To solve the spinor-spinor Bethe-Salpeter equation in Euclidean space we propose a novel method related to the use of hyperspherical harmonics. We suggest an appropriate extension to form a new basis of spin-angular harmonics that is suitable for a representation of the vertex functions. We present a numerical algorithm to solve the Bethe-Salpeter equation and investigate in detail the properties of the solution for the scalar, pseudoscalar and vector meson exchange kernels including the stability of bound states. We also compare our results to the non relativistic ones and to the results given by light front dynamics.
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.
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 …
The applications of the consistently coupled Schwinger-Dyson and Bethe-Salpeter approach to the eta-eta' complex are extended to the two-photon transition form factors of eta and eta' for spacelike transferred momenta. We compare our predictions with experiment and some other theoretical approaches.
Describing the lightest pseudoscalar mesons as bound states of quark and antiquark within the framework of an instantaneous Bethe-Salpeter formalism constructed such as to retain (in contrast to Salpeter's equation) as much information on the relativistic effects provided by the full quark propagator as conceivable allows for a surprisingly simple implementation of their near masslessness mandatory for their interpretability as pseudo-Goldstone bosons related to the spontaneous breaking of the chiral symmetries of quantum chromodynamics.
The instantaneous Bethe-Salpeter equation, derived from the general Bethe-Salpeter formalism by assuming that the involved interaction kernel is instantaneous, represents the most promising framework for the description of hadrons as bound states of quarks from first quantum-field-theoretic principles, that is, quantum chromodynamics. Here, by extending a previous analysis confined to the case of bound-state constituents with vanishing masses, we demonstrate that the instantaneous Bethe-Salpeter equation for bound-state constituents with (definitely) nonvanishing masses may be converted into a…
Several numerical investigations of the Salpeter equation with static confining interactions of Lorentz-scalar type revealed that its solutions are plagued by instabilities of presumably Klein-paradox nature. By proving rigorously that the energies of all predicted bound states are part of real, entirely discrete spectra bounded from below, these instabilities are shown, for confining interactions of harmonic-oscillator shape, to be absent for a reduced version of an instantaneous Bethe-Salpeter formalism designed to generalize the Salpeter equation towards an approximate inclusion of the exac…
We present a numerical solution of the four-quark Bethe-Salpeter equation for a scalar tetraquark. We find that the four-body equation dynamically generates pseudoscalar poles in the Bethe-Salpeter amplitude. The sensitivity to the pion poles leads to a light isoscalar tetraquark mass M ~ 400 MeV, which is comparable to that of the sigma/f0(500). The masses of its multiplet partners kappa and a0/f0 follow a similar pattern, thereby providing support for the tetraquark interpretation of the light scalar nonet.
We suggest a framework based on the rainbow approximation to the Dyson-Schwinger and Bethe-Salpeter equations with effective parameters adjusted to lattice QCD data to calculate the masses of the ground and excited states of pseudo-scalar glueballs. The structure of the truncated Bethe-Salpeter equation with the gluon and ghost propagators as solutions of the truncated Dyson-Schwinger equations is analysed in Landau gauge. Both, the Bete-Salpeter and Dyson-Schwinger equations, are solved numerically within the same rainbow-ladder truncation with the same effective parameters which ensure consi…
Pion and kaon observables are calculated using a Dyson-Schwinger Bethe-Salpeter formalism. It is shown that an infrared finite gluon propagator can lead to quark confinement via generation of complex mass poles in quark propagators. Observables, including electromagnetic form factors, are calculated entirely in Euclidean metric for spacelike values of bound state momentum and final results are extrapolated to the physical region.
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 simp…
Within our description of Goldstone-type pseudoscalar mesons as almost massless bound states of quark and antiquark by a three-dimensional bound-state equation of Bethe-Salpeter origin, taking into account the pointwise behaviour of the full light-quark propagators enables to characterize the effective interquark interactions more precisely than earlier studies exploiting just specific aspects.
I will discuss the gap and Bethe-Salpeter equations for quantum chromodynamics in Coulomb gauge under a leading order truncation scheme. Within this scheme, the role of charge conservation and the cancellation of infrared divergences becomes particularly clear. The quark gap equation exhibits not only chiral symmetry breaking, but explicitly reproduces the heavy quark limit. The corresponding gluon equation has a massive solution with interesting nonperturbative renormalization properties. I will further discuss various aspects of the Bethe-Salpeter equation for pseudoscalar and vector mesons …
To solve the spinor-spinor Bethe-Salpeter equation in Euclidean space we propose a novel method related to the use of hyperspherical harmonics. We suggest an appropriate extension to form a new basis of spin-angular harmonics that is suitable for a representation of the vertex functions. We present a numerical algorithm to solve the Bethe-Salpeter equation and investigate in detail the properties of the solution for the scalar, pseudoscalar and vector meson exchange kernels including the stability of bound states. We also compare our results to the non relativistic ones and to the results give…
Starting from a phenomenologically successful, numerical solution of the Dyson-Schwinger equation that governs the quark propagator, we reconstruct in detail the interaction kernel that has to enter the instantaneous approximation to the Bethe-Salpeter equation to allow us to describe the lightest pseudoscalar mesons as quark-antiquark bound states exhibiting the (almost) masslessness necessary for them to be interpretable as the (pseudo) Goldstone bosons related to the spontaneous chiral symmetry breaking of quantum chromodynamics.
Results for the light meson mass spectrum from a Bethe-Salpeter approach are presented. The results obtained in the standard framework are Poincaré covariant and compare favourably with lattice results. Using a more sophisticated scheme, the pseudoscalar, vector and $1^{++}$ ($a_1/f_1$) axialvector charge eigenstate masses are unaltered whereas the $1^{+-}$ ($b_1/h_1$) axialvector meson mass is raised.
In this work, I studied the spectrum of scalar mesons and axial-vector mesons via Dyson-Schwinger equation and Bethe-Salpeter equation approach in the rainbow-ladder approximation. An interaction model with a repulsive term added to the one used for the pseudoscalar and vector mesons is proposed. My results are consistent with the experiment data and other model results, which shows that this interaction model is effective for all the heavy, heavy-light, and light scalar and axial-vector mesons.
The combined Dyson-Schwinger and Bethe-Salpeter equations in rainbow-ladder approximation are used to search for Regge trajectories of mesons in the pseudo-scalar and vector channels. We focus on the often employed Alkofer-Watson-Weigel kernel which is known to deliver good results for the ground state meson spectra; it provides linear Regge trajectories in the $J^P= 0^-$ channel.
The Bethe-Salpeter equation for three bosons with zero-range interaction is solved for the first time. For comparison the light-front equation is also solved. The input is the two-body scattering length and the outputs are the three-body binding energies, Bethe-Salpeter amplitudes and light-front wave functions. Three different regimes are analyzed: ({\it i}) For weak enough two-body interaction the three-body system is unbound. ({\it ii}) For stronger two-body interaction a three-body bound state appears. It provides an interesting example of a deeply bound Borromean system. ({\it iii}) For e…
We investigate capabilities of the effective interaction in a rainbow-ladder truncated meson model of QCD within a covariant Landau-gauge Bethe-Salpeter-equation approach. Based upon past success for the light- as well as heavy-quark domains, we discuss the range of applicability and features of an effort with comprehensive phenomenological claim and goals.
The scalar three-body Bethe-Salpeter equation, with zero-range interaction, is solved in Minkowski space by direct integration of the four-dimensional integral equation. The singularities appearing in the propagators are treated properly by standard analytical and numerical methods, without relying on any ansatz or assumption. The results for the binding energies and transverse amplitudes are compared with the results computed in Euclidean space. A fair agreement between the calculations is found.
Masses for vector 1--, axialvector 1+-, 1++, and exotic-vector 1-+ mesons are calculated within a quantum field theoretic framework based on the Dyson-Schwinger equations using a model form of the quark-antiquark interaction that is separable. The model provides an excellent description of pion and K-meson observables and the flavor-octet ground-state meson spectrum. With no adjustments to model parameters, a numerical solution of the Bethe-Salpeter equation yields two exotic 1-+ mesons with masses of 1439 and 1498 MeV.
The Bethe-Salpeter equation and leptonic decay constants for pseudoscalar and vector quark-antiquark mesons with arbitrary quark masses are studied in Coulomb gauge, under a leading order truncation. As input, we use a pure linear rising potential, supplemented by a contact term arising from the conservation of total color charge. It is shown how the equations can be written in terms of manifestly finite functions, despite the infrared singular interaction. The resulting equations are solved numerically. Both the pattern of dynamical chiral symmetry breaking and the leading order heavy quark l…
We propose a method to reconstruct the Bethe-Salpeter amplitude in Minkowski space given the Euclidean Bethe-Salpeter amplitude -- or alternatively the Light-Front wave function -- as input. The method is based on the numerical inversion of the Nakanishi integral representation and computing the corresponding weight function. This inversion procedure is, in general, rather unstable, and we propose several ways to considerably reduce the instabilities. In terms of the Nakanishi weight function, one can easily compute the BS amplitude, the LF wave function and the electromagnetic form factor. Th…
The off-mass shell scattering amplitude, satisfying the Bethe-Salpeter equation for spinless particles in Minkowski space with the ladder kernel, is computed for the first time.
A three-dimensional reduction of the homogeneous Bethe-Salpeter equation retaining, in contrast to the Salpeter equation, the exact propagators (crucial for, e.g., a proper incorporation of dynamical chiral symmetry breakdown) is proposed. This formalism may be easily extended to Bethe-Salpeter equations for bound states composed of particles that are not, or not all, identical to spin-1/2 fermions as well as to three-dimensional reductions of the Bethe-Salpeter equation different from Salpeter's equation.
We investigate the relation between the rank I separable potential for the covariant Bethe-Salpeter equation and the one-boson-exchange potential. After several trials of the parameter choices, it turns out that it is not always possible to reproduce the phase-shifts calculated from a single term of the one-boson-exchange potential especially of the $σ$-exchange term, separately by the rank I separable potential. Instead, it is shown that the separable potential is useful to parameterize the total nucleon-nucleon interaction.
The transition form factor for electrodisintegration of a two-body bound system is calculated in the Bethe-Salpeter framework. For the initial (bound) and the final (scattering) states, we use our solutions of the Bethe-Salpeter equation in Minkowski space which were first obtained recently. The gauge invariance, which manifests itself in the conservation of the transition electromagnetic current Jq = 0, is studied numerically. It results from a cancellation between the plane wave and the final state interaction contributions. This cancellation takes place only if the initial bound state BS am…
The quantitative investigation of the scalar Bethe-Salpeter equation in Minkowski space, within the ladder-approximation framework, is extended to include the excited states. This study has been carried out for an interacting system composed by two massive bosons exchanging a massive scalar, by adopting (i) the Nakanishi integral representation of the Bethe-Salpeter amplitude, and (ii) the formally exact projection onto the null plane. Our analysis, on one hand, confirms the reliability of the method already applied to the ground state and, on the other one, extends the investigation from the …
We present a method to directly solving the Bethe-Salpeter equation in Minkowski space, both for bound and scattering states. It is based on a proper treatment of the singularities which appear in the kernel, propagators and Bethe-Salpeter amplitude itself. The off-mass shell scattering amplitude for spinless particles interacting by a one boson exchange is computed for the first time.
Consequent application of the instantaneous approximation to both the interaction and all propagators of the bound-state constituents allows us to forge, within the framework of the Bethe-Salpeter formalism for the description of bound states, an instantaneous form of the Bethe-Salpeter equation with exact (i.e., full) propagators of the bound-state constituents. This instantaneous equation generalizes the well-known Salpeter equation the derivation of which needs the additional assumption of free propagation of the bound-state constituents.
In this work the $B\bar{B}^*/D\bar{D}^*$ system is studied in the Bethe-Salpeter approach with quasipotential approximation. In our calculation both direct and cross diagrams are included in the one-boson-exchange potential. The numerical results indicate the existence of a isoscalar bound state $D\bar{D}^*$ with $J^{PC}=1^{++}$, which may be related to the $X(3872)$. In the isovector sector, no bound state is produced from the interactions of $D\bar{D}^*$ and $B\bar{B}^*$, which suggests the molecular state explanations for $Z_b(10610)$ and $Z_c(3900)$ are excluded.
The Bethe-Salpeter Equation for a two-scalar, S-wave bound system, interacting through a massive scalar, is investigated within the ladder approximation. By assuming a Nakanishi integral representation of the Bethe-Salpeter amplitude, one can deduce new integral equations that can be solved and quantitatively studied, overcoming the analytic difficulties of the Minkowski space. Finally, it is shown that the Light-front distributions of the valence state, directly obtained from the Bethe-Salpeter amplitude, open an effective window for studying the two-body dynamics.
Achieving self-consistent simultaneous interpretations of pions and kaons as bound states of quark and antiquark and as the (almost) massless boson states related, according to Goldstone's theorem, to the dynamical (and explicit) breakdown of the chiral symmetries of QCD still represents a major challenge. Applying inversion techniques to conveniently simplified versions of the homogeneous Bethe-Salpeter equation, governing bound states in quantum field theory, enables us to get along a straightforward route a qualitative idea of how the underlying effective interaction might look like.
The general form of the Bethe-Salpeter wave functions for the bound states composed of two vector fields of arbitrary spin and definite parity is corrected. Using the revised general formalism, we investigate the observed \emph{Y}(3940) state which is considered as a molecule state consisting of $D^{*0}\bar{D}^{*0}$. Though the attractive potential between $D^{*0}$ and $\bar{D}^{*0}$ including one light meson ($σ$, $π$, $ω$, $ρ$) exchange is considered, we find that in our approach the contribution from one-$π$ exchange is equal to zero and consider SU(3) symmetry breaking. The obtained mass o…
The Bethe-Salpeter approach allows for quantum-field-theoretic descriptions of relativistic bound states; its inherent complexity, however, usually prevents to find its exact solutions. Under suitable simplifying assumptions about the systems discussed, we derive analytically some examples of rigorous solutions to the instantaneous homogeneous Bethe-Salpeter equation by relating tentative solutions to the interactions responsible for the formation of bound states.
The Salpeter equation, a standard tool in hadron physics, constitutes a well-defined three-dimensional approximation to the Bethe-Salpeter formalism for the description of bound states within quantum field theories. However, if confinement is implemented in a too careless manner, the Salpeter equation might predict unstable states where only truly bound states are expected. A systematic spectral analysis of the Salpeter equation might serve to single out those Bethe-Salpeter interaction kernels which indeed yield stable bound states, with associated energy eigenvalues belonging to a real discr…
We shortly review different methods to obtain the scattering solutions of the Bethe-Salpeter equation in Minkowski space. We emphasize the possibility to obtain the zero energy observables in terms of the Euclidean scattering amplitude.
The off-mass shell scattering amplitude, satisfying the Bethe-Salpeter equation for spinless particles in Minkowski space with the ladder kernel, is computed for the first time.
Based on lattice non-relativistic QCD (NRQCD) studies we present results for Bethe-Salpeter amplitudes for $Υ(1S)$, $Υ(2S)$ and $Υ(3S)$ in vacuum as well as in quark-gluon plasma. Our study is based on 2+1 flavor $48^3 \times 12$ lattices generated using the Highly Improved Staggered Quark (HISQ) action and with a pion mass of $161$ MeV. At zero temperature the Bethe-Salpeter amplitudes follow the expectations based on non-relativistic potential models. At non-zero temperatures, the interpretation of Bethe-Salpeter amplitudes turns out to be more nuanced, but consistent with our previous latti…
The possibility of solving the Bethe-Salpeter Equation in Minkowski space, even for fermionic systems, is becoming actual, through the applications of well-known tools: i) the Nakanishi integral representation of the Bethe-Salpeter amplitude and ii) the light-front projection onto the null-plane. The theoretical background and some preliminary calculations are illustrated, in order to show the potentiality and the wide range of application of the method.
Using the solutions of the Bethe-Salpeter equation in Minkowski space for bound and scattering states found in previous works, we calculate the transition electromagnetic form factor describing the electro-disintegration of a bound system.
Some results obtained by a new method for solving the Bethe-Salpeter equation are presented. The method is valid for any kernel given by irreducible Feynman graphs. The Bethe-Salpeter amplitude, both in Minkowski and in Euclidean spaces, and the binding energy for ladder + cross-ladder kernel are found. We calculate also the corresponding electromagnetic form factor.
For two bound-state equations derived as simplified forms of the Bethe-Salpeter equation with confining interaction, stability of all solutions is rigorously shown.
Salpeter equations with potential functions rising to infinity in configuration space do not automatically predict stable bound states. For this to happen, also the Lorentz behaviour of the involved Bethe-Salpeter kernels is crucial. At least for interaction potentials of harmonic-oscillator form analytic scrutinies of Salpeter equations with confining interactions may identify those Bethe-Salpeter kernels which describe bound states free from the notorious instabilities encountered in numerical evaluations. Such truly confining kernels can be singled out systematically by requesting the resul…
We show the light-front representation of the field theoretical Bethe-Salpeter equation (BSE) in the ladder approximation using the quasi potential reduction. We discuss the equivalence of the covariant ladder Bethe-Salpeter equation with an infinite set of coupled equations for the Green's functions of the different light-front Fock-states.
A Ward-Takahashi identity preserving Bethe-Salpeter kernel can always be calculated explicitly from a dressed-quark-gluon vertex whose diagrammatic content is enumerable. We illustrate that fact using a vertex obtained via the complete resummation of dressed-gluon ladders. While this vertex is planar, the vertex-consistent kernel is nonplanar and that is true for any dressed vertex. In an exemplifying model the rainbow-ladder truncation of the gap and Bethe-Salpeter equations yields many results; e.g., pi- and rho-meson masses, that are changed little by including higher-order corrections. Rep…
Studying the Bethe-Salpeter formalism for interactions instantaneous in the rest frame of the bound states described, we show that, for bound-state constituents of arbitrary masses, the mass of the ground state of a given spin may be calculated almost entirely analytically with high accuracy, without the (numerical) diagonalization of the matrix representation obtained by expansion of the solutions over a suitable set of basis states.
