Bethe-Salpeter Amplitude Research Articles

First radial excitations of baryons in a contact interaction: Mass spectrum

We compute masses of twenty positive parity first radial excitations of spin-1/2 and spin-3/2 baryons composed of u, d, s, c, and b quarks in a quark-diquark picture within a contact interaction model. These excitations comprise of two elements; one characterized by a zero in the Faddeev amplitude, representing a radial excitation of the quark-diquark system and the other marked by a zero in the diquark’s Bethe-Salpeter amplitude, corresponding to an intrinsic excitation of the diquark correlation. Wherever possible, we compare our results with other models and/or experiment. We verify that the masses obtained through our model conform to the spacing rules for all the baryons studied, whether light or heavy and whether of spin-1/2 or spin-3/2. The computed masses do not just offer a guide to the future experimental searches but also compare well with the existing candidates for the possible radial excitations of some heavy baryons. Published by the American Physical Society 2024

Electromagnetic and two-photon transition form factors of the pseudoscalar mesons: An algebraic model computation

We compute electromagnetic and two-photon transition form factors of ground-state pseudoscalar mesons: π,K,ηc,ηb. To this end, we employ an algebraic model based upon the coupled formalism of Schwinger-Dyson and Bethe-Salpeter equations. Within this approach, the dressed quark propagator and the relevant Bethe-Salpeter amplitude encode the internal structure of the corresponding meson. Electromagnetic properties of the meson are probed via the quark-photon interaction. The algebraic model employed by us unifies the treatment of all ground-state pseudoscalar mesons. Its parameters are carefully fitted performing a global analysis of existing experimental data including the knowledge of the charge radii of the mesons studied. We then compute and predict electromagnetic and two-photon transition form factors for a wide range of probing photon momentum-squared which is of direct relevance to the experimental observations carried out thus far or planned at different hadron physics facilities such as the Thomas Jefferson National Accelerator Facility (JLab) and the forthcoming Electron-Ion Collider. We also present comparisons with other theoretical models and approaches and lattice quantum chromodynamics. Published by the American Physical Society 2024

QCD anomalies in electromagnetic processes: A solution to the γ→3π puzzle

In this work, the γ→3π form factor is calculated within the Dyson-Schwinger equations framework using a contact interaction model within the so-called modified rainbow ladder truncation. The present calculation takes into account the pseudovector component in the pion Bethe-Salpeter amplitude (BSA) and π−π scattering effects, producing a γ→3π anomaly that is 1+6Rπ2 larger than the low energy prediction. Here Rπ is the relative ratio of the pseudovector and pseudoscalar components in the pion BSA; with our parameters’ input, this correction raises the γ→3π anomaly by around 10%. The main outcome of this work is the unveiling of the origin of such a correction, which could be a possible explanation of the discrepancy between the existing experimental data and the low energy prediction. Moreover, it is highlighted how the magnitude of the anomaly is affected in effective theories that require an irremovable ultraviolet cutoff. We find that for both the anomalous processes π→2γ and γ→3π, the missing contribution to the anomaly can be compensated by the additional structures related with the quark anomalous magnetic moment. Published by the American Physical Society 2024

Kaon structure in the nuclear medium within the light front approach

We study the properties of the charged kaon in symmetric nuclear matter using a Bethe-Salpeter amplitude to model the quark-antiquark bound state, which is well constrained by previous studies of its vacuum properties. The electromagnetic form factor, charge radius, decay constant, and the light-front valence component probability are investigated in symmetric nuclear matter. In order to describe the constituent up and antistrange quarks in nuclear matter, we adopt the “quark-meson coupling (QMC) model,” which has been widely applied to various hadronic and nuclear phenomena in the nuclear medium. Published by the American Physical Society 2024

Algebraic model to study the internal structure of pseudoscalar mesons with heavy-light quark content

The internal structure of all lowest-lying pseudoscalar mesons with heavy-light quark content is studied in detail using an algebraic model that has been applied recently, and successfully, to the same physical observables of pseudoscalar and vector mesons with hidden-flavor quark content, from light to heavy quark sectors. The algebraic model consists on constructing simple and evidence-based of the meson’s Bethe-Salpeter amplitude (BSA) and quark’s propagator in such a way that the Bethe-Salpeter wave function (BSWF) can then be readily computed algebraically. Its subsequent projection onto the light front yields the light front wave function (LFWF) whose form allows us a simple access to the valence-quark parton distribution amplitude (PDA) by integrating over the transverse momentum squared. We exploit our current knowledge of the PDAs of lowest-lying pseudoscalar heavy-light mesons to compute their generalized parton distributions (GPDs) through the overlap representation of LFWFs. From these three dimensional knowledge, different limits/projections lead us to deduce the related parton distribution functions (PDFs), electromagnetic form factors (EFFs), and impact parameter space GPDs (IPS-GPDs). When possible, we make explicit comparisons with available experimental results and earlier theoretical predictions. Published by the American Physical Society 2024

Unpolarized transverse-momentum dependent distribution functions of a quark in a pion with Minkowskian dynamics

The unpolarized twist-2 (leading) and twist-3 (subleading), T-even, transverse-momentum dependent quark distributions in the pion are evaluated for the first time by using the actual solution of a dynamical equation in Minkowski space. The adopted theoretical framework is based on the 4D homogeneous Bethe–Salpeter integral equation with an interaction kernel given by a one-gluon exchange, featuring an extended quark-gluon vertex. The masses of quark and gluon as well as the interaction-vertex scale have been chosen in a range suggested by lattice-QCD calculations, and calibrated to reproduce both pion mass and decay constant. The sum rules to be fulfilled by the transverse-momentum dependent distributions are carefully investigated, particularly the leading-twist one, that has to match the collinear parton distribution function, and hence can be scrutinized in terms of existing data as well as theoretical predictions. Noteworthy, the joint use of the Fock expansion of the pion state facilitates an in-depth analysis of the content of the pion Bethe–Salpeter amplitude, allowing to calculate the gluon contribution to the quark average longitudinal fraction, that results to be ∼6%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim 6\\%$$\\end{document}. The current analysis highlights the role of the gluon exchanges through quantitative analysis of collinear and transverse-momentum distributions, showing, e.g. for both leading and subleading-twists, an early departure from the widely adopted exponential fall-off, for |k⊥|2>m2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\ extbf{k}_\\perp |^2> m^2$$\\end{document}, with the quark mass ∼ΛQCD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim {\\varLambda }_{QCD}$$\\end{document}.

Electromagnetic form factors in a contact interaction: scalar and pseudoscalar mesons

The determination of fundamental properties of hadrons is important to clearly understand the experimental measurements such as the anomalous magnetic moment of the muon. However, since a first principle approach is hard to find, several groups among the world are approaching phenomenologically to such description of hadrons. In this poster, we present a comprehensive survey of electromagnetic form factors of all light, heavy and heavy-light ground-state pseudoscalar and scalar mesons. Specifically, in this work we are interested on the determination of Elastic Meson Form Factors (EFF) for Scalar (S) and Pseudoscalar (PS) mesons. However, the calculation of all EFF requires the computation of the quark propagator, the Bethe-Salpeter (BS) equation and the Bethe-Salpeter Amplitudes (BSA) of mesons, their masses as well as the knowledge of the quark-photon interaction at different probing momenta.

Exploring the 0− bound state with dressed quarks in Minkowski space

The Bethe-Salpeter equation for a pseudoscalar bound-system, with i) a ladder kernel with massive gluons, ii) dynamically-dressed quark mass function and iii) an extended quark-gluon vertex, is solved in Minkowski space by using the Nakanishi integral representation of the Bethe-Salpeter amplitude. The quark dressing is implemented through a phenomenological ansatz, which was tuned by lattice QCD calculations of the quark running mass. The latter were also used for assigning the range of the gluon mass and the parameter featuring the extended color density. This framework allows to investigate the gluon dynamics that manifest itself in the quark dressing, quark-gluon vertex and the binding, directly in the physical space. We present the first results for low-density pseudoscalar systems in order to elucidate the onset of the interplay between the above mentioned gluonic phenomena, and we discuss both static and dynamical quantities, like valence longitudinal and transverse distributions.

Chiral limit of a fermion-scalar (1/2)+ system in covariant gauges

The homogeneous Bethe-Salpeter equation (BSE) of a (1/2)$^+$ bound system, that has both fermionic and bosonic degrees of freedom, that we call a {\em mock nucleon}, is studied in Minkowski space, in order to analyse the chiral limit in covariant gauges. After adopting an interaction kernel built with a one-particle exchange, the $\chi$-BSE is numerically solved by means of the Nakanishi integral representation and light-front projection. Noteworthy, the chiral limit induces a scale-invariance of the model and consequently generates a wealth of striking features: i) it reduces the number of non trivial Nakanishi weight functions to only one; ii) the form of the surviving weight function has a factorized dependence on the two relevant variables, compact and non-compact one; iii) the coupling constant becomes an explicit function of the real exponent governing the power-law fall-off of the non trivial Nakanishi weight function. The thorough investigation at large transverse-momentum of light-front Bethe-Salpeter amplitudes, obtained with massive constituents, provides a confirmation of the expected universal power-law fall-off, with exponents predicted by our non-perturbative framework. Finally, one can shed light on the exponents that govern the approach to the upper extremum of the longitudinal-momentum fraction distribution function of the {\em mock nucleon}, when the coupling constant varies.

Evidence of the Schwinger Mechanism from Lattice QCD

In quantum chromodynamics (QCD), gluons acquire a mass scale through the action of the Schwinger mechanism. This mass emerges as a result of the dynamical formation of massless bound-states of gluons which manifest as longitudinally coupled poles in the vertices. In this contribution, we show how the presence of these poles can be determined from lattice QCD results for the propagators and vertices. The crucial observation that allows this determination is that the Schwinger mechanism poles induce modifications, called “displacements”, to the Ward identities (WIs) relating two- and three-point functions. Importantly, the displacement functions correspond precisely to the Bethe–Salpeter amplitudes of the massless bound-states. We apply this idea to the case of the three-gluon vertex in pure Yang–Mills SU(3). Using lattice results in the corresponding WI, we find an unequivocal displacement and show that it is consistent with the prediction based on the Bethe–Salpeter equation.

Light-front wave functions of vector mesons in an algebraic model

Inspired by the recent development of an algebraic model which provides an adequate and unified description of the internal structure of the lowest-lying pseudoscalar mesons, belonging both to the light quarks sector and to the one of heavy quarks, we perform its first extension to the vector-meson case. The algebraic model describes the meson's structure in terms of the spectral density function that appears in a Nakanishi integral representation of the covariant quark-antiquark bound-state amplitude, i.e., the Bethe-Salpeter amplitude. We compute the leading-twist light-front wave functions of the $\ensuremath{\rho}(770)$, $\ensuremath{\phi}(1020)$, $J/\ensuremath{\psi}$, and $\mathrm{\ensuremath{\Upsilon}}(1S)$ mesons through their connection with the parton distribution amplitudes. Among the results we present, the following are of particular interest: (i) transverse light-front wave functions can be obtained algebraically from the corresponding parton distribution amplitudes, whereas that is not the case for longitudinal light-front wave functions, which requires an intermediate step where a spectral density function must be derived from the particular parton distribution amplitude; (ii) the derived spectral density functions show marked differences between light and heavy vector mesons, the latter being narrower as compared to the former, and these are also nonpositive definite, although the integral over the entire curve is larger than zero as expected; and (iii) the longitudinal and transverse light-front wave functions of vector mesons with light quark content exhibit steep $x$ and ${p}_{\ensuremath{\perp}}^{2}$ dependence, while those of the $J/\ensuremath{\psi}$ and $\mathrm{\ensuremath{\Upsilon}}(1S)$ mesons are characterized by narrow distributions in the $x$ range but, comparatively, much more gradual falloffs with respect to the ${p}_{\ensuremath{\perp}}^{2}$ range depicted.

Strong two-meson decays of light and charmed vector mesons

We calculate the strong decay couplings for $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$, $\ensuremath{\phi}\ensuremath{\rightarrow}KK$, ${K}^{*}\ensuremath{\rightarrow}K\ensuremath{\pi}$, and ${D}^{*}\ensuremath{\rightarrow}D\ensuremath{\pi}$ in a unified and consistent approach based on the impulse approximation, nonperturbative solutions of the quark-gap equation and the Poincar\'e invariant Bethe-Salpeter amplitudes of vector and pseudoscalar mesons. In particular, we obtain the coupling ${g}_{{D}^{*}D\ensuremath{\pi}}=17.2{4}_{\ensuremath{-}2.30}^{+3.06}$ in very good agreement with the experimental value by CLEO, which corresponds to a strong effective coupling between heavy vector and pseudoscalar mesons to the pion of $\stackrel{^}{g}=0.5{8}_{\ensuremath{-}0.08}^{+0.10}$.

Nonfactorizable charming loops in FCNC B decays versus B -decay semileptonic form factors

We compare a nonfactorizable charming-loop correction to an exclusive FCNC $B$ decay given in terms of the three-particle Bethe-Salpeter amplitude (3BS) of the $B$ meson, $⟨0|\overline{q}(x){G}_{\ensuremath{\mu}\ensuremath{\nu}}(z)b(0)|B(p)⟩$, with the corresponding correction to the $B$-meson semileptonic form factor. In spite of certain similarities, these two corrections are shown to have substantial differences: the form-factor correction is dominated by the collinear light-cone configuration of 3BS (${z}_{\ensuremath{\mu}}=u{x}_{\ensuremath{\mu}}$, $0<u<1$, ${x}^{2}=0$); in contrast, the FCNC amplitude is dominated by a different configuration with noncollinear arguments [${x}^{2}=0$, ${z}^{2}=0$, but $(x\ensuremath{-}z{)}^{2}\ensuremath{\ne}0$ (i.e., ${z}_{\ensuremath{\mu}}\ensuremath{\ne}u{x}_{\ensuremath{\mu}}$)].

Pion inspired by QCD: Nakanishi and light-front integral representations

The pion structure in Minkowski space is explored using the Nakanishi integral representation. A general framework is developed for the pion Bethe-Salpeter amplitude based on the K\"allen-Lehmann representation of the dressed quarks with an ansatz for the pseudoscalar $\ensuremath{\pi}\ensuremath{-}q\overline{q}$ vertex fulfilling the axial Ward-Takahashi identity. The Nakanishi weight functions are derived for the scalar amplitudes associated with the decomposition of the pion Bethe-Salpeter amplitude in different operator bases in the Dirac spinor space in terms of the involved spectral densities. The approach is applied to an analytical model of the pion Bethe-Salpeter amplitude, which is combined with Landau gauge lattice QCD results for the quark running mass at spacelike momentum. From the Nakanishi integral representation several pion observables were calculated, such as the decay constant, the spin decomposition of the valence probabilities, longitudinal and transverse momentum distributions from the valence component of the light-front wave function.

Pion model with the Nakanishi integral representation

In the present work, we describe a model for the pion based on an analytic expression for the Bethe-Salpeter (BSA) amplitude, combined with some ingredients from Lattice QCD calculations. The running quark mass function M(p2), used here, reproduces well the results of Lattice QCD calculations. The analytical form of the running quark mass function contains a single time-like pole, which implies in time-like poles of the dressed quark propagator. Such a form allows to build the weight functions, Gi(γ, z), for the Nakanishi integral representation of each scalar function, χi(k, p), appearing in the decomposition of the Bethe-Salpeter amplitude in terms of Dirac operators. Such scalar amplitudes can also be used to obtain the pion valence light-front wave function.

Exposing the effect of the p -wave component in the pion triplet under a strong magnetic field

The static properties, masses and decay constants, of pseudoscalar meson triplet in a strongly magnetized medium are studied through the Dyson-Schwinger equation approach treatment of a contact interaction. Complementary to the usual vector-vector form, a symmetry-preserving formulation of couplings has been proposed in this work, without modifying the quark propagator, to control the strength of the $p$-wave component of Bethe-Salpeter amplitude. It is found that, with the help of flexible auxiliary interaction, our simple model is able to reproduce the observation in the lattice QCD simulation, where the spectra of the charged pseudo-scalar meson shows a non-monotonic behavior as the magnetic field grows. The discovery of this work implies the strong magnetic field affects the inner structure of mesons dramatically.

New form of kernel in equation for Nakanishi function

The Bethe-Salpeter amplitude $\Phi(k,p)$ is expressed, by means of the Nakanishi integral representation, via a smooth function $g(\gamma,z)$. This function satisfies a canonical equation $g=Ng$. However, calculations of the kernel $N$ in this equation, presented previously, were restricted to one-boson exchange and, depending on method, dealt with complex multivalued functions. Although these difficulties are surmountable, but in practice, they complicate finding the unambiguous result. In the present work, an unambiguous expression for the kernel $N$ in terms of real functions is derived. For the one-boson scalar exchange, the explicit formula for $N$ is found. With this equation and kernel, the binding energies, calculated previously, are reproduced. Their finding, as well as calculation of the Bethe-Salpeter amplitude in the Minkowski space, become not more difficult than in the Euclidean one. The method can be generalized to any kernel given by irreducible Feynman graph. This generalization is illustrated by example of the cross-ladder kernel.

Pion electromagnetic form factor with Minkowskian dynamics

The pion electromagnetic form factor has been calculated for the first time from the solutions of the Bethe-Salpeter equation, obtained directly in Minkowski space by using the Nakanishi integral representation of the Bethe-Salpeter amplitude. The one-gluon exchange kernel contains as inputs the quark and gluon masses, as well as a scale parameter featuring the extended quark-gluon vertex. The range of variability of these parameters is suggested by lattice calculations. After presenting a very successful comparison with the existing data in the whole range of the momentum transfer, we show how inserting the light-front (LF) formalism in our approach allows to achieve further interesting results, like the evaluation of the LF valence form factor and a first investigation of the end-points effects.

Energy loss of heavy quarkonia in hot QCD plasmas

We compute the energy loss of heavy quarkonia in high temperature QCD plasmas and investigate the energy loss effects on quarkonium suppression. Based on the effective vertex derived from the Bethe-Salpeter amplitude for quarkonium, the collisional and radiative energy loss are determined by quarkonium-gluon elastic scattering and the associated gluon-bremsstrahlung, respectively. In the energy regime $E<m_{\Upsilon}^2/T$ the collisional energy loss is dominant over the radiative one, and the total energy loss increases with the plasma temperature and the initial energy of quarkonium. Our numerical analysis indicates that the medium-induced energy loss of the $\Upsilon$(1S) results in stronger suppression at higher momentum, although the energy loss effects are found to be small compared with the previous estimates of quarkonium dissociation in heavy-ion collisions.

Rainbow modified-ladder approximation and degenerate pion

Correlation functions can be described by the corresponding equations, viz., the gap equation for the quark propagator and the inhomogeneous Bethe-Salpeter equation for the vector dressed-fermion-Abelian-gauge-boson vertex in which specific truncations have to be implemented. The general vector and axial-vector Ward-Green-Takahashi identities require these correlation functions to be interconnected; in consequence of this, truncations made must be controlled consistently. It turns out that, if the rainbow approximation is assumed in the gap equation, the scattering kernel in the Bethe-Salpeter equation can adopt the ladder approximation, which is one of the most basic attempts to truncate the scattering kernel. Additionally, a modified-ladder approximation is also found to be a possible symmetry-preserving truncation scheme. As an illustration of this approximation for application, a treatment of the pion is included. The pion mass and decay constant are found to be degenerate in ladder and modified-ladder approximations, even though the Bethe-Salpeter amplitudes are with apparent distinction. The justification for the modified-ladder approximation is examined with the help of the Gell-Mann-Oakes-Renner relation.