Description of hadrons with Covariant Quark Model

Covariant quark model represents an effective field approach to hadronic interactions. It is based on a non-local Lagrangian, which provides full Lorentz invariance and has limited number of parameters. The model is suitable for multi-quark state description and successfully describes numerous experimental measurements. In this text we give a brief introduction to the model and refer to the achieved results.


Introduction
Hadron physics, nowadays in center of interest thanks to heavy-quark factories, lacks solid firstprinciple theoretical explanation for the wide range of the measured data. The perturbative approach is no longer applicable and other approaches with small model dependence (lattice QCD, ChPT) are usually restrained to a specific phase-space region.
To cover the hadronic experimental data in their large variety a model-depended approach is needed. The covariant quark model (CQM) can be applied to wide spectra of hadronic processes and results can be calculated using standard quantum-field theory techniques.

Covariant quark model in a nutshell
The CQM introduces an effective quark-hadron interaction (no gluons) which is governed by following Lagrangian (density) where the current J H (x) for different hadrons (meson, baryon, tetraquark) is Here C is charge conjugation matrix and F is a non-local vertex function where we require the barycenter of the quark system to correspond to the hadron position We use so-called compositeness condition to achieve an appropriate description of hadrons as quark bound states and eliminate quark-hadron couplings as free parameters. This topic was studied already decades ago [1,2] and the out-coming condition can be formulated as follows: the renormalization constant Z 1/2 M can be interpreted as the matrix element between a physical state and the corresponding bare state. Requiring Z 1/2 M = M bare |M dressed = 0 then implies that the physical state does not contain bare state and is therefore properly described as bound. The condition can be expressed via the derivative of the hadron mass operator, which can be calculated within the CQM The already mentioned cut-off parameter λ was introduced into the model in order to provide an infrared cut-off and ensure stability of heavy hadrons. Unity in form of delta function 1 = is introduced when evaluating Feynman diagrams, where α i corresponds to Schwinger parameters (used to express quark propagators). Then a multidimensional improper integral can be transformed into a form where only one improper integration remains. The cut-off is afterwards applied on the upper integration limit In this way Π becomes a smooth function where thresholds in the quark loop diagrams and corresponding branch points are removed. The integration is done numerically. The objects (mostly form factors) needed to predict observable properties of hadrons come from evaluation of corresponding Feynman graphs and few smart operator identities can be used to make the evaluation effective. They can be found, e.g., in [3].
When a weak hadron decay is concerned, we use an effective theory with Wilson coefficients and four-quark vertex to describe the flavour transition.

Overview of results
The following non-exhaustive overview of chosen results demonstrates the wide application range of the CQM.

Mesons
To illustrate results of the model in the domain of mesons one can mention the non-leptonic B s decays. In paper [4] we studied the processes B s → J/ψ + η ( ) . We obtained values for branching fractions in good agreement with experimental ones [5,6] B CQM (J/ψ η) = 4.67 B Belle (J/ψ η) = 5.10 ± 1.12 (8) B CQM J/ψ η = 4.04 B Belle J/ψ η = 3.71 ± 0.95 (9) and we have shown the importance of the non-trivial contribution of the model calculations to the corresponding form factors Branching fractions for different non-leptonic B s decays were studied in [7]. Although the results (Tab. 1) are mostly outside the 1σ measurement uncertainties, they describe the data behavior fairly well.

Summary and conclusion
Covariant quark model is a successful model with wide application spectra. It can be used for heavy and light hadrons, for strong and weak decays and for hadrons with different quark multiplicities (meson, baryon and tetraquark).