Bethe-Salpeter equations: mesons beyond the rainbow-ladder truncation

We investigate masses of light mesons from a coupled system of Dyson–Schwinger （DSE） and Bethe– Salpeter equations （BSE）,taking into account dominant non-Abelian,sub-leading Abelian,and dominant pion cloud contributions to the dressed quark-gluon vertex.The axial-vector Ward-Takahashi identity preserving Bethe-Salpeter kernel is constructed and the spectrum of light mesons calculated.Our model goes significantly beyond the rainbow-ladder.We find that sub-leading Abelian corrections are further dynamically suppressed,and that our results supersede early qualitative predictions from simple truncation schemes.     

Radial excited heavy mesons

In this study, the first radial excited heavy pseudoscalar and vector mesons (\begin{document}$\eta_c(2S)$\end{document}, \begin{document}$\psi(2S)$\end{document}, \begin{document}$B_c(2S)$\end{document}, \begin{document}$B^*_c(2S)$\end{document}, \begin{document}$\eta_b(2S)$\end{document}, and \begin{document}$\varUpsilon(2S)$\end{document}) are investigated using the Dyson-Schwinger equation and Bethe-Salpeter equation approach. It is shown that the effective interactions of the radial excited states are harder than those of the ground states. With the interaction well determined by fitting the masses and leptonic decay constants of \begin{document}$\psi(2S)$\end{document} and \begin{document}$\varUpsilon(2S)$\end{document}, the first radial excited heavy mesons could be quantitatively described in the rainbow ladder approximation. The masses and leptonic decay constants of \begin{document}$\eta_c(2S)$\end{document}, \begin{document}$B_c(2S)$\end{document}, \begin{document}$B^*_c(2S)$\end{document}, and \begin{document}$\eta_b(2S)$\end{document} are predicted.     

Cross-ladder effects in Bethe-Salpeter and light-front equations

The Bethe-Salpeter (BS) equation in Minkowski space for scalar particles is solved for a kernel given by a sum of ladder and cross-ladder exchanges. The solution of corresponding light-front (LF) equation, where we add the time-ordered stretched boxes, is also obtained. Cross-ladder contributions are found to be very large and attractive, whereas the influence of stretched boxes is negligible. Both approaches --BS and LF-- give very close results.     

Hadron properties and Dyson-Schwinger equations

An overview of the theory and phenomenology of hadrons and QCD is provided from a Dyson-Schwinger equation viewpoint. Following a discussion of the definition and realization of light-quark confinement, the nonperturbative nature of the running mass in QCD and inferences from the gap equation relating to the radius of convergence for expansions of observables in the current-quark mass are described. Some exact results for pseudoscalar mesons are also highlighted, with details relating to the U_A(1) problem, and calculated masses of the lightest J=0,1 states are discussed. Studies of nucleon properties are recapitulated upon and illustrated: through a comparison of the ln-weighted ratios of Pauli and Dirac form factors for the neutron and proton; and a perspective on the contribution of quark orbital angular momentum to the spin of a nucleon at rest. Comments on prospects for the future of the study of quarks in hadrons and nuclei round out the contribution.     

Space-Like Form of Bethe-Salpeter Equation: Bound State

According to Dirac's principle, we apply the space-like consistency conditions in a relativistic theory to two-particle system and then define the space-Like wavefunctions through introducing a space-like factor, which is equivalent to Bethe-Salpeter wavefunction in physical content. The space-like form of Bethe-Salpeter equation of bound states is derived in terms of the universal rearranging technology of interaction kernel. Its advantages are of explicit Lorentz-covariant form and the difficulty of ghost states is automatically overcome. We also discuss the normalization condition of the space-like function.     

Solving Bethe-Salpeter equation in Minkowski space

We develop a new method of solving the Bethe-Salpeter (BS) equation in Minkowski space. It is based on projecting the BS equation on the light-front (LF) plane and on the Nakanishi integral representation of the BS amplitude. This method is valid for any kernel given by the irreducible Feynman graphs. For massless ladder exchange, our approach reproduces analytically the Wick-Cutkosky equation. For massive ladder exchange, the numerical results coincide with the ones obtained by Wick rotation.     

Bethe-Salpeter dynamics for two-photon processes

Recent experimental data on single hadron production by two-photon beams inpetra andpep have provided a unique opportunity for testing specific models of confinement through a study of one of their cleanest predictionsviz the γγ →H amplitudes. Motivated by this new facility, aqcd-oriented Bethe-Salpeter model of harmonic confinement, which has already been found to describe rather well several classes of hadronic data (from mass spectra to electromagnetic and pionic couplings), is now employed for a detailed comparison of its predictions onP → γγ andT → γγ couplings with the data. The agreement is quite good for all cases except one (η → γγ). This paper is offered as a “Festschrift” in honour of Dr Raja Ramanna on the occasion of his sixtieth birthday. The subject is theoretical in content but seeks to exploit an entirely new window opened by the latest experimental technology (on two-photon physics). As such it is appropriately dedicated to one of the main architects of nuclear science in India.     

Spectra of Free Diquark in the Bethe-Salpeter Approach

In this work, we employ the Bethe-Salpeter （B-S） equation to investigate the spectra of free diquarks and their B-S wave functions. We find that the B-S approach can be consistently applied to study the diqaurks with two heavy quarks or one heavy and one light quarks, but for two light-quark systems, the results are not reliable. There are a few free parameters in the whole scenario which can only be fixed phenomenologically. Thus, to determine them, one has to study baryons which are composed of quarks and diquarks.     

Spectra of Free Diquark in the Bethe-Salpeter Approach

In this work, we employ the Bethe-Salpeter (B-S) equation to investigate the spectra of free diquarks and their B-S wave functions. We find that the B-S approach can be consistently applied to study the diqaurks with two heavy quarks or one heavy and one light quarks, but for two light-quark systems, the results are not reliable. There are a few free parameters in the whole scenario which can only be fixed phenomenologically. Thus, to determine them, one has to study baryons which are composed of quarks and diquarks.     

Space-Like Form of Bethe-Salpeter Equation: Scatter State

According to Dirac, and based on our space-like form of Bethe-Salpeter equation of bound state, we extend the definition of the space-like wavefunctions and derive out the spacelike form of Bethe-Salpeter equation for scatter state. The solution of the inhomogeneous term is also obtained in detail so as to clarify the significance of inhomogeneous space-like form of Bethe-Salpeter equation for scatter state. Moreover the translation method is finally discussed.     

Space-Like Form of Bethe-Salpeter Equation: Many Particles

The Gell-Mann and Low's technology and Bethe-Salpeter equation for manyparticles are first discussed. Then the space-like form of Bethe-Salpeter equation involving both the bound state and scatter state are extended to many-particle case. Consequently we build the formalism of the space-like equation with explicit Lorentz covariant form and without the difficulty of ghost states automatically.     

Exploring the light-quark interaction

Two basic motivations for an upgraded JLab facility are the needs: to determine the essential nature of light-quark confinement and dynamical chiral symmetry breaking (DCSB); and to understand nucleon structure and spectroscopy in terms of QCD's elementary degrees of freedom. During the next ten years a programme of experiment and theory will be conducted that can address these questions. We present a Dyson-Schwinger equation perspective on this effort with numerous illustrations, amongst them: an interpretation of string-breaking; a symmetry-preserving truncation for mesons; the nucleon's strangeness σ-term; and the neutron's charge distribution.     

Exploring the light-quark interaction

Two basic motivations for an upgraded JLab facility are the needs： to determine the essential nature of light-quark confinement and dynamical chiral symmetry breaking （DCSB）; and to understand nucleon structure and spectroscopy in terms of QCD＇s elementary degrees of freedom. During the next ten years a programme of experiment and theory will be conducted that can address these questions. We present a Dyson- Schwinger equation perspective on this effort with numerous illustrations, amongst them： an interpretation of string~breaking; a symmetry-preserving truncation for mesons; the nucleon＇s strangeness σ-term; and the neutron＇s charge distribution.     

单胶子交换梯形近似下Bethe－Salpeter方程赝标解

将在动量空间具有积分形式的单胶子交换梯形近似下Ｂｅｔｈｅ－Ｓａｌｐｅｔｅｒ方程化为微分方程，求出该方程在四动量为零时的赝标解全部分量，其中第一分量为已知的Ｇｏｌｄｓｔｅｉｎ解．     

The generalized parton distribution of the pion in a relativistic Bethe-Salpeter quark model

We present a Poincaré covariant calculation of the generalized parton distribution of the pion. Results for different values of the kinematical parameters are shown and discussed.     

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.     

Calculation of Kaon Electromagnetic Form Factor in the Framework of Coupled Schwinger—Dyson and Bethe—Salpeter Formulation

The kaon electromagnetic form factor is calculated in the framework of coupled Schwinger-Dyson equation in rainbow approximation and Bethe-Salpeter equation in ladder approximation with the modified flat-bottom potential,which is the combination of the flat-bottom potential with considerations for the infrared and ultraviolet asymptotic behaviours of the effective quark-gluon coupling.All our numerical results give good fit to experimental values and other theoretical results.     

New Closed Expression of Interaction Kernel in Bethe-Salpeter Equation for Quark-Antiquark Bound States

The interaction kernel in the Bethe-Salpeter equation for quark-antiquark bound states is derived newly from QCD in the case where the quark and the antiquark are of different flavors. The technique of the derivation is the usage of the irreducible decomposition of the Green‘s functions involved in the Bethe-Salpeter equation satisfied by the quark-antiquark four-point Green‘s function. The interaction kernel derived is given a closed and explicit expression which shows a specific structure of the kernel since the kernel is represented in terms of the quark, antiquark and gluon propagators and some kinds of quark, antiquark and/or gluon three, four, five and six-point vertices. Therefore,the expression of the kernel is not only convenient for perturbative calculations, but also suitable for nonperturbative investigations.     

A seven-mode truncation of the plane incompressible Navier-Stokes equations

A model obtained by a seven-mode truncation of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is studied. This model, extending a previously studied five-mode one, exhibits a very rich and varied phenomenology including some remarkable properties of hysteresis (i.e., coexistence of attractors). A stochastic behavior is found for high values of the Reynolds number, when no stable fixed points, closed orbits, or tori are present.     

Investigating S-wave bound states composed of two pseudoscalar mesons

In this study, we systematically investigated two-pseudoscalar meson systems with the Bethe-Salpeter equation in the ladder and instantaneous approximations. By solving the Bethe-Salpeter equation numerically with the kernel containing the one-particle exchange diagrams, we found that the \begin{document}$ K\bar{K} $\end{document}, \begin{document}$ DK $\end{document}, \begin{document}$ B\bar{K} $\end{document}, \begin{document}$ D\bar{D} $\end{document}, \begin{document}$ B\bar{B} $\end{document}, \begin{document}$ BD $\end{document}, \begin{document}$ D\bar{K} $\end{document}, \begin{document}$ BK $\end{document}, and \begin{document}$ B\bar{D} $\end{document} systems with \begin{document}$ I=0 $\end{document} can exist as bound states. We also studied the contributions from heavy meson (\begin{document}$ J/\psi $\end{document} and \begin{document}$\Upsilon $\end{document}) exchanges and found that the contributions from heavy meson exchanges cannot be ignored.