Bethe-Salpeter Wave Function Research Articles

Charmonium dissociation temperatures in lattice QCD with a finite volume technique

Dissociation temperatures of J/ψ, ψ′ and χc states play key roles in the sequential J/ψ suppression scenario for high-energy heavy-ion collisions. We report on a study of charmonium dissociation temperatures in quenched lattice QCD. On anisotropic lattices, we first subtract the effects of the constant mode in finite temperature meson correlators, which have led to unphysical results in previous studies. We then extract ground and first exited state masses by diagonalizing correlation functions among different source and sink operators. To distinguish bound states from scattering states, we first compare the charmonium mass spectra under different spatial boundary conditions, and examine the shape and the volume dependence of their Bethe–Salpeter wavefunctions. From these studies, we found, so far, no sign of scattering states up to about 2.3Tc.

Decay Constants of Vector Mesons

The light vector mesons are studied within the framework of the Bethe–Salpeter equation with the vector-vector-type flat-bottom potential. The Bethe–Salpeter wavefunctions and the decay constants of the vector mesons are obtained. All the obtained results, fρ, fφ, and fK*, are in agreement with the experimental values, respectively.

Strong decays of heavy baryons in the Bethe-Salpeter formalism

In this paper we study the properties of diquarks (composed of $u$ and/or $d$ quarks) in the Bethe-Salpeter (BS) formalism under the covariant instantaneous approximation. We calculate their BS wave functions and study their effective interaction with the pion. Using the effective coupling constant among the diquarks and the pion, in the heavy quark limit ${m}_{Q}\ensuremath{\rightarrow}\ensuremath{\infty}$, we calculate the decay widths of ${\ensuremath{\Sigma}}_{Q}^{(*)}$ ($Q=c$, $b$) in the BS formalism under the covariant instantaneous approximation and then give predictions of the decay widths $\ensuremath{\Gamma}({\ensuremath{\Sigma}}_{b}^{(*)}\ensuremath{\rightarrow}{\ensuremath{\Lambda}}_{b}+\ensuremath{\pi})$.

THE BETHE–SALPETER EQUATION APPROACH TO BOUND STATE: SCALAR–FERMION CASE AND SCALAR–SCALAR CASE

The general form of the Bethe–Salpeter wave functions for bound states comprising one scalar constituent and one fermion, or two scalars is presented. Using the reduced Salpeter equation obtained, we can work out the effective nonrelativistic potentials. And one new version of reduced Bethe–Salpeter equation is proposed by extending Gross approximation.

Instanton modifications of the bound state singularity in the Schwinger model

We consider the quark-antiquark Green's function in the Schwinger model with instanton contributions taken into account. Thanks to the fact that this function may analytically be found, we draw out singular terms, which arise due to the formation of the bound state in the theory--the massive Schwinger boson. The principal term has a pole character. The residue in this pole contains contributions from various instanton sectors: 0, {+-}1, {+-}2. It is shown that the nonzero ones change the factorizability property. The formula for the residue is compared to the Bethe-Salpeter wave function found as a field amplitude. Next, it is demonstrated, that apart from polar part, there appears in the Green's function also the weak branch point singularity of the logarithmic and dilogarithmic nature. These results are not in variance with the universally adopted S-matrix factorization.

COULOMB WAVE FUNCTIONS OF NUCLEON AND ROPER ON THE LATTICE

By applying a spatially extended sink operator, the nucleon two-point functions are calculated in Coulomb gauge from quenched lattice QCD with overlap fermions. The Bethe-Salpeter wave functions of nucleon and Roper resonance are extracted at different quark masses. The typical size of nucleon is about 0.7 fm, which seems insensitive to the quark masses involved in this work. It is also found that the wave function of Roper has a radial node.

Decay constants of the pseudoscalar charmonium and bottomonium

In this Letter, we investigate the structures of the pseudoscalar charmonium and bottomonium in the framework of the coupled rainbow Schwinger–Dyson equation and ladder Bethe–Salpeter equation with the confining effective potential (infrared modified flat bottom potential). As the current masses are very large, the dressing or renormalization for the c and b quarks are tender, however, mass poles in the timelike region are absent. The Euclidean time Fourier transformed quark propagator has no mass poles in the timelike region which naturally implements confinement. The Bethe–Salpeter wavefunctions for those mesons have the same type (Gaussian type) momentum dependence and center around zero momentum with spatial extension to about q2=1 GeV2 which happen to be the energy scale for chiral symmetry breaking, the strong interactions in the infrared region result in bound states. The decay constants for those pseudoscalar heavy quarkonia are compatible with the values of experimental extractions and theoretical calculations.

NUCLEON AND ROPER WAVEFUNCTIONS FROM OVERLAP FERMIONS

By applying a spatially extended sink operator, the nucleon two-point functions are calculated in Coulomb gauge from quenched lattice QCD with overlap fermions. The Bethe-Salpeter wavefunctions of nucleon and Roper resonance are extracted at different quark masses. The typical size of nucleon is about 0.7 fm, which seems insensitive to the quark masses involved in this work. It is also found that the wavefunction of Roper has a radial node.

Decay constants of the pseudoscalar mesons in the framework of the coupled Schwinger–Dyson equation and Bethe–Salpeter equation

In this article, we investigate the structures of the pseudoscalar mesons ( π, K, D, D s , B and B s ) in the framework of the coupled rainbow Schwinger–Dyson equation and ladder Bethe–Salpeter equation with the confining effective potential (infrared modified flat bottom potential). The Schwinger–Dyson functions for the u, d and s quarks are greatly renormalized at small momentum region and the curves are steep at about q 2 = 1 GeV 2 which indicates an explicitly dynamical symmetry breaking. The Euclidean time Fourier transformed quark propagators have no mass poles in the time-like region which naturally implements confinement. As for the c and b quarks, the current masses are very large, the renormalization are more tender, however, mass poles in the time-like region are also absent. The Bethe–Salpeter wavefunctions for those mesons have the same type (Gaussian type) momentum dependence and center around small momentum which indicate that the bound states exist in the infrared region. The decay constants for those pseudoscalar mesons are compatible with the values of experimental extractions and theoretical calculations, such as lattice simulations and QCD sum rules.

Decay constants of the pion and B mesons with the Bethe–Salpeter equation

In this Letter, we investigate the under-structures of the π and B mesons in the framework of the Bethe–Salpeter equation with the confining effective potential (infrared modified flat bottom potential). In bare quark–gluon vertex approximation, we obtain the algebraic expressions for the solutions of the coupled rainbow Schwinger–Dyson equation and ladder Bethe–Salpeter equation. Firstly, we neglect the rainbow Schwinger–Dyson equation, take the bare quark propagator and solve the Bethe–Salpeter equation numerically alone. Although the bare quark propagator cannot embody dynamical chiral symmetry breaking and has a mass pole in the time-like region, it can give reasonable results for the values of decay constants fπ and fB compared with the values of experimental data and other theoretical calculations, such as lattice simulations and QCD sum rules. Secondly, we explore those mesons within the framework of the coupled rainbow Schwinger–Dyson equation and ladder Bethe–Salpeter equation. The Schwinger–Dyson functions for the u and d quarks are greatly renormalized at small momentum region and the curves are steep at about q2=1 GeV2 which indicates an explicitly dynamical symmetry breaking. The Euclidean time Fourier-transformed quark propagator has no mass poles in the time-like region which naturally implements confinement. As for the b quark, the current mass is very large, the renormalization is more tender, however, mass pole in the time-like region is also absent. The Bethe–Salpeter wavefunctions for both the π and B mesons have the same type (Gaussian type) momentum dependence as the corresponding wavefunctions with the bare quark propagator, however, the quantitative values are changed and the values for the decay constants fπ and fB are changed correspondingly.

Quark Schwinger-Dyson evaluation of thel1,l2coefficients in the chiral Lagrangian

Using a systematic expansion of the quark-antiquark Bethe-Salpeter wave functions in the relativistic quark model and working to ${O(P}^{4})$ in the chiral limit, we are able to derive theoretical expressions relating the coefficients of the chiral Lagrangian ${l}_{1}{,l}_{2}$ to the underlying quark-antiquark wave functions and interaction kernels. This is accomplished by using a novel technique based on a Ward identity for the quark-antiquark ladder kernel which greatly simplifies the required effort. Numerical evaluations are performed in two simple specific models.

The pion form factor in improved lattice QCD

We calculate the electromagnetic form factor of the pion in lattice gauge theory. The non-perturbatively improved Sheikoleslami–Wohlert lattice action is used together with the O(a) improved current. The form factor is compared to results for other choices for the current and features of the structure of the pion deduced from the ‘Bethe–Salpeter wave function’ are discussed.

Spectrum of Binding System for Heavy Quark (Q) with an Anti-sbottom or for a Sbottom and Anti–sbottom Pair**The project supported in part by National Natural Science Foundation of China and a Grant of the Chinese Academy of Sciences for Outstanding Young Scholars

Since long-lived light bottom squark (sbottom) and its anti-particle with a mass close to the bottom quark have not been excluded by experiments so far, so we would like to consider such a sbottom to combine with its anti-particle to form a color singlet meson-like bound state or to combine with a common anti-quark to form a fermion-like one, or accordingly their anti-particles to form an anti-particle bound system. Namely we calculate the low-lying spectrum of the systems specifically based on QCD inspired potential model. To be relativistic as much as possible, we start with the framework of Bethe–Salpeter (BS) equation even for non-relativistic binding systems. Finally, we obtain the requested spectrum by constructing general forms of the BS wave functions and solving the BS equations under instantaneous approximation.

QCD sum rules for heavy baryons

We construct heavy baryon currents from the heavy quark limit of baryonic Bethe-Salpeter wave functions. We discuss the one-loop renormalization of these heavy baryon currents as well as their two-point correlators up to the order ${1/m}_{Q}.$ For the special case of the spin (3/2,5/2) doublet, we extract mass values from a QCD sum-rule analysis.

The Spin Symmetry of Heavy Baryons in the Framework of the Bethe–Salpeter Equation

We study the baryons containing a heavy quark in the framework of Bethe–Salpeter (BS) equation. The most general forms of the BS wavefunctions are given. In the heavy-quark limit we simplify the BS equations and we show clearly that the spin symmetry exists in heavy baryon states.

Space-Like Form of Bethe–Salpeter Equation: Bound State**The project supported by National Natural Science Foundation of China, the Foundation of National Education Committee and University of Science and Technology of China for the Excellent Personnel Returned to the Country

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.

THE BOUND STATE EQUATION FOR TWO GLUONS AND ITS SOLUTION

We derive the bound state equation for two gluons in relativistic quantum field theory, i.e. the Bethe–Salpeter (BS) equation for two gluons. To solve it, we choose the kernel as the sum of a one-gluon exchange potential, a contact interaction and a linear confining potential. Under instantaneous approximation, this BS equation is solved numerically. The spectrum and the BS wave function of the glueballs are obtained in this framework. The numerical results are in agreement with that of recent lattice calculation.

Bethe-Salpeter equation for heavy baryonsωQ(*)in the diquark picture

In the heavy quark limit, the heavy baryons ${\ensuremath{\omega}}_{Q}^{(*)}$ $(\ensuremath{\omega}$ stands for $\ensuremath{\Sigma},$ $\ensuremath{\Xi}$ or $\ensuremath{\Omega}$ and $Q=b$ or $c)$ are regarded as composed of a heavy quark and an axial vector, light diquark with good spin and isospin quantum numbers. Based on this diquark picture we establish the Bethe-Salpeter (BS) equation for ${\ensuremath{\omega}}_{Q}^{(*)}$ in the limit where the heavy quark has infinite mass, ${m}_{Q}\ensuremath{\rightarrow}\ensuremath{\infty}$. It is found that in this limit there are three components in the BS wave function for ${\ensuremath{\omega}}_{Q}^{(*)}$. Assuming the kernel to consist of a scalar confinement term and a one-gluon-exchange term we derive three coupled integral equations for the three BS scalar functions in the covariant instantaneous approximation. Numerical solutions for the three BS scalar functions are presented, including a discussion of their dependence on the various input parameters. These solutions are applied to calculate the Isgur-Wise functions $\ensuremath{\xi}(\ensuremath{\omega})$ and $\ensuremath{\zeta}(\ensuremath{\omega})$ for the weak transitions ${\ensuremath{\Omega}}_{b}^{(*)}\ensuremath{\rightarrow}{\ensuremath{\Omega}}_{c}^{(*)}$. Using these we give predictions for the Cabibbo-allowed nonleptonic decay widths and up-down asymmetries for ${\ensuremath{\Omega}}_{b}\ensuremath{\rightarrow}{\ensuremath{\Omega}}_{c}^{(*)}$ plus a pseudoscalar or vector meson.

Octet and decuplet baryons in a covariant and confining diquark-quark model

The baryon octet and decuplet masses and Bethe-Salpeter vertex and wave functions are calculated in the ladder approximation to the quark exchange between a scalar or axialvector diquark and a constituent quark. These functions reflecting full Lorentz covariance are given in terms of an expansion in Gegenbauer polynomials. In the rest frame of the baryon, a complete partial wave decomposition of the Bethe-Salpeter wave function is performed. The confinement of quarks and diquarks is implemented via a parametrization of the corresponding propagators. We also discuss some aspects of the momentum routing in the ladder approximation to the Bethe-Salpeter equation. Numerical results for the octet and decuplet masses with broken flavor SU(3) in the conserved isospin limit are presented.

Two-photon widths from the Bethe–Salpeter wave function

Two-photon widths of pseudoscalar quarkonium are calculated using a Bethe–Salpeter wave function, recently proposed by Gromes. The results are compared to the standard, nonrelativistic quark-model predictions. The nonrelativistic quark model requires a phenomenological mass-dependent factor to bring its calculated pion width in agreement with the experimental data. We find that the Bethe–Salpeter approach, even using a very simple model wave function, gives results that are close to the experimental data for the pion and at the same time the successful nonrelativistic quark-model predictions for heavier systems are not disturbed.