Relativistic Constituent Quark Model Research Articles

Quarks and quantum chromodynamics

A written account of my talk delivered at the Harald Fritzsch Memorial Symposium in Munich is given. The emphasis is on Harald’s achievements specifically in quark physics and in quantum chromodynamics. Beyond that exemplary results for low-energy baryons are discussed, as obtained over the past years in the framework of the relativistic constituent-quark model based on Goldstone-boson-exchange dynamics.

Electric transitions of the charmed-strange mesons in a relativistic quark model

In the present work, we adopt a relativistic constituent quark model to depict the charmed strange meson spectroscopy, in which D_{s0}(2317) and D_{s1}(2460) are considered as the 1^3P_0 and 1P_1^prime charmed strange mesons, respectively. By using the wave function obtained from the relativistic quark model, we further investigate the electric transitions between charmed strange mesons. We find the long wave length approximation is reasonable for the charmed strange meson radiative decay by comparing the results with different approximations. The estimated partial widths are all safely under the upper limits of the experimental data. Moreover, we find the branching ratio of D_{s1}(2536) rightarrow D_s^*gamma /D_s gamma are large enough to be detected, which could be searched by further experiments in Belle II and LHCb.

Linking infrared and ultraviolet parameters of pion-like states in strongly coupled gauge theories

It has been shown previously that in a relativistic constituent-quark model, predictions for the electromagnetic form factor of the pi meson match not only experimental data but also, in the limit of large momentum transfers, the asymptotics derived from Quantum Chromodynamics (QCD). This is remarkable since no parameters are introduced to provide for this infrared-ultraviolet link. Here, we follow this approach, going beyond QCD. We obtain numerical relations between the gauge coupling constant, the decay constant and the charge radius of the pion-like meson in general strongly-coupled theories. These relations are compared to published lattice results for SU(2) gauge theory with two fermion flavours, and a good agreement is demonstrated. Further applications of the approach, to be explored elsewhere, include composite Higgs and dark-matter models.

Magnetic moment of the ρ meson in instant-form relativistic quantum mechanics

We derive an explicit analytical expression for the magnetic dipole moment of the rho meson, mu_rho, in a relativistic constituent-quark model. We adopt our relativistic approach to composite systems, modified instant-form (mIF) Relativistic Quantum Mechanics (RQM), that we used particularly to construct a unified pi&rho model (Phys. Rev. D93, 036007 (2016) = arXiv:1602.00907) describing electroweak properties of light mesons. This model provides a parameter-free calculation to give mu_rho=(2.16+-0.03) [e/2M_rho] which is in accordance with the conventional experimental data. The magnetic, quadrupole and charge form factors also are derived and presented. We consider the small uncertainty of our value of magnetic moment as one of undoubted advantages of the method. A comparison is made with recent lattice QCD results and previous calculations using a variety of methods.

Euclidean bridge to the relativistic constituent quark model

${\bf Background}$ Knowledge of nucleon structure is today ever more of a precision science, with heightened theoretical and experimental activity expected in coming years. At the same time, a persistent gap lingers between theoretical approaches grounded in Euclidean methods (e.g., lattice QCD, Dyson-Schwinger Equations [DSEs]) as opposed to traditional Minkowski field theories (such as light-front constituent quark models). ${\bf Purpose}$ Seeking to bridge these complementary worldviews, we explore the potential of a Euclidean constituent quark model (ECQM). This formalism enables us to study the gluonic dressing of the quark-level axial-vector vertex, which we undertake as a test of the framework. ${\bf Method}$ To access its indispensable elements with a minimum of inessential detail, we develop our ECQM using the simplified quark $+$ scalar diquark picture of the nucleon. We construct a hyperspherical formalism involving polynomial expansions of diquark propagators to marry our ECQM with the results of Bethe-Salpeter Equation (BSE) analyses, and constrain model parameters by fitting electromagnetic form factor data. ${\bf Results}$ From this formalism, we define and compute a new quantity --- the Euclidean density function (EDF) --- an object that characterizes the nucleon's various charge distributions as functions of the quark's Euclidean momentum. Applying this technology and incorporating information from BSE analyses, we find the dressing effect on the proton's axial-singlet charge to be small in magnitude and consistent with zero. ${\bf Conclusions}$ The scalar quark $+$ diquark ECQM is a step toward a realistic quark model in Euclidean space, and urges additional refinements. The small size we obtain for the impact of the dressed vertex on the axial-singlet charge suggests that models without this effect are on firm ground to neglect it.

Flavor Analysis of Nucleon, Delta , and Hyperon Electromagnetic Form Factors

By the analysis of the world data base of elastic electron scattering on the proton and the neutron (for the latter, in fact, on ^2H and ^3He) important experimental insights have recently been gained into the flavor compositions of nucleon electromagnetic form factors. We report on testing the Graz Goldstone-boson-exchange relativistic constituent-quark model in comparison to the flavor contents in low-energy nucleons, as revealed from electron-scattering phenomenology. It is found that a satisfactory agreement is achieved between theory and experiment for momentum transfers up to Q^2sim 4hbox { GeV}^2, relying on three-quark configurations only. Analogous studies have been extended to the Delta and the hyperon electromagnetic form factors. For them we here show only some sample results in comparison to data from lattice quantum chromodynamics.

Photo-absorption sum rulesσ−1and further questions about correlations of constituents inside bound hadrons

Within the non-relativistic approach to photo-absorption sum rules for the 3N(4N)-nuclei the new σ −1 sum rules are proposed which are based on general chargesymmetry (CS) consequences for the “CS-conjugated” triton and 3 He and the CS-self-conjugated 4 He . Combining the relativistic constituent quark model of the nucleon and the known spin-,isospin- and the dipole-moment fluctuation sum rules, the relevant moments of the quark coordinate distribution and correlation functions in the ground state of the nucleon are derived and discussed.

Towards a realistic description of hadron resonances

We report on our attempts of treating excited hadron states as true quantum resonances. Hitherto the spectroscopy of mesons, usually considered as quark-antiquark systems, and of baryons, usually considered as three-quark systems, has been treated through excitation spectra of bound states (namely, confined few-quark systems), corresponding to poles of the quantum-mechanical resolvent at real negative values in the complex energy plane. As a result the wave functions, i.e. the residua of the resolvent, have not exhibited the behaviour as required for hadron resonances with their multiple decay modes. This has led to disturbing shortcomings in the description of hadronic resonance phenomena. We have aimed at a more realistic description of hadron resonances within relativistic constituent-quark models taking into account explicitly meson-decay channels. The corresponding coupled-channels theory is based on a relativistically invariant mass operator capable of producing hadron ground states with real energies and hadron resonances with complex energies, the latter corresponding to poles in the lower half-plane of the unphysical sheet of the complex energy plane. So far we have demonstrated the feasibility of the coupled-channels approach to hadron resonances along model calculations producing indeed the desired properties. The corresponding spectral properties will be discussed in this contribution. More refined studies are under way towards constructing a coupled-channels relativistic constituent-quark model for meson and baryon resonances.

Strong three-meson couplings ofJ/ψandηc

We discuss the strong couplings $g_{PPV}$ and $g_{VVP}$ for vector ($V$) and pseudoscalar ($P$) mesons, at least one of which is a charmonium state $J/\psi$ or $\eta_c$. The strong couplings are obtained as residues at the poles of suitable form factors, calculated in a broad range of momentum transfers using a dispersion formulation of the relativistic constituent quark model. The form factors obtained in this approach satisfy all constraints known for these quantities in the heavy-quark limit. Our results suggest sizably higher values for the strong meson couplings than those reported in the literature from QCD sum rules.

Exclusive decaysJ/ψ→D(s)(*)−ℓ+νℓin a covariant constituent quark model with infrared confinement

We investigate the exclusive semileptonic decays $J/\psi \to D_{(s)}^{(*)-} {\ell}^+ \nu_{\ell}$, where $\ell=e,\mu$, within the Standard Model. The relevant transition form factors are calculated in the framework of a relativistic constituent quark model with built-in infrared confinement. Our calculations predict the branching fractions $\mathcal{B}(J/\psi \to D_{(s)}^{(*)-} {\ell}^+ \nu_{\ell})$ to be of the order of $10^{-10}$ for $D_s^{(*)-}$ and $10^{-11}$ for $D^{(*)-}$. Most of our numerical results are consistent with other theoretical studies. However, some branching fractions are larger than those calculated in QCD sum rules approaches but smaller than those obtained in the covariant light-front quark model by a factor of about $2-3$.

Constraining scenarios of the soft/hard transition for the pion electromagnetic form factor with expected data of 12 GeV Jefferson Lab experiments and of the electron-ion collider

It has been shown previously [S. V. Troitsky and V. E. Troitsky, Phys. Rev. D 88, 093005 (2013)] that a nonperturbative relativistic constituent-quark model for the $\ensuremath{\pi}$-meson electromagnetic form factor allows for a quantitative description of the soft/hard transition, resulting in the correct quantum-chromodynamical asymptotics, including normalization, from the low-energy data without further parameter tuning. This happens universally whenever the constituent-quark mass is switched off. The energy range where the transition happens is therefore determined by the quark-mass running at intermediate energies and is not tightly constrained theoretically. Here we consider possible ways to pin down this energy range with coming experimental data. We demonstrate that expected experimental uncertainties of the 12 GeV Jefferson Lab data are larger than the span of predictions of the model, so these data might be used for testing the model but not for determination of the soft/hard transition scale. Contrarily, the projected electron-ion collider will be capable of pinning down the scale.

The constituent-quark model — Nowadays

The present performance of the constituent-quark model as an effective tool to describe low-energy hadrons on the basis of quantum chromodynamics is exemplified along the relativistic constituent-quark model with quark dynamics relying on a realistic confinement and a Goldstone-boson-exchange hyperfine interaction. In particular, the spectroscopy of all known baryons is covered within a universal model in good agreement with phenomenology. The structure of the nucleons as probed under electromagnetic, weak, and gravitational interactions is produced in a correct manner. Likewise, the electroweak form factors of the baryons with u, d, and s flavor contents, calculated so far, result reasonably. Shortcomings still remain with regard to hadronic resonance decays.

Form factors for BK l+l- decay

The exclusive BV(P)l+l- and BV decays, with a vector (V=K*, , …) or pseudo scalar (P=K, , …) meson in the final state are important for the search for flavour-changing new physics. In this paper we study one of the exclusive rare decays of pseudo scalar B mesons: BKl+l- by using a relativistic constituent quark model, including infrared connement. We calculate the relevant form factors of the (BP) transitions in the full kinematical range of momentum transfer. The calculated form factors are used to evaluate differential decay rates and polarization observables. We present numerical results on these observables using the covariant quark model. This model can be viewed as an effective quantum field theory approach based on an interaction Lagrangian of hadrons interacting with their constituent quarks. Universal and reliable predictions for exclusive processes involving both mesons composed from a quark and antiquark and baryons composed from three quarks result from this approach.

Transition from a relativistic constituent-quark model to the quantum-chromodynamical asymptotics: A quantitative description of the pion electromagnetic form factor at intermediate values of the momentum transfer

We adopt a non-perturbative relativistic constituent-quark model for the pi-meson electromagnetic form factor, which have successfully predicted experimental results, and supplement it with the effective momentum-dependent quark mass to study quantitatively the transition to the perturbative QCD asymptotics. The required asymptotical behaviour (including both the Q^{-2} fall-off and the correct coefficient) settles down automatically when the quark mass is switched off; however, the present experimental data on the form factor suggest that this cannot happen at the values of the momentum transfer below ~10 GeV^2. The effective constituent-quark mass below this scale acquires substantial non-perturbative contributions.

EFFECTIVE DEGREES OF FREEDOM IN LOW-ENERGY QUANTUM CHROMODYNAMICS

Confinement and spontaneous breaking of chiral symmetry are assumed to generate the governing degrees of freedom of low-energy quantum chromodynamics. On this basis a relativistic constituent-quark model is constructed and formulated along an invariant mass operator within Poincaré-invariant quantum mechanics. The model is effectively applied to the spectroscopy of all known baryons of flavors u, d, s, c and b. The mass-operator eigenstates are furthermore tested with regard to the baryon electromagnetic and axial form factors. Through using the point form of relativistic quantum mechanics, these observables are obtained in a manifestly covariant manner. For all light and strange baryon ground states the electroweak structures are reproduced either in good agreement with phenomenology or, if no experimental data exist, in consistency with results available from lattice quantum chromodynamics. It is concluded that the relativistic constituent-quark model, relying on {QQQ} Fock states only, provides a universal framework for the description of low-energy baryons. The most important ingredients are spontaneous chiral-symmetry breaking and strict relativistic invariance.

Electromagnetic and Axial Form Factors of Octet and Decuplet Baryons

We report from a study of the elastic electromagnetic and axial form factors of all lowest baryon states with flavors up, down, and strange along relativistic constituent-quark models. We consider the baryons as relativistic bound states of three constituent quarks and solve the eigenvalue problem of the invariant mass operator. The corresponding eigenstates are employed to calculate manifestly covariant form factors within the point form of Poincare-invariant quantum mechanics. The electromagnetic and axial current operators are constructed along the spectator model in point-form relativistic dynamics. We have thus obtained covariant predictions for the electroweak form factors, for momentum transfers up to Q2 ~ 4 GeV2, as well as the electric radii, magnetic moments, and axial charges. The theoretical results in general agree very well with existing phenomenological data. In cases, where no experimental information is yet available, the results are well compatible with data from lattice quantum chromodynamics.

Treatment of broken symmetry in the Faddeev approach to the strange baryon spectra

By solving the Faddeev equations we calculate the mass of the strange baryons in the framework of a relativistic constituent quark model. The Goldstone-boson-exchange quark-quark interaction is derived from $SU(3{)}_{F}$ symmetry, which is explicitly broken as the strange quark is much heavier. This broken symmetry can nicely be accounted for in the Faddeev framework.

Relativistic Coupled-Channel Quark Model for Meson Resonances

We present results from a study of meson ground and resonant states within a relativistic coupled-channel constituent-quark model. Along such an approach in particular the resonance character of hadron excitations can be fully taken into account. Thus it becomes possible to describe the decay of a hadron resonance into a lower lying state more realistically than in usual single-channel approaches. In a simplified model we demonstrate the viability and the advantages of our method by producing relativistic invariant results for the finite decay width of a meson resonance.

Spectroscopy of Baryons as Relativistic Three-Quark Systems

We present results from a study of baryon spectral properties within a relativistic constituent-quark model. In particular, we demonstrate the performance of a universal quark model for all light-, strange-, and heavy-flavor baryons with regard to their spectroscopy. Thereby we produce insights into the effective interaction between constituent quarks of the various flavors up, down, strange, charm, and bottom. The relativistically invariant mass spectra are obtained by two different methods for calculating the microscopic three-quark systems: a stochastic variational method, solving the eigenvalue problem of the invariant mass operator expressed by differential equations, and a Faddeev integral-equation method, adapted to treating long-range interactions, such as the quark confinement. The corresponding results agree very well, generally within a few percents. Taking into account relativistic effects through Poincare invariance of the mass operator, or equivalently of the Hamiltonian, turns out to be of utmost importance.

Transversity studies with protons and light nuclei

A general formalism to evaluate time-reversal odd transverse momentum dependent parton distributions (T-odd TMDS) is reviewed and applied to the calculation of the leading-twist quantities, i.e., the Sivers and the Boer-Mulders functions. Two different models of the proton structure, namely a non relativistic constituent quark model and the MIT bag model, have been used. The results obtained in both frameworks fulfill the Burkardt sum rule to a large extent. The calculation of nuclear effects in the extraction of neutron single spin asymmetries in semi-inclusive deep inelastic scattering off transversely polarized 3He is also illustrated. In the kinematics of JLab, it is found that the nuclear effects described by an Impulse Approximation approach are under theoretical control.