Department of Physics, North China Electric Power University, Baoding 071003, China
zgwang@aliyun.com
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Received
Accepted
Published
2025-04-20
2025-06-09
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Revised Date
2025-09-18
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Abstract
We review the exotic states, such as the X, Y, Z, T and P states, and present their possible assignments based on the QCD sum rules. We present many predictions which can be confronted to the experimental data in the future to diagnose the exotic states. Furthermore, we also mention other theoretical methods.
In 1964, Gell−Mann suggested that multiquark states beyond the minimal valence quark constituents and might exist [1], a quantitative model for the tetraquark states with the quark constituents was developed by Jaffe using the MIT bag model in 1977 [2, 3]. Later, the five-quark baryons with the quark constituents were developed [4], while the name pentaquark was introduced by Lipkin [5]. Also in 1964, Dyson and Xuang studied the dibaryon or six-quark states based on the symmetry [6], for more literatures on this subject, one can consult Refs. [7, 8]. The QCD allows the existence of multiquark states and hybrid states which contain not only quarks but also gluonic degrees of freedom [9-11].
Before observation of the X(3872) by the Belle Collaboration in 2003 [12], the most promising and most hot subject is the nature of the light mesons below , are they traditional states, tetraquark states or molecular states? [2, 3, 13-16]. In fact, the observation of the X(3872) stimulates more motivations and curiosities in exploring the nature of the light scalar mesons [17-29], for more references, see the reviews [30, 31, 32]. However, it is an un-resoled problem until now. There have been several excellent reviews of the exotic states with emphasis on different aspects [33-47].
Firstly, let us see the experimental data on the exotic states pedagogically and dogmatically in the order X, Y, Z, T and P sequentially, and sort out of the exotic states according to the masses from low to high roughly. In fact, there are relations among those X, Y and Z states in one way or the other, it is impossible to sort out them distinctly, the most related ones are grouped together into one sub-section. Furthermore, we would like to emphasize the assignment based on the QCD sum rules at the end of every sub-section if there exists a room, as the predicted spectroscopy based on the QCD sum rules cannot accommodate all the exotic states. Other possible assignments would be presented in Sections 3−5 and 6.1.
1.1 X(3872)
In 2003, the Belle Collaboration observed a narrow charmonium-like state X(3872) with the mass near the threshold in the mass spectrum in the exclusive processes [12]. The evidences for the decay modes observed by the Belle and BaBar Collaborations imply the positive charge conjugation [48-50]. Angular correlations between final state particles analyzed by the CDF, Belle and LHCb Collaborations favor the assignment [51-54]. It is a possible candidate for the tetraquark state [55-62], (not) molecular state ([63]) [64-84], (not) traditional charmonium ([85]) [86-88], threshold cusp [89], etc. However, none of those available assignment has won an overall consensus, its nature is still under heated debates, the very narrow width and exotic branching fractions make it a hot potato, the door remains open for another binding mechanism. The QCD sum rules allow both the color -type and 11-type tetraquark assignments, see Tab.9 in Section 3.1.1 and Tab.44 in Section 4.1.
In 2005, the Belle Collaboration observed the Y(3940) in the B-decays [90], later, the BaBar Collaboration confirmed the Y(3940) in the mass spectrum with a mass about in the decays [91, 92].
In 2007, the Belle Collaboration observed the X(3940) in the process with the subprocess [93]. In 2008, the Belle Collaboration confirmed the X(3940) in the same process, and observed the X(4160) in the subprocess [94].
In 2010, the X(3915) was observed in the process by the Belle Collaboration [95], then the BaBar Collaboration determined its quantum numbers [96], it is a good candidate for the conventional charmonium . The Y(3940) and X(3915) could be the same particle, and is denoted as the with the assignment in The Review of Particle Physics [97].
In 2006, the Z(3930) was observed in the process by the Belle Collaboration [98], then confirmed by the BaBar Collaboration in the same process [99], the two Collaborations both determined its quantum numbers to be , and it is widely assigned as the [97].
In 2017, the Belle Collaboration performed a full amplitude analysis of the process , and observed a new charmonium-like state which decays to the pair, the measured mass and width are and , respectively [100]. The hypothesis is favored over the hypothesis at the level of , and the Belle Collaboration assigned the as an alternative state [100].
In 2020, the LHCb Collaboration performed an amplitude analysis of the decays and observed that it is necessary to include the and with the and respectively in the channel, and to include the and with the and respectively in the channel [101, 102]. The measured Breit−Wigner masses and widths are
and the and can be identified as the .
In 2023, the LHCb Collaboration announced the observation of the X(3960) in the mass spectrum in the decays, and the assignment is favored [103], the measured Breit−Wigner mass and width are MeV and MeV, respectively.
The thresholds of the and are and , respectively, which favors assigning the X(3960) and as the same particle. However, the ratio of the branching fractions [103],
implies the exotic nature of this state, as it is harder to excite an pair from the vacuum compared with the or pair, and the traditional charmonium states predominantly decay into the and states rather than into the and states. In addition, there is no room for the X(3860), so, at least one of the X(3860), X(3915), and X(3960) should be exotic state [104-107].
For example, the updated nonrelativistic potential model (NR) and Godfrey−Isgur relativized potential model (GI) indicate that the 2P charmonium states have the masses (NR; GI)[108],
even the assignments of the and are not comfortable enough.
We can identify the X(3915), Y(3940) and as the same particle tentatively, and assign it as the color -type scalar tetraquark state based on the QCD sum rules, see Tab.9 in Section 3.1.1, furthermore, there exists a room for the X(3860). On the other hand, we can assign the X(3960) as the color -type or 11-type tetraquark state, see Tab.10 in Section 3.1.1 and Tab.44 in Section 4.1.
1.3 , , ,
In 2024, the LHCb Collaboration explored the decays and , and observed four charmonium(-like) states , , and with the quantum numbers , , and respectively in the mass spectrum [109]. The measured Breit−Wigner masses and widths are
The updated nonrelativistic potential model (NR) and Godfrey−Isgur relativized potential model (GI) indicate that the 3S/2P/3P charmonium states have the masses (NR; GI) [108],
the assignments , and in Ref. [109] are rather marginal.
Based on the predictions of the QCD sum rules, we can assign the and as the color -type tetraquark states tentatively based on the QCD sum rules, see Tab.9 in Section 3.1.1.
In 2009, the CDF Collaboration observed the X(4140) in the mass spectrum in the decays with a significance larger than [110]. In 2011, the CDF Collaboration confirmed the Y(4140) in the decays with a significance greater than , and observed an evidence for the X(4274) with an approximate significance of [111]. In 2013, the CMS Collaboration also confirmed the X(4140) in the decays [112].
In 2016, the LHCb Collaboration performed the first full amplitude analysis of the decays, confirmed the X(4140) and X(4274) in the mass spectrum, and determined the spin-parity to be [113, 114]. Moreover, the LHCb Collaboration observed two new particles X(4500) and X(4700) in the mass spectrum, and determined the spin-parity to be [113, 114]. The measured masses and widths are
If they are tetraquark states, their quark constituents must be . The S-wave systems have the , , , while the P-wave systems have the , , , . The LHCb’s data rule out the or molecule assignments.
In 2021, the LHCb Collaboration performed an improved full amplitude analysis of the exclusive process , observed the X(4685) (X(4630)) in the mass spectrum with the () and Breit-Wigner masses and widths,
Furthermore, they observed the and with the in the mass spectrum and confirmed the four old particles [115].
In 2024, the LHCb Collaboration performed the first full amplitude analysis of the decays , and they developed an amplitude model with 53 components comprising 11 hidden-charm exotic states, for example, the X(4475), X(4650), X(4710) and X(4800) in the mass spectrum with the , , and , respectively, while the X(4800) is just an effective description of generic partial wave-function [116].
The X(4475), X(4650), X(4710) and X(4800) have the isospin , while the X(4500), X(4685), X(4700) and X(4630) have the isospin according to the final states and . A possible explanation is that those states are genuinely different states, if the X(4475) state is the isospin partner of the X(4500) interpreted as the state, we would generally expect a larger mass difference of rather than several MeV. The un-normal light-flavor breaking effects make them difficult to assign in the scenario of tetraquark states.
Based on the predictions of the QCD sum rules, we can assign the X(4140), X(4274), X(4500), X(4685) and X(4700) as the color -type tetraquark states with the positive parity tentatively, see Tab.10 in Section 3.1.1, and assign the X(4630) as the color -type tetraquark state with the negative parity tentatively, see Tab.23 in Section 3.1.3.
1.5 X(4350)
In 2010, the Belle Collaboration measured the process for the mass distributions and observed a narrow peak of events with a significance of [117]. The mass and width are and respectively. However, the X(4350) is not confirmed by other experiments.
1.6 , ,
In 2020, the LHCb Collaboration reported a narrow peak in the invariant mass spectrum in the decays [101, 102]. The peak could be reasonably parameterized in terms of two Breit−Wigner resonances:
They are the first exotic hadrons with fully open flavor, the valence quarks are [101, 102]. The narrow peak can be assigned as the color -type tetraquark state with the [118-121], its radial/orbital excitation [122], non-tetraquark state [123], molecular state [121, 124-128], triangle singularity [129], etc.
In 2023, the LHCb Collaboration observed the tetraquark candidates with the spin-parity in the processes and with the significance larger than [130, 131]. The measured Breit−Wigner masses and widths are
respectively, and they belong to a new type of open-charm tetraquark states with the and quarks. The , and can be accommodated in the light flavor symmetry sextet.
We can assign the , and as the color -type tetraquark states with the tentatively based on the QCD sum rules, see Section 6.1.
1.7 X(5568)
In 2016, the D0 Collaboration observed a narrow structure X(5568) in the decays with significance of [132]. The mass and natural width are and , respectively. The systems consist of two quarks and two antiquarks of four different flavors, just like the with the observed 7 years later in the mass spectrum by the LHCb Collaboration [130, 131]. The D0 Collaboration fitted the systems with the S-wave Breit-Wigner parameters, the favored assignments are , but the assignments cannot be excluded according to decays , where the low-energy photon is not detected. It can be assigned as a tetraquark state with the [133-137], however, the X(5568) is not confirmed by the LHCb, CMS, ATLAS and CDF Collaborations [138-141].
1.8 X(6600), X(6900), X(7300)
In 2020, the LHCb Collaboration reported evidences of two fully-charm tetraquark candidates in the mass spectrum [142]. They observed a broad structure above the threshold ranging from 6.2 to 6.8 GeV and a narrow structure at about 6.9 GeV with the significance of larger than . In addition, they also observed some vague structures around 7.2 GeV.
In 2023, the ATLAS Collaboration observed statistically significant excesses in the channel, which are consistent with a narrow resonance at about and a broader structure at much lower mass. And they also observed a statistically significant excess at about in the channel [143].
In 2024, the CMS Collaboration observed three resonant structures in the mass spectrum with the masses , and , respectively [144]. While in the no-interference model, the measured Breit−Wigner masses and widths are [144]
The two-meson pairs , , , , , and lie at , , , , , and , respectively [97], it is difficult to assign the X(6600), X(6900) and X(7300) as the charmonium−charmonium molecular states without introducing coupled channel effects.
We can assign the X(6600), X(6900) and X(7300) as the color -type tetraquark states tentatively based on the QCD sum rules, see Tab.41 in Section 3.3.
In 2005, the BaBar Collaboration studied the initial-state radiation process and observed the Y(4260) in the mass spectrum, the measured mass and width are and , respectively [145]. Subsequently the Y(4260) was confirmed by the Belle and CLEO Collaborations [146, 147], the Belle Collaboration also observed an evidence for a very broad structure Y(4008) in the mass spectrum.
In 2007, the Belle Collaboration studied the initial-state radiation process , and observed the Y(4360) and Y(4660) in the mass spectrum at with a width of and with a width of , respectively [148, 149], then the Y(4660) was confirmed by the BaBar Collaboration [150].
In 2008, the Belle Collaboration studied the initial-state radiation process , observed a clear peak Y(4630) in the mass spectrum just above the threshold, and determined the mass and width to be and , respectively [151]. Thereafter, the Y(4660) and Y(4630) are taken as the same particle according to the uncertainties of the masses and widths, for example, in The Review of Particle Physics [97].
In 2014, the BESIII Collaboration searched for the production of with , and observed a resonance in the cross section, the measured mass and width are and , respectively [152].
In 2016, the BESIII Collaboration measured the cross sections of the process , and observed two structures, the Y(4220) has a mass of and a width of respectively, and the Y(4390) has a mass of and a width of respectively [153].
Also in 2016, the BESIII Collaboration precisely measured the cross section of the process and observed two resonant structures, which agree with the Y(4260) and Y(4360), respectively. The first resonance has a mass of and a width of , while the second one has a mass of and a width of [154].
In 2022, the BESIII Collaboration observed two resonant structures in the mass spectrum, one is the Y(4230) and the other is the Y(4500), which was observed for the first time with the Breit−Wigner mass and width and , respectively [155].
In 2023, the BESIII Collaboration observed three enhancements in the mass spectrum in the Born cross sections of the process , the first and third resonances are the Y(4230) and Y(4660), respectively, while the second resonance has the Breit-Wigner mass and width and , respectively, and is roughly compatible with the Y(4500) [156].
Also in 2023, the BESIII Collaboration observed three resonance structures in the mass spectrum, the two significant structures are consistent with the and , respectively, while the third structure is new, and has the Breit−Wigner mass and width and , respectively, therefore is named as Y(4790) [157].
Also in 2023, the BESIII Collaboration observed a new resonance Y(4710) in the mass spectrum with a significance over , the measured Breit−Wigner mass and width are and , respectively [158].
In 2024, the BESIII Collaboration measured the Born cross sections for the processes and , and observed the well established in the mass spectrum [159]. In addition, they observed a new resonance in the mass spectrum, and measured the mass and width as and , respectively, which are also roughly compatible with the Y(4500).
Also in 2024, the BESIII Collaboration studied the processes and , and observed that the relatively large cross section for the process is mainly due to the enhancement about 4.75 GeV, which maybe indicate a potential structure in the cross section [160]. If the enhancement is confirmed in the future by enough experimental data, there maybe exist another Y state, the Y(4750).
We should bear in mind, in 2023, the BESIII Collaboration studied the process at twelve center-of-mass energies from 4.6119 to 4.9509 GeV, determined the Born cross sections and effective form-factors with unprecedented precision, and obtained flat cross sections about 4.63 GeV, which does not indicate the resonant structure Y(4630) [161].
The charmonium-like candidates Y(4260/4230), Y(4360/4320), Y(4390), Y(4500), Y(4660) and Y(4710/4750/4790) with the overwhelm the accommodating capacity of the traditional model, some of them should be multiquark states.
Based on the predictions of the QCD sum rules, we can assign the Y(4260/4230), Y(4360/4320), Y(4390) and Y(4750) as the color -type tetraquark states with an explicit P-wave between the diquark and antidiquark tentatively, see Tab.33 in Section 3.1.4, and assign the Y(4360), Y(4390), Y(4500), Y(4660), Y(4710) and Y(4790) as the color -type tetraquark states with an implicit P-wave in the diquark or antidiquark tentatively, see Tab.22 and Tab.23 in Section 3.1.3.
1.10 , , , ,
In 2013, the BESIII Collaboration studied the process at a center-of-mass energy of , and observed a structure in the mass spectrum with a mass of and a width of [162], at the same time, the Belle Collaboration studied the process using initial-state radiation, and observed a structure in the mass spectrum with a mass of and a width of [163]. Then this structure was confirmed by the CLEO Collaboration [164].
Also in 2013, the BESIII Collaboration studied the process at , and observed a structure near the threshold in the recoil mass spectrum [165]. The measured mass and width are and , respectively [165]. Slightly later, the BESIII Collaboration studied the process at from to , and observed a distinct structure in the mass spectrum, the measured mass and width are and , respectively [166].
In 2014, the BESIII Collaboration studied the process at , and observed a distinct charged structure in the mass spectrum [167]. The measured mass and width are and , respectively, and the angular distribution of the system favors the assignment [167].
We tentatively identify the and as the same particle according to the uncertainties of the masses and widths [60]. In 2017, the BESIII Collaboration established the spin-parity of the to be [168].
In 2021, the BESIII Collaboration observed an excess near the and thresholds in the recoil-mass spectrum with the significance of 5.3 in the processes [169]. The Breit-Wigner mass and width of the new structure were measured as and , respectively.
The and have similar production modes,
and they should be cousins and have similar properties.
Also in 2021, the LHCb Collaboration reported two new exotic states with the valence quarks in the mass spectrum in the decays [115]. The most significant state has a mass of , a width of , and the spin-parity , while the broader state has a mass of , a width of , and the spin-parity or (with a difference in favor of the first hypothesis) [115]. Considering the large difference between the widths, the and are unlikely to be the same particle.
In 2023, the BESIII Collaboration reported an excess of the candidate at a mass of with a significance of in the process [170]. The is consistent with the tetraquark state with the valence quarks , spin-parity-charge-conjugation , a mass and a width predicted in previous work based on the QCD sum rules [171].
The charmonium-like states , , , and have non-zero electric charge, and are excellent candidates for the tetraquark (molecular) states [171, 172].
Based on the predictions of the QCD sum rules, we can assign the , , and as the color -type tetraquark states, see Tab.9 and Tab.11 in Section 3.1.1, or 11-type tetraquark states, see Tab.44 in Section 4.1.
1.11 , ,
In 2008, the Belle Collaboration reported the first observation of two resonance-like structures and exceeding in the mass spectrum near in the exclusive decays [173]. The Breit−Wigner masses and widths are , , and , respectively. However, the BaBar Collaboration observed no evidence for the and states in the mass spectrum in the exclusive decays and [174].
In 2018, the LHCb Collaboration observed an evidence for the resonant structure with the significance larger than in a Dalitz plot analysis of the decays, the measured mass and width are and respectively [175]. The assignments and are both consistent with the experimental data. However, the is not confirmed by other experiments until now.
1.12 ,
In 2007, the Belle Collaboration observed a distinct peak in the mass spectrum in the decays , the mass and width are and , respectively [176]. In 2009, the Belle Collaboration observed a signal for the decay from a Dalitz plot analysis of the decays [177]. In 2013, the Belle Collaboration performed a full amplitude analysis of the decays to reach the favored assignments [178].
In 2014, the LHCb Collaboration analyzed the decays by performing a four-dimensional fit of the amplitude, and provided the first independent confirmation of the resonance and established its spin-parity . The measured Breit−Wigner mass and width are and , respectively [179], which excludes the possibility of assigning the as the molecular state with the spin-parity [180], although it lies near the threshold.
The Okubo−Zweig−Iizuka supper-allowed decays
are expected to take place easily, and the energy gaps have the relation , the can be assigned as the first radial excitation of the [56, 181, 182], which was proposed before the of the were determined by the BESIII Collaboration [168].
In 2019, the LHCb Collaboration performed an angular analysis of the weak decays , examined the versus the plane, and observed two possible resonant structures in the vicinity of the energies and , respectively [183], the structure has not been confirmed by other experiments yet. According to the mass gaps , we can tentatively assign the as the first radial excitation of the [184, 185].
Based on the predictions of the QCD sum rules, we can assign the and as the first radial excitations of the color -type tetraquark states, see Tab.9 in Section 3.1.1 and Tab.17 in Section 3.1.2.
1.13 , , ,
In 2014, the Belle Collaboration analyzed the decays and observed a resonance in the mass spectrum with a statistical significance more than , the measured mass and width are and , respectively, the preferred assignment is [186].
In 2019, the LHCb Collaboration performed an angular analysis of the decays , examined the versus the plane, and observed two structures in the vicinity of the energies and , respectively [183].
In 2024, the LHCb Collaboration performed the first full amplitude analysis of the decays , and they developed an amplitude model with 53 components comprising 11 hidden-charm exotic states, for example, the and in the mass spectrum with the ; the and in the mass spectrum with the , which might be the radial excitations of the in the scenario of tetraquark states with the valence quarks ; the , and are effective descriptions of generic partial wave-functions with the , and , respectively [116]. The spin-parity of the is determined to be for the first time with a significance exceeding .
We group the with the , and together into one-subsection as its assignment is still an open problem, and we would like to revisit this subject to discuss the possible assignment based on the QCD sum rules in Section 3.1.
1.14 , , Y(10750)
In 2011, the Belle Collaboration reported the first observation of the and in the and mass spectra associated with a single charged pion in the decays, the quantum numbers are favored [187]. Subsequently, the Belle Collaboration updated the measured parameters , , and , respectively [188]. In 2013, the Belle Collaboration observed the decays for the first time, and obtained the neutral in a Dalitz analysis of the decays to the final states [189].
In 2019, the Belle Collaboration observed a resonance structure Y(10750) in the () cross sections [190]. The Breit−Wigner mass and width are and , respectively. The Y(10750) is observed in the mass spectrum with , its quantum numbers are .
The BelleII Collaboration confirmed the Y(10750) in the processes , [191], , [192], and observed no evidence in the processes , [193], [192].
Based on the predictions of the QCD sum rules, we can assign the and as the color -type or 11-type tetraquark states tentatively, see Section 3.1.1 and Section 4.1, and assign the Y(10750) as the color -type tetraquark state with an explicit P-wave between the diquark and antidiquark tentatively, see Section 3.1.4.
1.15
In 2021, the LHCb Collaboration formally announced observation of the exotic state just below the threshold [194, 195]. The Breit−Wigner mass and width are below the threshold and [194, 195]. The exotic state is consistent with the ground isoscalar tetraquark state with the valence quarks and spin-parity , and exploring the mass spectrum disfavors interpreting the as an isovector state. The observation of the is a great breakthrough beyond the for hadron physics, and it is the first doubly-charmed tetraquark candidate with the typical quark configuration .
Based on the predictions of the QCD sum rules, we can assign the as the color -type or 11-type tetraquark state tentatively, see Section 3.2 and Section 4.2.
1.16 , , , , , ,
In 2015, the LHCb Collaboration observed two exotic structures and in the mass spectrum in the decays [196]. The has a mass of and a width of , while the has a mass of and a width of . The preferred spin-parity assignments of the and are and , respectively [196].
In 2019, the LHCb Collaboration studied the decays with a data sample, which is an order of magnitude larger than that previously analyzed, and observed a narrow pentaquark candidate in the mass spectrum. Furthermore, the LHCb Collaboration confirmed the pentaquark structure , and observed that it consists of two narrow overlapping peaks and [197]. The measured masses and widths are
In 2021, the LHCb Collaboration reported an evidence of a hidden-charm pentaquark candidate with the strangeness in the mass spectrum with a significance of in the decays [198], the Breit−Wigner mass and width are
and the spin-parity have not been determined yet up to now.
In 2022, the LHCb Collaboration observed an evidence for a structure in the and systems in the decays with a significance about depending on the hypothesis [199], the Breit−Wigner mass and width are
Its existence still needs confirmation and its spin-parity is not measured yet.
In 2023, the LHCb Collaboration observed an evidence for a new structure in the mass distribution in the decays [200], the measured Breit−Wigner mass and width are
and the favored spin-parity is .
The and are observed in the mass spectrum, they have the isospin , as the strong decays conserve isospin. The , , , , and lie slightly below or above the thresholds of the charmed meson-baryon pairs , , , , (, , ) and , respectively. It is difficult to identify the as the molecular state without resorting to the help of large coupled-channel effects due to lacking nearby meson-baryon thresholds. Or the and are the same particle, such a possibility cannot be excluded at the present time.
Based on the predictions of the QCD sum rules, we can assign the , , , , and as the color -type pentaquark states tentatively, see Tab.50 and Tab.52 in Section 5.1, and assign the , , , , and as the color 11-type pentaquark states, see Tab.54 in Section 5.2.
1.17
In 2014, the scientists in the WASA-at-COSY Collaboration and SAID data analysis center performed exclusive and kinematically complete high-statistics measurements of the polarized scattering through the quasifree process in the energy region of the narrow resonance-like structure with the , and confirmed their (WASA-at-COSY Collaboration) early observation of the in the double-pionic fusion channels, they produced a resonance pole in the coupled partial waves at MeV [201, 202], — in accordance with the dibaryon resonance [203-207]. And we will revisit this subject at the end of Section 4.2.
2 Theoretical foundations
In this section, we would like to review the typical theoretical methods and related possible assignments concisely, then focus on the QCD sum rules in the subsequent sub-sections, see Sections 2.2, 2.3, and 2.4.
2.1 Typical theoretical methods and possible assignments
There have been tremendous progresses on the hadron spectrum containing two heavy quarks experimentally since the observation of the X(3872). It is surprising that many resonant structures lie around thresholds of a pair of heavy hadrons. A natural conjecture is that they are possible deuteron-like two-particle bound states bound via attractive interactions induced by one-pion exchange or one-boson exchange [63, 65, 67, 72, 76, 208-215], it is only a possibility. In the heavy quark limit, the mesons and baryons have the antiquark-diquark symmetry, , therefore the and systems could be analyzed in the same theoretical scheme, in this sub-section, we would like to focus on the tetraquark systems. Someone maybe wonder: are they threshold cusps, triangle singularities or genuine resonances? As there always exist threshold cusps at the S-wave thresholds or triangle singularities near the thresholds. Firstly, let us see the outcomes based on the (non)relativistic effective field theory.
2.1.1 Threshold cusps, triangle singularities or genuine resonances
Not all peaks in the invariant mass distributions are genuine resonances, they often arise due to the nearby kinematical singularities of the transition amplitudes in the complex energy plane. Those singularities (or Landau singularities) occur when the intermediate particles are on the mass-shell. The simplest case is the cusp at the normal two-body threshold, there always exists a cusp at the S-wave threshold of two particles coupling to the final states, while a more complicated case is the so-called triangle singularity. They maybe produce observable effects if the involved interactions are strong enough, sometimes, even mimic the behavior of a resonance. It is important to distinguish kinematic singularities from genuine resonances. We would like to give an example concerning the exotic states Y(4260) and to illustrate their possible assignments in the scenarios of threshold cusps, triangle singularities and genuine resonances.
The threshold cusp is determined by masses of the involved particles, how strong the cusp depends on detailed dynamics and the cusp could be rather dramatic if there is a nearby pole, thus it plays an important role in studying the exotic states [216]. For example, the X(3872), , , and lie near the , , , and thresholds, respectively, their quantum numbers are the same as the corresponding S-wave meson pairs although the of the have not been fully determined yet [97].
The X(3872) was assigned to be a threshold cusp by Bugg [89], subsequently, he realized that the very narrow line shapes in the and channels could not be fitted with only a threshold cusp, and a resonance or virtual state pole was necessary [217].
In a modified threshold cusp model [218, 219], see the Feynman diagram shown in Fig.1 as an example, both the inelastic () and elastic (, ) decay modes were considered for the and , analogous discussions are applied to the and . A Gaussian form-factor was chosen for all the vertices including the tree-level ones. Then the experimental data for the and mass spectra for the and the and mass spectra for the could be fitted very good, exotic resonances are not required to account for the experimental data. However, the fitting quality depends crucially on the cutoff parameter in the Gaussian form-factor.
The triangle singularity is determined by the masses of the intermediate particles plus the invariant masses of the external ones, therefore the triangle singularities are sensitive to the kinematic variables, the peak position and peak shape change according to the variations of the external energies. More precisely, the triangle singularities are determined by the scalar triangle loop integral, which does not depend on the orbital angular momentum for each vertex, however, sharp triangle singularity peaks are constrained to the S-wave internal particles, as momentum power factor weakens the singular behavior in other cases [216].
In the initial single-pion emission (ISPE) mechanism, the triangle diagrams contribute to threshold cusps, see Fig.2 for a typical Feynman diagram. This mechanism was suggested firstly to study the exotic structures and , Chen and Liu introduced a dipole form-factor to accompany the exchanged B-meson propagator and took account of the , , and triangle loop diagrams, and produced sharp cusps right around the and structures in the and mass spectra, but observed no cusp at the threshold [220]. Similarly, Chen, Liu and Matsuki [221] took account of the , , and triangle loop diagrams and the intermediate and to study the decays , and observed two peaks, the and its reflection. And they studied other processes with possible triangle singularities [222].
In Ref. [223], Dong, Guo and Zou showed that the threshold cusp appears as a peak only for channels with attractive interaction, and the cusp’s width is inversely proportional to the reduced mass for the relevant threshold. There should be threshold structures at any threshold of a and () pair, which have attractive interaction at threshold, in the invariant mass distribution of a state and a () state coupling to the and () pair, and the structure becomes more pronounced if there is a near-threshold pole.
In Ref. [224], Liu and Li supposed the Y(4260) as a molecular state to study its decays, see Fig.3 as an example, and observe that under special kinematic configurations, the triangle singularity maybe occur in the re-scattering amplitude, which can change the threshold behavior significantly. Obvious threshold enhancements or narrow cusp structures appear quite naturally without introducing a genuine resonance, but cannot exclude existence of a genuine resonance, such a mechanism also works for the pentaquark structures [225, 226].
If the Y(4260) have a large molecular component, the decays can occur through the re-scattering process [227-229], see Fig.3. The singularity regions provide an ideal environment for forming bound states or resonances. Although the Y(4260) lies slightly below the threshold, the triangle singularity in the invariant mass distribution of the re-scattering amplitude is still near the physical boundary and can influence the invariant mass distribution around the threshold significantly [227]. Despite the importance of the triangle diagram contribution, it is insisted that a resonance was still needed in order to fit to the narrow peak observed in experiments [224, 227, 230, 231]. The diagrams similar to Fig.3 also play an important role in the hidden-bottom sector [232, 233].
The resonance pole can be incorporated by constructing a unitarized coupled-channel scattering T-matrix by fitting to the experimental data, the best fit still demands the T-matrix to have a resonant or virtual pole near the threshold, which can be interpreted as the [234]. The molecule assignment provides a natural explanation for the resonance-like structure in the Y(4260) decays [229, 234], the kinematical threshold cusp cannot produce a narrow peak in the invariant mass distribution in the elastic channel in contrast with a genuine S-matrix pole [235].
In a similar scenario, Szczepaniak suggested that the peak could be attributed to the loop instead of the loop, which is in the physical region by neglecting the width of the [236]. The triangle singularities can produce enhancement potentially in the amplitude consistent with the experimental data qualitatively.
In Ref. [237], Chen, Du and Guo performed a unified description of the and mass distributions for the and the mass distribution for the at 4.23 and 4.26 GeV. They take account of the open-charm meson loops containing triangle singularities, the - coupled-channel interaction respecting unitarity, and the strong - final-state interaction using dispersion relations, which lead to a precise determination of the pole mass and width MeV and MeV, respectively, and indicate the molecular and non-molecular components are of similar importance for the structure .
Precisely measuring the near threshold structures plays an important role in diagnosing the heavy-hadron interactions, therefore understanding the puzzling hidden-charm and hidden-bottom structures. Furthermore, it is important to search for the resonant structures in processes free of triangle singularities, such as the photo-production and pion-induced production processes in the and collisions [238-240]. For a recent review on the production of the exotic hadrons in the and nuclear collisions, see Ref. [241].
2.1.2 Dynamical generated resonances and molecular states
If we take the traditional heavy mesons as the elementary degrees of freedom, then we construct the heavy meson effective Lagrangian according to the chiral symmetry, hidden-local symmetry and heavy quark symmetry [242-244]. It is easy to obtain the two-meson scattering amplitudes V. Then we have three choices:
Firstly, we unitarize the amplitudes by taking account of the intermediate two-meson loops with the coupled channel effects through the Bethe-Salpeter or Lippmann-Schwinger equation with on-shell factorization [245, 246],
where the G is the loop function, see Fig.4 for a diagrammatical representation. Then we explore the analytical properties of the full amplitudes T, and try to find the poles in the complex Riemann sheets, such as the bound states, virtual states and resonances. Such discussions are applied to the baryon-meson systems directly.
Bound states appear as poles on the physical sheet, and only appear on the real -axis below the lowest threshold by causality. Virtual states also appear on the real -axis, however, on the unphysical Riemann sheet. Resonances appear as poles on an unphysical Riemann sheet close to the physical one with non-zero imaginary part, and they appear in conjugate pairs. For example, the loosely bound states X(3872), Y(3940), , , [70, 75, 247-254], the hidden-charm pentaquark resonances [255-259]. We usually apply Weinberg’s compositeness condition to estimate the hadronic molecule components [260-266].
Secondly, we take the scattering amplitudes V as interaction kernels, solve the quasi-potential Bethe-Salpeter or Lippmann−Schwinger equation with the coupled channel effects directly, then explore the analytical properties of the full amplitudes [267-273], or obtain the bound energies directly to estimate the bound states [76, 274-278].
Thirdly, we reduce the scattering amplitudes to interaction potentials in the momentum space in terms of the Breit approximation and introduce monopole form-factors associated with the exchanged particles. Generally, we should introduce form-factors in each interaction vertex, which reflects the off-shell effect of the exchanged meson and the structure effect, because the components of the molecular states and exchanged mesons are not point particles. Then we perform the Fourier transformation to obtain the potential in the coordinate space, finally we solve the Schrödinger equation directly to obtain the binding energy [63, 279-289].
2.1.3 states with coupled channel effects
In the famous GI model, the charmonium has the mass about [86], which is about above the X(3872) lying near the threshold, the strong coupling to the nearby threshold maybe lead to some molecular configuration. Furthermore, pure charmonium assignment cannot interpret the high decay rate [290].
We can extend the constituent quark models to include the meson−meson Fock components, and write the physical charmonium (bottomonium) states in terms of ,
where the are the eigenstates, the are or eigenstates, are the two-meson state with quantum numbers, and the is the relative wave function between the two mesons. Then we solve the Schrodinger equation directly [291-300].
2.1.4 Hybrid and tetraquark states in Born−Oppenheimer approximation
Due to the large ratio of the mass of a nucleus to that of the electron, the electrons respond almost instantaneously to the motion of the nuclei. The energy of the electrons combined with the repulsive Coulomb energy of the nuclei defines a Born−Oppenheimer potential. Accordingly, due to the large ratio of the heavy-quark mass to the energy scale associated with the gluon field, the gluons respond almost instantaneously to the motion of the heavy quarks Q and [301-306].
In the static limit, the Q and serve as two color-sources separated by a distance , the ground-state flavor-singlet Born−Oppenheimer potential is defined by the minimal energy of the gluonic configurations, whose small and large limiting behaviors are qualitatively compatible with the simple phenomenological Cornell potential, thus the describes the traditional heavy quarkonium states. The excited Born-Oppenheimer potentials are defined as the minimal energies of the excited configurations for the gluon and light-quark fields with the quantum numbers [301-303].
The hybrid states are energy levels of a heavy quark pair in the excited flavor-singlet Born-Oppenheimer potentials, the hybrid potentials. Juge, Kuti and Morningstar [307] calculated many flavor-singlet potentials using the quenched lattice QCD. There have been some works on the lowest lying hybrid potentials using lattice QCD with two flavors of dynamical Wilson fermions [308, 309]. At large , the hybrid potential is a flux-tube extending between the Q and . At small , the hybrid potential approaches the repulsive color-Coulomb potential between a Q and in a color-octet state. In the limit , the Q and sources reduce to a single local color-octet source. The energy levels of the flavor-singlet gluon and light-quark field configurations bound to a static color-octet source are called static hybrid mesons. The most effective pictorial representation of the hybrid states is the flux-tube model. Lattice QCD simulations show that two static quarks Q and at large distances are confined by approximately cylindrical regions of the color fields [307-309].
The tetraquark states are energy levels in the Born-Oppenheimer potentials with nonsinglet (excited singlet) flavor quantum numbers, the tetraquark potentials, which are distinguished by the quantum numbers, , , , , , etc. At large , the minimal-energy configuration consists of two static mesons localized near the Q and sources. At small , the minimal-energy configuration is the flavor-singlet potential accompanied by one or two pions (two or three pions), depending on the quantum numbers . There have been works on the energies of static adjoint mesons using the quenched lattice QCD [310]. The static adjoint mesons are energy levels of the light-quark and gluon fields with nonsinglet flavor quantum numbers bound to a static color-octet source.
The heavy quark motion is restored by solving the Schrodinger equation in each of those potentials, and many X, Y and Z mesons could be assigned as the bound states with the Born−Oppenheimer potentials [301-306].
2.1.5 Tetraquarks in diquark models
If we take the quarks in color triplet 3 as the basic constituents, then we could construct the hadrons according to the symmetry. For the traditional mesons,
For the traditional baryons,
For the tetraquark molecular states,
and we usually call the color 11 type structures as the molecular states. For the tetraquark states,
and we usually call the color type structures as the tetraquark states. For the pentaquark molecular states,
For the pentaquark states,
If we take the viewpoint of the quantum field theory, the scattering amplitude for one-gluon exchange is proportional to
where the is the Gell-Mann matrix, the , , , and are color indexes, the is the color number, and in the real world. The negative sign in front of the antisymmetric antitriplet indicates the interaction is attractive, which favors formation of the diquarks in color antitriplet, while the positive sign in front of the symmetric sextet 6 indicates the interaction is repulsive, which disfavors formation of the diquarks in color sextet.
In this sub-sub-section, we would like to focus on the tetraquark states, as the extension to the pentaquark states is straightforward. Now we define the color factor,
where the subscripts and denote the quarks, and for the and 6 diquark , respectively, and and for the 1 and 8 quark-antiquark , respectively. If we define , then and for the and type tetraquark states. It is feasible to take both the and diquark configurations to explore the tetraquark states, while the preferred or usually chosen configuration is of the type.
The color-spin Hamiltonian can be written as [311]
the color factor can be absorbed into the chromomagnetic couplings after taking matrix elements between the type tetraquark states.
In 2004, Maiani et al. [55] introduced the simple spin-spin Hamiltonian,
to study the hidden-charm tetraquark states in the diquark model, where the is the charmed diquark mass. They took the X(3872) with the as the basic input and predicted a mass spectrum for the type hidden-charm tetraquark states with the , and . Maiani et al. [312] assigned the charged resonance in the decays as the X(3882) or Z(3882) according to the BESIII and Belle data, however, there is no evidence for the lower resonance X(3754) or Z(3754). In fact, in the Type-I diquark model, see the Hamiltonian in Eq. (22) [55], the predicted masses and for the states are smaller than that of the tetraquark candidates and observed later, respectively [162, 163, 165-167].
In 2005, Maiani et al. [313] assigned the Y(4260) to be the first orbital excitation of the state by including the spin-orbit interaction, and obtained a crucial prediction that the Y(4260) should decay predominantly in the channel. The decay model has not been observed yet up to now. In 2009, Drenska, Faccini and Polosa [314] studied the tetraquark states with the , , , , , and by computing the mass spectrum and decay modes.
In 2014, Maiani et al. [56] restricted the dominant spin−spin interactions to the ones within each diquark, and simplify the effective Hamiltonian,
which could describe the hierarchy of the masses of the X(3872), , very well in the scenario of tetraquark states, furthermore, they introduced a spin−orbit interaction to interpret the Y states,
where the , and are parameters to be fitted experimentally. Then they sorted the tetraquark states in terms of , where the L is the angular momentum between the diquark and antidiquark, , , , , and assigned the Y(4008), Y(4260), Y(4290/4220) and Y(4630) to be the tetraquark states , , and , respectively. The effective Hamiltonian, see Eq. (23), is referred to as the Type-II diquark model. Then the mass spectrum of the tetraquark states was explored [315], and applied to study the LHCb’s resonances [316].
In 2017, Maiani, Polosa and Riquer introduced a hypothesis that the diquarks and antidiquarks in tetraquarks are separated by a potential barrier to answer the long standing questions challenging the diquark-antidiquark model of exotic resonances [317].
In 2018, Ali et al. [318] analyzed the P-wave hidden-charm tetraquark states in the diquark model using an effective Hamiltonian incorporating the dominant spin−spin, spin−orbit and tensor interactions,
where , the and are the spins of the diquark () and 3 antidiquark (), respectively, the , , and are parameters to be fitted experimentally. And their updated analysis indicate that it is favorable to assign the Y(4220), Y(4330), Y(4390), Y(4660) as the tetraquark states , , and , respectively.
In 2021, Maiani, Polosa and Riquer [319] suggested that the and are two different particles, and there exist two nonets with the and , respectively, thus they could assign the X(3872), , , and X(4140) consistently.
Again, let us turn to the chromomagnetic interaction model, see Eq. (21), and choose the plus configurations and 11 plus 88 configurations as two independent representations (or basis) respectively to explore the mass spectrum of the exotic states and their decay channels, and have obtained many successful descriptions [320-324].
In the dynamical diquark picture, Brodsky, Hwang and Lebed [57] assumed that the pair forms promptly at the production point, and rapidly separates due to the kinematics of the production process, as the diquark and antidiquark are colored objects, they cannot separate asymptotically far apart; they create a color flux tube or string between them. If sufficient energy is available, the string would break to create an additional pair, and rearrange into a baryon-antibaryon pair, for example, the pair. The overlap of the wave-functions between the quark and antiquark is suppressed greatly, due to the large spatial separation between the diquark and antidiquark pair, therefore, the transition rate is suppressed and leads to small exotic widths [57]. The exotic mass spectrum is calculated in this picture [325-331].
In the relativized quark model, the Hamiltonian can be written as
where the is the linear confining potential, the is the one-gluon exchange potential including a Coulomb and a hyperfine term. Then the type configurations or both the and type configurations are taken into account to solve the Schrodinger equations to obtain the mass spectrum [332-341].
In the quasipotential approach, Ebert et al. [59, 342-344] took the -type configurations to study the hidden-charm (hidden-bottom, charm-bottom or fully-heavy) tetraquark mass spectrum by solving the Schrödinger type equations, where an effective one-gluon exchange potential plus a linear confining potential are adopted.
In the constituent quark model, all possible quark configurations satisfying the Pauli principle are explored by solving the Schrodinger equation with the potential kernel containing the confinement plus one-gluon-exchange plus (or not plus) one-meson-exchange interactions [345-352], while in the color flux-tube model, a multi-body interacting confinement potential instead of a two-body interacting confinement potential is chosen [353, 354].
2.1.6 Tetraquark states with lattice QCD
Lattice QCD provides rather accurate and reliable calculations for the hadrons which lie well below strong-decay threshold and do not decay strongly, the physical information is commonly extracted from the discrete energy spectrum. The physical system with specified quantum numbers is created from the vacuum using an operator at time , then this system propagates for a time before being annihilated by an operator . The spectral decomposition is performed to express the correlators in terms of the energies and overlaps of the eigenstates ,
. The correlators are calculated on the lattice and their time-dependence is used to extract the and [355, 356]. The lattice QCD has been applied extensively to study the exotic states [357-364].
In the energy region near or above the strong decay thresholds, the masses of the bound states and resonances are inferred from the finite-volume scattering matrix of one-channel elastic or multiple-channel inelastic scattering. Various approaches with varying degrees of mathematical rigour have been used in the simulations [365]. The simplest example is a one-channel elastic scattering with the partial wave , where the scattering matrix satisfying unitarity is parameterized in terms of the phase shift ,
the phase shift for the S-wave scattering is extracted using the well-established and rigorous Luscher’s relation [366-368], which applies for the elastic scattering below and above threshold. The phase shifts provide copious information about the masses of resonances and bound states. In the vicinity of a hadronic resonance with a mass and a width , the cross section has a Breit−Wigner-type shape with the value ,
Below and above threshold, the can be expanded by the effective range approximation,
where the is the scattering length and the is the effective range. On the other hand, the bound state (B) is realized when the scattering amplitude has a pole at the value ,
the location of this shallow bound state can be obtained by parameterizing near the threshold and finding the which satisfies . Most of the exotic candidates are above several two-hadron thresholds, and have more than one decay channels, which requires determining the scattering matrix for the coupled-channel nonelastic scattering matrix elements [369, 370, 371, 372, 373].
The HALQCD Collaboration use the HALQCD approach to extract the coupled-channel scattering matrix [374, 375], the HALQCD approach is based on the lattice determination of the potential between different channels, then employs the Nambu-Bethe-Salpeter equation to extract the masses of the bound states.
2.2 Multiquark states with the QCD sum rules
The QCD sum rules were introduced by Shifman, Vainstein and Zakharov in 1979 to study the conventional mesons [376, 377], then they were extended to study the conventional baryons by Ioffe [378]. The QCD sum rules are analytic and fully relativistic, and approach the bound state problem in QCD from short distances and move to longer distances step by step by including the non-perturbative effects so as to extract information on the hadronic properties.
In the past years, the QCD sum rules have been applied widely to study the hadronic properties, such as the masses of the quarks; masses and decay constants of the light and heavy mesons and baryons; form-factors of the mesons and baryons; valence quark distributions and spin structure functions of the nucleons; structure functions of the photon, pseudoscalar, vector and axialvector mesons; hadronic matrix elements for the , , mixing; strong coupling constants and magnetic moments of the mesons and baryons; parameters of the effective field theories; spectroscopy and properties of the exotic states; hadrons in the nuclear matter; properties of hadronic matter at high temperature and density [46, 47, 379-388]. Especially since 2007, the QCD sum rules have been applied extensively to study the X, Y, Z, T and P states, which are the typical multiquark candidates [46, 47, 58, 60, 81]. For the early works on the exotic states, we can consult Refs. [389-400].
In this sub-section, we would like to illustrate the general procedure of the QCD sum rules for the masses of the conventional hadrons and multiquark states concisely.
At the beginning point, let us write down the general two-point vacuum correlation functions,
where the are the local currents consist of quark-gluon fields with specified quantum numbers, and the T denotes the time-ordering operation. For the conventional mesons and baryons, the currents have been explored extensively [381], for the multiquark states, the currents can be constructed straightforwardly.
At the large squared momentum region , the integral in Eq. (32) is dominated by small spatial distances and time intervals,
to avoid fast exponential oscillating, where the hadron size . If we set , then the hadron size . Therefore, at the condition of large hadron size, say , the local currents are questionable to interpolate the corresponding hadrons.
Now let us take it for granted that the exotic states have the size , just like the conventional mesons and baryons, could be interpolated by the local currents tacitly. For example, the charge radii of the , and are , and respectively from the Particle Data Group [97]. We extend the QCD sum rules on the conventional mesons and baryons to study the multiquark states directly with a simple replacement of the interpolating currents, and would like to come back to this subject again in Section 2.3.
A Lorentz invariant vacuum average can be expressed as
where the is a function. Then
The dominant contributions to the come from the region,
In the limit , we reach the light-cone , which is a necessary but not yet sufficient condition for the short-distance dominance, we have to constrain .
Now we focus on the quark-gluon degrees of freedom and calculate the correlation functions using Wilson’s operator product expansion to separate the physics of short and long distances,
where the are the Wilson’s coefficients encoding short-distance contributions, the are vacuum expectations of the local operators with dimension . The short-distance contributions at are encoded in the coefficients , the long-distance contributions at are absorbed into the vacuum condensates [385]. If , the Wilson coefficients depend only on short-distance dynamics, the vacuum condensates embody the long-distance effects. The lowest condensate is vacuum expectation of the unit operator associated with the perturbative contributions. The vacuum condensates with dimensions , 4, 5, 6, are quark condensate , gluon condensate , mixed condensate , four-quark condensate , , which parameterize the non-perturbative effects or soft gluons and quarks. We can consult Refs. [381, 385] for the basic techniques in performing the operator product expansion.
If there exist heavy quark lines and light-quark lines in the correlation functions , each heavy quark line emits a gluon and each light quark line contributes a quark-antiquark pair, we obtain a quark-gluon operator,
which is of dimension , we should perform the operator product expansion up to the vacuum condensates of dimension at least [60, 61, 81-83]. For example, and , we should calculate the vacuum condensates of dimension 18, see Fig.5. When , we obtain the conventional mesons, it is obvious that we should calculate the mixed condensate .
Then we obtain the Källen-Lehmann representation through dispersion relation at the quark-gluon degrees of freedom,
where the denotes the thresholds.
At the hadron degrees of freedom, we insert a complete set of intermediate hadronic states with the same quantum numbers as the currents into the correlation functions , and take account of the current-hadron couplings to obtain the analytical expressions, again we obtain the Källen−Lehmann representation through dispersion relation,
where
the subscript H denotes the hadron side.
According to the Quark−Hadron duality, we introduce the continuum threshold parameters , and match the QCD side with hadron side of the correlation functions ,
An important point is the choice of the continuum threshold , which is a physical parameter that should be determined from the hadronic spectrum. Then we perform the Borel transformation,
with to obtain the QCD sum rules,
where the is the Borel parameter. Some typical and useful examples of the Borel transformation are given in the Appendix. If only the ground state is taken, then
we obtain the QCD sum rules,
where the is the mass of the ground state of the conventional hadron or multiquark state, the is the pole residue.
It is obvious that the Borel transformation wipes out any eventual subtraction terms in the correlation functions and suppresses the continuum contributions exponentially, therefore, it improves the convergent behavior of the dispersion integral. Furthermore, it suppresses the higher-dimensional operators in the operator product expansion factorially, which contain inverse powers of the , see Eq. (481), thus justifies truncation of the operator product expansion and favors a good convergent behavior.
Finally, we eliminate the pole residue to obtain the QCD sum rules for the ground state mass,
In the QCD sum rules, we choose some phenomenological inputs which limit the accuracy of this method to be around [388].
2.3 Are QCD sum rules reliable to study multiquark states
Any color singlet four-quark and five-quark currents can be written as , where the and are color singlet clusters with , 2, 3, , for example,
for the five-quark current [402], where the components .
According to the pioneer works [403-405], in the coordinate space, we write the two-hadron-reducible contributions as
and the two-hadron-irreducible contributions as
At the phenomenological side of the correlation functions, we can write the two-hadron-reducible contributions as
where the couplings
and the could be estimated phenomenologically [403-406].
In the correlation functions for the color singlet-singlet type currents [407, 408], Lucha, Melikhov and Sazdjian assert that the Feynman diagrams can be divided into factorizable and nonfactorizable diagrams in the color space, the contributions at the order with , which are factorizable in the color space, are exactly canceled out by the meson-meson scattering states at the hadron side, the nonfactorizable diagrams, if have a Landau singularity, begin to make contributions to the tetraquark (molecular) states, the tetraquark (molecular) states begin to receive contributions at the order , see Fig.6.
In Ref. [409], we examine the assertion of Lucha, Melikhov and Sazdjian in details and use two examples for the currents and to illustrate that the Landau equation is of no use in the QCD sum rule for the tetraquark molecular states, where
Firstly, we cannot assert that the factorizable Feynman diagrams in color space are exactly canceled out by the meson-meson scattering states, because the meson-meson scattering state and tetraquark molecular state both have four valence quarks, which can be divided into two color-neutral clusters. We cannot distinguish which Feynman diagrams contribute to the meson-meson scattering state or tetraquark molecular state based on the two color-neutral clusters [409].
Secondly, the quarks and gluons are confined objects, they cannot be put on the mass-shell, it is questionable to assert that the Landau equation is applicable for the quark-gluon bound states [410].
If we insist on applying the Landau equation to study the Feynman diagrams, we should choose the pole masses rather than the masses to warrant mass poles. As the tetraquark (molecular) states begin to receive contributions at the order [407, 408], it is reasonable to take the pole masses as
See Refs. [411, 412] for the explicit expressions of the and . If the Landau equation is applicable for the tetraquark (molecular) states, it is certainly applicable for the traditional charmonium and bottomonium states. In the case of the -quark (-quark), the pole mass () from the Particle Data Group [411], the Landau singularity appears at the -channel (). It is unreliable that the masses of the charmonium (bottomonium) states lie below the threshold () for the and ( and ) [409].
Thirdly, the nonfactorizable Feynman diagrams which have the Landau singularities begin to appear at the order rather than at the order , and make contributions to the tetraquark molecular states, if the assertion (only nonfactorizable Feynman diagrams which have Landau singularities make contributions to the tetraquark molecular states) of Lucha, Melikhov and Sazdjian is right.
The nonfactorizable contributions appear at the order due to the operators , which come from the Feynman diagrams shown in Fig.7. If we insist on choosing the pole mass and applying the landau equation to study the diagrams, we obtain a sub-leading Landau singularity at the -channel . From the operators , we obtain the vacuum condensate , where the is absorbed into the vacuum condensate. The nonfactorizable Feynman diagrams appear at the order or , not at the order asserted in Refs. [407, 408].
In fact, for the triply-heavy dibaryon-type currents and ,
even in the lowest order Feynman diagrams, there are both connected and disconnected contributions in the color space [413], see Fig.8. From the first diagram in Fig.8, we can obtain both connected and disconnected Feynman diagrams, the connected contributions appear due to the quark-gluon operators [413], which are of the order and come from the Feynman diagrams shown in Fig.9.
Fourthly, the Landau equation serves as a kinematical equation in the momentum space, and does not depend on the factorizable and nonfactorizable properties of the Feynman diagrams in the color space.
In the leading order, the factorizable Feynman diagrams shown in Fig.10 can be divided into two color-neutral clusters, however, in the momentum space, they are nonfactorizable diagrams, the basic integrals are of the form,
If we choose the pole masses, there is a Landau singularity at , which is just a signal of a four-quark intermediate state. We cannot assert that it is a signal of a two-meson scattering state or a tetraquark molecular state, because the meson−meson scattering state and tetraquark molecular state both have four valence quarks, , , and , which form two color-neutral clusters.
Fifthly, only formal QCD sum rules for the tetraquark (molecular) states are obtained based on the assertion of Lucha, Melikhov and Sazdjian in Refs. [407, 408], no feasible QCD sum rules are obtained up to now.
Sixthly, we carry out the operator product expansion in the deep Euclidean space, , then obtain the physical spectral densities at the quark-gluon level through dispersion relation [376, 377],
where the denotes the correlation functions. The Landau singularities require that the squared momentum in the Feynman diagrams, see Fig.10 and Eq. (57), it is questionable to perform the operator product expansion.
Seventhly, we choose the local four-quark or five-quark currents, while the traditional mesons and baryons are spatial extended objects and have mean spatial sizes , for example, , , from the lattice QCD, where the subscripts E and M stand for the electric and magnetic radii, respectively [414], , , , , from the self-consistent chiral quark-soliton model [415], , from the light-front quark model [416], , from the screened potential model [88]. Local currents couple potentially to the compact objects having the average spatial sizes as that of the typical heavy mesons and baryons, not to the two-particle scattering states with average spatial size , which is too large to be interpolated by the local currents [417, 418].
Now we take a short digression to give a short notice. In the QCD sum rules, as we choose the local currents, the four-quark and five-quark states are all compact objects, they are -type, -type, 11-type or 88-type tetraquark states, and -type or 11-type pentaquark states, although we usually call the 11-type states as the molecular states.
Now, let us write down the correlation functions and for the two currents shown in Eq. (54),
If we assume the four-quark currents couple potentially both to the two-meson scattering states and molecular states, then we can express the and in terms of the heavy meson fields,
where we have taken the standard definitions for the decay constants of the traditional mesons and pole residues of the tetraquark molecular states,
, the , and have the , and , respectively, the are the polarization vectors, we introduce the superscripts to denote the parity.
It is straightforward to obtain the hadronic representation,
where
we project out the components and by introducing the operators and respectively,
where , , , , , , , , , and .
The traditional hidden-flavor mesons have the normal quantum numbers, , , , , , , , , . The component receives contributions with the exotic quantum numbers , the component receives contributions with the normal quantum numbers . We choose the component with the exotic quantum numbers and discard the component with the normal quantum numbers . Thereafter, we will neglect the superscript in the for simplicity.
According to the assertion of Lucha, Melikhov and Sazdjian [407, 408], all contributions of the order with are exactly canceled out by the two-meson scattering states, we set
at the hadron side [409]. Then let us take the quark−hadron duality below the continuum threshold and perform Borel transformation with respect to the variable to obtain the QCD sum rules:
the explicit expressions of the QCD spectral densities and are given in Ref. [409]. We introduce the parameter to measure the deviations from 1, if , we could get the conclusion tentatively that the two-meson scattering states can saturate the QCD sum rules. Then we differentiate Eqs. (72) and (73) with respect to , and obtain two additional QCD sum rules,
Thereafter, we will denote the QCD sum rules in Eqs. (74) and (75) as the QCDSR I, and the QCD sum rules in Eqs. (72) and (73) as the QCDSR II.
On the other hand, if the two-meson scattering states cannot saturate the QCD sum rules, we have to introduce the tetraquark molecular states to saturate the QCD sum rules,
then we differentiate Eq. (76) with respect to , and obtain two QCD sum rules for the masses of the tetraquark molecular states,
In Fig.11, we plot the values of the with variations of the Borel parameters with the continuum threshold parameters and for the and two-meson scattering states, respectively. From Fig.11, we can see explicitly that the values of the increase monotonically and quickly with the increase of the Borel parameters , no platform appears, which indicates that the QCD sum rules in Eqs. (72) and (73) obtained according to the assertion of Lucha, Melikhov and Sazdjian are unreasonable. Reasonable QCD sum rules lead to platforms flat enough or not flat enough, rather than no evidence of platforms, the two-meson scattering states cannot saturate the QCD sum rules.
We saturate the hadron side of the QCD sum rules with the tetraquark molecular states alone, and study the QCD sum rues shown in Eqs. (76) and (77). In Fig.12, we plot tetraquark molecule masses with variations of the Borel parameters . From the Fig.12, we observe that there appear Borel platforms in the Borel windows indeed, the relevant results are shown explicitly in Tab.1, we adopt the energy scale formula to choose the best energy scales of the QCD spectral densities. The tetraquark molecular states alone can satisfy the QCD sum rules [409]. We obtain the prediction and , which are consistent with the observed by the BESIII and Belle Collaborations [162, 163] and the X(4630) observed one-year latter by the LHCb Collaboration [115]. If we have taken account of the light-flavor breaking effects of the energy scale formula, the fit between the theoretical calculation and experimental measurement would be better for the X(4630).
The local currents do suppress but do not forbid the couplings between the four-quark currents and two-meson scattering states, as the overlaps of the wave-functions are very small [417], furthermore, the quantum field theory does not forbid the couplings between the four-quark currents and two-meson scattering states if they have the same quantum numbers. We study the contributions of the intermediate meson−meson scattering states , , , etc besides the tetraquark molecular state to the correlation function as an example,
where . We choose the bare quantities and to absorb the divergences in the self-energies , , , etc. The renormalized energies satisfy the relation , where the overlines above the self-energies denote that the divergent terms have been subtracted. As the tetraquark molecular state is unstable, the relation should be modified, , and . The renormalized self-energies contribute a finite imaginary part to modify the dispersion relation,
If we assign the to be the tetraquark molecular state with the [81, 82, 83], the physical width from the Particle Data Group [411].
We take account of the finite width effect by the simple replacement of the hadronic spectral density,
where
Then the hadron sides of the QCD sum rules in Eq. (76) and Eq. (77) undergo the following changes,
for the value . We can absorb the numerical factors and into the pole residue with the simple replacement safely, the intermediate meson-loops cannot affect the mass significantly, but affect the pole residue remarkably, which are consistent with the fact that we obtain the masses of the tetraquark molecular states from a fraction, see Eq. (77).
We obtain the conclusion confidently that it is reliable to study the multiquark states with the QCD sum rules, the contaminations from the two-particle scattering states play a tiny role [409].
2.4 Energy scale dependence of the QCD sum rules
In calculating the Feynman diagrams, we usually adopt the dimensional regularization to regularize the divergences, and resort to wave-function, quark-mass and current renormalizations to absorb the ultraviolet divergences, and resort to the vacuum condensate redefinitions to absorb the infrared divergences. Thus, the correlation functions are free of divergences. And we expect to calculate the at any energy scale at which perturbative calculations are feasible, and the physical quantities are independent on the specified energy scale . Roughly speaking, the correlation functions are independent on the energy scale approximately,
at least the bare are independent on the energy scale.
We write down the correlation functions for the hidden-charm (or hidden-bottom) four-quark currents, the most commonly chosen currents in studying the X, Y and Z states, in the Källen−Lehmann representation,
In fact, there are subtraction terms neglected at the right side of Eq. (85), which could be deleted after performing the Borel transformation. The should be independent on the energy scale we adopt to perform the operator product expansion, but which does not mean
due to the two features inherited from the QCD sum rules:
● Perturbative corrections are neglected, even in the QCD sum rules for the traditional mesons, we cannot take account of the radiative corrections up to arbitrary orders, for example, we only have calculated the radiative corrections up to the order for the pseudoscalar mesons up to now [419-421]. The higher dimensional vacuum condensates are factorized into lower dimensional ones based on the vacuum saturation, for example,
where , or , , the subscripts denote the Dirac spinor and color spaces, respectively, therefore the energy scale dependence of the higher dimensional vacuum condensates is modified.
● Truncations which are physical quantities determined by the experimental data set in, the correlations between the thresholds and continuum thresholds are unknown. Quark−hadron duality is just an assumption.
After performing the Borel transformation, we obtain the integrals
which are sensitive to the Q-quark mass , in other words, the energy scale . Variations of the energy scale can lead to changes of integral ranges of the variable besides the QCD spectral densities , therefore changes of the Borel windows and predicted hadron masses and pole residues. The strong fine-structure appears even in the tree-level,
where . Thus we have to deal with the energy scale dependence of the QCD sum rules.
Let us take a short digression and perform some phenomenological analysis. We can describe the heavy four-quark systems by a double-well potential with two light quarks lying in the two wells, respectively. In the heavy quark limit, the Q-quark serves as a static well potential and attracts with the light quark to form a heavy diquark in color antitriplet,
or attracts with the light antiquark to form a meson-like color-singlet cluster (meson-like color-octet cluster),
the -quark serves as another static well potential and attracts with the light antiquark to form a heavy antidiquark in color triplet,
or attracts with the light quark to form a heavy meson-like color-singlet cluster (meson-like color-octet-cluster),
Then
the two heavy quarks Q and stabilize the four-quark systems , just as in the case of the molecule in QED [57].
We can also describe the hidden-charm (or hidden-bottom) five-quark systems by a double-well potential. In the heavy quark limit, the Q-quark (-quark) serves as a static well potential, the diquark and quark lie in the two wells, respectively,
or
where the denotes the heavy triquark in the color triplet. Then
Now we can obtain the conclusion tentatively that the heavy tetraquark states are characterized by the effective heavy quark masses (or constituent quark masses) and the virtuality (or bound energy not as robust) [81, 422, 423, 424].
In summary, the QCD sum rules have three typical energy scales , , . It is natural to set the energy scales as,
which works very well for the tetraquark states and pentaquark states. It can improve the convergence of the operator product expansion remarkably and enhance the pole contributions remarkably.
We usually set the small and quark masses to be zero, and take account of the light-flavor breaking effects by introducing an effective -quark mass , thus we reach the modified energy scale formula,
to choose the suitable energy scales of the QCD spectral densities [61, 82, 83, 172, 425], where the , 1 and 2 denote the numbers of the -quarks, the and have universal values to be commonly used elsewhere.
We can rewrite the energy scale formula in Eq. (99) in the following form,
where the Constants have the values [426]. As we cannot obtain energy scale independent QCD sum rules, we conjecture that the predicted multiquark masses and the pertinent energy scales of the QCD spectral densities have a Regge-trajectory-like relation, see Eq. (101), where the Constants are free parameters and fitted by the QCD sum rules. Direct calculations have proven that the Constants have universal values and work well. We take account of the light-flavor breaking effects, and write down the modified energy scale formula [418],
In Ref. [60], we take the X(3872) and as the hidden-charm tetraquark with the and , respectively, and explore the energy scale dependence of the QCD sum rules for the exotic states for the first time. In Fig.13, we plot the mass of the with variations of the Borel parameter and energy scale for the continuum threshold parameter . From the figure, we can see clearly that the mass decreases monotonously with increase of the energy scale. The energy scale is the lowest energy scale to reproduce the experimental data [162, 163].
There are three schemes to choose the input parameters at the QCD side of the QCD sum rules:
Scheme I. We take the energy scale formula and its modifications, see Eqs. (99) and (100), to choose the energy scales of the QCD spectral densities in a consistent way.
Scheme II. We take the (modified-minimal-subtraction) masses for the heavy quarks , and take the light quark masses and vacuum condensates at the energy scale .
Scheme III. We take all the input parameters at the energy scale .
The Scheme II is adopted in most QCD sum rules [46, 47, 58, 180, 427-461] ([462-474]), where the masses are usually smaller than (or equal to) the values from the Particle Data Group (with much smaller pole contributions) [97].
The Scheme III was adopted in early works of Wang and his collaborators in 2009−2011 [475-486], where many elegant four-quark currents were constructed originally.
In Ref. [487], we study the pentaquark molecular states in the three schemes in details and examine their advantages and shortcomings, in Schemes I and III, we truncate the operator product expansion up to the vacuum condensates of , while in Scheme II, we truncate the operator product expansion up to the vacuum condensates of , which is commonly adopted in this case.
We write down the correlation functions , and firstly,
where the currents , , , ,
We separate the contributions of the molecular states with the , and unambiguously, then we introduce the weight functions and to obtain the QCD sum rules at the hadron side,
where the are the continuum threshold parameters and the are the Borel parameters, the and with the , and are hadronic spectral densities, the are the pole residues.
We perform the operator product expansion to obtain the analytical QCD spectral densities and through dispersion relation, then we take the quark-hadron duality below the continuum thresholds and introduce the weight functions and to obtain the QCD sum rules:
For the technical details, one can consult Ref. [487] or Section 5.1.
We differentiate Eq. (106) with respect to , then eliminate the pole residues to obtain the molecule masses,
where and .
We show the Borel windows , continuum threshold parameters , energy scales of the QCD spectral densities and pole contributions of the ground states explicitly in Tab.2. In the Scheme III, the pole contributions are less than , which are too small, and the Scheme III could be abandoned. On the other hand, the convergent behaviors of the operator product expansion have relation Scheme I > Scheme II > Scheme III.
At last, we take account of all uncertainties of the input parameters, and obtain the masses and pole residues of the molecular states, which are shown explicitly in Tab.3 and Fig.14 and Fig.15.
In Fig.14 and Fig.15, we plot the masses at much larger ranges of the Borel parameters than the Borel windows. The predicted masses in the Scheme I decrease monotonously and quickly with increase of the Borel parameters at the region , then reach small platforms and increase slowly with increase of the Borel parameters. The predicted masses in the Scheme II increase monotonously and quickly with increase of the Borel parameters at the region , then increase slowly and steadily with increase of the Borel parameters. It is obvious that the flatness of the platforms have relation Scheme I > Scheme II.
Both the predictions in the Schemes I and II support assigning the as the molecular state with the , assigning the as the molecular state with the , assigning the as the molecular state with the or the molecular state with the . However, if we take account of the vacuum condensates of the dimensions of and 13 in the Scheme II, we would obtain a mass about larger than the corresponding one given in Tab.3, so we prefer the Scheme I [487].
3 3 type tetraquark states
3.1 Hidden-heavy tetraquark states
The scattering amplitude for one-gluon exchange is proportional to,
the negative (positive) sign in front of the antisymmetric antitriplet (symmetric sextet 6) indicates the interaction is attractive (repulsive), which favors (disfavors) formation of the diquarks in color (6). We usually construct the type color-singlet four-quark currents to interpolate the tetraquark states,
where the and are the quarks, the and are the Dirac -matrixes. The color factor has the relation,
With the simple replacement,
we obtain the corresponding type currents ,
If the type currents satisfy the Fermi−Dirac statistics, the quantum field theory does not forbid their existence. In fact, in the potential quark models, we usually take both the and diquark configurations, see Section 2.1.5. The type currents are also widely used in literatures [34, 42, 121, 134, 185, 437, 488-493].
Or we construct the 88 type currents directly to interpolate the exotic states [455, 459, 494-497], although the one-gluon exchange induced interaction is repulsive in this channel, see Section 2.1.5. In Ref. [496], we take the as the 88 type axial-vector molecule-like state, and construct the 88 type axial-vector current to study its mass and width with the QCD sum rules, the numerical results support assigning the as the 88 type molecule-like state with the . Furthermore, we discuss the possible assignments of the , and Z(4430) as the type tetraquark states with the .
3.1.1 Tetraquark states with positive parity
The one-gluon-exchange induced attractive interactions favor formation of the diquarks in the color antitriplet, flavor antitriplet and spin singlet [311], while the most favored configurations are the scalar and axialvector diquark states [498-504]. The QCD sum rules indicate that the heavy-light scalar and axialvector diquark states have almost degenerate masses [498, 499, 500], while the masses of the light axialvector diquark states lie about above that of the light scalar diquark states [501-504], if they have the same valence quarks.
The diquarks in the have five structures in the Dirac spinor space, where , C, , and for the scalar, pseudoscalar, vector, axialvector and tensor diquarks, respectively. In the non-relativistic quark model, a P-wave changes the parity by contributing a factor with the angular momentum . The and diquark states have the spin-parity and , respectively, the corresponding C and diquark states have the spin-parity and , respectively, the effects of the P-waves are embodied in the underlined in the and . The tensor diquark states have both the and components, we project out the and components explicitly, and introduce the symbols and to represent them respectively. We would like to give an example on the heavy-light tensor diquarks to illustrate how to perform the projection.
Under parity transformation , the tensor diquarks have the properties,
where the four vectors and . We introduce the four vector and project out the and components explicitly [505],
where , , , . We can also introduce the P-wave explicitly in the and diquarks and obtain the vector diquarks and tensor diquarks , where the derivative embodies the P-wave effects.
We can also adopt the covariant derivative with the simple replacement , then the four-quark currents are gauge covariant, however, gluonic components are introduced, we have to deal with both valence quarks and gluons.
Now let us construct the -type four-quark currents to interpolate the hidden-charm tetraquark states with the , , and ,
where the subscripts ± denote the positive and negative charge conjugation, respectively, the superscripts or subscripts P, S, A() and V() denote the pseudoscalar, scalar, axialvector and vector diquark and antidiquark operators, respectively [61]. With the simple replacements,
where , , we obtain the corresponding currents for the states [172]. Again, with the simple replacements,
we obtain the corresponding currents for the states [425, 506, 507].
We introduce the symbols,
for simplicity.
Under parity transformation , the currents have the properties,
where and , and we have neglected other superscripts and subscripts.
The currents , and have the symbolic quark constituent with the isospin and , other currents in the isospin multiplets can be constructed analogously, for example, we write down the isospin singlet current for the directly,
In the isospin limit, the currents with the symbolic quark constituents , , , couple potentially to the hidden-charm tetraquark states with degenerated masses, the currents with the isospins and 0 lead to the same QCD sum rules. Thereafter, we will denote the states as the isospin triplet, and the X states as the isospin singlet,
Accordingly, the states with the symbolic quark constituents and isospin have degenerated masses.
The currents with the symbolic quark constituents and have definite charge conjugation. We would like to assume that the type tetraquark states have the same charge conjugation as their charge-neutral cousins.
Under charge conjugation transformation , the currents , and have the properties,
where we have neglected other superscripts and subscripts.
Now we write down the correlation functions , and ,
At the hadron side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the currents , and into the correlation functions , and to obtain the hadronic representation [376, 377], and isolate the ground state hidden-charm tetraquark contributions,
where . We add the superscripts ± in the , and to denote the positive and negative charge conjugation, respectively, and add the superscripts (subscripts) ± in the (the and components) to denote the positive and negative parity, respectively. And the and have the same tensor structures. The pole residues are defined by
where , the and are the polarization vectors. We choose the components and to study the hidden-charm tetraquark states.
In Tab.4, we present the quark constituents and corresponding currents explicitly.
At the QCD side, we carry out the operator product expansion for the correlation functions , and . For example, we contract the quark fields in the with the Wick’s theorem, and obtain the results,
where the full quark propagators, ,
and ,
with , [60, 381, 508], and the correspond to the currents and , respectively. In Eq. (129), we retain the terms and come from the Fierz transformation of the to absorb the gluons emitted from the other quark lines to form and to extract the mixed condensate and four-quark condensate , respectively [60], where , or . The condensate comes from the , and rather than comes from the radiative corrections to the .
In fact, the method of adopting the full propagators in Eq. (128) implies vacuum saturation implicitly, the factorization of the higher dimensional vacuum condensates, for example, see Eq. (87), is already performed. We calculate the higher dimensional vacuum condensates using the formula rigorously.
Let us see Eq. (128) again, there are two Q-quark propagators and two -quark propagators, if each Q-quark line emits a gluon and each -quark line contributes a quark−antiquark pair, we obtain an operator (or or ), which is of dimension 10, see the Feynman diagram in Fig.7 for example. We should take account of the vacuum condensates at least up to dimension 10. The higher dimensional vacuum condensates are associated with the , or , which manifest themselves at small Borel parameter and play an important role in determining the Borel windows, where they play a minor important role. Therefore, we take account of the vacuum condensates , , , , , , , and , which are vacuum expectations of the quark-gluon operators of the order with . We truncate the operator product expansion in such a way consistently and rigorously. The condensates , , have the dimensions 6, 8, 9 respectively, but they are vacuum expectations of the quark-gluon operators of the order , , respectively, and discarded. Furthermore, direct calculations indicate such contributions are tiny indeed [509].
We accomplish the integrals in Eqs. (128)−(130) sequentially by simply setting with to regularize the divergences, such a simple scheme misses many subtraction terms, which make no contribution in obtaining the imaginary parts through . Finally, we obtain the QCD spectral densities through dispersion relation. Now we take a short digression to give an example,
with . We set and , then we obtain,
In the case and , we obtain,
and
where , , and .
Then we match the hadron side with the QCD side of the components and of the correlation functions , and below the continuum thresholds and perform Borel transformation with respect to to obtain the QCD sum rules:
We derive Eq. (135) with respect to , and obtain the QCD sum rules for the masses of the hidden-charm tetraquark states or X,
Now let us begin to perform numerical analysis, and write down the energy-scale dependence of the input parameters,
where , , , , , and for the flavors , 4 and 3, respectively [510, 511]. Because the -quark is concerned, we choose the flavor number to be .
At the beginning points, we adopt the commonly-used values , , , , , at the typical energy scale [376, 377, 381, 385], and adopt the (modified-minimal-subtraction) masses and from the Particle Data Group [510].
We apply the modified energy scale formula to choose the suitable energy scales of the QCD spectral densities [61, 172, 425], where the and are the effective and -quark masses respectively, and have universal values to be commonly used elsewhere. We adopt the updated value [512], and take the collective light-flavor -breaking effects into account by introducing an effective -quark mass () for the S-wave (P-wave) tetraquark states [82, 83, 171, 172, 418, 425, 513-515].
The continuum threshold parameters are not completely free parameters, and cannot be determined by the QCD sum rules themselves completely. We often consult the experimental data in choosing the continuum threshold parameters. The can be assigned to be the first radial excitation of the according to the analogous decays,
and the analogous mass gaps and from the Particle Data Group [56, 181, 182, 411].
We tentatively choose the continuum threshold parameters as and vary the continuum threshold parameters and Borel parameters to satisfy the following four criteria:
● Pole dominance at the hadron side;
● Convergence of the operator product expansion;
● Appearance of the Borel platforms;
● Fulfillment of the modified energy scale formula,
via trial and error.
Thereafter, such criteria are adopted for all the hidden-charm (hidden-bottom) tetraquark (molecular) states, hidden-charm (hidden-bottom) pentaquark (molecular) states, doubly-charm (doubly-bottom) tetraquark (molecular) states, doubly-charm (doubly-bottom) pentaquark (molecular) states, etc.
The pole dominance at the hadron side and convergence of the operator product expansion at the QCD side are two basic criteria, we should satisfy them to obtain reliable QCD sum rules. In the QCD sum rules for the hidden-charm tetraquark and pentaquark states, the largest power of the energy variable in the QCD spectral densities and , respectively, which make the integrals,
converge more slowly compared to the traditional hadrons, it is very difficult to satisfy the two basic criteria at the same time. We define the pole contributions (PCs) by
while we define the contributions of the vacuum condensates of dimension by
or
sometimes we would like to use the notation in stead of . Compared to the criterion in Eq. (142), the criterion in Eq. (141) is strong and leads to larger Borel parameter . If we only study the ground state contributions, Eq. (141) is preferred.
After trial and error, we obtain the Borel windows, continuum threshold parameters, energy scales of the QCD spectral densities, pole contributions, and contributions of the vacuum condensates of dimension 10, which are shown explicitly in Tab.5 and Tab.6.
At the hadron side, the pole contributions are about (40−60)%, the pole dominance is well satisfied. It is more easy to obtain the pole contributions (40−60)% with the help of the modified energy scale formula. In Tab.6, we provide two sets of data for the scalar tetraquark states, one set data is based on the continuum threshold parameters about with the central values (this criterion is also adopted for the tetraquark states with the , and ), the other set data is based on the continuum threshold parameters about with the central values , which will be characterized by the additional symbol in Tab.6, Tab.8 and Tab.10.
At the QCD side, the contributions of the vacuum condensates of dimension 10 are or except for for the tetraquark state with the , the convergent behavior of the operator product expansion is very good.
In Fig.16, we plot the contributions of the vacuum condensates with variation of the Borel parameter for the current . From the figure, we can see clearly that the main contributions come from the , , , and , while the largest contribution comes from the . Therefore, if we would like to calculate the radiative corrections, we should calculate the radiative corrections to the , not just to the . At the present time, only the radiative corrections to the are partially calculated [128, 452, 456, 461, 516-519].
We take account of all the uncertainties of the input parameters and obtain the masses and pole residues of the scalar, axialvector and tensor hidden-charm tetraquark states, which are shown explicitly in Tab.7 and Tab.8. From Tab.5–Tab.8, we could see clearly that the modified energy scale formula is well satisfied. In Fig.17, we plot the masses of the and tetraquark states having the spin-parity with variations of the Borel parameters at much larger ranges than the Borel widows as an example, there appear platforms in the Borel windows indeed.
In Tab.9 and Tab.10, we present the possible assignments of the ground state hidden-charm tetraquark states, and revisit the assignments based on the tetraquark scenario, there are rooms for the X(3860), X(3872), X(3915), X(3960), X(4140), X(4274), X(4500), X(4685), X(4700), , , , , and .
It is not difficult to reproduce the mass of the X(3872) in the scenario of tetraquark state [58, 60, 61], it is a great challenge to reproduce the tiny width from the Particle Data Group consistently [97]. In Ref. [62], we study the strong decays , , , and via the QCD sum rules according to rigorous quark-hadron duality, and obtain the total decay width about , it is the first time to reproduce the tiny width of the X(3872) via the QCD sum rules. The thresholds of the and are and respectively, which are larger than the value of the mass of the X(3872) from the Particle Data Group [97] and also lead to the possibility that it may be a or molecular state [84], the decays and take place through the virtual and mesons, respectively, we introduce form-factors to parameterize the off-shell-ness, however, the arbitrariness in choosing the form-factors would impair the predictive ability.
As far as the is concerned, it is not difficult to reproduce its mass and width via the QCD sum rules [60, 61, 520]. It is the benchmark for the type hidden-charm tetraquark states in the simple diquark models [56, 301, 312, 313, 521].
In Tab.9, there are enough rooms to accommodate the and in the scenario of tetraquark states, as the central values of the predicted tetraquark masses happen to coincide with the experimental data from the LHCb Collaboration [109]. We should bear in mind that the predictions were made before the experimental observation, therefore the calculations are robust enough [61]. While in the traditional charmonium scenario, the and have the masses and , respectively [108], there exist discrepancies about .
The predictions and for the 1S and tetraquark states with the respectively support assigning the X(4140) and X(4685) as the 1S and 2S or tetraquark states respectively. In Refs. [507, 522], we obtain the prediction for the 1S tetraquark state with the in the old scheme, which supports assigning the X(4140) as the tetraquark state.
The prediction for the 1S tetraquark state supports assigning the X(4274) as the tetraquark state with the . The calculations in Refs. [507, 522] are updated, as where the light-flavor mass-breaking effects in the energy scale formula are not taken into account.
In Ref. [523], the predictions support assigning the X(4274) as the tetraquark state with a relative P-wave between the diquark and antidiquark pairs, such an assignment does not suffer from shortcomings in sense of treating scheme. The X(4274) maybe have two important Fock components.
The X(4140) and X(4274) are also play an important role in establishing the hidden-charm tetraquark states, especially the X(4140). There have been several possible assignments of the X(4140), such as the tetraquark state [315, 316, 332, 333, 425, 507, 522, 524-527], hybrid state [477, 478, 528] or rescattering effect [529].
In Tab.10, we assign the X(3960) and X(4500) as the 1S and 2S tetraquark states with the , the X(4700) as the 1S tetraquark state with the ; or assign the X(4700) as the 2S or tetraquark state with the . In Ref. [530], we assign the X(4500) as the tetraquark state with the , where the denotes the diquark with an explicit P-wave. In Ref. [527], Chen et al. assigned the X(4500) and X(4700) as the D-wave tetraquark states with the quark content and : the X(4500) consists of one D-wave diquark and one S-wave antidiquark, with the antisymmetric color structure ; the X(4700) consists of similar diquarks but with the symmetric color structure .
However, there is no room for the X(4350), we should introduce mixing effect to interpret its nature [484].
In fact, the predictions of the QCD sum rules have arbitrariness, and depend on the interpolating currents, truncations of the operator product expansion, convergent behavior, pole contributions, input parameters, etc. Therefore, the predictions maybe quite different, we have to perform systematic investigations with uniform criterion to outcome the shortcomings.
We suggest to study the two-body strong decays to diagnose those hidden-charm tetraquark states [61, 425], for example,
The LHCb Collaboration observed the and in the mass spectrum [109], see the two-body strong decays of the shown in Eq. (144).
As the tetraquark states are concerned, we present the predictions based on the direct calculations plus light-flavor -breaking effects in Tab.11 [171, 172]. In Ref. [171], we tentatively assign the as the -type hidden-charm tetraquark state with the according to the analogous properties of the and , and study the -type tetraquark states without strange, with strange and with hidden-strange via the QCD sum rules in a consistent way. Then we study the hadronic coupling constants of the tetraquark states without strange and with strange via the QCD sum rules based on rigorous quark-hadron duality, and obtain the total decay widths
,
and suggest to search for the state in the mass spectrum of the , , , , . Slightly later, the BESIII Collaboration observed an evidence for the [170].
With the simple replacement,
we obtain the corresponding QCD sum rules for the hidden-bottom tetraquark states. And we would like to present the results from the QCD sum rules in Ref. [531] directly, see Tab.12 and Tab.13.
In calculations, we use the energy scale formula with the effective -quark mass to determine the ideal energy scales of the QCD spectral densities [532], and choose the continuum threshold parameters as a constraint to extract the masses and pole residues from the QCD sum rules. The predicted masses and for the tetraquark states support assigning the and to be the hidden-bottom tetraquark states with the , more theoretical and experimental works are still needed to assign the and unambiguously according to the partial decay widths.
In the diquark-model, the and are also assigned as the -type and -type hidden-bottom tetraquark states, respectively [533, 534]. And we could extend this subsection to study the -like tetraquark states [458, 535, 536, 537].
3.1.2 Tetraquark states with the first radial excitations
In this sub-section, we would like to study the first radial excitations of the hidden-charm tetraquark states and make possible assignments of the exotic states. In Ref. [538], Maior de Sousa and Rodrigues da Silva suggested a theoretical scheme to study the double-pole QCD sum rules, and study the quarkonia , , and , the predicted hadron masses are not good enough, they adopt the experimental masses except for the to study the decay constants. In Ref. [182], we extend this scheme to study the and as the ground state and its first radial excitation, respectively, and observed that it is impossible to reproduce the experimental masses at the same energy scale, just as in the case of the , and [538], we should resort to the energy scale formula, see Eq. (99), to choose the optimal energy scales independently.
We adopt the correlation functions and [184], see Eq. (125), and set , , , and , , and , see Eq. (116).
At the hadron side, if we only take account of the ground state hidden-charm tetraquark states, we obtain the QCD sum rules:
Thereafter, we will refer the QCD sum rules in Eq. (149) and Eq. (150) as QCDSR I.
If we take into account the contributions of the first radially excited tetraquark states in the hadronic representation, we can obtain the QCD sum rules,
where the is continuum threshold parameter, then we introduce the notations , , and resort to the subscripts 1 and 2 to represent the ground state and first radially excited state respectively for simplicity. We rewrite the QCD sum rules as
here we introduce the subscript to represent the QCD representation. We derive the QCD sum rules in Eq. (152) in regard to to obtain
From Eqs. (152) and (153), we obtain the QCD sum rules,
where the indexes . Let us derive the QCD sum rules in Eq. (154) in regard to to obtain
The squared masses satisfy the equation,
where
the indexes and . Finally we solve above equation analytically to obtain two solutions [182, 538],
Thereafter, we will denote the QCD sum rules in Eq. (151) and Eqs. (158) and (159) as QCDSR II. In calculations, we observe that if we specify the energy scales of the spectral densities in the QCD representation, only one solution satisfies the energy scale formula in the QCDSR II, we have to abandon the other solution, i.e., the mass (). It is the unique feature of our works [182, 184], which is in contrast to Ref. [538].
The Okubo−Zweig−Iizuka supper-allowed decays,
are expected to take place easily. The energy gaps maybe have the relations and . The charmonium masses are , and from the Particle Data Group [411], , , we can choose the continuum threshold parameters to be and tentatively and vary the continuum threshold parameters and Borel parameters to satisfy the four criteria in Section 3.1.1.
After trial and error, we obtain the continuum threshold parameters, Borel windows, best energy scales, and contributions of the ground states for the QCDSR I, see Tab.14. In Tab.14, we write the continuum threshold parameters as rather than as for the and tetraquark states to remain the same form as in Ref. [512]. Again we obtain the parameters for the QCDSR II using trial and error, see Tab.15. We take the energy scale formula to obtain the ideal energy scales of the spectral densities [182, 184].
From Tab.14 and Tab.15, we can see clearly that the contributions of the single-pole terms are about (40−60)% for the QCDSR I, the contributions of the two-pole terms are about (70−80)% for the QCDSR II, which satisfy the pole dominance very well. In the QCDSR II, the contributions of the ground states are about (30−45)%, which are much less than the ground state contributions in the QCDSR I, we prefer the QCDSR I for the ground states.
We take account of all the uncertainties of the parameters, and obtain the masses and pole residues, which are shown in Tab.16 and Tab.17. From those Tables, we can see that the ground state tetraquark masses from the QCDSR I and the radially excited tetraquark masses from the QCDSR II satisfy the energy scale formula , where the updated value of the effective -quark mass is adopted [512]. In Tab.17, we also present the central values of the ground state masses and pole residues extracted from the QCDSR II at the same energy scales. If we examine Tab.17, we would observe the ground state masses cannot satisfy the energy scale formula, and should be discarded.
For example, in Fig.18, we plot the ground state masses from the QCDSR I and the first radially excited tetraquark masses from the QCDSR II with variations of the Borel parameters for the and states. From the figure, we observe that there indeed appear very flat platforms in the Borel windows.
We present the possible assignments explicitly in Tab.9. The predicted mass for the 2S tetraquark state exhibits very good agreement with the experimental data from the LHCb Collaboration [179], which is in favor of assigning the as the first radial excitation of the tetraquark state with the [182, 184].
The predicted mass for the 2S tetraquark state and for the 2S tetraquark state both exhibit very good agreement with the experimental data from the LHCb Collaboration [183], and the predicted mass for the 1S tetraquark state is also compatible with the experimental data. In summary, there are three tetraquark state candidates with the for the .
The scheme of the QCDSR II and its modification have been applied extensively to study the 2S tetraquark states, such as the [182, 184, 185, 464, 539], [184, 185], X(4500) (as the 2S state of the X(3915)) [540, 541], X(4685) (as the 2S state of the X(4140)) [522], and and [542].
3.1.3 Tetraquark states with negative parity
In Section 3.1.1, we study the hidden-heavy tetraquark states with the positive parity, in this subsection, we would like to study the hidden-heavy tetraquark states with the negative parity. Again we choose the diquark operators without explicit P-waves as the elementary building blocks, for detailed discussions about the diquark operators, see the beginning of Section 3.1.1. Compared to the vector tetraquark states, it is easy to analyze the pseudoscalar tetraquark states, and we would like to present the results directly.
Again, let us adopt the correlation functions and defined in Eq. (125), and write down the currents
the subscripts ± denote the positive and negative charge conjugation, respectively, the superscripts P, S, A() and V() denote the pseudoscalar, scalar, axialvector and vector diquark and antidiquark operators, respectively [424, 505, 512, 543, 544]. With the simple replacement,
we reach the corresponding currents for the states [424, 544, 545].
Under parity transformation , the current operators and have the properties,
according to the properties of the diquark operators,
where , , , , and . For , , , , , , we obtain , , , , , . And we rewrite Eq. (167) in more explicit form,
where , , 2, 3.
Under charge conjugation transformation , the currents and have the properties,
The currents and have the symbolic quark structure with the isospin and . In the isospin limit, the currents with the symbolic quark structures
couple potentially to the hidden-charm tetraquark states with degenerated masses, and the currents with the isospin and 0 lead to the same QCD sum rules. Only the currents with the symbolic quark structures and have definite charge conjugation, again we assume that the tetraquark states have the same charge conjugation as their neutral partners.
According to the current-hadron duality, we obtain the hadronic representation,
where the pole residues are defined by
, , the are the polarization vectors, we add the superscripts/subscripts ± to denote the positive and negative parity, respectively, and add the wide-tilde and superscript in the to denote the currents and , respectively. We choose the components and to explore the negative parity hidden-charm tetraquark states [543, 545].
At the QCD side, we calculate the vacuum condensates up to dimension 10 and take into account the vacuum condensates , , , , , , , and , which are vacuum expectations of the quark-gluon operators of the order with [543, 545].
We match the hadron side with the QCD side of the components and below the continuum thresholds with the help of the spectral representation, and perform Borel transformation with respect to to obtain the QCD sum rules:
the explicit expressions of the QCD spectral densities are neglected for simplicity.
We derive Eq. (175) with respect to , and obtain the QCD sum rules for the masses of the vector hidden-charm tetraquark states ,
With a simple replacement , we obtain the corresponding QCD sum rules for the hidden-bottom tetraquark states directly.
We perform the same procedure as in the previous subsections, and obtain the Borel windows, continuum threshold parameters, suitable energy scales of the QCD spectral densities and pole contributions, which are shown explicitly in Tab.18 and Tab.19. From the Tables, we can see clearly that the pole contributions are about (40−60)%, the central values are larger than 50%, the pole dominance is well satisfied. In calculations, we observe that the main contributions come from the perturbative terms, the higher dimensional condensates play a minor important role and the contributions , the operator product expansion converges very good.
We take account of all the uncertainties of the relevant parameters and obtain the masses and pole residues of the hidden-charm (and hidden-strange) tetraquark states with the and , which are shown explicitly in Tab.20 and Tab.21.
In Tab.22 and Tab.23, we present the possible assignments of the hidden-charm tetraquark states with the and obtained in Refs. [543, 545]. Considering the large uncertainties, it is possible to assign the X(4630) as the state with the , which has a mass , see Tab.23. In Ref. [409], we prove that it is feasible and reliable to study the multiquark states in the framework of the QCD sum rules, and obtain the prediction for the mass of the molecular state with the exotic quantum numbers , , which was obtained before the LHCb data and is compatible with the LHCb data. The X(4630) maybe have two important Fock components.
After Ref. [543] was published, the Y(4500) was observed by the BESIII Collaboration [155, 156, 159]. At the energy about , we obtain three hidden-charm tetraquark states with the , the , and tetraquark states have the masses , and , respectively [543]. In Ref. [546], we study the two-body strong decays systematically, i.e., we obtain thirty QCD sum rules for the hadronic coupling constants based on rigorous quark−hadron duality, then obtain the partial decay widths, therefore the total widths approximately, which are compatible with the experimental data of the Y(4500) from the BESIII Collaboration, see Section 7.1 for details. In Ref. [547], we take the Y(4500) as the tetraquark state, and study the three-body strong decay with the light-cone QCD sum rules, see Section 7.2 for details.
If only the mass is concerned, the Y(4660) can be assigned as the , , or tetraquark state, see Tab.22-Tab.23. In other words, the Y(4660) maybe have several important Fock components, we have to study the strong decays in details to diagnose its nature. For example, if we assign the Y(4660) as the or state, then the strong decays , , , , , , , and are Okubo-Zweig-Iizuka super-allowed, considering the intermediate process . Up to now, only the decays , , and have been observed [97], which cannot exclude the assignments or , as the decay can take place through the re-scattering mechanism. We can investigate or search for the neutral tetraquark states with the and through the two-body or three-body strong decays,
From Tab.22 and Tab.23, we observe that there is no room for the Y(4260/4220). In Ref. [544], we choose the currents,
to interpolate the Y states, and fit the correlation functions,
by taking the as free parameters. We obtain the best values, which are shown in Tab.24, at the pertinent energy scales for the current and for the current , the values of the pole residue are very small. Without introducing explicit P-waves, we cannot produce the experimental mass of the Y(4260/4220) in the scenario of tetraquark state.
The Y(4660) has been studied extensively via the QCD sum rules [424, 431, 440, 444, 485, 512, 543-545, 548], however, no definite conclusion can be obtained, more works are still needed to decipher its structure.
Now let us turn to the pseudoscalar tetraquark states and write down the local currents,
with , , , , the superscripts ± symbolize the positive and negative charge-conjugation, respectively, the subscripts P, S, V, A and T stand for the pseudoscalar, scalar, vector, axialvector and tensor diquark operators, respectively [515].
Under parity transformation , the have the property,
Under charge-conjugation transformation , the have the property,
and we can prove that the current through performing the Fierz-transformation. Again, we take the isospin limit , the four-quark currents with the symbolic quark structures,
couple potentially to the pseudoscalar tetraquark states with degenerated masses. And the four-quark currents with the symbolic quark structures,
also couple potentially to the pseudoscalar tetraquark states with degenerated masses according to the isospin symmetry. And we obtain the QCD sum rules routinely.
In Tab.25, we present the Borel windows, continuum threshold parameters, energy scales of the QCD spectral densities and pole contributions. From the Table, we can see distinctly that the pole contributions are about (40−60)% at the hadron side, while the central values are larger than 50%, the pole dominance criterion is satisfied very good. On the other hand, the higher vacuum condensates play a minor important role, the operator product expansion converges very well.
We take all the uncertainties of the parameters into account and acquire the masses and pole residues, see Tab.26. From Tab.25 and Tab.26, we can see distinctly that the modified energy scale formula with , 1 or 2 is satisfied, where we subtract the small -quark mass approximately to account for the small light-flavor mass-breaking effects, which is slightly different from Eq. (100).
As can be seen distinctly from Tab.26 that the lowest mass of the pseudoscalar hidden-charm tetraquark state with the symbolic quark constituents is about , which is much larger than the value from the LHCb Collaboration [179]. In 2014, the LHCb Collaboration provided the first independent confirmation of the existence of the in the mass spectrum and established its spin-parity to be [179]. Furthermore, the LHCb Collaboration observed a weak evidence for an additional resonance, the , in the mass spectrum with the preferred spin-parity and the Breit−Wigner mass and width , respectively with large uncertainties [179]. If the is confirmed by further experiments in the future, it is an excellent candidate for the hidden-charm tetraquark state with the , and we should revisit the QCD sum rules for the discrepancy.
In Ref. [437], Chen and Zhu studied the hidden-charm tetraquark states with the symbolic quark constituents with the QCD sum rules, and obtained the ground state masses for the tetraquark states with the , the masses , , for the tetraquark states with the . The present predictions are consistent with their calculations, again, we should bear in mind that their interpolating currents and schemes in treating the operator product expansion and input parameters at the QCD side differ from the present work remarkably. Any current with the same quantum numbers and same quark structure as a Fock state in a hadron couples potentially to this hadron, so we can construct several currents to interpolate a hadron, or construct a current to interpolate several hadrons.
From Tab.26, we can see explicitly that the central values of the masses of the tetraquark states with the symbolic quark constituents , , are about , and , respectively, the central values of the masses of the tetraquark states with the symbolic quark constituents , and are about , and , respectively. We obtain the conclusion tentatively that the currents , and ( and ) couple potentially to three (two) different pseudoscalar tetraquark states with almost degenerated masses, or to one pseudoscalar tetraquark state with three (two) different Fock components. As the currents with the same quantum numbers couple potentially to the pseudoscalar tetraquark states with almost degenerated masses, the mixing effects cannot improve the predictions remarkably if only the tetraquark masses are concerned. All in all, we obtain reasonable predictions for the masses of the pseudoscalar tetraquark states without strange, with strange and with hidden-strange, the central values are about , and , respectively.
The following two-body strong decays of the pseudoscalar hidden-charm tetraquark states,
can take place through the Okubo−Zweig−Iizuka super-allowed fall-apart mechanism, we suggest to search for the pseudoscalar hidden-charm tetraquark states in those channels.
The QCD sum rules obtained in this sub-section can be extended directly to study the tetraquark states in the bottom sector with the simple replacements and .
3.1.4 Tetraquark states with an explicit P-wave
In the type-II diquark model [56], Maiani et al assign the Y(4008), Y(4260), Y(4290/4220) and Y(4630) as four tetraquark states with the based on the effective spin-spin and spin-orbit interactions, see Eq. (24). In Ref. [318], A. Ali et al incorporate the dominant spin-spin, spin-orbit and tensor interactions, see Eq. (25), and observe that the preferred assignments of the tetraquark states with the are the Y(4220), Y(4330), Y(4390), Y(4660). In the diquark model, the quantum numbers of the Y states are shown explicitly in Tab.27, where the L is the angular momentum between the diquark and antidiquark, , , and denotes the explicit P-wave.
We take the isospin limit, and construct the interpolating currents according to the quantum numbers shown in Tab.27,
where embodies the explicit P-wave.
Under charge conjugation transformation , the currents and have the property,
the currents have definite charge conjugation.
We choose the currents , , and and resort to the correlation functions in Eq. (125) to study the vector tetraquark states using the modified energy scale formula,
to determine the ideal energy scales of the QCD spectral densities, and reexamine the possible assignments of the Y states [549, 550]. The numerical results are shown explicitly in Tab.28 and Tab.29.
The predicted mass of the tetraquark state is in excellent agreement with the experimental data from the BESIII Collaboration [154], or from the Particle Data Group [411], which supports assigning the Y(4260/4220) as the type vector tetraquark state. We obtain the vector hidden-charm tetraquark state with the lowest mass up to now.
There have been other possible assignments for the Y(4260) states, such as the hybrid states [303, 551, 552, 553, 554], molecular state [227, 229, 280, 555, 556], baryonium states [557, 558], hadro-charmonium state [559], and interference effect [560, 561].
From Tab.22 and Tab.29, we can see explicitly that there are no rooms for the Y(4008) and Y(4750) in the hidden-charm tetraquark scenario. If we assign the Y(4220/4230/4260) as the ground state, then we could assign the Y(4750) as its first radial excitation according to the mass gap [562].
We carry out the calculations routinely to obtain two QCD sum rules,
where the and correspond to the ground states Y and first radial excitations , respectively [562].
We adopt the QCDSR II, see Eq. (151) and Eqs. (158) and (159), to study the radially excited states, and obtain the Borel windows, continuum threshold parameters, suitable energy scales and pole contributions, which are shown explicitly in Tab.30 and Tab.31. From the tables, we can see explicitly that the pole contributions of the 1P states (the 1P plus 2P states) are about (40−60)% ((67−85)%), the pole dominance is satisfied very well. On the other hand, the contributions from the highest dimensional condensates play a minor important role, or ( or ) for the 1P states (the 1P plus 2P states), the operator product expansion converges very good and better than that in our previous work [550], see Tab.28.
The predicted masses and pole residues are presented in Tab.32. From Tab.31 and Tab.32, we can see explicitly that the modified energy scale formula, see Eq. (189), can be well satisfied, and the relations and are hold, see Eq. (160).
In Fig.19, we plot the masses of the 1P and 2P hidden-charm tetraquark states with the . From the figure, we can see explicitly that there appear flat platforms in the Borel windows, the uncertainties come from the Borel parameters are rather small.
In Tab.33, we present the possible assignments of the vector tetraquark states [562]. From the table, we can see explicitly that there is a room to accommodate the Y(4750), i.e., the Y(4220/4260) and Y(4750) can be assigned as the ground state and first radial excited state of the type tetraquark states with the , respectively [562].
We can study the corresponding hidden-bottom tetraquark states with the simple replacement in Eq. (187). In Ref. [563], we observe that the Y(10750) observed by the Belle Collaboration [190] can be assigned as the type hidden-bottom tetraquark state with the .
3.2 Doubly heavy tetraquark states
In 2016, the LHCb Collaboration observed the doubly-charmed baryon state in the mass spectrum and measured the mass, but did not determine the spin [564]. The maybe have the spin or , we can take the diquark as basic constituent to construct the current
or
to study it with the QCD sum rules [565]. The observation of the doubly-charmed baryon state has led to a renaissance on the doubly-heavy tetraquark spectroscopy [566-568]. For a system, if the two Q-quarks are in long separation, the gluon exchange induced force between them would be screened by the two -quarks, then a loosely type bound state is formed. If the two Q-quarks are in short separation, the pair forms a compact point-like color source in heavy quark limit, and attracts a pair, which serves as another compact point-like color source, then an exotic type tetraquark state is formed. The existence and stability of the tetraquark states have been extensively discussed in early literatures based on the potential models [569-574] and heavy quark symmetry [575].
In Ref. [576], we choose the currents and to study the doubly heavy tetraquark states with the , where
, again, we adopt the correlation functions in Eq. (125). The tetraquark states are spatial extended objects, not point-like objects, however, we choose the local currents to interpolate them and take all the quarks and antiquarks as the color sources, and neglect the finite size effects.
We rewrite the current as
according to the identity in the color space. The current is of type in the color space, we can also construct the current satisfying the Fermi−Dirac statistics,
which is of type in the color space, and differs from the corresponding current constructed in Ref. [577] slightly.
The color factor defined in Eq. (20) has the values, and for the and 6 diquark , respectively. If we define , then and for the and type tetraquark states. The one-gluon exchange induced attractive (repulsive) interaction favors (disfavors) formation of the (6) diquark state (), while the -type tetraquark states are expected to have much smaller masses than that of the -type tetraquark states according to the . Furthermore, the color magnetic interaction, see Eq. (21), leads to mixing between the and -type tetraquark states. In Ref. [577], Du et al. obtained degenerate masses for the and -type tetraquark states based on the QCD sum rules. We should study this subject further.
After tedious but straightforward calculations, we obtain the QCD sum rules for the doubly-heavy tetraquark states [576], which are named as the T states in The Review of Particle Physics [97]. According to the energy scale formula in Eq. (99), we suggest an energy scale formula,
to determine the optimal energy scales of the QCD spectral densities.
There was no experimental candidate for the doubly heavy tetraquark state when performing the calculations [576]. After careful examinations, we choose the effective heavy quark masses and , and take account of the breaking effect by subtracting the . In Tab.34, we present the Borel windows , continuum threshold parameters , optimal energy scales , pole contributions of the ground states, where the same parameters as the ones in the QCD sum rules for the are chosen, see the last line.
We take into account all the uncertainties of the relevant parameters, and obtain the values of the masses and pole residues of the , which are shown explicitly in Tab.34 and Fig.20. From Fig.20, we can see explicitly that there appear platforms in the Borel windows shown in Tab.34 indeed. And we suggested to search for the states in the Okubo−Zweig−Iizuka super-allowed two-body strong decays
and weak decays through at the quark level,
In 2021, the LHCb Collaboration observed the exotic state just below the threshold [194, 195]. The Breit−Wigner mass and width are below the threshold and [194, 195]. While the Particle Data Group fit the Breit−Wigner mass and width to be and , respectively [97]. The prediction is in excellent agreement with the LHCb data.
Before the LHCb data, several theoretical groups had made predictions for the masses [276, 285, 336, 342, 345, 351, 354, 360, 361, 566, 567, 576-596], the predicted masses differ from each other in one way or the other.
In Ref. [578], we resort to the correlation functions and in Eq. (125) and currents
where
to study the mass spectrum of the , and doubly charmed tetraquark states systematically, where the subscripts 0, 1 and 2 denote the spins.
At the phenomenological side, we obtain the hadronic representation and isolate the ground state contributions,
where , the pole residues and are defined analogous to Eq. (127) with the simple replacements and , and , .
We perform the calculations routinely, and obtain the QCD sum rules for the masses and pole residues from the components , , and , respectively, where and . Again, we adopt the modified energy scale formula in Eq. (197) to determine the best energy scales of the QCD spectral densities.
After trial and error, we obtain the Borel windows , continuum threshold parameters , optimal energy scales , pole contributions, see Tab.35. From the Table, we can see clearly that the pole dominance can be well satisfied. In calculations, we observe that for the tetraquark states, the operator product expansion is well convergent, while in the case of the , and tetraquark states, the contributions of the vacuum condensates of dimensions have the hierarchy , the operator product expansion is also convergent. At last, we take account of all uncertainties of the relevant parameters, and obtain the values of the masses and pole residues, which are shown explicitly in Tab.35.
The centroids of the masses of the type tetraquark states are
which are slightly larger than the centroids of the masses of the corresponding type tetraquark states,
the lowest states are the type tetraquark states, which is consistent with our naive expectation that the axialvector (anti)diquarks have larger masses than the corresponding scalar (anti)diquarks. The lowest centroids and originate from the spin splitting, in other words, the spin-spin interaction between the doubly heavy diquark and light antidiquark. In fact, the predicted masses have uncertainties, the centroids of the masses are not the super values, all values within uncertainties make sense.
The QCD sum rules indicate that the masses of the light axialvector diquark states lie about above that of the light scalar diquark states [501-504], if they have the same valence quarks. Therefore, the centroids of the masses of the type tetraquark states should be larger than , the present calculations maybe under-estimate the doubly-heavy tetraquark masses. If we take the simple replacement in the modified energy scale formula in Eq. (197), the predictions should be improved, about .
After observation of the , several new works on the in the type tetraquark scenario appear [324, 472, 597-601]. Roughly speaking, the centroid of the type tetraquark states maybe lie about above the corresponding type tetraquark states, and more works are still needed.
The doubly-charmed tetraquark states with the , and lie near the corresponding charmed meson pair thresholds, the decays are Okubo−Zweig−Iizuka super-allowed,
but the available phase spaces are very small, thus the decays are kinematically depressed, the doubly charmed tetraquark states with the , and maybe have small widths. On the other hand, the doubly charmed tetraquark states with the lie above the corresponding charmed meson pair thresholds, the decays are Okubo−Zweig−Iizuka super-allowed,
and the available phase spaces are large, thus the decays are kinematically facilitated, the doubly charmed tetraquark states with the should have large widths.
3.3 Fully heavy tetraquark states
The exotic states , , , , , , , , , , are excellent candidates for the multiquark states, which consist two heavy quarks and two or three light quarks, we have to deal with both the heavy and light degrees of freedom of the dynamics. If there are multiquark configurations consisting of fully heavy quarks, the dynamics is much simple at first glance, and the tetraquark states have been studied extensively before the LHCb data [323, 333, 346, 349, 493, 569, 602-613].
The quarks have color symmetry, we can construct the tetraquark states according to the routine ,
For the diquarks, only the operators and could exist due to Fermi−Dirac statistics, and we usually take the operators to construct the four-quark currents and [610, 611], where , , and
then resort to the correlation functions and shown in Eq. (125) to obtain the QCD sum rules.
In Ref. [493], Chen et al. constructed the currents , and with , and , to interpolate the tetraquark states with the , and , respectively,
where , , , , , , , the and are color indexes. The , C, are antisymmetric, while the , , are symmetric. The currents , and are in the color representation, while the currents , and are in the color representation. For more currents and predictions in Scheme II, we can consult Ref. [493].
At the hadron side, we isolate the ground state contributions and obtain the results [610, 611],
where , we add the superscripts 1 and 2 to denote the spins, and define the pole residues and with analogous to Eq. (127) with the simple replacements , and .
If we take into account the first radial excited states, we obtain
Again we take the quark-hadron duality below the continuum thresholds and , respectively, and perform the Borel transformation with respect to the variable to obtain the QCD sum rules:
where the QCD spectral densities , , and [610, 611, 614].
In Fig.21, we plot the masses of the tetraquark states with variations of the energy scales and Borel parameters for the threshold parameters and . The predicted masses decrease monotonously and slowly with increase of the energy scales, the QCD sum rules are stable with variations of the Borel parameters at the energy scales . We take the largest energy scale in Ref. [610]. In Refs. [610, 611], we obtain the predictions for the masses of the ground states, see Tab.36, where .
In 2020, the LHCb Collaboration observed a broad structure above the threshold ranging from 6.2 to 6.8 GeV and a narrow structure at about 6.9 GeV in the mass spectrum, and they also observed some vague structures around 7.2 GeV [142]. Accordingly, we obtain the masses of the 2S and 2P states from the QCDSR II in Eq. (217), and obtain the 3/4S and 3/4P masses by fitting the Regge trajectories,
where the and are constants, see Tab.37 and Tab.38, which support assigning the broad structure from 6.2 to 6.8 GeV in the di- mass spectrum as the 2S or 2P tetraquark state, and assigning the narrow structure at about 6.9 GeV in the di- mass spectrum as the 3S tetraquark state [614].
In 2023, the ATLAS Collaboration observed a narrow resonance at about and a broader structure at much lower mass in the channel, moreover, they observed a statistically significant excess at about in the channel [143]. In 2024, the CMS Collaboration observed three resonant structures in the mass spectrum,
In Ref. [615], we update the calculations by taking the energy scale , the lower bound in Fig.21, and adopt the relation,
and obtain the predictions, see Tab.39 and Tab.40, and make the possible assignments, see Tab.41. From Tab.41, we can see explicitly that the lowest state lies at about 6.2 GeV, which is consistent with the recent coupled-channel analysis [616].
The thresholds of the , and are , and , respectively [97], we cannot obtain a simple molecule scenario to interpret those X states without introducing complex coupled-channel effects [617-620], or just assign them as the and type tetraquark states [334, 337, 344, 352, 460, 595, 621-631], or with gluonic constituent [632].
We perform Fierz transformation for the currents and , and obtain particular superpositions of a series of color 11 type currents,
with , the components couple potentially to the molecular states, where the and stand for some Dirac -matrixes. Therefore the type tetraquark states have some important 11 type Fock components, which would decay to their constituents via the Okubo−Zweig−Iizuka super-allowed fall apart mechanism if they are kinematically permitted. There exists a term , the decay to the di- is super-allowed, which is consistent with the observations of the ATLAS, CMS and LHCb experiments.
If we insist on that the di- system should have positive charge conjugation, we would like to construct a cousin of currents, and ,
which couple potentially to the fully-charm tetraquark states with the and , respectively.
In Ref. [633], we introduce a relative P-wave to construct the doubly-charmed vector diquark operator , then construct the scalar and tensor four-quark currents,
to study the scalar, axialvector and tensor fully-charm tetraquark states with the QCD sum rules. And we observe that the ground state type tetraquark states and the first radial excited states of the type tetraquark states have almost degenerated masses.
We can extend this subsection directly to study the type fully heavy pentaquark states and type fully heavy hexaquark states [634, 635].
4 11 type tetraquark states
The X, Y, Z, T and P states always lie near the two-particle thresholds, such as
naively, we expect that they consist of two color-neutral clusters, and they are molecular states, more precisely, they are the 11 type hidden-charm or doubly-charmed tetraquark or pentaquark states [636]. The establishment of the of the Y(4140) by the LHCb Collaboration [113, 114] excludes its assignment as the molecular state with the [432, 435, 477, 478, 637, 638], however, which does not mean non-existence of the molecular state with the .
The type four-quark currents could be reformed in a series of 11 type four-quark currents through Fierz transformation [424, 639], some useful examples are given explicitly in the appendix, see Eqs. (482)−(489).
According to the quark-hadron duality, the and type local currents couple potentially to the and type tetraquark states, respectively. The type tetraquark states could be taken as a particular superposition of a series of the type tetraquark states, while the type tetraquark states could decay through the Okubo−Zweig−Iizuka super-allowed fall-apart mechanism. We usually use the identities in Eqs. (482)−(489) to analyze the strong decays [424, 639]. For example, the current in Eq. (482) couples potentially to the , in the nonrelativistic and heavy quark limit, the component can be reduced to the form,
where the , and are two-component spinors of the quark fields, the are the three-vectors of the quark fields, the are the pauli matrixes, and the are the spin operators. It is obvious that the currents and couple potentially to the and states. However, the strong decays are kinematically forbidden.
4.1 Hidden heavy tetraquark states
Again, let us adopt the correlation functions , and defined in Eq. (125) and write down the currents
and
and , . The subscripts , , and stand for the two color-neutral clusters; especially, the subscripts , and correspond to the two color-neutral clusters , and , respectively, etc.
Again, we take the isospin limit, the currents with the symbolic quark structures , , , couple potentially to the hidden-charm molecular states with degenerated masses, the currents with the isospin and 0 lead to the same QCD sum rules [81-83, 422].
Under parity transformation , the currents , and have the properties,
where and . Under charge conjugation transformation , the currents , and have the properties,
At the hadron side, we obtain the hadronic representation and isolate the ground state hidden-charm molecule contributions [82, 83],
where the Z represents the molecular states , , , etc. We add the subscripts ± in the hidden-charm molecular states and the components and to represent the positive and negative parity contributions, respectively, and define the pole residues analogous to Eq. (127) with . We choose the components and to study the scalar, axialvector and tensor hidden-charm molecular states. According to the discussions in Section 2.3, we have no reason to worry about the contaminations from the two-meson scattering states.
At the QCD side, we carry out the operator product expansion up to the vacuum condensates of dimension 10, and take account of the vacuum condensates , , , , , , , and , where , or quark, just like in Section 3.1.1.
According to analogous routine of Section 3.1.1, we obtain the QCD sum rules:
and
In the heavy quark limit, we describe the systems by a double-well potential model, the heavy quark Q () serves as a static well potential and attracts the light antiquark () to form a color-neutral cluster. We introduce the effective heavy quark mass and divide the molecular states into both the heavy and light degrees of freedom, i.e., and , respectively.
Analysis of the and mass spectrum with the famous Cornell potential or Coulomb-potential-plus-linear-potential leads to the constituent quark masses and [640], we can set the effective -quark mass as the constituent quark mass . The old value and updated value , which are fitted for the hidden-charm molecular states, are all consistent with the constituent quark mass [81, 422, 641]. We choose the value to determine the ideal energy scales of the QCD spectral densities, and add an uncertainty to account for the difference between the values and . Furthermore, we take the modified energy scale formula
with , 1, 2 and to account for the light flavor breaking effects.
After trial and error, we obtain the Borel windows, continuum threshold parameters, energy scales of the QCD spectral densities, pole contributions, and contributions of the vacuum condensates of dimension 10, which are shown explicitly in Tab.42. At the hadron side, the pole contributions are about (40−60)%, while the central values are larger than 50%, the pole dominance condition is well satisfied. At the QCD side, the contributions of the vacuum condensates of dimension 10 are or , the convergent behaviors of the operator product expansion are very good.
We take account of all the uncertainties of the relevant parameters, and obtain the masses and pole residues of the molecular states without strange, with strange and with hidden-strange, which are shown explicitly in Tab.43. From Tab.42 and Tab.43, it is obvious that the modified energy scale formula in Eq. (235) is well satisfied [82, 83].
In Fig.22, we plot the masses of the axialvector molecular states , , and with variations of the Borel parameters at much larger ranges than the Borel widows as an example. From the figure, we can see plainly that there appear very flat platforms in the Borel windows indeed, where the regions between the two short vertical lines are the Borel windows.
In Fig.22, we also present the experimental values of the masses of the , , and [169, 642], the predicted masses are in excellent agreement with the experimental data. The calculations support assigning the , , and to be the , , and tetraquark molecular states with the quantum numbers , , and , respectively. In Tab.44, we present the possible assignments of the ground state hidden-charm molecular states. However, the lattice QCD calculations do not favor the existence of the [357, 370, 643, 644].
We could reproduce the experimental masses of the , , , and both in the scenarios of tetraquark states and molecular states, see Tab.9-Tab.11 and Tab.44, the tetraquark scenario can accommodate much more exotic X and Z states than the molecule scenario. Even in the tetraquark scenario, there are no rooms to accommodate the X(3940), X(4160), and without resorting to fine tuning. The X(3940) and X(4160) might be the conventional and states with the , respectively. The might be a mixing scalar tetraquark state with the , and the might be an axialvector color 88 type tetraquark state with the . For detailed discussions about this subject, one can consult Ref. [61].
The and have almost degenerated masses but quite different decay widths [162, 163, 167], they are taken as the same particle by the Particle Data Group [97] (also Ref. [60]), however, it is difficult to explain the large ratio,
from the BESIII Collaboration [167]. If we assign the as the type tetraquark state, and assign the as the molecular state, it is easy to explain the large ratio .
The observed by the BESIII Collaboration near the and thresholds in the recoil-mass spectrum in the processes , its Breit-Wigner mass and width are and respectively with an assignment of the spin-parity [169]. The observed in the mass spectrum by the LHCb Collaboration has a mass of , a width of , and the spin-parity [115]. The and are two quite different particles, although they have almost degenerated masses.
In the mass spectrum from the LHCb Collaboration, there is a hint of a dip at the energy about [115], which maybe due to the observed by the BESIII Collaboration with a mass about [170]. More experimental data are still needed to obtain a precise resolution.
After Ref. [82] was published, the Belle Collaboration observed weak evidences for two structures in the invariant mass spectrum in the two-photon process , one at with a width , and another at with a width [645]. The first structure is consistent with the X(3915) or , the second one might be an exotic charmonium-like state. We present its possible assignment in Tab.44, see X2(4014), which cannot be accommodated in the scenario of tetraquark state.
In Tab.43, the central values of the predicted molecule masses lie at the thresholds of the corresponding two-meson scattering states, where we have taken the currents having two color-neutral clusters, in each cluster, the constituents and (or Q and ) are in relative S-wave, see Eq. (226).
Now let us see the possible outcomes, if one of the color-neutral clusters has a relative P-wave between the constituents and (or Q and ), as one of the possible assignments of the Y(4260) is the molecular state with the [227, 229, 280, 555, 556].
For example, in the isospin limit, we write the valence quarks of the and molecular states symbolically as
the isospin triplet and singlet have degenerate masses. We take the isospin limit to study those molecular states [641].
Again we resort to the correlation functions in Eq. (125) with the currents , , and to study the and molecular states, where
Under charge conjugation transformation , the currents have the properties,
According to the quark-hadron duality, we isolate the ground state contributions of the vector molecular states,
where the are the pole residues.
We carry out the operator product expansion up to the vacuum condensates of dimension 10 in a consistent way routinely [82, 83, 641], and obtain the QCD sum rules for the masses and pole residues.
We adopt the energy scale formula in Eq. (235) to choose the best energy scales of the QCD spectral densities, and take the updated value [641].
After trial and error, we obtain the Borel parameters, continuum threshold parameters, pole contributions and energy scales, see Tab.45, where the central values of the pole contributions are larger than 50%, the pole dominance is well satisfied. In the Borel windows, the contributions , the operator product expansion is well convergent.
We take account of all the uncertainties of the relevant parameters, and obtain the values of the masses and pole residues, which are also shown in Tab.45 [641].
The prediction is consistent with the experimental data from the BESIII Collaboration within uncertainties [153], while the predictions , and are much larger than upper bound of the experimental data and [153], moreover, they are much larger than the near thresholds , , , [646]. The present predictions only support assigning the Y(4390) to be the molecular state. Or the Y(4260) has sizable non component [647].
In Ref. [434], Zhang and Huang studied the type scalar, vector and axialvector molecular states with the QCD sum rules systematically by calculating the operator product expansion up to the vacuum condensates of dimension 6. The predicted molecule masses and are consistent with the Y(4220) and Y(4390), respectively. However, they do not distinguish the charge conjugation of the molecular states and neglect the higher dimensional vacuum condensates.
In Ref. [430], Lee, Morita and Nielsen distinguished the charge conjugation, and calculated the operator product expansion up to the vacuum condensates of dimension 6 including dimension 8 partly. They obtained the mass of the molecular state with the , , which differs from the prediction significantly [434].
In Refs. [430, 434], Scheme II is chosen, some higher dimensional vacuum condensates are neglected, which are associated with , , in the QCD spectral densities and manifest themselves at small values of the , thus we have to choose large values of to warrant convergence of the operator product expansion. The higher dimensional vacuum condensates, see Fig.23, play an important role in determining the Borel windows therefore the ground state masses and pole residues, we should take them into account consistently. All in all, we observe that the QCD sum rules favor much larger masses than the two-meson thresholds if there exists a P-wave in one constituent, and we should bear in mind that the continuum threshold parameters should not be large enough to include contaminations from the higher resonances [648], i.e., if a bound state really exists, the continuum threshold should be less than in the case of the hidden-charm four-quark systems.
Besides the Y(4260), it is also difficult to reproduce the mass of the Y(4660) with a type current, however, we can reproduce its mass with a type current [482, 485], i.e., it might be a bound state [482, 485, 649, 650].
With the simple replacement , we obtain the corresponding QCD sum rules for the hidden-bottom tetraquark molecular states from Eq. (234). We can reproduce the experimental masses of the and as the and molecular states respectively with the via the QCD sum rules [81, 422, 443, 445, 451], although those QCD sum rules suffer from shortcomings in one way or the other. It is more easy to reproduce a weak bound state if there exist weak attractive interactions between the two constituents [76, 211, 271, 274, 282, 651-655].
In Ref. [557], Qiao assigned the Y(4260) as the baryonium state with the . In Ref. [558], Qiao included the constituent, and suggests a triplet and a singlet baryonium states,
and
then he assigns the as the 2S state with the , and the Y(4360) (Y(4660)) as the 2S () baryonium state with the .
We study the and type hidden-charm baryonium states via the QCD sum rules consistently by carrying out the operator product expansion up to the vacuum condensates of dimension 16 according to the counting roles in Sections 2.2 and 3.1.1, and observe that the state with the and states with the , lie at the corresponding baryon-antibaryon thresholds, respectively, while the states with the , and states with the , lie above the corresponding baryon-antibaryon thresholds, respectively, whose masses are all much larger than the Y(4260) [656]. In Ref. [657], Wan, Tang and Qiao study the states with the , , and via the QCD sum rules by taking account of the vacuum condensates up to dimension 12, and observe that only the baryonium states with the and could exist, they also obtain masses much larger than the corresponding thresholds, respectively.
In Ref. [658], we construct the color singlet-singlet type six-quark current,
with or , to study the hexaquark molecular state by calculating the vacuum condensates up to dimension 14, the predicted mass supports assigning the X(7200) to be the hexaquark molecular state with the . However, the pole contribution is not larger than , which would weaken the predictive ability.
In Ref. [659], we construct the color singlet-singlet-singlet type six-quark current with the to study the system via the QCD sum rules by considering the contributions of the vacuum condensates up to dimension-16, and observe that there indeed exists a resonance state which lies above the threshold, and suggest to search for it in the mass spectrum.
Such subjects need further studies to obtain definite conclusion, at the present time, they are open problems.
4.2 Doubly heavy tetraquark states
Before and after the observation of the doubly-charmed tetraquark candidate , especially after the observation, there have been many works on the doubly-charmed tetraquark (molecular) states with different theoretical approaches [371-373, 375, 471, 472, 513, 660-669].
If we perform Fierz rearrangements for the four-quark axialvector current [576], see Eq. (194), we obtain a special superposition of the color singlet-singlet type currents,
The currents , , and couple potentially to the color singlet-singlet type tetraquark states or two-meson scattering states. In fact, there exist spatial separations between the diquark and antidiquark pair, the currents should be modified to ,
where the four-vector . The spatial distance between the diquark and antidiquark pair maybe frustrate the Fierz rearrangements or recombination, although we usually take the local limit , we should not take it for granted that the Fierz rearrangements are feasible [401], we cannot obtain the conclusion that the has the Fock component according to the component in Eq. (244). According to the predictions in Tab.34, we can only obtain the conclusion tentatively that the has a diquark−antidiquark type tetraquark Fock component with the spin-parity and isospin [576]. It is interesting to explore whether or not there exists a color 11 type Fock component indeed.
Again, we resort to the correlation functions , and defined in Eq. (125), and construct the currents , and ,
and , , the subscripts , , and stand for the two color-neutral clusters, we add the subscripts L and H to distinguish the lighter and heavier states in the same doublet due to the mixing effects, as direct calculations indicate that there exists such a tendency.
Under parity transformation , the currents have the properties,
where and . We rewrite Eq. (253) in more explicit form,
where the space indexes , , 2, 3. There are both positive and negative components, and they couple potentially to the axialvector/tensor and pseudoscalar/vector molecular states, respectively. We will introduce an superscript to denote the negative parity.
According to the quark−hadron duality, we obtain the hadronic representation and isolate the ground state contributions of the scalar, axialvector and tensor molecular states [83, 513],
where the pole residues and are defined analogous to Eq. (127) with the simple replacements and , and and , we introduce the superscripts 0, 1 and 2 denote the spins of the molecular states. Thereafter, we choose the components and .
We accomplish the operator product expansion up to the vacuum condensates of dimension 10 and take account of the vacuum condensates , , , , , , and with the vacuum saturation in a consistent way [670], where , or [60, 61, 82, 83, 171, 172, 418, 425, 513-515]. Again, we neglect the and quark masses, and take account of the terms according to the light-flavor breaking effects.
Then we match the hadron side with the QCD side below the continuum thresholds , perform the Borel transformation, and obtain the QCD sum rules for the masses and pole residues. With a simple replacement , we obtain the corresponding QCD sum rules for the doubly-bottom tetraquark states, the calculations are straightforward.
We take the modified energy scale formula shown in Eq. (235) and the effective -quark masses and to determine the optimal energy scales of the QCD spectral densities. After trial and error, we obtain the Borel parameters, continuum threshold parameters, energy scales of the QCD spectral densities, pole contributions and contributions of the vacuum condensates of dimension 10, which are shown in Tab.46 [83, 513]. The pole contributions are about (40−60)% at the hadron side, the central values are larger than 50%, the pole dominance is satisfied very good. Moreover, the contributions of the vacuum condensates of dimension 10 are or at the QCD side, the operator product expansion converges very good [83, 513].
We take account of all the uncertainties of the relevant parameters, and obtain the masses and pole residues of the doubly-charmed molecular states without strange, with strange and with doubly-strange, which are presented explicitly in Tab.47 [83, 513].
In Fig.24, as an example, we plot the masses of the axialvector and molecular states with variations of the Borel parameters at much larger ranges than the Borel widows. There appear very flat platforms in the Borel windows, the regions between the two short perpendicular lines.
There exist both a lighter and a heavier state for the and molecular states, the lighter state with the isospin has a mass , which is in excellent agreement with the mass of the doubly-charmed tetraquark candidate from the LHCb Collaboration [194, 195], and supports assigning the to be the molecular state, as the has the isospin . In other words, the exotic state maybe have a Fock component. The heavier state with the isospin has a mass , the central value lies slightly above the threshold, the strong decays to the final states are kinematically allowed but with small phase-space. If we choose the same input parameters, the molecular state with the isospin has slightly larger mass than the corresponding molecule with the isospin , it is indeed that the isoscalar molecular state is lighter.
For the , , and molecular states, there exists a P-wave in the color-singlet constituents, the P-wave is embodied implicitly in the underlined in the scalar currents , and axialvector currents , , as multiplying to the pseudoscalar currents , and vector currents , changes their parity. We should introduce the spin-orbit interactions to account for the large mass gaps between the lighter and heavier states , i.e., , [83, 513].
From Tab.47, we can see explicitly that the , , , and molecular states have much larger masses than the corresponding two-meson thresholds, just like in the case of the hidden-charm molecular states studied in the previous sub-section, where there exist a relative P-wave in one of the color-neutral clusters.
Beyond those color 11 tetraquark states, there maybe also exist some corresponding hexaquark states. The QCD sum rules indicate that there exist the , and dibaryon states [413, 656, 657], the type triply-charmed hexaquark states [671], and the color type triply-charmed hexaquark molecular states [426]. However, for the light dibaryon/baryonium states, we can only obtain very small pole contributions at the hadron side or bad convergent behaviors of the operator product expansion at the QCD side [205, 672, 673], which weakens the predictive ability. If we adopt the truncation rule in Sections 2.2 and 3.1.1, i.e., each heavy quark line emits a gluon and each light quark contributes a quark−antiquark pair, which leads to a quark−gluon operator to reach the highest dimensional vacuum condensates, the two basic criteria of the QCD sum rules are difficult to satisfy. Such subjects need further studies.
On the other hand, Lattice calculations indicate that there maybe exist the , , , , dibaryon states [674-679]. While the heavy-antiquark−diquark symmetry implies that there exists a model-independent relation between the spin-splitting in the masses of the hidden-charm pentaquark states and corresponding splitting for the triply-charmed dibaryon states [680].
5 Hidden heavy pentaquark states
If a baryon current has the spin-parity , then the current would have the spin-parity , as multiplying changes the parity of the current [681]. In 1993, Bagan et al. [682] took the infinite heavy quark limit to separate the contributions of the positive and negative parity heavy baryon states unambiguously. In 1996, Jido et al. [683] introduced a novel approach to separate the contributions of the negative-parity light-flavor baryons from the positive-parity ones.
At first, we write down the correlation functions ,
where we add the subscripts ± to denote the positive and negative parity, respectively, . We decompose the correlation functions ,
according to Lorentz covariance, because
The currents couple potentially to both the positive- and negative-parity baryons [681],
where the denote the positive and negative parity baryons, respectively.
According to the relation in Eq. (260), we obtain the hadronic representation [683],
where the are the baryon masses, the pole residues are defined by , the are the Dirac spinors.
If we take , then
where
the contribution () contains contributions from the positive parity (negative parity) states only.
We carry out the operator product expansion at large region, then use the dispersion relation to obtain the spectral densities and at the quark-gluon level. At last, we introduce the weight functions , , and obtain the QCD sum rules,
where the and are the threshold and continuum threshold respectively, the is the Borel parameter [683]. Thereafter, such semi-analytical method was applied to study the heavy, doubly-heavy and triply-heavy baryons with the and [684-689].
As the procedure introduced in Ref. [683] is semi-analytical, in 2016, we suggested an analytical procedure to study the pentaquark states [423]. For a correlation function , at the hadron side, we obtain the spectral densities through the dispersion relation,
where the subscript H denotes the hadron side, then we introduce the weight function to obtain the QCD sum rules at the hadron side,
where the are the continuum threshold parameters. We separate the contributions of the negative parity pentaquark states from that of the positive parity ones unambiguously [423]. The calculations at the QCD side are analytical as we do not set .
For the early works on the pentaquark states, one can consult the QCD sum rules on the , where the currents [690], [691] and [692],
were constructed to interpolate it. The current was studied in the semi-analytical method [691].
5.1 type pentaquark states
Now, let us turn to the pentaquark states completely and write down the correlation functions , and ,
where
we choose the and diquarks in the color , the most stable diquark configurations, as the basic constituents to construct the diquark−diquark−antiquark type currents , and with the spin-parity , and , respectively, which are expected to couple potentially to the lowest pentaquark states [402, 423, 514].
In the currents , and , there are diquark constituents , , , , with , . If we use the and to represent the spins of the light and heavy diquarks, respectively, the light diquarks , and have the spins , 1 and 1, respectively, the heavy diquarks and have the spins and 1, respectively. A light diquark and a heavy diquark form a charmed tetraquark in the color with the angular momentum , which has the values , 1 or 2. The -quark operator has the spin-parity , the -quark operator has the spin-parity due to the factor . The total angular momentums are with the values , or , which are shown explicitly in Tab.48. In Tab.48, we present the quark structures of the currents explicitly [402]. For example, in the current , there are a scalar diquark with the spin-parity , an axialvector diquark with the spin-parity , and an antiquark with the spin-parity , the total angular momentum is . For more intuitive and simple diquark models for the pentaquark states, one can consult Refs. [693-698].
Although the currents , and have negative parity, but they couple potentially to the pentaquark states with positive parity, as multiplying to the currents , and changes their parity [681-689, 699-701].
Now we write down the current-pentaquark couplings explicitly,
where the superscripts ± denote the positive parity and negative parity, respectively, the subscripts , and denote the spins of the pentaquark states, the , and are the pole residues.
The spinors satisfy the Dirac equations , while the spinors and satisfy the Rarita−Schwinger equations and , and relations , , , , , respectively [423, 565].
At the hadron side, we insert a complete set of intermediate pentaquark states with the same quantum numbers as the currents , , , , and into the correlation functions , and to obtain the hadronic representation, and isolate the lowest states of the hidden-charm pentaquark states with negative and positive parity [402, 423, 514]:
where we have used the summations of the spinors [702],
and on mass-shell. We study the components , , , , and to avoid possible contaminations from other pentaquark states with different spins.
We obtain the spectral densities at the hadron side through dispersion relation,
where , , , the subscript H denotes the hadron side, then we introduce the weight functions and to obtain the QCD sum rules at the hadron side,
where the are the continuum threshold parameters, and the are the Borel parameters. We distinguish the contributions of the negative and positive parity pentaquark states unambiguously, and there is no contamination.
Now we briefly outline the operator product expansion. Firstly, we contract the , and quark fields in the correlation functions , and with Wick theorem, for example,
for the current , where the , and are the full , and quark propagators, respectively, see Eqs. (129) and (130). Then we compute all the integrals in the coordinate and momentum spaces sequentially to obtain the , and at the quark-gluon level, and finally we obtain the QCD spectral densities through dispersion relation,
where , , . In computing the integrals, we draw up all the Feynman diagrams from Eq. (282) and calculate them one by one. In Eq. (282), there are two -quark propagators and three light quark propagators, if each -quark line emits a gluon and each light quark line contributes a quark−antiquark pair, we obtain an operator according to the counting role in Eq. (38), which is of dimension 13, see Fig.25. We should take account of the vacuum condensates at least up to dimension 13 in stead of dimension 10, which is adopted in most literatures. The vacuum condensates , , are of dimension 11 and 13 respectively, and come from the Feynman diagrams shown in Fig.25 [402]. Those vacuum condensates are associated with the , and , which manifest themselves at the small and play an important role in choosing the Borel windows. We take the truncations and in a consistent way, the quark-gluon operators of the orders with and dimension are discarded.
In Refs. [423, 703-706], we take the truncations and and discard the quark-gluon operators of the orders with and dimension . Sometimes we also neglected the vacuum condensates , , , , , , which are not associated with the , and to manifest themselves at the small . Such an approximation would impair the predictive ability.
Then let us match the hadron side with the QCD side of the correlation functions, take the quark-hadron duality below the continuum thresholds , and obtain the QCD sum rules:
where .
We derive Eq. (284) with respect to , then eliminate the pole residues and obtain the QCD sum rules for the masses of the hidden-charm pentaquark states,
With a simple replacement , we obtain the corresponding QCD sum rules for the hidden-bottom pentaquark states.
According to the discussions in Section 2.4, we take the energy scale formula,
to determine the best energy scales of the QCD spectral densities [402, 423, 703-706], and choose the updated value of the effective -quark mass [512].
After trial and error, we obtain the Borel windows , continuum threshold parameters , ideal energy scales of the QCD spectral densities, pole contributions of the ground states and contributions of the vacuum condensates of dimension 13, which are shown explicitly in Tab.49 [402].
In Fig.26, we plot the contributions of the vacuum condensates of dimensions 11 and 13 with variation of the Borel parameter for the hidden-charm pentaquark state (0, 0, 0, ) with the central values of the parameters shown in Tab.49 as an example. The vacuum condensates of dimension 13 manifest themselves at the region , we should choose the value . While the vacuum condensates of dimension 11 manifest themselves at the region , which requires a larger Borel parameter to warrant the convergence of the operator product expansion. The higher dimensional vacuum condensates play an important role in choosing the Borel windows, where they play an minor important role as the operator product expansion should be convergent, we should take them into account in a consistent way, for example, the is less than [402].
In Fig.27, we plot the mass of the hidden-charm pentaquark state (0, 0, 0, ) with variation of the Borel parameter for truncations of the operator product expansion up to the vacuum condensates of dimensions 10 and 13, respectively [402]. The vacuum condensates of dimensions 11 and 13 play an important role to obtain stable QCD sum rules, we should take them into account.
In Tab.49, the pole contributions are about (40−60)% and the contributions of the vacuum condensates of dimension 13 are or , the pole dominance and convergence of the operator product expansion are all satisfied, the two basic criteria of the QCD sum rules are satisfied in the case of the hidden-charm pentaquark states for the first time.
We take account of all uncertainties of the relevant parameters, and obtain the masses and pole residues of the hidden-charm pentaquark states with negative parity, which are shown explicitly in Tab.50.
The predicted masses for the ground state (0, 0, 0, ) pentaquark state and for the ground state (1, 1, 0, ) pentaquark state are both in excellent agreement with the experimental data from the LHCb Collaboration [197], and support assigning the to be the hidden-charm pentaquark state with the .
After the work was published [402], the LHCb Collaboration observed an evidence for a structure in the and systems with a significance about depending on the hypothesis [199], the Breit−Wigner mass and width are and respectively. The can be assigned as the ground state (1, 1, 0, ) pentaquark state with the mass .
The predicted masses for the ground state (0, 1, 1, ) pentaquark state, for the ground state (1, 0, 1, ) pentaquark state and for the ground state (0, 1, 1, ), (1, 1, 2, ), (0, 1, 1, ) pentaquark states are in excellent agreement (or compatible with) the experimental data from the LHCb Collaboration [197], and support assigning the to be the hidden-charm pentaquark state with , or .
The predicted masses for the ground state (0, 1, 1, ) pentaquark state, for the ground state (1, 0, 1, ) pentaquark state and for the ground state (1, 0, 1, ) pentaquark states are in excellent agreement the experimental data from the LHCb Collaboration [197], and support assigning the to be the hidden-charm pentaquark state with or .
In Tab.50, we present the possible assignments of the , and explicitly as a summary. In Tab.51, we compare the present predictions with our previous calculations [423, 703-705], where the vacuum condensates of dimension 11 and 13 were neglected, sometimes the vacuum condensates , and were also neglected. From the Tab.51, we can see that in some cases the predicted masses change remarkably, while in other cases the predicted masses change slightly. In Ref. [423], we construct the current to interpolate the hidden-charm tetraquark state with the , which should be updated.
In Ref. [704], we construct the with the spin-parity to study the hidden-charm pentaquark states with the according to the rules,
where the denotes the additional P-wave and is embodied by a in constructing the currents, the subscripts , , denote the quark constituents. The quark and antiquark have opposite parity, we usually take it for granted that the quarks (antiquarks) have positive (negative) parity, and the -quark has the .
We write down the currents explicitly,
where the superscripts and are the spins of the light and heavy diquarks, respectively.
We take the isospin limit and classify the currents which couple potentially to the pentaquark states with degenerate masses into the following 8 types,
and perform the operator product expansion up to the vacuum condensates of dimension 10 to obtain the QCD sum rules, the predictions for the hidden-charm pentaquark states with the spin-parity are presented in Table 61 in the Appendix.
From Table 61, we can see that the two-body strong decays to the ,
for example,
could take place, but the decay widths are rather small due to the small available phase-spaces; on the other hand, the two-body strong decays,
for example,
can take place more easily, the decay widths are larger due to the larger available phase-spaces; furthermore, the two-body strong decays
for example,
can take place fluently, the decay widths are rather large due to the large available phase-spaces. We can search for those pentaquark states in the and mass spectra in the decays of the bottom baryons to the final states and associated with the light vector mesons or pseudoscalar mesons [693, 707, 708], for example,
In Ref. [705], we construct the currents to interpolate the hidden-charm tetraquark states, where the superscripts and are the spins of the light and heavy diquarks, respectively, , the is the spin of the heavy antiquark, the subscripts , , are the light quark constituents , or . We write down the currents explicitly,
We take the isospin limit and classify the thirty currents couple potentially to the hidden-charm pentaquark states with degenerate masses into the following twelve types,
and perform the operator product expansion up to the vacuum condensates of dimension 10 to obtain the QCD sum rules, the predictions are presented in Table 62 in the Appendix.
According to Fig.26 and Fig.27, the vacuum condensates of the dimensions 11 and 13 play an important role, we should update the old calculations, just like in Ref. [402].
In Ref. [514], we extend our previous works [402, 703] to explore the possible assignment of the as the (0, 0, 0, ) pentaquark state with the spin-parity . We choose the current and carry out the operator product expansion up to the vacuum condensates of dimension 13, where
In calculations, we take the modified energy scale formula to constraint the QCD spectral densities.
We obtain the Borel window via trial and error, the pole contribution is about , which is large enough to extract the pentaquark mass reliably. In Fig.28, we plot the contributions of the higher dimensional vacuum condensates with variation of the Borel parameter . The higher dimensional vacuum condensates manifest themselves at the region , we should choose the value . Their values decrease monotonously and quickly with the increase of the Borel parameter, in the Borel window , the contributions of the higher dimensional vacuum condensates are , , , , , the convergent behavior is very good [514].
In Fig.29, we plot the predicted pentaquark mass with variation of the Borel parameter with the truncations of the operator product expansion up to the vacuum condensates of dimensions and 13, respectively. From the figure, we can see explicitly that the vacuum condensates of dimensions 11 and 13 play an important role to obtain the flat platform, we should take them into account in a consistent way, just like in the case of Ref. [402], where no -quark is present. All in all, the higher dimensional vacuum condensates play an important role in obtaining the flat platform, especially those associated with the inverse Borel parameters , , and .
At last, we obtain the mass and pole residue [514],
The predicted mass is in excellent agreement with the experimental data from the LHCb Collaboration [198], and supports assigning the as the (0, 0, 0, ) pentaquark state with the spin-parity . In Ref. [402], we observe that the can be assigned to be the (0, 0, 0, ) pentaquark state with the spin-parity . The light-flavor mass-breaking effect is about , the estimations presented in Tab.52 are reasonable and reliable, where we take the experimental value of the mass of the . In Refs. [704, 705], we study the and hidden-charm pentaquark states with the strangeness , , and in a systematic way by carrying out the operator product expansion up to the vacuum condensates of dimension 10 and choosing the old value , and observe that the light-flavor mass-breaking effects are for the negative-parity pentaquark states. The new analysis supports a larger light-flavor mass-breaking effect.
We can extend this section directly to study the type triply-heavy pentaquark states [709].
5.2 11 type pentaquark states
In Ref. [710], Chen et al. constructed the color singlet-singlet type currents,
for the first time, and take the currents and to study the and by carrying out the operator product expansion up to the vacuum condensates of dimension 8. Thereafter, the 11 type currents were used to interpolate the pentaquark molecular states [418, 487, 711-719]. However, in those works, the isospins of the currents are not specified and should be improved, as the two-body strong decays to the final states and conserve isospins.
The and quarks have the isospin , i.e., and , where the is the isospin operator. Then the , , , , , , , , , , , , , , , and correspond to the eigenstates , , , , , , , , , , , , , , , and in the isospin space , respectively. We construct the color-singlet currents to interpolate them,
Accordingly, we construct the 11 type five-quark currents to interpolate the , , and type pentaquark sates, where the , , , and represent the color-singlet clusters having the same quantum numbers as the physical states , , , and respectively except for the masses,
the subscripts , , 0 and 1 represent the isospins I [418, 720-722]. They are the isospin eigenstates , , or .
For example, from Eq. (327), we obtain the relations,
the currents and have both the isospin and components, and couple potentially to the pentaquark molecular states with the isospins and , which decay to the final states and , respectively. As the , , , and are observed in the mass spectrum, it is better to choose the current with the definite isospin, we prefer the color singlet−singlet type currents with the definite isospins [418, 720-722], and thereafter those currents are adopted in Refs. [723-725]. Although the mass splitting between the isospin cousins is of several 10-MeV in most cases, in some cases, the mass splitting can be as large as 150 MeV, see Tab.54. Phenomenologically, the molecule-type P states have been studied extensively with the help of heavy quark symmetry [214, 257, 268, 269, 277, 287-289, 726-738], it is more easy to apply isospin eigenstates in the effective field theory than in QCD.
We resort to the correlation functions , and in Eq. (269), and perform analogous analysis to obtain the QCD sum rules for the masses and pole residues like Eqs. (284)−(285),
where , and , the pole residues are defined analogous to Eqs. (272)−(274).
As we study the 11 type pentaquark states, it is better to choose the updated values and to determine the optimal energy scales of the QCD spectral densities with the formula [641],
After trial and error, we obtain the Borel windows, continuum threshold parameters, energy scales and pole contributions, see Tab.53 [720-722], the pole contributions are about or slightly larger than (40−60)%, we obtain the largest pole contributions up to now. In the Borel windows, the highest dimensional condensate contributions and are approximately zero, the most important contributions are mainly from the lowest order contributions , and . The operator product expansion converges very well.
Then we calculate the uncertainties of the masses and pole residues, which are shown in Tab.54 [720-722]. The central value of the mass of the molecular state with the is GeV, it is only about 10 MeV below the threshold, we assign it as the tentatively. For the molecular state with the , the central value of the mass is GeV, which is about 10 MeV above the threshold, we tentatively assign it as the resonance, the isospin cousin of the .
In a similar way, we assign the , and as the , and molecular states with the , and , respectively. For the molecular states (resonances) with the , with the and with the , the central values of the masses are about 20 MeV, 10 MeV and 90 MeV above the corresponding meson-baryon thresholds, respectively.
If we choose the same input parameters, the 11 type pentaquark states with the isospin (1) have slightly larger masses than the corresponding ones with the isospin (0).
The , , and can be assigned as the , , and molecular states with the isospin respectively, since the two-body strong decays conserve isospin. If the assignments are robust, there exist four slightly higher molecular states , , and with the isospin , we can search for the four resonances in the mass spectrum, as the two-body strong decays also conserve isospin, the , and have the isospins , and , respectively.
The and molecular states lie about and above the respectively, the is unlikely to be the or molecular state. The mass of the molecular state with the isospin is , which is near the value , but lies slightly above the corresponding meson-baryon threshold, it is a resonance, from the decay channel [198], the isospin of the is zero, which excludes assigning the as the molecular state with the isospin . The mass of the molecular state with the isospin is , which is in very good agreement with the , it is very good to assign the as the molecular state with the isospin and the spin-parity . The predications also favor assigning the as the molecular state with the spin-parity and isospin .
The predictions support assigning the as the molecular state with the spin-parity and isospin , the observation of its cousin with the isospin in the mass spectrum would decipher the inner structure of the . However, there exists no candidate for the [199].
Beyond the type currents, in Ref. [739], Pimikov, Lee and Zhang construct the color type currents and to interpolate the hidden-charm pentaquark states,
where the quark fields have the flavors and color , the and are some Dirac matrixes. Then, they calculate the mass spectrum by taking account of the vacuum condensates , , , , , and .
6 Singly-heavy exotic states
6.1 Singly-heavy tetraquark states
The and observed in the mass spectrum are the first exotic structures with fully open flavor [101, 102], they have the valence quarks . The and are observed in the and mass spectra, respectively [130, 131], they have the valence quarks and , respectively.
Based on the predicted masses of the -type scalar tetraquark states from the QCD sum rules [61, 740],
we estimate the mass of the -type tetraquark state crudely,
which is consistent with the mass of the within uncertainties [119].
In Ref. [119], we construct the and -type scalar four-quark currents,
to study the tetraquark states with the correlation function , see Eq. (125). We carry out the operator product expansion up to the vacuum condensates of dimension-11 and assume vacuum saturation for the higher dimensional vacuum condensates according to the routine in Section 2.2 and Section 3.1.1. As there are three -quark lines and one Q-quark line, if each Q-quark line emits a gluon and each -quark line contributes a quark-antiquark pair, we obtain a quark-gluon operator , which is of dimension 11, and leads to the vacuum condensates and .
We obtain the QCD sum rules routinely, at the QCD side, we choose the flavor numbers and the typical energy scale . After trial and error, we obtain the Borel windows and continuum threshold parameters, therefore the pole contributions of the ground states and convergent behaviors of the operator product expansion, see Tab.55. In the Borel windows, the pole contributions are about , the central values exceed . The contributions of the vacuum condensates are about and for the and -type tetraquark states, respectively.
At last, we obtain the values of the masses and pole residues, which are also shown in Tab.55. In Fig.30, we plot the masses of the and -type scalar tetraquark states with variations of the Borel parameters in much larger ranges than the Borel windows, there appear platforms in the Borel windows indeed.
The predicted mass is consistent with the experimental value from the LHCb Collaboration [101, 102], and supports assigning the to be the -type tetraquark state with the spin-parity . While the predicted mass lies above the experimental value [101, 102].
In Ref. [741], we construct the -type currents to study the ground state mass spectrum of the tetraquark states with strange and doubly-strange via the QCD sum rules to verify the inner structures of the , where
the superscripts 0, 1 and 2 denote the spins. With a simple replacement , we obtain the corresponding currents in the same isospin multiplets. In the isospin limit, the tetraquark states in the same multiplets have the same masses.
Again, we resort to the correlation functions and , see Eq. (125), and obtain the ground state contributions according to the hadron representations in Eqs. (202)−(204), and obtain the QCD sum rules routinely, again we take the flavor numbers and the typical energy scale .
After trial and error, we obtain the Borel windows, continuum threshold parameters and pole contributions, see Tab.56, where the pole contributions are and the central values for the six states are larger than 50%, furthermore, the contributions of the vacuum condensates show a descending trend . At last, we obtain the masses and pole residues, which are also shown in Tab.56 [741].
In Tab.56, the predicted mass of the state , , is in very good agreement with the experimental values and from the LHCb Collaboration [130, 131], and supports assigning the to be the -type tetraquark states with the spin-parity .
Those typical singly-charmed tetraquark candidates, which lie at , have attracted many theoretical works, and are assigned as the type tetraquark states [118-121, 742], or their radial/orbital excitations [122], or non-tetraquark states [123], molecular states [121, 124-128, 743, 744, 745], triangle singularities [129, 746, 747], etc. We could only obtain a mass about for the singly-charmed tetraquark states at the cost of sacrificing the pole dominance [748, 749, 750].
At the bottom sector, the singly-bottom tetraquark candidate X(5568) is not confirmed [751, 752]. The lowest mass of the ground state might have a mass , which lies above the X(5568).
6.2 Singly-heavy pentaquark states
The experimental candidates for the singly-charmed pentaquark states are not as robust as that of the singly-charmed tetraquark states, the assignments in the scenario of pentaquark states are only conjectures.
In 2017, the LHCb Collaboration observed five narrow structures , , , and [753]. Also in 2017, the Belle Collaboration confirmed the , , and in the decay mode [754]. In 2023, the LHCb Collaboration observed the and in the mass spectrum [755], which lie near the and thresholds, respectively, the measured Breit−Wigner masses and decay widths are
As early as 2005, the Belle Collaboration tentatively assigned the , and in the mass spectra as the isospin triplet states with the spin-parity [756], the measured masses and decay widths are
In 2008, the BaBar Collaboration observed the in the mass spectrum with the possible spin-parity [757], the mass and decay width are
In 2007, the BaBar Collaboration observed the in the invariant mass spectrum [758]. Subsequently, the Belle Collaboration verified the in the decay mode [759]. In 2017, the LHCb Collaboration determined the spin-parity of the to be by analyzing the process [760]. The measured masses and decay widths are
In 2023, the Belle Collaboration studied the decays and found a new structure in the mass spectrum [761], the mass and decay width are
The , and lie near the thresholds, and they might be the molecular states. For example, we can assign the as the molecular (bound) state [762-765], and assign the as the molecular state [763, 765-770]. We would like to study the 11 type charmed pentaquark states and explore the possible assignments in the scenario of molecular states.
Again, we resort to the correlation functions , and defined in Eq. (269), and write down the currents,
and
the subscripts , , and are isospin indexes , we choose the convention [771].
The currents , and couple potentially to the , and singly-charmed molecular states , , and , respectively, for more details, see Section 5.1. At the hadron side, we isolate the ground state contributions,
We choose the components , , , , and to explore the spin-parity , and molecular states, respectively. Again see Section 5.1 for technical details in tensor analysis.
Again we obtain the spectral densities through dispersion relation,
where , , , the subscript H denotes the hadron side, then we introduce the weight functions and to obtain the QCD sum rules at the hadron side,
which are free from contaminations of the positive-parity molecular states.
At the QCD side, we accomplish the operator product expansion with the full light/heavy-quark propagators up to the vacuum condensates of dimension 13 in a consistent way, and obtain the QCD spectral densities through dispersion relation,
where , , . See Section 3.1.1 for technical details.
At last, we obtain the QCD sum rules,
We differentiate Eq. (360) with respect to , then eliminate the pole residues to obtain the QCD sum rules for the masses,
where the spectral densities and .
We take the quark flavor number and resort to the modified energy scale formula
to choose the ideal energy scales of the QCD spectral densities, where , the effective -quark mass and effective -quark mass , the is the number of the -quark in the currents/states, see Section 4.1 for details.
Routinely, we obtain the Borel windows, continuum threshold parameters, energy scales of the QCD spectral densities and contributions of the , see Tab.57, where the pole dominance is well satisfied. On the other hand, the highest dimensional vacuum condensate contributions have the relation , the operator product expansion is convergent very well.
At last, we obtain the masses and pole residues of the pentaquark molecular states, see Tab.58 [771]. From Tab.57 and Tab.58, it is obvious that the modified energy scale formula is satisfied in most cases. The energy scales for the , and states are smaller than the energy scales for the corresponding , and states, respectively, we choose the same energy scales for those cousins, as larger masses correspond to larger energy scales naively.
The predicted masses and for the and molecular states (respectively) with the support assigning the and as the and molecular states, respectively. The predicted mass for the molecular state with the supports assigning the as the molecular state. The predicted mass for the molecular state is consistent with the within uncertainties, which does not exclude the possibility that they are molecular states.
In Ref. [767], Zhang studied the and as the S-wave and molecule candidates respectively with the QCD sum rules, obtains the masses and for the S-wave state with the and S-wave state with the , respectively, which are somewhat bigger than the experimental data for the and , respectively, and differ from the present calculations significantly. We should bear in mind that Zhang chose Scheme II while we choose Scheme I.
Taking the () pentaquark molecular states with the () as an example, we obtain a symmetric isotriplet , , and an antisymmetric isosinglet . The () is a good candidate for the () molecular state with the isospin . We expect to search for the molecular states with the isospin and in the and mass spectra respectively to shed light on the nature of the ().
Other interpretations also exist, the could be assigned as overlap of the and [772] or P-wave charmed baryon state with the [773-776]. Cheng et al suggest that the is not possible to be a charmed baryon state [777]. The is most likely to be the or (2P) state [776-779]. And the is probably the (2P) [780] or (1P) state [781]. The lies in the region of the 2S state [782, 783], while the is a good candidate for the 2S state [783] or D-wave baryon state with the [782, 784, 785].
With a simple replacement , we obtain the corresponding QCD sum rules for the singly-bottom pentaquark molecular states, see Eqs. (360) and (361). In Ref. [786], we choose the current,
to interpolate the molecular states with the , the prediction supports assigning the as the pentaquark molecular state with the .
In Ref. [787], we construct the type five-quark currents,
to interpolate the singly-charmed pentaquark states with the . With a simple replacement,
we obtain the isospin singlet current, the expressions of the QCD sum rules survive. The predictions support assigning the excited states from the LHCb Collaboration as the or -type pentaquark states with the and the mass about . We group the quark flavors as to construct the currents, if we group the quark flavors as , the -type current will vanish due to Fermi-Dirac statistics, but the -type current survives and leads to almost degenerated mass according to the light-flavor symmetry.
In Ref. [788], we study the type singly-charmed pentaquark states with the via the QCD sum rules. We distinguish the isospins in constructing the interpolating currents via two clusters, a diquark () plus a triquark (), which have the properties,
with , , , the is the isospin operator. The diquark clusters and triquark clusters have the isospins , or 0. We could obtain some mass relations based on the breaking effects of the , , quarks, the mass relation among the diquark clusters is , while the mass relation among the triquark clusters is , if the hidden-flavor (for and ) isospin singlet is excluded. Moreover, the isospin triplet and isospin singlet are expected to have degenerate masses, which can be inferred from the tiny mass difference between the vector mesons and . In fact, if we choose the currents and to interpolate the and , respectively, we obtain the same QCD sum rules.
We write down the currents explicitly,
and study the singly-charmed pentaquark states and with the QCD sum rules in details. Then we estimate the masses of the singly-charmed pentaquark states , , and with to be according to the light-flavor breaking effects, which is compatible with the experimental values of the masses of the , , , .
7 Strong decays of exotic states
In Refs. [520, 789], we suggest rigorous quark-hadron duality to calculate the hadronic coupling constants in the two-body strong decays of the tetraquark states with the QCD sum rules. At first, we write down the three-point correlation functions ,
where the currents interpolate the tetraquark states A, the currents and interpolate the conventional mesons B and C, respectively,
the , and are the pole residues or decay constants.
At the hadron side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the currents , , into the three-point correlation functions and isolate the ground state contributions to obtain the result [789],
where , the are the hadronic coupling constants defined by
the four functions , , and have complex dependence on the transitions between the ground states and the higher resonances or continuum states.
We rewrite the correlation functions at the hadron side as
through triple-dispersion relation, where the are the hadronic spectral densities,
where the , and are the thresholds, the , , are the continuum thresholds.
Now we carry out the operator product expansion at the QCD side, and write the correlation functions as
through double-dispersion relation, where the are the QCD spectral densities,
As the QCD spectral densities do not exist,
because
Thereafter we will write the QCD spectral densities as for simplicity.
We match the hadron side with the QCD side of the correlation functions, and accomplish the integral over firstly to obtain the rigorous quark-hadron duality [520],
the denotes the thresholds . Now we write down the quark-hadron duality explicitly,
No approximation is needed, we do not need the continuum threshold parameter in the channel, as we match the hadron side with the QCD side below the continuum thresholds and to obtain rigorous quark-hadron duality, and we take account of the continuum contributions in the channel.
In Eq. (380), we introduce the parameters , , and to parameterize the net effects,
In numerical calculations, we take the relevant functions and as free parameters, and choose suitable values to eliminate the contaminations from the higher resonances and continuum states to obtain the stable QCD sum rules with the variations of the Borel parameters .
According to the discussions in Section 2.3, the quantum field theory does not forbid the couplings between the four-quark currents and two-meson scattering states , if they have the same quantum numbers. The local currents have direct non-vanishing couplings to the two-meson scattering states , although the overlaps of the wave-functions are very small [417], which leads to a finite width to modify the dispersion relation, see Eqs. (78) and (79).
There exists another term at the hadron side beyond that shown in Eq. (380),
where
Such terms shown in Eq. (382) could be absorbed into the parameters with the simple replacement,
In Ref. [446, 790], Nielsen et al approximate the hadron side of the correlation functions as
then match them with the QCD side below the continuum threshold by taking the chiral limit and sequentially, where the stands for the pole-continuum transitions, and we have rewritten their notations into the present form for convenience. Although Nielsen et al take account of the continuum contributions by introducing a parameter in the channel phenomenologically, they neglect the continuum contributions in the channel at the hadron side by hand. Such an approximation is also adopted in Refs. [47, 532, 791, 792].
There is another scheme to study the strong hadronic coupling constants with the correlation functions ,
At the QCD side, a double-dispersion relation,
with is adopted [465, 472]. However, we cannot obtain the QCD spectral densities . Such an scheme is also adopted in the case of pentaquark states [793-796].
Let us turn to Eq. (380) again. If the B are charmonium or bottomonium states, we set and perform the double Borel transformation with respect to the variables and , respectively to obtain the QCD sum rules,
where the and are the Borel parameters. If the B are open-charm or open-bottom mesons, we set and perform the double Borel transformation with respect to the variables and , respectively to obtain the QCD sum rules,
where . Or set , just like in the first case. The scheme based on the rigorous quark-hadron duality is adopted in Refs. [62, 171, 507, 523, 797-806].
7.1 Strong decays of the Y(4500) as an example
In this sub-section, we would like to use a typical example to illustrate the procedure in details.
After Ref. [543] was published, the Y(4500) was observed by the BESIII Collaboration [155, 156, 159]. At the energy about , we obtain three hidden-charm tetraquark states with the , the , and tetraquark states have the masses , and , respectively [543], see Tab.22. In Ref. [546], we study their two-body strong decays systematically with the three-point correlation functions,
With the simple replacement , we obtain the corresponding correlation functions for the current . And with the simple replacements and , we obtain the corresponding correlation functions for the current , where the currents
According to quark-hadron duality, we obtain the hadron representation and isolate the ground state contributions explicitly,
other ground state contributions are given in Ref. [546], where
with the simple replacements , and , we obtain the hadronic representation for the , and channels, respectively,
With the simple replacements and , we obtain the corresponding components for the currents and , except for the component ,
We introduce the collective notations to simplify the expressions,
With the simple replacements and , we obtain the collective notations for the currents and , except for the , , , where
and
We adopt the standard definitions for the decay constants and pole residues [546], and we define the hadronic coupling constants,
In Eq. (423), there are contributions coming from the and tetraquark states, and we cannot choose the pertinent structures to exclude the contaminations from the tetraquark state X, so we include it at the hadron side. The unknown parameters, , , , etc., parameterize the complex interactions among the excitations in the channels and the ground state charmed meson pairs or charmonium plus . It is difficult to choose the pertinent tensor structures in Eqs. (406)−(423) to obtain good QCD sum rules without contaminations, and we have to reach the satisfactory results via trial and error.
At the QCD side, we accomplish the operator product expansion up to the vacuum condensates of dimension 5 and take account of both the connected and disconnected Feynman diagrams in the color space, see Fig.31 (in Refs. [47, 446, 532, 790, 791], only the connected Feynman diagrams are taken into account, i.e., the D, E and I in Fig.31), and choose the components to study the hadronic coupling constants based on the rigorous quark-hadron duality [520, 789].
Then we set in the components , and carry out the double Borel transformation with respect to the variables and respectively, and set to obtain thirty QCD sum rules,
with the simple replacements , and , we obtain the QCD sum rules for the , and channels, respectively,
With the simple replacements and , we obtain the QCD sum rules for the and , except for the channel,
the explicit expressions of the QCD side are given in Ref. [546].
We take the unknown parameters , , , as free parameters, and adjust the suitable values to obtain flat Borel platforms for the hadronic coupling constants [546],
the Borel windows are shown explicitly in Tab.59. We obtain uniform flat platforms , where the max and min denote the maximum and minimum, respectively. In calculations, we choose quark flavor numbers , and evolve all the input parameters to the energy scale . For detailed information about the parameters, one can consult Ref. [546].
In Fig.32, we plot the with variation of the Borel parameter at a large interval as an example, in the Borel window, there appears very flat platform indeed.
We estimate the uncertainties of the hadronic coupling constants routinely. For an input parameter , , the left side can be written as , , , where
. We set , , approximately.
After taking into account the uncertainties, we obtain the values of the hadronic coupling constants, which are shown explicitly in Tab.59, then we obtain the partial decay widths directly, and show them explicitly in Tab.60.
At last, we saturate the total widths with the summary of partial decay widths,
The widths of the Y(4484), Y(4469) and Y(4544) are , and , respectively, from the BESIII Collaboration [155, 156, 159], which are compatible with the theoretical predictions in magnitude.
From Tab.60, we obtain the typical decay modes. For the state, the decays,
have the largest partial decay width ; while the decay,
has zero partial decay width. For the state, the decays,
have the largest partial decay width ; while the decay,
has zero partial decay width. For the state, the decay,
has the largest partial decay width ; while the decay,
has the partial decay width . We can search for the Y(4500) in those typical decays to diagnose its nature.
7.2 Light-cone QCD sum rules for the Y(4500) as an example
The light-cone QCD sum rules have been applied extensively to study the two-body strong decays of the tetraquark (molecular) states [127, 462, 467, 807-811], where only the ground state contributions are isolated, and an unknown parameter is (or not) introduced by hand to parameterize the higher resonance contributions, then the Ioffe-Smilga-type trick,
is (or not) adopted to subtract this parameter [812, 813]. The Ioffe−Smilga-type trick was suggested to deal with the traditional hadrons, where there exists a triangle Feynman diagram. In the case of tetraquark (molecular) states, we deal with two disconnected loop diagrams approximately, see Fig.31, we would like not to resort to the Ioffe−Smilga trick, and write down the higher resonance contributions explicitly.
We would like to use an example to illustrate how to study the strong decays of the tetraquark states via the light-cone QCD sum rules, and write down the three-point correlation function ,
where
interpolate the Y(4500), and , respectively [547], the is the external state.
At the hadron side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the interpolating currents into the three-point correlation function, and isolate the ground state contributions,
where , we adopt the standard definitions for the decay constants , , , and define the hadronic coupling constants and ,
the , and are polarization vectors of the Y(4500), and , respectively. In the isospin limit, , and .
We multiply Eq. (452) with the tensor and obtain
Again, we take the isospin limit, then , and we write down the relevant components explicitly,
where
Then we choose the tensor structures and to study the hadronic coupling constants and , respectively. We obtain the hadronic spectral densities through triple dispersion relation,
where the , and are the thresholds, and we add the subscript H to represent the hadron side.
We carry out the operator product expansion up to the vacuum condensates of dimension 5 and neglect the tiny gluon condensate contributions [520, 789],
where , , , , . And we have used the light-cone distribution functions [814],
and the approximation,
for the twist-3 quark-gluon light-cone distribution functions with the value at the energy scale [814, 815]. Such terms proportional to and their contributions are greatly suppressed, and we also neglect the twist-4 light-cone distribution functions due to their small contributions. According to the Gell−Mann−Oakes−Renner relation , we take account of the Chiral enhanced contributions fully in Eqs. (458) and (459).
In Fig.33, we draw the lowest order Feynman diagrams as an example to illustrate the operator product expansion.
In the soft limit , , we can set , then we obtain the QCD spectral densities through double dispersion relation,
we add the superscript (subscript) to stand for the QCD side.
We match the hadron side with the QCD side below the continuum thresholds and to acquire rigorous quark-hadron duality [520, 789],
and we carry out the integral over firstly, then
where , and we introduce the parameters to parameterize the contributions concerning the higher resonances and continuum states in the channel,
As the strong interactions among the ground states , , and excited states are complex, and we have no knowledge about the corresponding four-hadron contact vertex. In practical calculations, we can take the unknown functions as free parameters and adjust the values to acquire flat platforms for the hadronic coupling constants with variations of the Borel parameters. Such a method works well in the case of three-hadron contact vertexes [520, 789], and we expect it also works here.
In Eq. (452) and Eq. (454), there exist three poles in the limit , and . According to the relation , we set and perform double Borel transformation with respect to the variables and respectively, then we set to obtain two QCD sum rules,
where , and .
In calculations, we fit the free parameters as and to acquire uniform flat Borel platforms . The Borel windows are and , where the subscripts and Y represent the corresponding channels. In Fig.34, we plot the hadronic coupling constants and with variations of the Borel parameters. In the Borel windows, there appear very flat platforms indeed.
We obtain the hadronic coupling constants routinely,
by setting
It is direct to obtain the partial decay width,
where .
The partial decay width is much smaller than the total width from the BESIII Collaboration [156].
The three-body strong decays of the Y(4230) are also studied in this scheme [816], this scheme could be applied to the two-body strong decays of the tetraquark (molecular) states straightforwardly.
We write down the two-point correlation functions ,
where the currents and interpolate the tetraquark (molecular) states and traditional mesons, respectively, the are external states.
At the hadron side, we obtain
where , and we rewrite the correlation functions as
through double-dispersion relation, where the are the hadronic spectral densities,
where the and are the thresholds, the and are the continuum thresholds.
Then we carry out the operator product expansion (not necessary on the light-cone) at the QCD side, and write the correlation functions as
through single-dispersion relation, where the are the QCD spectral densities,
As the QCD spectral densities do not exist,
because
And we will write the QCD spectral densities as for simplicity.
We match the hadron side with the QCD side of the correlation functions, and accomplish the integral over firstly to obtain the rigorous quark-hadron duality [520],
And we write down the quark-hadron duality explicitly,
Again, we introduce the parameters to parameterize the net effects. In numerical calculations, we take the as free parameters, and choose suitable values to obtain the stable QCD sum rules with the variations of the Borel parameters .
8 Conclusion and perspective
At the present time, we can only say confidently that the tetraquark and pentaquark states are established in sense of that there are four and five valence quarks, respectively. The under-structures are still under hot debates, more experimental and theoretical works are still needed before reaching definite conclusion. The QCD sum rules method is a reliable and powerful theoretical tool in studying the multiquark states and has given many successful descriptions, however, the predictions have arbitrariness depending on the treating schemes, only comprehensive and systematic works would work.
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