Four-quark States Research Articles

Charged charmonium molecules

We make use of a self-consistent quark-model based study of four-quark charmonium-like states to interpret recent charmonium experimental data. We conclude that there exists a ${D}^{*}{\overline{D}}^{*}$ meson-meson molecule with quantum numbers $({I}^{G}){J}^{PC}=({1}^{\ensuremath{-}}){2}^{++}$. Our study confirms the presence of charged charmonium-like resonances on the excited charmonium spectrum. We find support from recent experimental data by the Belle Collaboration [R. Mizuk et al. (Belle Collaboration), Phys. Rev. D 78, 072004 (2008)]. Confirmation of the experimental data by the Belle Collaboration and the determination of the quantum numbers of the new structures would help in discriminating among different theoretical models and would give further support to the theoretical analysis of T. Fern\'andez-Caram\'es, A. Valcarce, and J. Vijande [Phys. Rev. Lett. 103, 222001 (2009)].

Molecular and compact four-quark states

We study charmonium (cc̄nn̄), bottomonium (bb̄nn̄) and exotic (ccn̄n̄ and bbn̄n̄) four-quark states by means of a standard non-relativistic quark potential model. We look for possible bound states. Among them we are able to distinguish between meson-meson molecules and compact four-quark states.

S-waveQQq¯q¯state in the constituent quark model

Many proposals have been put forward to explore four-quark states $QQ\overline{q}\overline{q}$ ($Q=s$, $c$, $b$; $q=u$, $d$) by experiment, so a systematic study of $QQ\overline{q}\overline{q}$ spectrum with different constituent quark models by a high precision, few-body method, the Gaussian expression method, is useful. Three quark models: the Bhaduri, Cohler, Nogami quark model, the chiral quark model (ChQM), and the quark delocalization color screening model are all employed for a systematic calculation of the $S$-wave $QQ\overline{q}\overline{q}$ spectrum with different color structures, using the Gaussian expression method. The results show that only the $bb\overline{q}\overline{q}$ state with $(I,J)=(0,1)$ is bound in different color structures within the different quark models. The binding energy varies from several MeV for a di-meson structure to over 100 MeV for a diquark-antidiquark structure. For the $cc\overline{q}\overline{q}$ system, the state with $(I,J)=(0,1)$ is bound in a di-meson structure, and also bound in a diquark-antidiquark structure if pseudoscalar meson exchanges are accounted for. All are weakly bound states. The mixture of diquark-antidiquark and molecular structures is discussed in the framework of quark models for the first time; $cc\overline{q}\overline{q}$ with $(I,J)=(0,1)$ is below the threshold in addition to $bb\overline{q}\overline{q}$ in both the ChQM and the Bhaduri, Cohler, Nogami quark model. In the same channel, $ss\overline{q}\overline{q}$ is also a possible bound state with mass around 1.4 GeV in ChQM.

Charmonium Spectroscopy above Thresholds

We present a systematic and self-consistent analysis of four-quark charmonium states and applied it to study compact four-quark systems and meson-meson molecules. Our results are robust and should serve to clarify the situation of charmonium spectroscopy above the threshold production of charmed mesons.

Probabilities in nonorthogonal bases: Four-quark systems

Four-quark states may exist as colorless meson-meson molecules or compact systems with two-body colored components. We derive an analytical procedure to expand an arbitrary four-quark wave function in terms of nonorthogonal color singlet-singlet vectors. Using this expansion we develop the necessary formalism to evaluate the probability of physical components with an arbitrary four-quark wave function. Its application to characterize bound and unbound four-quark states as meson-meson, molecular, or compact systems is discussed

An estimate of the branching fraction ofτ→πη′ντ

We calculate the expected branching fraction of the second-class-current decay $\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\pi}{\ensuremath{\eta}}^{\ensuremath{'}}{\ensuremath{\nu}}_{\ensuremath{\tau}}$, motivated by a recent experimental upper-limit determination of this quantity. The largest contribution to the branching fraction is due to the intermediate ${a}_{0}(980)$ scalar meson, assuming it is a $\overline{u}d$ state. Smaller contributions arise from ${a}_{0}(1450)$, $\ensuremath{\rho}(770)$, and $\ensuremath{\rho}(1450)$. Our calculated values are substantially below the experimental upper limit, and are smaller still if the ${a}_{0}(980)$ is a four-quark state, as often suggested. Thus, a precise measurement or tight upper limit has the potential to determine the nature of the ${a}_{0}(980)$, as well as provide information about new scalar interactions.

New States and Charmonium Spectroscopy at BaBar

Several charmonium-like states above D D ¯ threshold have been discovered at the Belle and BaBar B -factories. Some are produced via Initial State Radiation (e.g. Y ( 4260 ) and Y ( 4350 ) ), and some are observed in B meson decays (e.g. X ( 3872 ) , Y ( 3940 ) , and Z ( 4430 ) − ). The Z ( 4430 ) − state has generated a great deal of interest, having minimum quark content ( c c ¯ d u ¯ ) and, thus, representing the unequivocal manifestation of a four-quark meson state. Here we summarize recent BaBar results on the Y ( 4260 ) , X ( 3872 ) , Y ( 3940 ) , and on a search for the Z ( 4430 ) − .

Light four-quark states and QCD sum rule

The relations among four-quark states, diquarks and QCD sum rules are discussed. The situation of the existing, but incomplete studies of four-quark states with QCD sum rules is analyzed. Masses of some diquark clusters were attempted to be determined by QCD sum rules, and masses of some light tetraquark states were obtained in terms of the diquarks.

Exotic meson-meson molecules and compact four-quark states

We present an exact calculation of $S$ and $P$ wave $QQ\overline{n}\overline{n}$ states using different standard nonrelativistic quark-quark potentials. We explore in detail the charm and bottom sectors looking for bound states that could be measured within existing facilities. Against the proliferation of four-quark states sometimes predicted in the literature, we found a small number of candidates to be stable. We analyze their properties in a trial to distinguish between compact and molecular states. Possible decay modes are discussed.

Formula omitted] light exotic mesons in QCD

We systematically re-examine the extraction of the masses and couplings of the 1−+ hybrid, four-quark and molecule mesons from QCD spectral sum rules (QSSR). To NLO for the perturbative and power corrections, the hybrid mass is MH=1.81(6) GeV and MH⩽2.2(2) GeV from the positivity of the spectral function. In the same way, but to LO, the four-quark state mass is M4q=1.70(4) GeV and M4q⩽2.4(1) GeV, while the molecule mass is 1.3(1) GeV. The observed π1(1400) and π1(1600) could be explained by a two-component mixing with the set of input masses (1.2–1.3; 1.70–1.74) GeV and with a mixing angle θ≃−(11.7±2.2)°, which slightly favours a molecule/four-quark mixing, and which eventually suggests that the π1(2015) is mostly an hybrid meson. Isospin and non-exotic partners of the previous states and some of their radial excitations are also expected to be found in the energy region around 2 GeV. Further tests of this phenomenological scenario are required.

Tetraquarks with charm in coupled-channel formalism

The relativistic four-quark equations are found in the framework of coupled-channel formalism. The dynamical mixing of the meson-meson states with the four-quark states is considered. The approximate solutions of these equations using the method based on the extraction of leading singularities of the amplitudes are obtained. The four-quark amplitudes of cryptoexotic mesons including the quarks of three flavours (u,d,s) and the charmed quark are constructed. The poles of these amplitudes determine the masses of tetraquarks. The mass values of low-lying tetraquarks are calculated.

ππcontribution to the QCD sum rules for the light tetraquark

We perform a QCD sum rule analysis for the ${f}_{0}$ light tetraquark taking into account the contribution arising from the two pion intermediate state. With the interpolating currents of the different chiral combinations of scalar and pseudoscalar diquarks, it is demonstrated that the interpolating current with maximum chirality has a large coupling to the two pion state, but the current with zero chirality interacts only weekly with this state. Taking into account the form factor in the ${f}_{0}$-two pion vertex, it is shown that the ${f}_{0}$ coupling to the two pion state leads to an increase of the lightest tetraquark mass by a value of about 100 MeV. The analysis of the resulting sum rule shows that the $\ensuremath{\sigma}({f}_{0}(600))$-meson state might be treated as the four-quark bound state in the instanton field which has a rather strong coupling to the two pion state.

Chiral SU(3) Quark Model Study of Tetraquark States: cnn̄s̄/css̄s̄

The new members of the charm-strange family D*sJ(2317), DsJ(2460), and Ds(2632), which have the surprising properties, are challenging the present models. Many theoretical interpretations have been devoted to this issue. Most authors suggest that they are not the conventional cs̄ quark model states, but possibly are four-quark states, molecule states, or mixtures of a P-wave cs̄ and a four-quark state. In this work, we follow the four-quark-state picture, and study the masses of cnn̄s̄/css̄s̄ states (n is u or d quark) in the chiral SU(3) quark model. The numerical results show that the mass of the mixed four-quark state (cnn̄s̄/css̄s̄) with spin parity JP = 0+ might not be Ds(2632). At the same time, we also conclude that D*sJ (2317) and DsJ (2460) cannot be explained as the pure four-quark state.

Mixings of four-quark components in light nonsinglet scalar mesons in QCD sum rules

Mixings of four-quark components in the nonsinglet scalar mesons are studied in the QCD sum rules. We propose a formulation to evaluate the cross correlators of $q\overline{q}$ and $qq\overline{q}\overline{q}$ operators and to define the mixings of different Fock states in the sum rule. It is applied to the nonsinglet scalar mesons, ${a}_{0}$ and ${K}_{0}^{*}$. It is found that the four-quark operators predict lower masses than the $q\overline{q}$ operators and that the four-quark states occupy about 70%--90% of the lowest mass states.

Masses and mixing ofcqq¯q¯tetraquarks using Glozman-Riska hyperfine interaction

In this paper we perform a detailed study of the masses and mixing of the single charmed scalar tetraquarks: $cq\overline{q}\overline{q}$. We also give a systematic analysis of these tetraquark states by weight diagrams, quantum numbers, and flavor wave functions. Tetraquark masses are calculated using four different fits. The following $\mathrm{SU}(3{)}_{\mathrm{F}}$ representations are discussed: ${\overline{15}}_{S}$, ${\overline{3}}_{S}$, ${6}_{A}$, and ${\overline{3}}_{A}$. We use the flavor-spin Glozman-Riska interaction Hamiltonian with SU(3) flavor symmetry breaking. There are 27 different tetraquarks composed of a charm quark $c$ and of the three light flavors $u$, $d$, $s$: 11 cryptoexotic $(3{\mathrm{D}}_{s}^{+},4{\mathrm{D}}^{+},4{\mathrm{D}}^{0})$ and 16 explicit exotic states. We discuss ${\mathrm{D}}_{s}$ and its isospin partners in the same multiplet, as well as all the other four-quark states. Some explicit exotic states appear in the spectrum with the same masses as ${\mathrm{D}}_{s}^{+}(2632)$ in ${\overline{15}}_{S}$ and with the same masses as ${\mathrm{D}}_{s}^{+}(2317)$ in ${6}_{A}$ representation, which confirm the tetraquark nature of these states.

Once more about theKK¯molecule approach to the light scalars

We show that the recent paper [Eur. Phys. J. A 24, 437 (2005)], claiming that the radiative decays $\ensuremath{\phi}\ensuremath{\rightarrow}{a}_{0}(980)\ensuremath{\gamma}$ and $\ensuremath{\phi}\ensuremath{\rightarrow}{f}_{0}(980)\ensuremath{\gamma}$ should be of the same order of magnitude regardless of whether the ${a}_{0}(980)$ and ${f}_{0}(980)$ are compact four-quark states or extended $K\overline{K}$ molecule states, is misleading.

Do the QCD sum rules support four-quark states?

We test the validity of the QCD sum rules applied to the light scalar mesons, the charmed mesons ${D}_{s0}(2317)$ and ${D}_{s1}(2460)$, and the $X(3872)$ axial meson, considered as tetraquark states. We find that, with the studied currents, it is possible to find an acceptable Borel window only for the $X(3872)$ meson. In such a Borel window we have simultaneously a good operator product expansion convergence and a pole contribution which is bigger than the continuum contribution. These results may indicate that light many-quark states cannot be considered as resonances separated from the continuum.

Diquark and light four-quark states

Four-quark states with different internal clusters are discussed within the constituent quark model. It is pointed out that the diquark concept is not meaningful in the construction of a tetraquark interpolating current in the QCD sum-rule approach, and hence existing sum-rule studies of four-quark states are incomplete. An updated QCD sum-rule determination of the properties of diquark clusters is then used as input for the constituent quark model to obtain the masses of light ${0}^{++}$ tetraquark states (i.e. a bound state of two diquark clusters). The results support the identification of $\ensuremath{\sigma}(600)$, ${f}_{0}(980)$, and ${a}_{0}(980)$ as the ${0}^{++}$ light tetraquark states, and seem to be inconsistent with the tetraquark state interpretation of the new BES observations of the near-threshold $p\overline{p}$ enhancements, $X(1835)$ and $X(1812)$, with the possible exception that $X(1576)$ may be an exotic first orbital excitation of ${f}_{0}(980)$ or ${a}_{0}(980)$.

Testing the nature of the [formula omitted] and [formula omitted] states using QCD sum rules

We use the QCD sum rules to analize the masses and partial hadronic decays widths of the mesons D s J + ( 2317 ) and X ( 3872 ) , in the hypotesis that these mesons can be identified as four-quark states. We use a diquak-antidiquark current and work to the order of m s in full QCD. Our results indicate that a four-quark structure is acceptable for D s J + ( 2317 ) , but it is dificult to explain the very narrow width of the meson X ( 3872 ) .

QCD sum rule approach to the new mesons and the gDsJ DK coupling constant

We use diquark-antidiquark currents to investigate the masses and partial decay widths of the recently observed mesons D+sJ(2317), D*0(0)(2308) and X(3872), considered as four-quark states, in a QCD sum rule approach. In particular we investigate the coupling constant gDsJ DK. We found that the gDsJ DK obtained in this four-quark scenario is smaller than the coupling constant obtained when D+sJ(2317) is considered as a conventional cs state.