Four-quark States Research Articles

Light scalar quarkonia from Laplace sum rule at NLO

We review our results on light scalar quarkonia (q¯q and four-quark states) from (inverse) QCD Laplace sum rules (LSR) and their ratios R within stability criteria and including higher order perturbative (PT) corrections up to the (estimated) O(αs5). As the Operator Product Expansion (OPE) usually converges for D⩽6−8, we evaluated the QCD spectral functions at Lowest Order (LO) of PT QCD and up to the D=6 dimension vaccum condensates. We request that the optimal results obey the constraint: Pole (Resonance) contribution to the spectral integral is larger than the QCD continuum one which excludes an on-shell mass around (500−600)MeV obtained for values of the QCD continuum threshold tc⩽(1∼1.5)GeV2. Our results for the different assignments of the scalar mesons are compiled in Tables 1 to 3.

Investigating the composition of the K[formula omitted] state with π±K[formula omitted] correlations at the LHC

The first measurements of femtoscopic correlations with the particle pair combinations π±KS0 in pp collisions at s=13 TeV at the Large Hadron Collider (LHC) are reported by the ALICE experiment. Using the femtoscopic approach, it is shown that it is possible to study the elusive K0⁎(700) particle that has been considered a tetraquark candidate for over forty years. Source and final-state interaction parameters are extracted by fitting a model assuming a Gaussian source to the experimentally measured two-particle correlation functions. The final-state interaction in the π±KS0 system is modeled through a resonant scattering amplitude, defined in terms of a mass and a coupling parameter, The extracted mass and Breit–Wigner width, derived from the coupling parameter, of the final-state interaction are found to be consistent with previous measurements of the K0⁎(700). The small value and increase of the correlation strength with increasing source size support the hypothesis that the K0⁎(700) is a four-quark state, i.e. a tetraquark state of the form (q1,q2‾,q3,q3‾) in which q1, q2 and q3 indicate the flavor of the valence quarks of the π and KS0. This latter trend is also confirmed via a simple geometric model that assumes a tetraquark structure of the K0⁎(700) resonance.

First Observation of a Three-Resonance Structure in e+e−→Nonopen Charm Hadrons

We report the measurement of the inclusive cross sections for e+e−→nOCH (where nOCH denotes non-open charm hadrons) with improved precision at center-of-mass (c.m.) energies from 3.645 to 3.871 GeV. We observe three resonances: R(3760), R(3780), and R(3810) with significances of 8.1σ, 13.7σ, and 8.8σ, respectively. The R(3810) state is observed for the first time, while the R(3760) and R(3780) states are observed for the first time in the nOCH cross sections. Two sets of resonance parameters describe the energy-dependent line shape of the cross sections well. In set I [set II], the R(3810) state has mass (3805.7±1.1±2.7) [(3805.7±1.1±2.7)] MeV/c2, total width (11.6±2.9±1.9) [(11.5±2.8±1.9)] MeV, and an electronic width multiplied by the nOCH decay branching fraction of (10.9±3.8±2.5) [(11.0±3.4±2.5)] eV. In addition, we measure the branching fractions B[R(3760)→nOCH]=(25.2±16.1±30.4)%[(6.4±4.8±7.7)%] and B[R(3780)→nOCH]=(12.3±6.6±8.3)%[(10.4±4.8±7.0)%] for the first time. The R(3760) state can be interpreted as an open-charm (OC) molecular state, but containing a simple four-quark state component. The R(3810) state can be interpreted as a hadrocharmonium state. Published by the American Physical Society 2024

Hidden-flavor four-quark states in the charm and bottom region

We discuss the spectrum and the internal composition of ground and excited four-quark states in the charm and bottom energy region. To this end we extend previous calculations within the framework of the relativistic four-body Faddeev-Yakubovsky equation to include quantum numbers with JPC=0++,0−+,1−−,1+− and 1++ and study their internal composition in terms of heavy-light meson pairs, hadroquarkonia and diquark-antidiquark clusters. We observe similar patterns in the charm and bottom energy region with different compositions of the four-quark states depending on JPC quantum numbers. Most notably, we find that all states with C·P=+1 are dominated by heavy-light meson contributions, whereas for axial-vector states with JPC=1+− including the Zc(3900) we find a much more complicated picture depending on the flavor content. We systematically compare our results for the spectrum with existing experimental results and provide predictions for future analyses. Published by the American Physical Society 2024

Formula omitted] from QCD Laplace sum rule

We present a review of our recent works on heavy-light exotic states, with new compact integrated expressions of the spectral functions at lowest order (LO) of perturbative (PT) QCD and up to d=6 condensates of the Operator Product Expansion (OPE). Including higher order (HO) PT corrections, which we have estimated by assuming the factorization of the molecule/four-quark spectral functions, we evaluate masses and couplings with the Laplace sum rules (LSR) and finite energy (FESR) QCD spectral sum rules (QSSR) approaches. Our optimal results based on stability criteria are summarized and compared with experimental candidates and some LO QSSR results. We conclude that, our predicted spectra, do not confirm that the X(5568) is a pure molecule or a four-quark state. The masses of the XZ observed states are compatible with (almost) pure JPC=1+±,0++ molecule or/and four-quark states. The SU3 corrections on the meson masses are tiny: ≤10% (respectively ≤3%) for the c (respectively b)-quark channel but can be large for the couplings (≤20%). We expect that the resonance near the D−K+ threshold can be originated from the 0+(D−K+) molecule and/or D−K+ scattering. The X0(2900), XJ(3150) (if J = 0), X1(2900) and the XJ(3350) (if J = 1) can emerge from a minimal mixing model between tetramoles and first radial excitation. The Zc(3900,4025,4040,4430) and Zcs(3983) spectra are well reproduced by the molecules and tetramoles candidates, or radial excitation.

Formula omitted] and TQQ within QCD Laplace sum rules

We present a review of our results for the masses and couplings of the scalar fully heavy four-quarks and TQQq¯q¯′(JP=0±,1±) tetraquarks states from (inverse) QCD Laplace sum rule (LSR), their ratios R and double ratio of sum rules (DRSR) within stability criteria and including Factorized Next-to-Leading Order (FNLO) Perturbative (PT) corrections. As the Operator Product Expansion (OPE) usually converges for d⩽6−8, we evaluated the QCD spectral functions at Lowest Order (LO) of PT QCD and up to 〈G3〉. Our results for the 0++ fully heavy four-quark states may explain the LHCb broad structure around (6.2−6.7)GeV which can be due to ηc‾ηc,χc1‾χc1 and J/ψ‾J/ψ molecules or/and their analogue ScSc,AcAc and VcVc tetraquarks. The peak at (6.8−6.9)GeV can be identified to the χc0‾χc0 molecule or/and the PcPc tetraquark state. Then, combining R and DRSR we focus on the analysis of the four-quark nature of Tccq¯q¯′1± and 0± states. We show that combining R and DRSR can provide more precise results: MTcc1+=3886(6)MeV and MTcc0+=3883(3)MeV. In a similar way, one obtain the improved mass predictions: MTccs¯u¯1+=3931(7)MeV, MTccs¯u¯0+=3983(7)MeV and MTccs¯s¯0+=3993(11)MeV. From our estimates of the masses of the Pseudoscalar and Vector Tccq¯q¯′ states, we observe that the interpolating currents lead to two classes: Class H (Heavy) states with masses around 6GeV and Class L (Light) states around (3.8−4.4)GeV where the pseudoscalar (resp. all vector states) are below the D‾D0,D‾sDs0 (resp. D‾D1,D‾sDs1) open charm thresholds. Finally, we extend the whole study to the bottom sector and confront our results with the ones from different LSR predictions and some other approaches (lattice, potential models, ⋯) in the literature.

Scrutinizing the light scalar quarkonia from LSR at higher orders

We scrutinize, improve some determinations of the masses and couplings of light scalar quarkonia (q¯q and four-quark states) and present new results for the π+π−,K+K−,… molecules using QCD Laplace Sum Rule (LSR) truncated at the D=6 dimension vacuum condensates. We pay a special attention on the higher order perturbative (PT) corrections up to the (estimated) O(αs5) which improve the quality of the analysis. We request that the optimal results obey the rigorous constraint: Resonance ≥ QCD continuum contributions (RP/C≥1) in the LSR which excludes a Breit-Wigner/on-shell (not to be confused with a complex pole) scalar meson mass around (500-600) MeV obtained for values [tc≤(1∼1.5) GeV2] of the QCD continuum threshold. Mass-splittings due to SU(3) breakings are small. We discuss the different assignments of the observed scalar mesons below 2 GeV in the Conclusions where the I=0 states are compared with the scalar gluonia. The results are compiled in Tables 3 to 6. We hope that the systematic analysis done in this paper helps to clarify the complex spectra of the light scalar mesons.

The scalar exotic resonances X(3915), X(3960), X_0(4140)

The scalar resonances X(3915), X(3960), X_0(4140) are considered as exotic four-quark states: cqbar{c} bar{q}, csbar{c} bar{s}, csbar{c}bar{s}, while the X(3863) is proved to be the cbar{c}, 2,^3P_0 state. The masses and the widths of these resonances are calculated in the framework of the Extended Recoupling Model, where a four-quark system is formed inside the bag and has relatively small size (lesssim 1.0 fm). Then the resonance X(3915) appears due to the transitions: J/psi omega into D^{*+}D^{*-} (or D^{*0}bar{D}^{*0}) and back, while the X(3960) is created due to the transitions D_s^+D_s^- into J/psi phi and back, and the X_0(4140) is formed in the transitions J/psi phi into D_s^{*+}D_s^{*-} and back. The characteristic feature of the recoupling mechanism is that this type of resonances can be predominantly in the S-wave decay channels and has J^P=0^+. In two-channel case the resonance occurs to be just near the lower threshold, while due to coupling to third channel (like the cbar{c} channel) it is shifted up and lies by (20–30) MeV above the lower threshold. The following masses and widths are calculated: M(X(3915))=3920 MeV, Gamma (X(3915))=20 MeV; M(X(3960))=3970 MeV, Gamma (X(3960)=45(5) MeV, M(X_0(4140))= 4120(20) MeV, Gamma (X_0(4140))=100 MeV, which are in good agreement with experiment.

Tcs and Tcs¯ Family Production in Multi-Production Processes

The production mechanism of multiquark exotic hadrons in high energy multiproduction processes lies in the structure of the relevant exotic hadrons as well as in some important aspects of high energy scattering, such as multi-parton interactions, underlying events, etc. At mass pole around 2900 MeV, a family of open charm tetraquarks, Tcss and Tcs¯s, are observed in B decay. They are also suitable for study in multiproduction processes to obtain more information on their structure. If these resonances are produced as compact four-quark states, one can predict the production properties based on the similarities in their production mechanism to those of Ξc, Σc, and Λc. Physics implies that the colour and baryon number fluctuations of the preconfinement system in high energy scattering can enhance both the baryon and four-quark state production rates via ‘diquark fragmentation’. We calculate the production properties of the tetraquark family Tcss and Tcs¯s at LHC energy for the forthcoming LHC measurements.

QCD spectral sum rules 2022

We present a compact review of the status of QCD spectral sum rules until 2022. We emphasize the recent progresses for determining the QCD input parameters (αs, running quark masses, quark and gluon condensates) where their correlations have been taken into account. Some selected phenomenological uses of the sum rules in different channels (light and heavy quarks, gluonia/glueballs, hybrids and four-quark states) are briefly reviewed and commented. The estimate of the 1−+ light hybrid mass is revised which confirms the hybrid nature of the π1(1600) but not the π1(2050).

Femtoscopy measurements of two-kaon combinations in Au+Au collisions at the STAR experiment

Abstract. Relativistic heavy-ion collisions can study the properties of nuclear matter in high-energy experiments like the STAR experiment. One of the methods to learn about the properties of the bulk matter is the femtoscopy technique, which relies on information carried by the particles produced during the collisions. The emission source parameters, like space-time characteristics, can be extracted using the technique. High statistics data from RHIC make it possible to study the correlations between strange particles, like charged and neutral kaons. The pair-wise interactions between the identical kaons that form the basis for femtoscopy are quantum statistics and the Coulomb interaction for K±K±, and quantum statistics and the final-state interaction (FSI) through the f0(980)/a0(980) threshold resonances for KS0KS0. The interactions between non-identical kaons pairs of KS0K± are essential, as the strong FSI is described only by the a0(98O) resonance, which could be a four-quark state. These proceedings will present the femtoscopic measurements of strange particles with charged and neutral kaons correlations in Au+Au collisions at the RHIC energy.

Molecular interpretation of X(3960) as D_s^+ D_s^- state

We study D_s^+ D_s^- and D bar{D} states assuming that they are hadronic molecules with J^{PC}=0^{++} quantum number. We use two-point QCD sum rule formalism and extract the mass and decay constant values of these states. We take into account contributions of various quark, gluon, and mixed vacuum condensates up to dimension eight. The extracted mass and decay constant values of D bar{D} and D_s^+ D_s^- states read as M_{D bar{D}} = 3795^{+85}_{-82}~{textrm{MeV}} , f_{D bar{D}} = 1.70^{+0.33}_{-0.29} times 10^{-2} ~{textrm{GeV}}^5, and M_{D_s^+ D_s^-} = 3983^{+93}_{-88} ~{textrm{MeV}}, f_{D_s^+ D_s^-} = 2.52^{+0.64}_{-0.54} times 10^{-2} ~{textrm{GeV}}^5, respectively. The predicted mass of D_s^+ D_s^- state is in good agreement with the recent LHCb observation and supports quantum number and molecular picture assignments. A possible observation of D bar{D} state would help for establishing the lowest four-quark state in charmonium sector.

Analysis of the X(3960) and related tetraquark molecular states via the QCD sum rules

In this work, we study the Dbar{D}, DD, Dbar{D}_s, DD_s, D_sbar{D}_s and D_sD_s tetraquark molecular states with the J^{PC}=0^{++} via the QCD sum rules. The prediction M_{D_sbar{D}_s} = 3.98pm 0.10, text {GeV} is in very good agreement with the experimental value M_{X(3960)} = 3956 pm 5pm 10 ,text {MeV} from the LHCb collaboration and supports assigning the X(3960) as the D_s^+D_s^- molecular state with the J^{PC}=0^{++}. We take account of our previous works on the four-quark states consisting of two color-neutral clusters and acquire the mass spectrum of the ground state hidden-charm and doubly-charm tetraquark molecular states.

The two-channel exotic charmonium-like resonances in the mass region (3900–4700) MeV

Numerous resonances in the c{{bar{c}}} and 4q systems, containing c{{bar{c}}} plus s{{bar{s}}} quarks (or light q{{bar{q}}}) were observed during last decades and recently the LHCb has found a remarkable narrow peak T_{cc}(3875) in the D D^* system. Besides there are several highly excited charmonium-like resonances, which can be treated as the shifted standard charmonium states. We analyze all these groups of the resonances in the mass region (3900–4700) MeV, using relativistic strong coupling theory with possible channel coupling phenomena. For the shifted charmonium states conventional charmonium spectrum is presented, being calculated with the relativistic string Hamiltonian, which does not contain fitting parameters, while for high excitations the universal flattened confining potential is used. It is shown that X(4274), X(4500), X(4700) can be identified as 3,^3P_1, 4,^3P_0, 5,^3P_0 states. The group of exotic states are considered using the Extended Recoupling Model, where two mesons m_1,m_2 transfer into another pair of mesons m_3, m_4 and back (infinite number of times), creating the four-quark resonance. Within this approach the resonances – T_{cc}(3875), Z_c(3900), X(3915), Z_{cs}(3985), X(4014), and X(4140) – can be explained as the exotic four-quark states in the S-wave decay channels.

Covariant tetraquark equations in quantum field theory

We derive general covariant coupled equations of QCD describing the tetraquark in terms of a mix of four-quark states $2q2\bar q$, and two-quark states $q\bar q$. The coupling of $2q2\bar q$ to $q\bar q$ states is achieved by a simple contraction of a four-quark $q\bar q$-irreducible Green function down to a two-quark $q\bar q$ Bethe-Salpeter kernel. The resulting tetraquark equations are expressed in an exact field theoretic form, and are in agreement with those obtained previously by consideration of disconnected interactions; however, despite being more general, they have been derived here in a much simpler and more transparent way.

Four-quark states with charm quarks in a two-body Bethe–Salpeter approach

We study the internal structure of a range of four-quark states with charm quark contributions using a two-body Bethe–Salpeter equation. Thereby, we examine charmonium-like states with hidden charm and quark content cbar{c}qbar{q}, open-charm states with quark content ccbar{q}bar{q} and all-charm states with ccbar{c}bar{c}. In particular we study the internal competition between meson–meson components and diquark–antidiquark components in the wave functions of these states. Our results indicate that the chi _{c1}(3872) and the Z_c(3900) are predominantly Dbar{D}^* states and that the recently discovered open-charm state T_{cc}^+ is dominated by an internal DD^* component. In both cases the diquark components are negligible. For the all-charm state X(6900) with as yet unknown quantum numbers we identify candidates in the excitation spectra of 0^+ and 1^+ states. Furthermore, our framework serves to provide predictions for further, yet undiscovered open and hidden charm four-quark states.

Light scalars: Four-quark versus two-quark states in the complex energy plane from Bethe-Salpeter equations

We study the dynamical generation of scalar mesons in the light quark sector ($q\in\{u,d,s\}$) and calculate the masses and widths for the $f_0(500),a_0(980)$ and $f_0(980)$. To this end we study the mixing of conventional $q\bar{q}$ and `exotic' $q\bar{q}q\bar{q}$ states via a coupled set of two-body Bethe-Salpeter equations based on a symmetry-preserving truncation of the underlying Dyson-Schwinger equations. This allows us to determine the dominant components of each state. Furthermore, we extend our previous framework into the complex energy plane such that we can study the analytic structure of the states in question and extract their width. At the physical point of small quark masses, the $\sigma$ meson is predominantly a $\pi\pi$ resonance. Consequently, its mass and width is driven by the effects of chiral symmetry breaking. At larger quark masses, however, the conventional $q\bar{q}$ components take over. Furthermore, we find a strong molecular $K\bar{K}$ component for both, the strange-light $f_0(980)$ and the $a_0(980)$.

Newly observed exotic doubly charmed meson [formula omitted

In this work, we treat the newly observed doubly charmed four-quark state Tcc+ as an axial-vector tetraquark with content ccu‾d‾, and calculate its spectroscopic parameters and width. The mass and current coupling of the tetraquark Tcc+ are found by means of the QCD two-point sum rule method by taking into account quark, gluon and mixed condensates up to dimension 10. The width of the Tcc+ is evaluated using partial widths of decay processes Tcc+→T˜π0 and Tcc+→Tcc;u‾u‾0π+, where T˜=ccu‾d‾ and Tcc;u‾u‾0 are scalar tetraquarks. To compute the partial width of the first process, we apply the QCD three-point sum rule approach and exact numerical value of the strong coupling g that corresponds to the vertex Tcc+T˜π0. The width of the second decay is estimated using isospin symmetry and the prediction obtained for the first channel. Our results for the mass m=(3868±124)MeV and width Γ=(489±92)keV of the tetraquark Tcc+ are in a nice agreement with recent measurements of the LHCb collaboration.

Modifications on the Properties of \(D^*_{s0}\)(2317) as Four-quark State in Thermal Medium

Since the low mass of $D^*_{s0}(2317)$ has still been a problem to the conventional quark model, one can consider other options regarding as multi-quark system. Therefore we investigate the scalar open-charm state $D^*_{s0}(2317)$ by Thermal QCD Sum Rules (TQCDSR) method using the two-point correlation function together with contributions of the non-perturbative condensates up to dimension six. Deriving and numerically analyzing thermal mass and pole residue sum rules, we accomplish the effects on the properties of $D^*_{s0}(2317)$ resonance in hot medium. Our numerical evaluations indicate that the variations in mass and pole residue values are stable through the growing temperature up to $T\cong100~\mathrm{MeV}$, but they begin to fall after this point. At critical temperature, the values of mass and pole residue change up to $91\%, 70\% $ of their values in vacuum in the molecular scenario and $ 91\%, 69\% $ in the diquark-antidiquark scenario. Our results does not give any definite information as to whether $D^*_{s0}(2317)$ resonance has molecular or diquark-antidiquark structure since they are very close to each other to differentiate them. Besides, we predict the hadronic parameters of the bottom partner of the $D^*_{s0}(2317)$ resonance in both molecular and diquark-antidiquark pictures. This bound state is worth investigating in future experiments. Also the detailed search of hot medium effects on the hadronic parameters of open-charm meson $D^*_{s0}(2317)$ and the bottom partner could have some implications to define the QCD phase diagram obtained from heavy-ion collision experiments. Moreover these results can be useful in distinguishing conventional quark model mesons from exotica.

Hunting for tetraquarks in ultraperipheral heavy ion collisions

Ultra-peripheral heavy ion collisions constitute an ideal setup to look for exotic hadrons because of their low event multiplicity and the possibility of an efficient background rejection. We propose to look for four-quark states produced by photon-photon fusion in these collisions at center-of-mass energy per nucleon pair $\sqrt{s_{\rm NN}}=5.5~{\rm TeV}$. In particular, we focus on those states that would represent a definite smoking gun for the compact tetraquark model. We show that the $X(6900)$, a likely $cc\bar{c}\bar{c}$ compact state, is a perfect candidate for this search, and estimate a production cross section ranging from around $250$ nb to $1150$ nb, depending on its quantum numbers. Furthermore, we discuss the importance of ultra-peripheral collisions to the search for the scalar and tensor partners of the $X(3872)$ predicted by the diquarkonium model, and not yet observed. The completion of such a flavor-spin multiplet would strongly speak in favor of the compact tetraquark model.