Productions of X(1835) as baryonium with sizable gluon content

X(1835) has been treated as a baryonium with sizable gluon content, and to be almost flavor singlet. This picture allows us to rationally understand X(1835) production in J/ψ radiative decays, and its large couplings with pˉ, η^′ππ. The processes ϒ(1S) → γX(1835) and J/ψ → ωX(1835) have been examined. It has been found that Br(ϒ(1S) → γX(1835))Br(X(1835) → pˉ) < 6.45×10^-7, which is compatible with CLEO's recent experimental result (Phys. Rev. D 73, 032001 (2006) hep-ex/0510015). The branching fractions Br(J/ψ → ωX(1835)), Br(J/ψ → ρX(1835)) with X(1835) → pˉ and X(1835) → η^′π⁺π^- have been estimated by the quark-pair creation model. We show that they are heavily suppressed, so the signal of X(1835) is very difficult, if not impossible, to be observed in these processes. The experimental checks for these estimations are expected. The existence of the baryonium nonet is conjectured, and a model-independent derivation of their production branching fractions is presented.     

Analysis of the X（1835） as a Baryonium State with the Bethe-Salpeter Equation

We take the X（1835） as a pseudoscalar baryonium state, and calculate the mass spectrum of the baryon-antibaryon bound states pp, ΣΣ, ΞΞ, and ΛΛ in the framework of the Bethe—Salpeter equation with a phenomenological potential. The numerical results indicate the pp, ΣΣ and ΞΞ bound states may exist, and the X（1835） can be tentatively identified as the pp bound state.     

Heavy-quark bound states in potentials with the Bethe-Salpeter equation

The spinless Bethe—Salpeter equation is solved for three attractive static quark-antiquark potentials of the form V(r)=–ar^–+br+c, 01, and the effective non-Coulombic power-law potential of the formV(r)=ar ^0.1+c to obtain the spin-averaged energy levels in bottomonium (b ) and charmonium (c ) families. The shifted 1/N expansion technique is used. Calculations of the energy eigenvalues are carried out up to third order and parameters of each potential are adjusted to obtain the best agreement with the experimental spin-averaged data (SAD). Flavor-dependent and flavor-independent cases are considered in this work.     

Bethe-Salpeter equation with the sine-Gordon interaction

An attempt is made to study the interaction Hamiltonian,H _int =Gψ ²(x)U(φ(x)) in the Bethe-Salpeter framework for the confined states of theψ particles interactingvia the exchange of theU field, whereU(φ) = cos (gφ). An approximate solution of the eigenvalue problem is obtained in the instantaneous approximation by projecting the Wick-rotated Bethe-Salpeter equation onto the surface of a four-dimensional sphere and employing Hecke’s theorem in the weak-binding limit. We find that the spectrum of energies for the confined states,E =2m+B (B is the binding energy), is characterized byE ∼n ⁶, wheren is the principal quantum number.     

X(1835): A Possible Baryonium

We point out that the large p(p) coupling of X(1835) and the suppression of the three-body strange final states strongly indicate that X(1835) may be a p(p) baryonium. We also point out that the branching ratio of X(1835) →ηππ should be bigger than that of X(1835) →η′ππ. We suggest BES search X(1835) in the ηππ channel Finally, X(1835)may be used a tetraquark generator if X(1835) is really established as a baryonium state.     

Skyrme模型中的质子-反质子束缚态

BES(北京谱仪)发现了奇特粒子态X(1835). 述评近年来把X(1835)解释为Skyrme模型中的NN-重子偶素的工作. 有2个证据支持这种解释: 1) 存在NN-经典Skyrmion解,其结合能为～10MeV; 2) 这种Skyrmion-重子偶素的衰变是由X(1835)中的p-p经由量子隧穿而湮没所引起的, 因此最可几的衰变道是X(1835)→η4π 或 X→η' 2π. 这些导致了对BES实验数据的合理解释, 特别是关于X(1835)最可几的衰变模式的预言. 该预言对实验有价值.     

Some properties of the newly observed X(1835) state at BES

Recently the BES collaboration has announced the observation of a resonant state in the π⁺π^-η’ spectrum in J/ψ→γπ⁺π^-η’ decay. Fitting the data with a 0^-+ state, the mass is determined to be 1833.7 MeV with 7.7σ statistic significance. This state is consistent with the one extracted from previously reported pp̄ threshold enhancement data in J/ψ→γpp̄. We study the properties of this state using QCD anomaly and QCD sum rules, assuming X(1835) to be a pseudoscalar, and we show that it is consistent with the data. We find that this state has a sizeable matrix element , leading to branching ratios of (2.61–7.37)×10^-3 and (2.21–10.61)×10^-2 for J/ψ→γG_p and for G_p→π⁺π^-η^′, respectively. Combining the calculated branching ratio of J/ψ→γG_p and data on threshold enhancement in J/ψ→γpp̄, we determine the coupling for the G_p–p–p̄ interaction. We finally study the branching ratios of the other J/ψ→γ+three mesons decay modes. We find that J/ψ→γG_p→γ(π⁺π^-η,KKπ⁰) can provide useful tests for the mechanism proposed. PACS 11.55.Hx; 12.39.Fe; 12.39.Mk; 13.25.Gv     

Spinor Bethe-Salpeter equation with generalized ladder kernels and the Goldstein problem

The spinor Bethe-Salpeter equation is investigated for tightly bound fermion pairs. The covariant interaction kernel contains contributions of vector and axial-vector gluons within the framework of the Stückelberg formalism. The free gluon propagators of the strict ladder approximation are replaced by a convenient spectral form. This generalized ladder model can be extended to a large class of gauge field theories by specifying the spectral functions. The model-independent O(4) analysis of the Wick-rotated wave functions is carried out by using a complete set of four-dimensional scalar, vector, and tensor spherical harmonics. At vanishing center-of-mass energy, the radial Bethe-Salpeter equations can be classified in six disconnected sectors. All these equations are recorded in a general form which provides a study of the gauge dependence of the wave functions at short distances. Illustrative calculations are based on a simple Abelian field theory. In two Goldstein equations the leading singular term of the kernel may be absent by cancellation. In addition, one obtains a generalized Goldstein equation in which the kernel includes a gauge-independent marginally singular term. It is discussed how corrections of the large-distance behavior of the singular Goldstein kernel can lead to normalizable bound-state solutions without introducing a short-distance cutoff. Exact and numerical solutions are presented by using a simple parametrization of the kernel. In other sectors, the noncanonical angular behavior of the solutions may be avoided by prescribing a complex mass for the Stückelberg ghosts.     

Application of the Riemann method to the Bethe-Salpeter equation

The Bethe-Salpeter equation describing the interaction of two scalar particles via the exchange of a third scalar particle with mass 0 is in configuration space a hyperbolic partial differential equation of fourth order which will be studied with the help of the Riemann method. This method yields two Volterra equations the solutions of which are special solutions of the Bethe-Salpeter equation. The wave function is a superposition of the special solutions. For the coefficients one gets a system of two integral equations. The Fredholm determinant of the system is the generalization of the nonrelativistic Jost function.     

Numerical solution of the spinor Bethe-Salpeter equation and the goldstein problem

The spinor Bethe-Salpeter equation describing bound states of a fermion-antifermion pair with massless-boson exchange reduces to a single (uncoupled) partial differential equation for special combinations of the fermion-boson couplings. For spinless bound states with positive or negative parity this equation is a generalization to nonvanishing bound-state masses of the equations studied by Kummer and Goldstein, respectively. In the tight-binding limit the Kummer equation has a discrete spectrum, in contrast to the Goldstein equation, while for loose binding only the generalized Goldstein equation has a nonrelativistic limit. For intermediate binding energies the equations are solved numerically. The generalized Kummer equation is shown to possess a discrete spectrum of coupling constants for all bound-state masses. For the generalized Goldstein equation a discrete spectrum of coupling constants is found only if the binding energy is smaller than a critical value.     

Analysis of the charmonium-like states X~*(3860), X(3872), X(3915),X(3930) and X(3940) according to their strong decay behaviors

Inspired by the newly observed state X~*(3860), we analyze the strong decay behaviors of some charmoniumlike states, X~*(3860), X(3872), X(3915), X(3930) and X(3940), with the ~3P_0 model. We carry out our work based on the hypothesis that all of these states are charmonium systems. Our analysis indicates that, as a 0~(++) charmonium state, X~*(3860) can reproduce the experimental data. As for X(3872), it can tentatively be interpreted as the mixture0 of a cc system and a D~*D~0 molecular state. If we consider X(3940) as a 3~1 S_0 state, its total width in the present work is much lower than the experimental result. Thus, the 3~1 S_0 charmonium state seems not to be a good candidate for X(3940). Furthermore, our analysis implies that it is reasonable to assign X(3915) and X(3930) to be the same state, 2~(++). However, combining our analysis with that in Refs.[14, 71], we speculate that X(3915)/X(3930) might also be the mixture of a cc system and a molecular state.     

Hölder-Banach space analysis of the Bethe-Salpeter equation

We find a Hölder Banach space in which the Bethe-Salpeter equation is a compact integral equation as it stands. We study the properties of the solution in preparation for an analysis of linear field theory models of 3-body amplitudes. In particular we study the properties of the Regge poles of the solution and prove the existence and uniqueness of on mass shell scattering amplitudes.     

Approximation methods in the context of the bound-state Bethe-Salpeter equation

Numerical approximation schemes of the Wick-rotated scalar Bethe-Salpeter equation are discussed for general local potentials with special emphasis on mesh-point methods. Convergence properties are obtained by considering the analytic properties of the kernel. To this end, the four-dimensional partial wave equations are formulated in a new representation-independent way. The close relationship of variational and mesh-point methods is demonstrated and the difficulties which arise if singular potentials are introduced are discussed. For marginal singular potentials those difficulties are overcome in a new way by redefining the corresponding two-particle Green's function. Numerical examples for this case are given.     

Complete reduction of fermion-antifermion Bethe-Salpeter equation with static kernel. II

Recent work with the Bethe-Salpeter (BS) equation for a $\text{spin}\text{-}\text{1}\text{2}$ fermion-antifermion system is developed further and extended in a number of ways. Static kernels are considered, as previously, but in the present case, the structure of the kernels serves to suppress completely all pair terms. The reduction of the 16 coupled integro-differential equations involving the BS bispinor wavefunctions can be carried out much more simply than before. The effective Hamiltonians of the eigenvalue equations which result for single scalar and vector wavefunctions are not only in formally closed operator form, correct to all orders in the coupling constant and $\text{v}\text{c}$, but have an explicit and simple structure. They depend at most linearly on the static potential, and also on powers of momentum and simple functions of the relativistic free-particle energies.Previous work is further generalized by the consideration of static four-vector interactions (vector gluons or photons) in an arbitrary “gauge” for equal mass particles and antiparticles on the one hand, and the fourth (scalar) component of a vector potential for unequal mass particle and antiparticle on the other hand, the latter in order to study the mass dependence of the effective interactions and of the wavefunctions.Spin-orbit, spin-spin, and tensor interactions are expected to arise from the Breit-like four-vector interactions in conventional $\text{v}\text{c}$ expansions, but the present results are correct to all orders in $\text{v}\text{c}$. Furthermore, such types of interactions also arise, as previously, for the case of a Coulomb-like potential, for which they are not, at first guess, expected. In the unequal mass case, an interesting new type of effective tensor interaction is also generated which serves to mix triplet and singlet states.Various special “gauges” are considered in the equal mass case. The general structure of the BS amplitude is also discussed, and specifically developed in some detail in perturbation theory for the ground state of positronium.While the current work should be considered as a general examination of the properties of a large class of BS equations of the Coulomb-Dirac or Breit-type, two areas for specific applications immediately come to mind: the calculation of positronium energy levels to very high order and the study of quark-antiquark binding involving both like and unlike quarks.     

Application of spectral representations to the nonrelativistic and the relativistic Bethe-Salpeter equation

The eigenvalue problem of the scalar Bethe-Salpeter equation is solved by application of the vertical Dyson representation. The method of solution is developed in complete analogy to the solution of Schrödinger's equation by a Stieltjes representation in the case of a Yukawa potential. The eigenvalues are zeros of a characteristic determinant, which can be understood as a generalization of the nonrelativistic Jost function.     

Relativistic remnants in the reduction of the Bethe-Salpeter equation to the Schrödinger equation

Following Salpeter, the Bethe-Salpeter equation for the bound system of two oppositely charged particles is reduced to a Schrödinger equation for each of the following cases: (a) both particles are spin 1/2 particles, (b) one particle is a spinor while the other is spinless, and (c) both particles are spinless. It is shown that ife is the magnitude of charge carried by each of the particles whose masses are set equal to the electron and proton masses then, strictly speaking, only in case (a) do we obtain the familiar Schrödinger equation for the hydrogen atom. The latter equation is recovered in the other two cases only if relativistic remnants—terms of the order of 10^?5 and smaller—are neglected in comparison with unity. Attention is drawn to a situation where such remnants may not be negligibly small, viz. the problem of confinement of quarks.     

O(4) Expansion of the Ladder Bethe-Salpeter Equation

The Bethe-Salpeter amplitude is expanded on a hyperspherical basis thereby reducing the original four-dimensional integral equation into an infinite set of coupled one-dimensional ones. It is shown that this representation offers a highly accurate method to determine numerically the bound state solutions. For generic cases only a few hyperspherical waves are needed to achieve convergence, both for the ground state as well as for radially or orbitally excited states. The wave function is reconstructed for several cases and in particular it is shown that it becomes independent of the relative time in the nonrelativistic regime. Received July 18, 1996; accepted in final form October 2, 1996