional to the X(3872) binding energy. However, in Ref. [131], the authors argued that the breakup cross-section can be approximated by the probability-weighted sum of the cross-sections for the scattering of comoving pions with constituent charmed mesons inside X(3872). This is because for X(3872) with such a small binding momentum γX, the comoving particles can easily probe the internal structure of X(3872) and thus scatter directly with the constituent particles. Consequently, the breakup cross-section should be insensitive to the X(3872) binding energy. By modifying the comover interaction model, the LHCb data can be described well under the hadronic molecular picture of X(3872), as shown in Fig. 10 [131].

Fig. 10Full-screen View

(Color online) Ratios of the X(3872) and ψ(2S) cross-sections (blue: from prompt productions; red: from b-hadron decays) measured in the $J/\psi {\pi }^{+}{\pi }^{-}$ channel as functions of the multiplicity dN/dy. The data were measured by LHCb [98], and the curves with uncertainty bands were obtained from fitting to the LHCb data in Ref. [131]. From Ref. [131]

3.2.2

Light-flavor hadron production

In previous discussions, exotic hadronic states such as charm and beauty quarks were primarily examined within systems featuring heavy-flavor quarks. There are also many hadronic states in the light-flavor sector with properties that do not align with traditional quark model predictions. For example, the mass of Λ(1405) is lower than that of non-strange N(1535) with the same JP quantum numbers, even though Λ(1405) contains a heavier s quark from the perspective of the three-quark baryon model. In the 1950s, Λ(1405) was predicted as a bound state of $K\overline{N}$ by Dalit and Tuan [132] before its discovery. Since the 1990s, Λ(1405) has been considered as a molecular state of $K\overline{N}$ [133], accompanied by another nearby pole in the coupled-channel (π∑-$\overline{K}N$) scattering amplitudes, known as the two-pole structure for Λ(1405) [134]. The second and lower poles are now listed as Λ(1380) in the latest version of the RPP [22]. Similarly, in a light-meson system, the lowest scalar octet is considered a ground state family of tetraquarks [135], and f₀(980) is a good candidate for the $K\overline{K}$ molecule [127, 128].

It is also important to investigate the possible dibaryon states to better understand baryon interactions beyond protons and neutrons. To date, the only well-established dibaryon molecular state is the deuteron state, whereas the internal structure of another candidate, d^*(2380) [136], discovered over a decade ago, remains unclear. A common explanation for this is that d^*(2380) is a double-Δ state [137-139]. A resonance peak was recently observed below the threshold in the $\overline{K}NN$ system, with preliminary studies indicating $\text{Λ}N$ is the main decay channel [140].

In low-to-moderate energy pp and $p\overline{p}$ collisions, specifically with c.m. energies below 3.5 GeV and especially near the thresholds of certain rare decay channels, exclusive measurements can provide detailed information on many reactions, facilitating sophisticated theoretical modeling. There are large experimental data samples for pp and $p\overline{p}$ collisions in this energy region, for example, the IUCF, HADES, CELSIUS, and COSY experiments (for reviews, see Refs. [141, 142]). Here, we illustrate how the study of final-state systems in few-body reactions from pp and $p\overline{p}$ collisions can be used to search for baryon excited states, dibaryons, and meson excited states and discuss how to identify more exotic states and study their properties. There are three types of tree diagrams, as shown in Fig. 11, where (a), (b), and (c) indicate the main mechanisms for searching for the baryon excited states, dibaryons, and meson excited states, respectively.

Fig. 11Full-screen View

Illustration of the tree diagrams of the NN reaction to few-body final states, where the solid lines represent nucleons, dashed lines represent mesons, and double lines represent baryon and dibaryon resonances

The search for baryon excited states relies primarily on different final state combinations. By applying selection rules based on isospin conservation, regions with significant contributions from certain particles can be identified for further study. For example, in the $pp\to pn{\pi }^{+}$ process, although N(1440)Nσ has a significant impact [143], the largest contribution originates from $\text{Δ}{\left(1232\right)}^{++}\to p{\pi }^{+}$, making peaks such as N(1440) less prominent. Conversely, in $\overline{p}p\to \overline{p}n{\pi }^{+}$, the contribution of $\text{Δ}{\left(1232\right)}^{++}$ is absent, and the cross-section for other charged $\text{Δ}(1232)$ states decreases by a factor of nine because of the isospin factor, leading to increased visibility of peaks such as N(1440) [144]. Another example is in the $pp\to n{\displaystyle \sum {}^{+}}{K}^{+}$ process [145, 146], where ${\displaystyle \sum {}^{+}}{K}^{+}$ can only form isospin-3/2 states, thus providing a good venue for detecting Δ baryon excited states. In previous experiments, measurements were performed for many final states, such as $pp\to pp\pi$, ppη, $pp\varphi$, $pK+{\displaystyle \sum {}^0}$, and $pK+\text{Λ}$. It is important to consider the interaction between the initial and final states by using various methods to handle the final state interaction [141]. Despite the complexity of these theoretical methods, they can only explain a portion of experimental data. For example, in $pp\to pp\eta$, only data for the c.m. energy of 12 MeV above the ppη threshold can be well described, whereas the double-peak structure in the pη invariant mass spectrum at higher energies cannot be explained by considering only known excited nucleons [147]. Furthermore, polarization data cannot be explained by existing models [141]. Another issue is that because of this complication, only a single reaction process is typically analyzed, such as the analysis of $pp\to pK\text{Λ}$ in Ref. [148]. However, when considering the KΛ resonance states, although the threshold starts from approximately 1.6 GeV, which covers the N(1650) region, N(1535) could also make a significant contribution, owing to the presence of hidden strangeness in this state [149]. A better approach involves a combined analysis of multiple final states to extract information on multiple resonances. Similarly, the hidden-charm Pc states in $pp\left(\overline{p}\right)\to pp\left(\overline{p}\right)J/\psi$ and $pp\left(\overline{p}\right)\to pp\left(\overline{p}\right){\eta }_c$ reactions may be investigated [150].

pp and pd collisions offer good locations to search for dibaryon states. In reactions with either single-pion or double-pion production or with KΛ/K∑ in the final states, apart from the threshold cusp effect in K∑, no dibaryon states were observed [151] until 2014 when the Wide Angle Shower Apparatus (WASA) experiment at the Cooler Synchrotron (COSY) established a narrow resonant structure d^* in the $pn\to NN\pi \pi$ reaction, with $I\left(J^P\right)=0\left(3^{+}\right)$ and a width of 70 MeV [136]. It was proposed to be a double-Δ molecular state [137-139], although whether this corresponds to a genuine resonance is still under debate [152]. For theoretical predictions prior to the experimental observations, we refer to the review [151]. In particular, the quark model prediction for an isospin-3/2 dibaryon in Refs. [153, 154] agreed well with the measured mass. In theory, a notable feature of d^*(2380) is that its width is considerably smaller than that of two Δ baryons, implying the possible presence of many compact six-quark components in d^*(2380) [151, 155], although there is still no definitive conclusion. This discovery opens up a new direction in the search for dibaryon states. For pp reactions, as shown in Fig. 11 (b), new dibaryon states may be formed by emitting one or more mesons. Recently, in the J-PARC E15 experiment [140], a resonance peak structure with a width of 100 MeV was found in the $\text{Λ}+p$ final state in the reaction $K^{-}{+}^3\text{H}e\to \left(K^{-}+pp\right)+n\to (\text{Λ}+p)+n$ below the threshold of $\overline{K}NN$ by approximately 40 MeV, indicating the existence of a $\text{Λ}N$ dibaryon state. The existence of this resonance peak can be verified in the $pp\to K\text{Λ}p$ reaction; however, the energy at the COSY is only at the threshold for producing this peak structure [156]. Previous theoretical calculations for this reaction that did not consider the influence of the peak structure must be revised [157]. For further discussion on the production of dibaryon states in pp collisions, refer to [158-160].

To produce excited meson states in the low-energy region, the contribution presented in the diagram in Fig. 11 (c) in exchanging two particles is expected to be smaller than that in the other two diagrams. However, in the high-energy region, the intermediate exchanged meson can be replaced by a pomeron, and the fusion of the two exchanged pomerons to produce mesons plays a crucial role in producing double-pion and double-kaon resonant particles through diffraction processes [161, 162]. Owing to its gluon-rich environment, this reaction is often used to search for another type of exotic state, that is, glueballs, in high-energy pp or $p\overline{p}$ reactions. For calculations of the production of PC=++ mesons, such as f₀(980), f₂(1270), and f₀(1500), considering the pomeron–pomeron fusion, we refer to Refs. [163-165].

Finally, note that considering only the tree diagram mechanisms in Fig. 11 to estimate the production of light hadrons in pp collisions offers a rough approximation. The loop contributions in hadronic reactions are often crucial. For reactions with multiple hadrons in the final state, the final-state interactions and coupled-channel effects may be crucial and can significantly affect the extraction of resonance properties. Therefore, it is essential to use comprehensive coupled-channel models for few-body systems and a combined analysis of various related reactions to extract the resonance poles. Relevant efforts have been made in, for example, the Jülich-Bonn model [166], Argonne–Osaka model [167], and three-body unitary models [168, 169]. However, such models must include many parameters, and obtaining convincingly determined model parameters requires a significant amount of experimental input. Therefore, additional experimental data are required.

Production in heavy-ion collisions

In this subsection, we briefly review recent studies on the production of X(3872) and $T_{cc}^{+}$ in HICs. Earlier reviews on related topics can be found in Refs. [15, 170-172].

As discussed in Sect. 3.1, the only reported signal of prompt X(3872) production in HICs originated from the CMS collaboration [85]. The yields, evolution, and distributions of X(3872) as well as those of other exotic hadrons produced in HICs are complicated by the surrounding QCD medium/nuclear matter/pion gas.

Various coalescence models [73, 173-175] and statistical hadronization models (SHMs) [176] have been used to estimate the yields of X(3872) in hadronic molecular and compact tetraquark pictures. The former depend on a suitable wave function in the coordinate space to encode structural information (for discussions on the subtlety of the short-distance part, see Sect. 3.2.1). In the instantaneous coalescence model, there are still several unclear parameters, such as the volume size at which coalescence occurs, available light quark number at the hadronization temperature, and oscillator frequency of the Wigner function. The SHM assumes that hadrons are in thermal and chemical equilibrium, that is, with a charm-quark fugacity factor to ensure charm-quark conservation. The yield of X(3872) in this model depends only on its mass and not on its internal structure.

In Ref. [73], the authors used both coalescence and statistical models to estimate the yields of various hadrons at the RHIC, and the model results are summarized in Fig. 12. As shown in the figure, compared to the yield of normal hadrons, that of a compact tetraquark is typically one order of magnitude smaller, and that of a hadronic molecule is a factor of two or more. However, the coalescence model considered does not include the evolution effect in the medium, and the statistical model depends only on the masses of the hadrons instead of their internal structures.

Fig. 12Full-screen View

Ratios between the yields in the coalescence and statistical models [73]. The gray band is the region for normal hadrons. From Ref. [73]

Further information to deepen our understanding of the nature of exotic hadrons includes information on their various distributions, such as, the centrality, rapidity, and transverse momentum distributions. As reported in Ref. [175], a multiphase transport model was used to estimate the yield of X(3872) in Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=2.76\text{TeV}$. Considering that the fireball volume is crucial in the production of X(3872), they concluded that the yield of X(3872) in the molecular picture was two orders of magnitude larger than that in the compact picture, and significant centrality dependence was obtained (see Fig. 13). In addition, the corresponding rapidity and transverse momentum spectra, as well as the elliptical flow coefficient v₂ versus the transverse momentum for X(3872) in the molecular picture, were also predicted. The same method was used to estimate the production of the double-charm tetraquark state $T_{cc}^{+}$ [177] in Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=2.76\text{TeV}$ in the molecular picture. The ratio between the yields of $T_{cc}^{+}$ and X(3872) decreased with centrality, which agrees with the experimental fact that the yield of $T_{cc}^{+}$ is approximately two orders of magnitude smaller than that of X(3872).

Fig. 13Full-screen View

(Color online) Centrality dependence of X(3872) in Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=2.76\text{TeV}$ in both the molecular (red solid boxes) and compact tetraquark (blue shaded boxes) pictures obtained in Ref. [175]. The uncertainties are purely statistical. From Ref. [175]

When the wave function among the constituents is considered, for example, the Wigner function described in Ref. [178], the coalescence probability decreases owing to the strict constraints on the relative momentum between constituents, despite the large geometric size of hadronic molecules [178]. In this case, Ref. [178] showed that the total yield of a compact tetraquark X(3872) is several times larger than that of the molecular picture for Pb-Pb collisions. Simultaneously, the effect of fireball volume on centrality dependence in the molecular picture is not as significant as that described in Ref. [175]. In addition to the distribution of X(3872), the evolution of charm quarks in QGP was also explored in Ref. [178] using the Langevin equation. The yields in both scenarios decreased with evolution time. The same method has been applied to a double-charm tetraquark state [179].

There was also an early study [180] based on the SU(4) effective Lagrangians that considered the evolution of X(3872) by calculating the corresponding production and absorption cross-sections to estimate the hadronic effects on the X(3872) meson abundance in HICs. The absorption cross-sections of the X(3872) meson by pions and ρ mesons during the hadronic stage of HICs were evaluated. They estimated the yield of X(3872) in HICs using both the statistical and coalescence models. They found that the absorption cross-section was two orders of magnitude larger than the production cross-section, and the time evolution of X(3872) abundance in the HICs was stable (see Fig. 14), which is in contrast to the conclusions of Ref. [178].

Fig. 14Full-screen View

Time evolution of X(3872) in the hadronic molecular and compact tetraquark pictures in central Au-Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV with quantum number $J^P{=1}^{+}$ from the calculations in Ref. [180]. From Ref. [180]

The production of X(3872) in HICs is due to a large number of heavy quarks, as many as 20 $c\overline{c}$ pairs per unit rapidity, in Pb-Pb collisions at LHC energies [181, 73]. Charm quarks are free to move through large volumes and can coalesce to form bound states at the end of the QGP phase or their mixtures. A thermally averaged cross-section was proposed in Refs. [181, 182], with the dissociation and production processes estimated using $X+\pi \to \overline{D}+D$, $X+\pi \to {\overline{D}}^{*}+D^{*}$, $D+\overline{D}\to X+\pi$ and ${\overline{D}}^{*}+D^{*}\to \pi +X$, respectively. Because these studies were based on the coalescence model, their conclusions were similar. That is, the cross-sections in the molecular picture were larger than those in the compact tetraquark picture [181, 182], owing to geometrical arguments [181]. The detailed numbers were scheme-dependent. However, the geometrical estimate of cross-sections cannot be used in high-energy collisions because the particles that scatter with X(3872) can have sufficient momentum to probe into X(3872) and thus the size of X(3872) is no longer important [131]. The same method was also applied to the double-charm tetraquark $T_{cc}^{+}$ in both the LHC and RHIC energy regions within both the molecular and compact tetraquark pictures [183].

In Ref. [184], a transport calculation of X(3872) through the fireball formed during Pb-Pb collisions at $\sqrt{s_{\text{NN}}}=5\text{TeV}$ was reported. The formation and dissociation of X(3872) depend on two transport parameters: its inelastic reaction rate and thermal equilibrium limit in an evolving hot QCD medium. The latter was controlled by the charm production cross-section in primordial nucleon–nucleon collisions. They found that the yield of the loosely bound molecule, assumed to have formed later in the fireball evolution than in the tetraquark, was a factor of two smaller than that of the compact tetraquark. The same bulk-medium evolution used for charmonium and bottomonium transport was implemented, which was approximated by a cylindrically expanding fireball volume with a transverse flow profile of the blast-wave type. They used evolution parameters that reproduced the fits to empirical light-hadron spectra (pions, kaons, and protons) at thermal freeze-out temperatures. This resulted in the relative yields for the hadronic molecule and compact tetraquark pictures in Ref. [184] being qualitatively different from most coalescence model calculations [73, 173-175]. In Ref. [184], the initial hadronic abundance of the molecular configuration was assumed to be zero, which is different from the instantaneous coalescence models described in Refs. [73, 175], where the assumption of a wave function provides a large phase space and results in a significant surpassing of the equilibrium limit. The difference in these results may be regarded as a showcase of the model dependence for estimating the production of exotic hadrons in HICs, owing to the complications of the QCD medium.

In Ref. [185], the authors studied the properties of X(3872) in a hot-pion bath based on its molecular picture. They found that its width became a few tens of MeV at temperatures of $100-150\text{MeV}$ and its normal mass moved above the DD^* threshold. Their calculation was based on the SU(4) effective Lagrangians and imaginary time formalism in a self-consistent approach. The peak associated with X(3872) became significantly wider with increasing temperature. This is because of the appearance of a finite imaginary part of the amplitude at the pole position when temperature effects were included. Their results indicated that at typical kinetic freeze-out temperatures at the RHIC and LHC, X(3872) cannot be considered a loosely bound state. In contrast, a compact tetraquark-type state, which should couple weakly to $D{\overline{D}}^{*}$ because otherwise, a large molecular component is unavoidable, would barely change its behavior under the same circumstances.

Recently, a more rigorous treatment of the thermal corrections from the hot pion gas to the propagator of a loosely bound charm-meson molecule was presented using a zero-range effective field theory (ZREFT) [186]. One might simply expect that ZREFT cannot be applied at a high temperature, which is characterized by the kinetic freeze-out temperature and is orders of magnitude larger than the binding energy of a loosely bound state. Fortunately, the authors of Ref. [186] illustrated that ZREFT can be applied to such a system by first integrating out thermal pions, leaving the ZREFT parameters temperature-dependent. The only correction to the binding energy is a small temperature-dependent correction to the complex binding momentum. It is noticed that the thermal corrections to the binding energy of the molecule only appear at the next-to-leading order, and the results are shown by Fig. 15. These results indicate that loosely bound molecules, such as X(3872) and $T_{cc}^{+}$, can survive in the thermal environment of a hadron gas at sufficiently low temperatures. This conclusion is consistent with the large rate of X(3872) production in Pb-Pb collisions obtained by the CMS Collaboration.

Fig. 15Full-screen View

(Color online) Real parts of the poles of two-body propagators (left-panel) and the thermal widths (right panel) for X(3872) (red solid curves), and $T_{cc}^{+}$ (blue solid curves) in the pion gas as a function of temperature T [186]. The dashed lines in the left panel are the thresholds of the constituent charm-meson pair, and those in the right panel are the sum of the decay widths of the constituents. From Ref. [186]

Very recently, in Ref. [187], the authors proposed that the production rate of a loosely bound hadronic molecule, such as X(3872), in HICs can be expressed in terms of the short-distance contact density at the kinetic freeze-out of the hadron gas, which approaches a nonvanishing limit as the binding energy decreases to 0.

Because both X(3872) and ψ(2S) can be reconstructed in the $J/\psi {\pi }^{+}{\pi }^{-}$ channel, the production of $X(3872)$ is often compared to that of ψ(2S) (see, for example, Fig. 5). For instance, the prompt X(3872) to ψ(2S) yield ratio was found to be [85] $\mathcal{R}=\frac{N_{X\left(3872\right)}}{N_{\psi }\left(2S\right)}=1.08\pm 0.49(\text{stat})\pm 0.52(\text{syst})\mathrm{,}$ (5) with a central value approximately one order of magnitude higher than the 0.09 observed for pp collisions [98]. In Ref. [188], the authors used a method similar to that in Refs. [181, 182] to estimate the thermally averaged cross-sections for the production and absorption of ψ(2S) and used them in the rate equation to determine the time evolution of $N_{\psi \left(2S\right)}$. The yield ratio $N_X/N_{\psi \left(2S\right)}$ was predicted as a function of the centrality of the c.m. energy and charged hadron multiplicity measured in the mid-rapidity region $\left[\text{d}{N}_{\text{ch}}/\text{d}\eta (\eta <0.5)\right]$.

Similarly, Ref. [189] presents a phenomenological model for the partonic medium attenuation effects on X(3872) and ψ(2S) production in both pp and Pb-Pb collisions. A medium-assisted enhancement effect was proposed for X(3872) production, which was argued to be dominant at high parton densities and large medium sizes. Its competition with the absorption-induced suppression leads to a specific pattern of the $N_X/N_{\psi \left(2S\right)}$ ratio, which first decreases and then increases when the partonic medium evolves from a small to a large colliding system. A comparison of the results obtained in Ref. [189] using the data [85, 98,190] is shown in Fig. 16.

Fig. 16Full-screen View

(Color online) Yield of X(3872) relative to that of ψ(2S) obtained in Ref. [189] in comparison with the LHCb pp collisions data at $\sqrt{s_{\text{NN}}}=8\text{TeV}$ (red circle) [98], LHCb pPb collisions data at $\sqrt{s_{\text{NN}}}=8.16\text{TeV}$ (orange triangle) [190], and CMS Pb-Pb collisions data at $\sqrt{s_{\text{NN}}}=5.02\text{TeV}$ (green box) [85]. From Ref. [189]

In addition to the measurements of the centrality, transverse momentum, and rapidity distributions of exotic hadrons discussed in the previous subsection, an additional important observable that has recently received intensive interest is the momentum correlation between two hadrons. Because the correlation function contains information about the hadron–hadron final state interaction, exotic hadrons that couple to these two hadrons can be studied.

The momentum correlation function can be expressed in terms of the single-particle emission function $S\left(x_i\mathrm{,}{p}_i\right)$, which describes the probability of emitting a particle in space-time xi with momentum p_i through convolution with the squared relative two-particle wave function: $C(q\mathrm{,}P)=\frac{{\displaystyle \int }{d}^4{x}_1{d}^4{x}_2{S}_1\left(x_1\mathrm{,}{p}_1\right)S_2\left(x_2\mathrm{,}{p}_2\right){\left|{\phi }^{(-)}(r\mathrm{,}q)\right|}^2}{{\displaystyle \int }{d}^4{x}_1{S}_1\left(x_1\mathrm{,}{p}_1\right){\displaystyle \int }{d}^4{x}_2{S}_2\left(x_2\mathrm{,}{p}_2\right)}\mathrm{.}$ The relative wave function ${\left|{\phi }^{(-)}\left(r\mathrm{,}q\right)\right|}^2$ is for the hadron pair in the outgoing state that contains information about the hadron–hadron interaction. If the two hadrons are completely independent of each other, the momentum correlation function should be equal to unity. The correlation function is often approximated as follows (see, for example, Ref. [191]) $C(q\mathrm{,}P)={\displaystyle \int }{d}^{3}r{S}_{12}\left(r\right){\left|{\phi }^{(-)}\left(q\mathrm{,}r\right)\right|}^2\mathrm{.}$ (6) Typically, a spherical Gaussian source function with a given source radius is assumed, which ignores the dynamic properties of the particle emission sources. Low-energy scattering observables for various systems, including scattering lengths and effective ranges, have been extracted using this method. Many studies have been conducted along this direction, for example Refs. [170-172, 192]. For a recent review, see Ref. [15]. However, note that the results of the hadron–hadron interactions extracted using this method depend on the assumption of the source profile and source size. Although low-energy scattering observables correspond to long-distance physics, their source profile is short. On the one hand, the source function serves as a UV regulator (a form factor) for the production of the hadron pair, and the source size acts as a cutoff in the form factor. However, the correlation function measured in the experiments as physical observables should not depend on the UV regulator. From this perspective, work still needs to be performed to obtain the hadron–hadron interactions in a model-independent manner. It is worth noting that recent studies by ALICE [193, 194], which modeled the source considering a Gaussian profile and an exponential tail owing to strongly decayed resonances, suggested a common source for the meson–meson and meson–baryon sectors in pp collisions at the LHC.