In this article, we take the X(1835) as a pseudoscalar baryonium state, and calculate the mass spectrum of the baryon-antibaryon bound states $p\bar{p}$, $Σ\barΣ$, $Ξ\barΞ$, and $Λ\barΛ$ in the framework of the Bethe-Salpeter equation with a phenomenological potential. The numerical results indicate the $p\bar{p}$, $Σ\barΣ$ and $Ξ\barΞ$ bound states maybe exist, and the $X(1835)$ can be tentatively identified as the $p\bar{p}$ bound state.
In this article, we study the mass spectrum of the baryon-antibaryon bound states $p\bar{p}$, $Σ\barΣ$, $Ξ\barΞ$, $Λ\barΛ$, $p\bar{N}(1440)$, $Σ\barΣ(1660)$, $Ξ\barΞ^\prime$ and $Λ\barΛ(1600)$ with the Bethe-Salpeter equation. The numerical results indicate that the $p\bar{p}$, $Σ\barΣ$, $Ξ\barΞ$, $p\bar{N}(1440)$, $Σ\barΣ(1660)$, $Ξ\barΞ^\prime$ bound states maybe exist, and the new resonances X(1835) and X(2370) can be tentatively identified as the $p\bar{p}$ and $p\bar{N}(1440)$ (or $N(1400)\bar{p}$) bound states respectively with some gluon constituents, and the new resonance X(2120) may b…
Rare radiative leptonic decay $B\to l\bar ν_l γ$ is studied in the instantaneous Bethe-Salpeter approach. The results are compared to other relativistic quark model predictions.
In this article, we take into account the one-pion exchange force besides the one-gluon exchange force to study the mass difference of the $π$ and $ρ$ mesons with the Bethe-Salpeter equation. After projecting the Bethe-Salpeter equation into an simple form, we can see explicitly that the bound energy $|E_π|\gg |E_ρ|$.
We study the $s$-wave kaon-nucleon bound state with the strangeness $S=-1$ in the Bethe-Salpeter formalism in the ladder and instantaneous approximations. We solve the Bethe-Salpeter equation of the bound state and obtain the Bethe-Salpeter amplitude. It is shown that the $K^-p$ bound state exists in this formalism. We also study the decay width of the bound state based on the Bethe-Salpeter techniques. The mass of this bound state is 1422 MeV and its decay width is obviously smaller than that of $Λ(1405)$. These results indicate that there may be some other structures in the observed resonanc…
The Bethe--Salpeter equation for the pion in chiral symmetric models is studied with a special care to consistency with low-energy relations. We propose a reduction of the rainbow Schwinger--Dyson and the ladder Bethe--Salpeter equations with a dressed gluon propagator. We prove that the reduction preserves the Ward--Takahashi identity for the axial-vector current and the PCAC relation.
The Schwinger-Dyson Bethe-Salpeter approach to the bound state problem is applied to the spin zero glueball spectrum. Although a moderately successful description of the two point functions and the glueball spectrum can be obtained, further work is required to reduce sensitivity to model truncations.
The parametrization of the realistic Bethe-Salpeter amplitude for the deuteron is given. Eight components of the amplitude in the Euclidean space are presented as an analytical fit to the numerical solution of the Bethe-Salpeter equation in the ladder approximation. An applicability of the parametrization to the observables of the deuteron is briefly discussed.
We present the Minkowski space solutions of the inhomogeneous Bethe-Salpeter equation for spinless particles with a ladder kernel. The off-mass shell scattering amplitude is first obtained.
The electromagnetic form factors calculated through Euclidean Bethe-Salpeter amplitude and through the light-front wave function are compared with the one found using the Bethe-Salpeter amplitude in Minkowski space. The form factor expressed through the Euclidean Bethe-Salpeter amplitude (both within and without static approximation) considerably differs from the Minkowski one, whereas form factor found in the light-front approach is almost indistinguishable from it.
We evaluate the complete spectrum of the Bc mesons, below the open flavor BD threshold, in a Bethe-Salpeter model. We make predictions for the radiative decay widths of the Bc excited states. We compare our results with those of other models.
The Bethe-Salpeter equation for Wick-Cutkosky like models is solved in dressed ladder approximation. The bare vertex truncation of the Dyson-Schwinger equations for propagators is combined with the dressed ladder Bethe-Salpeter equation for the scalar S-wave bound state amplitudes. With the help of spectral representation the results are obtained directly in Minkowski space. We give a new analytic formula for the resulting equation simplifying the numerical treatment. The bare ladder approximation of Bethe-Salpeter equation is compared with the one with dressed ladder. The elastic electromagne…
We solve the Bethe-Salpeter equation in order to determine the spectrum of pseudoscalar and vector meson bound states for light as well as heavy quarks. The fermion propagators are obtained by solving the Schwinger-Dyson equation consistently with the Bethe-Salpeter equation, a procedure necessary for demonstrating the Goldstone nature of the pion in the chiral limit. Our results agree qualitatively and quantitatively with expectations both from current algebra for light quarks and from composite models for heavy quarks.}
In a recent comment [1] Blankleider and Kvinikhidze claim that our work [2] is based on a set of inconsistent Bethe-Salpeter equations for the coupling of four-quark to two-quark states. Here we demonstrate that their argument is insubstantial.
We use the LSZ reduction theorem and interpolating fields, alongwith the heavy quark effective theory, to investigate the structure of the Bethe-Salpeter amplitude for heavy hadrons. We show how a simple form of this amplitude, used extensively in heavy hadron decay calculations, follows naturally upto $O(1/M)$ from these field theoretic considerations.
In this paper we derive a hierarchy of integral equations from the 4PI effective action which have the form of Bethe-Salpeter equations. We show that, together with the equation of motion for the self-consistent 4-vertex, these integral equations are closed, and that their expansions give infinite series of connected diagrams which have the correct symmetry.
The new method of solution of the Bethe-Salpeter equation for quark-antiquark pseudoscalar bound state is proposed. With the help of integral representation the results are directly obtained in Minkowski space. Dressing of Greens functions is naturally considered providing thus the correct inclusion of the running coupling constant and the quark propagators as well as. The first numerical results are presented for a simplified ladder approximation.
We present a calculation of the spectrum of light and heavy quark bound states in the rainbow-ladder truncation of Dyson-Schwinger/Bethe-Salpeter equations. By extending the formalism include the case of total angular momentum J=3, we are able to explore Regge trajectories and make prediction of tensor bound states for light and heavy quarkonia.
A new method of solution of the Bethe-Salpeter equation for a pseudoscalar quark-antiquark bound state is proposed. With the help of an integral representation, the results are directly obtained in Minkowski space. Dressing of Green's functions is naturally taken into account, thus providing the possible inclusion of a running coupling constant as well as quark propagators. First numerical results are presented for a simplified ladder approximation.
We interpret the $X_1(2900)$ as an $S$-wave $\bar{D}_1K$ molecular state in the Bethe-Salpeter equation approach with the ladder and instantaneous approximations for the kernel. By solving the Bethe-Salpeter equation numerically with the kernel containing one-particle-exchange diagrams and introducing three different form factors (monopole, dipole, and exponential form factors) in the verties, we find the bound state exists. We also study the decay width of the decay $X_1(2900)$ to $D^-K^+$.
We present a systematic algebraic and numerical investigation of the instantaneous Bethe-Salpeter equation. Emphasis is placed on confining interaction kernels of the Lorentz scalar, time component vector, and full vector types. We explore stability of the solutions and Regge behavior for each of these interactions, and conclude that only time component vector confinement leads to normal Regge structure and stable solutions.
We study the scalar bound states of $D^{\ast}\bar{D}^{\ast}$ and $B^{\ast}\bar{B}^{\ast}$ in the Bethe-Salpeter formalism, with the effective interaction kernel extracted from the chiral perturbative theory and the heavy quark effective theory in the ladder approximation and the covariant instantaneous approximation. The results show that, in the scalar case ($J=0$), there can only exist $I = 0$ bound states for parameters in proper range, while there cannot exist the $I = 1$ bound states in the whole reasonable parameter range, due to more constraints arising from our definition of the Bethe-…
The Bethe-Salpeter equation is used to comprehensively study mesons with J=0,1 and equal-mass constituents for quark masses from the chiral limit to the b-quark mass. The survey contains masses of the ground states in all corresponding J^{PC} channels including those with "exotic" quantum numbers. The emphasis is put on each particular state's sensitivity to the low- and intermediate-momentum, i.e., long-range part of the strong interaction.
In this work, we study the $B\bar{K}$ molecule in the Bethe-Salpeter (BS) equation approach. With the kernel containing one-particle-exchange diagrams and introducing two different form factors (monopole form factor and dipole form factor) in the vertex, we solve the BS equation numerically in the covariant instantaneous approximation. We investigate the isoscalar and isovector $B\bar{K}$ systems, and we find $X(5568)$ cannot be a $B\bar{K}$ molecule.
We evaluate the $π^0γ^* \to γ$ transition form factor and $γγ$ decay widths for $π^0, η_c$ and $η_b$ treated as $q\bar{q}$ bound states in the Bethe-Salpeter formalism, incorporating the dynamical chiral symmetry breaking and the Goldstone nature of the pion. In the chiral limit, the Abelian axial anomaly is incorporated analytically in our coupled Schwinger-Dyson and Bethe-Salpeter approach, which is also capable of quantitatively describing systems of heavy quarks, concretely $η_c$ and possibly $η_b$.
We discuss the possibility that the X(3872) can be a $D\bar{D}^*$ molecular bound state in the Bethe-Salpeter equation approach in the ladder and instantaneous approximations. We show that the $D\bar{D}^*$ bound state with quantum numbers $J^{PC}=1^{++}$ exists. We also calculate the decay width of $X(3872) \rightarrow γJ/ψ$ channel and compare our result with those from previous calculations.
We study a method for solving the homogeneous Bethe-Salpeter equation. By introducing a `fictitious' eigenvalue $λ$ the homogeneous Bethe-Salpeter equation is interpreted as a linear eigenvalue equation, where the bound state mass is treated as an input parameter. Using the improved ladder approximation with the constant fermion mass, we extensively study the spectrum of the fictitious eigenvalue $λ$ for the vector bound states and find the discrete spectrum for vanishing bound state mass. We also evaluate the bound state masses by tuning appropriate eigenvalues $λ$ to be unity, and find massl…
In this work, we assume that the observed state $Ξ(1620)$ is a $s$-wave $Λ\bar{K}$ or $Σ\bar{K}$ bound state. Based on this molecule picture, we establish the Bethe-Salpeter equations for $Ξ(1620)$ in the ladder and instantaneous approximations. We solve the Bethe-Salpeter equations for the $Λ\bar{K}$ and $Σ\bar{K}$ systems numerically and find that the $Ξ(1620)$ can be explained as $Λ\bar{K}$ and $Σ\bar{K}$ bound states with $J^P=1/2^-$, respectively. Then we calculate the decay widths of $Ξ(1620)\rightarrowΞπ$ in these two different molecule pictures systems, respectively.
We calculate the form factors for the semileptonic decays of the $B$ meson to $D$ and $D^*$ mesons in a Bethe-Salpeter model. We show that our model is consistent with the constraints of Heavy Quark Effective Theory (HQET) and we extract the matrix elements that represent the $1/m_Q$ corrections to the form factors in HQET. With available data, we obtain $V_{cb} $=$ (31.9 \pm 1.4)\times10^{-3}$.
CP asymmetry in some exclusive decays of charged B meson are calculated in a Bethe-Salpeter approach. Hadronic final state interactions are ignored. Complex decay amplitudes are assumed to arise entirely from perturbative quark-antiquark loops. Calculations are done both with and without the gluon quark-antiquark vacuum polarization loops. The effects of neglecting the imaginary parts arising from the diagonal quark-antiquark loops are also studied.
The light quark-antiquark scattering Green's function is considered near a meson resonance peak. The Bethe-Salpeter equation is used to write formal expressions for the resonance width/mass ratio. Arguments are made concerning to what extent this ratio can be calculated perturbatively, and an upper bound is placed on the growth of this ratio as a function of radial excitation. Certain mesons and their radial excitations are considered, as well as the more general issue of classifying mesons in the quark model.
We review in detail modern numerical methods used in the determination and solution of Bethe-Salpeter and Dyson--Schwinger equations. The algorithms and techniques described are applicable to both the rainbow-ladder truncation and its non-trivial extensions. We discuss pedagogically the steps involved in constructing conventional mesons and baryons as systems of two- and three-quarks respectively. As further application we describe the self-consistent calculation of form-factors and highlight the challenges that remain therein.
The heavy baryon $Λ_b $ is regarded as composed of a heavy quark and a scalar diquark which has good spin and isospin quantum numbers. In this picture, we calculate the electromagnetic (EM) form factors of $Λ_b$ in the Bethe-Salpeter equation approach. We find that the shapes of the EM form factors of $Λ_b$ are similar to those of $Λ$, which have a peak at about $ω=1$, but the amplitudes are much smaller than those of $Λ$.
In the heavy quark limit, we establish the Bethe-Salpeter equations for the ground state $QQ^{\prime} \bar u \bar d$ containing one heavy diquark $QQ^{\prime}$ ($Q, Q^{\prime}=b$ or $c$) and one light antidiquark $\bar u \bar d$. We solve the Bethe-Salpeter equations numerically in the covariant instantaneous approximations with the kernels containing a scalar confinement term and a one-gluon-exchange term. Numerical solutions for the Bethe-Salpeter wave functions are presented. The results show that the masses of $bb \bar u \bar d$, $bc \bar u \bar d$, and $cc \bar u \bar d$ bound states lie …
In this article, we study the structure of the ρmeson in the framework of the coupled rainbow Schwinger-Dyson equation and ladder Bethe-Salpeter equation with a confining effective potential. The u and d quark propagators get significantly modified, the mass poles are absent in the timelike region, which implements confinement naturally. The Bethe-Salpeter amplitudes of the ρmeson center around zero momentum and extend to the energy scale about $q^2=1GeV^2$, which happens to be the energy scale of chiral symmetry breaking, strong interactions in the infrared region result in bound state. The n…
We employ the framework of Bethe-Salpeter equation under Covariant Instantaneous Ansatz to study the leptonic decays of pseudoscalar mesons. The Dirac structure of hadron-quark vertex function $Γ$ is generalized to include various Dirac covariants besides $γ_5 $ from their complete set. The covariants are incorporated in accordance with a power counting rule, order by order in powers of the inverse of the meson mass. The decay constants are calculated with the incorporation of leading order covariants. Most of the results are dramatically improved.
We apply our Bethe-Salpeter model for mesons to the $B_c$ family with parameters fixed in our previous investigation. We evaluate the mass of the pseudo-scalar $B_c$ meson as 6.356 GeV/$c^2$ and 6.380 GeV/$c^2$ and the lifetime as 0.47 ps and 0.46 ps respectively in two reductions of the Bethe-Salpeter Equation, in good agreement with the recently reported mass of 6.40 $\pm$ 0.39 (stat.) $\pm$ 0.13 (syst.) GeV/$c^2$ and lifetime of $0.46^{+0.18}_{-0.16}$ (stat.) $\pm$ 0.03 (syst.) ps by the CDF Collaboration. We evaluate the decay constant of the $B_c$ meson and compare different contributions…
The Bethe-Salpeter equations for the light-quark composite systems, q q-bar, are written in terms of spectral integrals. For the q q-bar -mesons characterized by the mass M, spin J and radial quantum number n, the equations are presented for the following (n,M^2)-trajectories: pi_J, eta_J, a_J, f_J, rho_J, omega_J, h_J and b_J.
We study the possible bound states of the $K\bar K$ system in the Bethe-Salpeter formalism in the ladder and instantaneous approximations. We find that the bound states exist. However, these bound states have very small decay widths. Therefore, besides the possible $K\bar K$ component, there may be some other structures in the observed $f_0(980)$ and $a_0(980)$ .
We derive a confining $ q \bar{q}$ Bethe--Salpeter equation starting from the same assumptions on the Wilson loop integral already adopted in the derivation of a semirelativistic heavy quark potential. We show that, by standard approximations, an effective meson squared mass operator can be obtained from our BS kernel and that, from this, by ${1\over m^2}$ expansion, the corresponding Wilson loop potential is recovered, spin--dependent and velocity--dependent terms included. We also show, that, on the contrary, neglecting spin--dependent terms, relativistic flux tube model is reproduced.
We investigate the Bethe--Salpeter description of mesons in the limit where one of the constituing quarks is infinitely heavy. To recover the non--relativistic quark model out of the Bethe--Salpeter formalism it is usual to assume that the potential is instantaneous and the Fock space is reduced to $Q\bar{q}$ states. We show that in the Feynman gauge in perturbation theory and for a heavy quark being on--mass shell the instantaneity of the potential is valid in the rest--frame of the meson. Furthermore we argue that to first order in the velocity exchange during a heavy meson transition, the e…
The nuclear effects in the spin-dependent structure functions $g_1^D$ and $b_2^D$ are calculated in the relativistic approach based on the Bethe-Salpeter equation with a realistic meson-exchange potential. The results of calculations are compared with the non-relativistic calculations. The problem of extraction of the neutron spin structure function, $g_1^n$, from the deuteron data is discussed.
Dynamical breaking of chiral symmetry in effective models of QCD is studied. Introducing a cut-off function or a non-local interaction, the Noether current must be modified and thus the Ward-Takahashi identity and the PCAC relation are modified accordingly. We point out that the pion decay constant must be defined consistently with the Noether current so that the low-energy relations are satisfied. We define the proxy of the Noether current for general effective models, which is consistent with loop expansion of the Cornwall-Jackiw-Tomboulis effective action. A general formula for the pion dec…
We solved the instantaneous Bethe-Salpeter equation for heavy pseudoscalars in different kernels, where the kernels are obtained using linear scalar potential plus one gluon exchange vector potentials in Feynman gauge, Landau gauge, Coulomb gauge and time-component Coulomb gauge. We obtained the mass spectra of heavy pseudoscalars, and compared the results between different kernels, found that using the same parameters we obtain the smallest mass splitting in time-component Coulomb gauge, the similar largest mass splitting in Feynman and Coulomb gauges, middle size splitting in Landau gauge.
In this paper we study the properties of diquarks (composed of $u$ and/or $d$ quarks) in the Bethe-Salpeter 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 $m_Q\to\infty$, we calculate the decay widths of $Σ_Q^{(*)}$ ($Q=c,b$) in the BS formalism under the covariant instantaneous approximation and then give predictions of the decay widths $Γ(Σ_b^{(*)}\toΛ_b+π)$.
We employ the framework of Bethe-Salpeter equation under Covariant Instantaneous Ansatz (CIA) to study the leptonic decays of vector mesons. The structure of hadron-quark vertex function Gamma is generalized to include various Dirac covariants (other than i gamma dot epsilon) from their complete set. They are incorporated in accordance with a naive power counting rule order by order in powers of the inverse of the meson mass. The decay constants for rho omega and phi mesons are calculated with the incorporation of leading order covariants.
A new method for solving the Bethe-Salpeter equation is developed. It allows to find the Bethe-Salpeter amplitudes both in Minkowski and in Euclidean spaces and, as a by product, the light-front wave function. The method is valid for any kernel given by irreducible Feynman graphs. Bethe-Salpeter and Light-Front equations for scalar particles with ladder + cross-ladder kernel are solved.
We give a derivation of the particle-hole RPA equations for an interacting multi-fermion system by applying the instantaneous approximation to the amputated two-fermion propagator of the system. In relativistic field theory the same approximation leads from the fermion-antifermion Bethe-Salpeter equation to the Salpeter equation. We show that RPA equations and Salpeter equation are indeed equivalent.
A nontrivial instantaneous variant of the Bethe-Salpeter formalism allows us to discuss massless pseudoscalar mesons from a configuration-space-potential point of view.
The light pseudoscalar mesons play a twofold role: they may or have to be regarded both as low-lying bound states of the fundamental degrees of freedom of quantum chromodynamics as well as the (pseudo-) Goldstone bosons of the spontaneously broken chiral symmetries of quantum chromodynamics. We interrelate these aspects in a single quantum-field-theoretic approach relying on the Bethe-Salpeter formalism in instantaneous approximation by very simple means: the shape of the pseudoscalar-meson Bethe-Salpeter wave function dictated by chiral symmetry is used in Bethe-Salpeter equations for bound s…
The Bethe-Salpeter equation for bound states of a fermion-antifermion pair in the instantaneous approximation for the involved interaction kernel is converted into an equivalent matrix eigenvalue problem with explicitly (algebraically) given matrices.
Inversion enables the construction of interaction potentials underlying - under fortunate circumstances even analytic - instantaneous Bethe-Salpeter descriptions of all lightest pseudoscalar mesons as quark-antiquark bound states of Goldstone-boson nature.
I review, in a personal perspective, the history of the theory of non-relativistic bound states in QED and QCD from the Bethe-Salpeter equation to the construction of effective field theories.
We present a new method for solving the two-body Bethe-Salpeter equation in Minkowski space. It is based on the Nakanishi integral representation of the Bethe-Salpeter amplitude and on subsequent projection of the equation on the light-front plane. The method is valid for any kernel given by the irreducible Feynman graphs and for systems of spinless particles or fermions. The Bethe-Salpeter amplitudes in Minkowski space are obtained. The electromagnetic form factors are computed and compared to the Euclidean results.
The challenge to obtain from the Euclidean Bethe--Salpeter amplitude the amplitude in Minkowski is solved by resorting to un-Wick rotating the Euclidean homogeneous integral equation. The results obtained with this new practical method for the amputated Bethe--Salpeter amplitude for a two-boson bound state reveals a rich analytic structure of this amplitude, which can be traced back to the Minkowski space Bethe--Salpeter equation using the Nakanishi integral representation. The method can be extended to small rotation angles bringing the Euclidean solution closer to the Minkowski one and could…
We propose a new numerical method to compute parton distribution amplitude(PDA) from the Euclidean Bethe-Salpeter wave function. The essential step is to extract the weight function in the Nakanishi representation of the Bethe-Salpeter wave function in Euclidean space, which is an ill-posed inversion problem, via the maximum entropy method(MEM). The Nakanishi weight function as well as the corresponding light-front PDA can be well determined. We confirm the previous works on PDA computation therein the different method has been performed.
Within the Bethe-Salpeter formalism for instantaneous interactions, we describe, along a totally analytic route, the lightest pseudoscalar mesons by quark-antiquark bound states which show at least three indispensable general features, namely, the (almost) masslessness required for pions and kaons to be interpretable as (pseudo) Goldstone bosons, the suitable asymptotic behaviour in the limit of large spacelike relative momenta as determined by the relationship between quark mass function and Bethe-Salpeter amplitudes, and a pointwise behaviour for finite spacelike relative momenta suited for …
For a two-particle bound-state equation closer to its Bethe-Salpeter origins than Salpeter's equation, with effective-interaction kernel deliberately forged such as to ensure, in the limit of zero mass of the bound-state constituents, the vanishing of the arising bound-state mass, we scrutinize the emerging features of the lightest pseudoscalar mesons for their agreement with the behaviour predicted by a generalization of the Gell-Mann-Oakes-Renner relation.
A popular three-dimensional reduction of the Bethe-Salpeter formalism for the description of bound states in quantum field theory is the Salpeter equation, derived by assuming both instantaneous interactions and free propagation of all bound-state constituents. Numerical (variational) studies of the Salpeter equation with confining interaction, however, observed specific instabilities of the solutions, likely related to the Klein paradox and rendering (part of the) bound states unstable. An analytic investigation of this problem by a comprehensive spectral analysis is feasible for the reduced …
