Analysis of the tetraquark and hexaquark molecular states with the QCD sum rules

In this article, we construct the color-singlet-color-singlet type currents and the color-singlet-colorsinglet-color-singlet type currents to study the scalar D*■*, D*D* tetraquark molecular states and the vector D*D*■*, D*D*D* hexaquark molecular states with the QCD sum rules in details. In calculations, we choose the pertinent energy scales of the QCD spectral densities with the energy scale formula ■for the tetraquark and hexaquark molecular states respectively in a consistent way. We obtain stable QCD sum rules for the scalar D*■*, D*D*tetraquark molecular states and the vector D*D*■* hexaquark molecular state, but cannot obtain stable QCD sum rules for the vector D*D*D* hexaquark molecular state. The connected(nonfactorizable)Feynman diagrams at the tree level(or the lowest order) and their induced diagrams via substituting the quark lines make positive contributions for the scalar D*D* tetraquark molecular state, but make negative or destructive contributions for the vector D*D*D* hexaquark molecular state. It is of no use or meaningless to distinguish the factorizable and nonfactorizable properties of the Feynman diagrams in the color space in the operator product expansion so as to interpret them in terms of the hadronic observables, we can only obtain information about the short-distance and long-distance contributions.     

Analysis of hidden-charm pentaquark molecular states with and without strangeness via the QCD sum rules

In this study, we investigate the \begin{document}$\bar{D}\Sigma_c$\end{document}, \begin{document}$\bar{D}\Xi^\prime_c$\end{document}, \begin{document}$\bar{D}\Sigma_c^*$\end{document}, \begin{document}$\bar{D}\Xi_c^*$\end{document}, \begin{document}$\bar{D}^{*}\Sigma_c$\end{document}, \begin{document}$\bar{D}^{*}\Xi^\prime_c$\end{document}, \begin{document}$\bar{D}^{*}\Sigma_c^*$\end{document}, and \begin{document}$\bar{D}^{*}\Xi_c^*$\end{document} pentaquark molecular states with and without strangeness via the QCD sum rules in detail, focusing on the light flavor, \begin{document}$SU(3)$\end{document} , breaking effects, and make predictions for new pentaquark molecular states besides assigning \begin{document}$P_c(4312)$\end{document}, \begin{document}$P_c(4380)$\end{document}, \begin{document}$P_c(4440)$\end{document}, \begin{document}$P_c(4457)$\end{document} , and \begin{document}$P_{cs}(4459)$\end{document} self-consistently. In the future, we can search for these pentaquark molecular states in the decay of \begin{document}$\Lambda_b^0$\end{document}, \begin{document}$\Xi_b^0$\end{document} , and \begin{document}$\Xi_b^-$\end{document} . Furthermore, we discuss high-dimensional vacuum condensates in detail.     

Analysis the f 0(980) and a 0(980) mesons as four-quark states with the QCD sum rules

In this article, we take the point of view that the scalar mesons f₀(980) and a₀(980) are the diquark-antidiquark states , and we devote our attention to the determination of their masses in the framework of the QCD sum rule approach with the interpolating currents constructed from scalar-scalar type and pseudoscalar-pseudoscalar type diquark pairs respectively. The numerical results indicate that the scalar mesons f₀(980) and a₀(980) may have two possible diquark-antidiquark substructures.Received: 27 January 2005, Revised: 22 March 2005, Published online: 31 May 2005PACS: 12.38.Lg; 13.25.Jx; 14.40.Cs     

Analysis of Pcs(4338) and related pentaquark molecular states via QCD sum rules

In this study, we tentatively identify \begin{document}$ P_{cs}(4338) $\end{document} as the \begin{document}$ \bar{D}\Xi_c $\end{document}molecular state and distinguish the isospins of current operators to explore in detail the\begin{document}$ \bar{D}\Xi_c $\end{document}, \begin{document}$ \bar{D}\Lambda_c $\end{document}, \begin{document}$ \bar{D}_s\Xi_c $\end{document}, \begin{document}$ \bar{D}_s\Lambda_c $\end{document}, \begin{document}$ \bar{D}^*\Xi_c $\end{document}, \begin{document}$ \bar{D}^*\Lambda_c $\end{document}, \begin{document}$ \bar{D}^*_s\Xi_c $\end{document}, and \begin{document}$ \bar{D}^*_s\Lambda_c $\end{document} molecular states without strange, with strange, and with double strange in the framework of QCD sum rules. The present exploration favors identifying \begin{document}$ P_{cs}(4338) $\end{document} (\begin{document}$ P_{cs}(4459) $\end{document}) as the \begin{document}$ \bar{D}\Xi_c $\end{document} (\begin{document}$ \bar{D}^*\Xi_c $\end{document}) molecular state with the spin-parity \begin{document}$ J^P={\dfrac{1}{2}}^- $\end{document} (\begin{document}$ {\dfrac{3}{2}}^- $\end{document}) and isospin \begin{document}$ (I,I_3)=(0,0) $\end{document}, and the observation of their cousins with the isospin \begin{document}$ (I,I_3)=(1,0) $\end{document} in the \begin{document}$ J/\psi\Sigma^0/\eta_c\Sigma^0 $\end{document} invariant mass distributions would decipher their inner structures.     

Analysis of the strong decays of Pc(4312) as a pentaquark molecular state with QCD sum rules

In this article, we tentatively assign \begin{document}$P_c(4312)$\end{document} to be the \begin{document}$\bar{D}\Sigma_c$\end{document} pentaquark molecular state with the spin-parity \begin{document}$J^P={\frac{1}{2}}^-$\end{document} , and discuss the factorizable and non-factorizable contributions in the two-point QCD sum rules for the \begin{document}$\bar{D}\Sigma_c$\end{document} molecular state in detail to prove the reliability of the single pole approximation in the hadronic spectral density. We study its two-body strong decays with the QCD sum rules, and special attention is paid to match the hadron side with the QCD side of the correlation functions to obtain solid duality. We obtain the partial decay widths \begin{document}$\Gamma\left(P_c(4312)\to \eta_c p\right)=0.255\,\,{\rm{MeV}}$\end{document} and \begin{document}$\Gamma\left(P_c(4312)\to J/\psi p\right)=9.296^{+19.542}_{-9.296}\,\,{\rm{MeV}}$\end{document} , which are compatible with the experimental value of the total width, and support assigning \begin{document}$P_c(4312)$\end{document} to be the \begin{document}$\bar{D}\Sigma_c$\end{document} pentaquark molecular state.     

Thermal QCD sum rules study of vector charmonium and bottomonium states

We calculate the masses and leptonic decay constants of the heavy vector quarkonia, J/ψ and ϒ mesons at finite temperature. In particular, considering the thermal spectral density as well as additional operators coming up at finite temperature, the thermal QCD sum rules are acquired. Our numerical calculations demonstrate that the masses and decay constants are insensitive to the variation of temperature up to T ≅ 100 MeV, however after this point, they start to fall altering the temperature. At deconfinement temperature, the decay constants attain roughly to 45% of their vacuum values, while the masses are diminished about 12%, and 2.5% for J/ψ and ϒ states, respectively. The obtained results at zero temperature are in good consistency with the existing experimental data as well as predictions of the other nonperturbative models.     

Analysis of (1/2)~+ baryon states containing fourth-family quarks from QCD sum rules

When the fourth generation of quarks have sufficiently small mixing with ordinary standard-model quarks, the hadrons made up from these quarks can be long-lived enough. We analyze the (1/2)⁺ baryon states containing fourth-generation quarks and standard-model quarks, i.e. the charm or bottom quarks, in the QCD sum rules approach. Considering the perturbative and two gluon condensate contributions in the calculation, we give the numerical results of the masses and pole residues.     

QCD sum rules with finite masses

The concept of QCD sum rules is extended to bound states composed of particles with finite mass such as scalar quarks or strange quarks. It turns out that mass corrections become important in this context. The number of relevant corrections is analyzed in a systematic discussion of the IR- and UV-divergencies, leading in general to a finite number of corrections. The results are demonstrated for a system of two massless quarks and two heavy scalar quarks.We wish to thank Dr. Lech Mankiewicz for very helpful discussions. This work was supported by DFG (G. Hess program).     

Analysis of the <Emphasis Type="Italic">Y</Emphasis>(4140) and related molecular states with QCD sum rules

In this article, we assume that there exist scalar D^*[`(D)]<sup>*</sup>{D}^{\ast}{\bar {D}}^{\ast}, D<sub>s</sub><sup>*</sup>[`(D)]_s^*{D}_{s}^{\ast}{\bar{D}}_{s}^{\ast}, B^*[`(B)]<sup>*</sup>{B}^{\ast}{\bar {B}}^{\ast} and B<sub>s</sub><sup>*</sup>[`(B)]_s^*{B}_{s}^{\ast}{\bar{B}}_{s}^{\ast} molecular states, and study their masses using the QCD sum rules. The numerical results indicate that the masses are about (250–500) MeV above the corresponding D ^*–[`(D)]<sup>*</sup>{\bar{D}}^{\ast}, D <sub>s</sub> <sup>*</sup>–[`(D)]_s^*{\bar {D}}_{s}^{\ast}, B ^*–[`(B)]<sup>*</sup>{\bar{B}}^{\ast} and B <sub>s</sub> <sup>*</sup>–[`(B)]_s^*{\bar {B}}_{s}^{\ast} thresholds, the Y(4140) is unlikely a scalar D_s^*[`(D)]<sub>s</sub><sup>*</sup>{D}_{s}^{\ast}{\bar{D}}_{s}^{\ast} molecular state. The scalar D<sup>*</sup>[`(D)]^*D^{\ast}{\bar{D}}^{\ast}, D_s^*[`(D)]<sub>s</sub><sup>*</sup>D_{s}^{\ast}{\bar{D}}_{s}^{\ast}, B<sup>*</sup>[`(B)]^*B^{\ast}{\bar{B}}^{\ast} and B_s^*[`(B)]<sub>s</sub><sup>*</sup>B_{s}^{\ast}{\bar{B}}_{s}^{\ast} molecular states maybe not exist, while the scalar D￠<sup>*</sup>[`(D)]^￠*{D'}^{\ast}{\bar{D}}^{\prime\ast}, D_s^￠*[`(D)]<sub>s</sub><sup>￠*</sup>{D}_{s}^{\prime\ast}{\bar{D}}_{s}^{\prime\ast}, B<sup>￠*</sup>[`(B)]^￠*{B}^{\prime\ast}{\bar{B}}^{\prime\ast} and B_s^￠*[`(B)]_s^￠*{B}_{s}^{\prime\ast}{\bar{B}}_{s}^{\prime\ast} molecular states maybe exist.     

Analysis of the QCD sum rules for the heavy quarkonium

We study phenomenologically the QCD sum rules given by Shifman et al. for the heavy quarkonium. In the charmonium sum rules, we find that the contribution of the physical continuum to the moment ?_n is consistent with that of the effective one. As for the bottonium, the sum rules corrected by the Coulomb-like interaction are saturated very well by the four resonances observed at CESR. It is predicted that the³S₁ ground state of the toponium must exist in the range of \(M_{t\bar t} \) =30–44GeV, if the sum rules for the top quark are assumed.     

Analysis of the scalar doubly charmed hexaquark state with QCD sum rules

In this article, we study the scalar-diquark–scalar-diquark–scalar-diquark type hexaquark state with the QCD sum rules by carrying out the operator product expansion up to the vacuum condensates of dimension 16. We obtain a lowest hexaquark mass of \(6.60^{+0.12}_{-0.09}\,\mathrm {GeV}\), which can be confronted with the experimental data in the future.     

0~+ tetraquark states from improved QCD sum rules:delving into X(5568)

In order to investigate the possibility of the recently observed X(5568) being a 0~+ tetraquark state, we make an improvement to the study of the related various configuration states in the framework of the QCD sum rules. Particularly, to ensure the quality of the analysis, condensates up to dimension 12 are included to inspect the convergence of operator product expansion(OPE) and improve the final results of the studied states. We note that some condensate contributions could play an important role on the OPE side. By releasing the rigid OPE convergence criterion, we arrive at the numerical value 5.57+0.35 for the scalar-scalar diquark-antidiquark 0~+ -0.23 Ge Vstate, which agrees with the experimental data for the X(5568) and could support its interpretation in terms of a 0~+ tetraquark state with the scalar-scalar configuration. The corresponding result for the axial-axial current is calculated to be 5.77+0.44 still consistent with the mass of X(5568) in view of the uncertainty. The feasibility of-0.33 Ge V, which isX(5568) being a tetraquark state with the axial-axial configuration therefore cannot be definitely excluded. For the pseudoscalar-pseudoscalar and the vector-vector cases, their unsatisfactory OPE convergence make it difficult to find reasonable work windows to extract the hadronic information.     

Analysis of (1/2)+ baryon states containing fourth-family quarks from QCD sum rules

When the fourth generation of quarks have sufficiently small mixing with ordinary standard-model quarks, the hadrons made up from these quarks can be long-lived enough. We analyze the (1/2)⁺ baryon states containing fourth-generation quarks and standard-model quarks, i.e. the charm or bottom quarks, in the QCD sum rules approach. Considering the perturbative and two gluon condensate contributions in the calculation, we give the numerical results of the masses and pole residues.     

QCD sum rules and pion wave functions

We consider light cone sum rules for a vertex function involving a pion state. These incorporate radiative corrections, continuum effects and power corrections; the latter depend on the non-asymptotic form of a higher twist component of the pion wave function. We derive restrictions on this component from sum rules for two-point functions and propose a model wave function for it. Finally we analyse our vertex sum rules in the light of this information and find results for the lowest twist component in good agreement with those already obtained from two-point function sum rules with derivative currents.