The Covariant Spectator Theory (CST) is used to calculate the mass spectrum and vertex functions of heavy-light and heavy mesons in Minkowski space. The covariant kernel contains Lorentz scalar, pseudoscalar, and vector contributions. The numerical calculations are performed in momentum space, where special care is taken to treat the strong singularities present in the confining kernel. The observed meson spectrum is very well reproduced after fitting a small number of model parameters. Remarkably, a fit to a few pseudoscalar meson states only, which are insensitive to spin-orbit and tensor forces and do not allow to separate the spin-spin from the central interaction, leads to essentially the same model parameters as a more general fit. This demonstrates that the covariance of the chosen interaction kernel is responsible for the very accurate prediction of the spin-dependent quark-antiquark interactions.
The Covariant Spectator Theory is applied to the description of quarks and the pion. The dressed quark mass function is calculated dynamically in Minkowski space and used in the calculation of the pion electromagnetic form factor. The effects of the mass function on the pion form factor and the different quark-pole contributions to the triangle diagram are analyzed.
We propose a model for the quark-antiquark interaction in Minkowski space using the Covariant Spectator Theory. We show that with an equal-weighted scalar-pseudoscalar structure for the confining part of our interaction kernel the axial-vector Ward-Takahashi identity is preserved and our model complies with the Adler-zero constraint for pi-pi-scattering imposed by chiral symmetry.
We introduce a covariant approach in Minkowski space for the description of quarks and mesons that exhibits both chiral-symmetry breaking and confinement. In a simple model for the interquark interaction the quark mass function is obtained and used in the calculation of the pion form factor. We study the effects of the mass function and of the different quark pole contributions on the pion form factor.
The pion electromagnetic form factor at spacelike momentum transfer is calculated in relativistic impulse approximation using the Covariant Spectator Theory. The same dressed quark mass function and the equation for the pion bound-state vertex function as discussed in the companion paper are used for the calculation, together with a dressed quark current that satisfies the Ward-Takahashi identity. The results obtained for the pion form factor are in agreement with experimental data, they exhibit the typical monopole behavior at high-momentum transfer, and they satisfy some remarkable scaling r…
The pion form factor is calculated in the framework of the charge-conjugation invariant Covariant Spectator Theory. This formalism is established in Minkowski space and the calculation is set up in momentum space. In a previous calculation we included only the leading pole coming from the spectator quark (referred to as the relativistic impulse approximation). In this paper we also include the contributions from the poles of the quark which interacts with the photon and average over all poles in both the upper and lower half planes in order to preserve charge conjugation invariance (referred t…
Heavy and heavy-light tensor and axial-tensor mesons in the Covariant Spectator Theory
Centro de Ciências e Tecnologias Nucleares and Departamento de Engenharia e Ciências Nucleares, Instituto Superior Técnico, Universidade de Lisboa（里斯本大学技术研究所核科学与工程系及核科学与技术中心）；Departamento de Física, Universidade de Évora（埃武拉大学物理系）；Laboratório de Instrumentação e Física Experimental de Partículas—LIP（粒子实验物理与仪器实验室）
The pi-pi scattering amplitude calculated with a model for the quark-antiquark interaction in the framework of the Covariant Spectator Theory (CST) is shown to satisfy the Adler zero constraint imposed by chiral symmetry. The CST formalism is established in Minkowski space and our calculations are performed in momentum space. We prove that the axial-vector Ward-Takahashi identity is satisfied by our model. Then we show that, similar to what happens within the Bethe-Salpeter formalism, application of the axial-vector Ward-Takahashi identity to the CST pi-pi scattering amplitude allows us to sum…
We present the effective range expansions for the 1S_0 and 3S_1 scattering phase shifts, and the relativistic deuteron wave functions that accompany our recent high precision fits (with chi^2/N{data} approx 1) to the 2007 world np data below 350 MeV. The wave functions are expanded in a series of analytical functions (with the correct asymptotic behavior at both large and small arguments) that can be Fourier-transformed from momentum to coordinate space and are convenient to use in any application. A fortran subroutine to compute these wave functions can be obtained from the authors.
We use the covariant spectator theory with an effective quark-antiquark interaction, containing Lorentz scalar, pseudoscalar, and vector contributions, to calculate the masses and vertex functions of, simultaneously, heavy and heavy-light mesons. We perform least-square fits of the model parameters, including the quark masses, to the meson spectrum and systematically study the sensitivity of the parameters with respect to different sets of fitted data. We investigate the influence of the vector confining interaction by using a continuous parameter controlling its weight. We find that vector co…
Deep-inelastic structure functions are studied within a covariant scalar diquark spectator model of the nucleon. Treating the target as a two-body bound state of a quark and a scalar diquark, the Bethe-Salpeter equation (BSE) for the bound state vertex function is solved in the ladder approximation. The valence quark distribution is discussed in terms of the solutions of the BSE.
Using a covariant spectator quark model that describes the recent lattice QCD data for the $Δ$ electromagnetic form factors and all available experimental data on $γN \to Δ$ transitions, we analyze the charge and magnetic dipole distributions of the $Δ$ baryon and discuss its shape. We conclude that the quadrupole moment of the $Δ$ is a good indicator of the deformation and that the $Δ^+$ charge distribution has an oblate shape. We also calculate transverse moments and find that they do not lead to unambiguous conclusions about the underlying shape.
A modification of the one boson exchange (OBE) kernel for the covariant spectator theory (CST) is presented and discussed. When applied to the scattering of two identical particles, the previously used kernels either introduced spurious singularities, or removed them in an ad-hoc way. The new modification not only removes these singularities, but also maintains the convergence of the two-body CST equation (sometimes called the Gross equation) when used to describe the scattering of two identical scalar particles.
Nucleon structure functions, as measured in deep-inelastic lepton scattering, are studied within a covariant scalar diquark spectator model. Regarding the nucleon as an approximate two-body bound state of a quark and diquark, the Bethe-Salpeter equation (BSE) for the bound state vertex function is solved in the ladder approximation. The valence quark distribution is discussed in terms of the solutions of the BSE.
The deuteron magnetic moment is calculated using two model wave functions obtained from 2007 high precision fits to $np$ scattering data. Included in the calculation are a new class of isoscalar $np$ interaction currents which are automatically generated by the nuclear force model used in these fits. After normalizing the wave functions, nearly identical predictions are obtained: model WJC-1, with larger relativistic P-state components, gives 0.863(2), while model WJC-2 with very small P-state components gives 0.864(2) These are about 1\% larger than the measured value of the moment, 0.857 n.m…
Heavy and heavy-light mesons with arbitrary spin and parity in the Covariant Spectator Theory
Universidade de Évora（埃武拉大学）；Laboratório de Instrumentação e Física Experimental de Partículas—LIP（粒子物理实验与仪器实验室）；Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院）
As an application of the Covariant Spectator Theory (CST) we calculate the spectrum of heavy-light and heavy-heavy mesons using covariant versions of a linear confining potential, a one- gluon exchange, and a constant interaction. The CST equations possess the correct one-body limit and are therefore well-suited to describe mesons in which one quark is much heavier than the other. We find a good fit to the mass spectrum of heavy-light and heavy-heavy mesons with just three parameters (apart from the quark masses). Remarkably, the fit parameters are nearly unchanged when we fit to experimental …
The masses and vertex functions of heavy and heavy-light mesons, described as quark-antiquark bound states, are calculated with the Covariant Spectator Theory (CST). We use a kernel with an adjustable mixture of Lorentz scalar, pseudoscalar, and vector linear confining interaction, together with a one-gluon-exchange kernel. A series of fits to the heavy and heavy-light meson spectrum were calculated, and we discuss what conclusions can be drawn from it, especially about the Lorentz structure of the kernel. We also apply the Brodsky-Huang-Lepage prescription to express the CST wave functions fo…
Using the framework of the Covariant Spectator Theory (CST) [1] we are developing a covariant model formulated in Minkowski space to study mesonic structure and spectra. Treating mesons as effective $q\bar{q}$ states, we focused in [2] on the nonrelativistic bound-state problem in momentum space with a linear confining potential. Although integrable, this kernel has singularities which are difficult to handle numerically. In [2] we reformulate it into a form in which all singularities are explicitely removed. The resulting equations are then easier to solve and yield accurate and stable soluti…
The deuteron form factors are calculated using two model wave functions obtained from the 2007 CST high precision fits to $np$ scattering data. Included in the calculation are a new class of isoscalar $np$ interaction currents which are automatically generated by the nuclear force model used in these fits. If the nuclear model WJC2 is used, a precision fit ($χ^2$/datum $\eqsim1$) to the Sick Global Analysis (GA) of all $ed$ elastic scattering data can be obtained by adjusting the unknown off-shell nucleon form factors $F_3(Q^2)$ (discussed before) and $F_4(Q^2)$ (introduced in this paper), and…
We study the $γ^\ast Λ\to Σ^0$ transition form factors by applying the covariant spectator quark model. Using the parametrization for the baryon core wave functions as well as for the pion cloud dressing obtained in a previous work, we calculate the dependence on the momentum transfer squared, $Q^2$, of the electromagnetic transition form factors. The magnetic form factor is dominated by the valence quark contributions. The final result for the transition magnetic moment, a combination of the quark core and pion cloud effects, turns out to give a value very close to the data. The pion cloud co…
Expressions for the nucleon wave functions in the covariant spectator theory (CST) are derived. The nucleon is described as a system with a off-mass-shell constituent quark, free to interact with an external probe, and two spectator constituent quarks on their mass shell. Integrating over the internal momentum of the on-mass-shell quark pair allows us to derive an effective nucleon wave function that can be written only in terms of the quark and diquark (quark-pair) variables. The derived nucleon wave function includes contributions from S, P and D-waves.
Preliminary calculations using the Covariant Spectator Theory (CST) employed a scalar linear confining interaction and an additional constant vector potential to compute the mesonic mass spectra. In this work we generalize the confining interaction to include more general structures, in particular a vector and also a pseudoscalar part, as suggested by a recent study. A one-gluon-exchange kernel is also implemented to describe the short-range part of the interaction. We solve the simplest CST approximation to the complete Bethe-Salpeter equation, the one-channel spectator equation, using a nume…
In this work, we perform a covariant treatment of quark-antiquark systems. We calculate the spectra and wave functions using a formalism based on the Covariant Spectator Theory (CST). Our results not only reproduce very well the experimental data with a very small set of global parameters, but they also allow a direct test of the predictive power of covariant kernels.
A constituent quark model based on the spectator formalism is applied to the gamma N -> N* transition for the three cases, where N* is the nucleon, the Delta and the Roper resonance. The model is covariant, and therefore can be used for the predictions at higher four-momentum transfer squared, Q2. The baryons are described as an off-mass-shell quark and a spectator on-mass-shell diquark systems. The quark electromagnetic current is described by quark form factors, which have a form inspired by the vector meson dominance. The valence quark contributions of the model are calibrated by lattice QC…
We solve a Minkowski-space integral equation, derived in the Covariant Spectator Theory, for quark-antiquark bound states describing heavy and heavy-light mesons. The equation's kernel contains a one-gluon exchange interaction and a covariant generalization of a linear confining potential with a mixed scalar, pseudoscalar, and vector Lorentz structure, characterized by a continuous mixing parameter. We investigate to what extent the Lorentz structure of the confining kernel can be determined by fitting the mixing parameter to the meson spectrum.
Background: DIS on the polarized deuteron with detection of a proton in the nuclear breakup region (spectator tagging) represents a unique method for extracting the neutron spin structure functions and studying nuclear modifications. The tagged proton momentum controls the nuclear configuration during the DIS process and enables a differential analysis of nuclear effects. Such measurements could be performed with the future electron-ion collider (EIC) and forward proton detectors if deuteron beam polarization could be achieved. Purpose: Develop theoretical framework for polarized deuteron DIS …
We apply the covariant spectator quark model to the study of the electromagnetic structure of the $N^\ast(1520)$ state ($J^{P}= \frac{3}{2}^-$), an important resonance from the second resonance region in both spacelike and timelike regimes. The contributions from the valence quark effects are calculated for the $γ^\ast N \to N^\ast(1520)$ helicity amplitudes. The results are used to parametrize the meson cloud dominant at low $Q^2$.
Using a covariant spectator constituent quark model we predict an electric quadrupole moment -0.043 efm^2 and a magnetic octupole moment -0.0035 efm^3 for the Delta+ excited state of the nucleon.
We analyze the analytic structure of the Covariant Spectator Theory (CST) contribution to the self-energy amplitude for a scalar particle in a ϕ^2 χ-theory. To this end we derive dispersion relations in 1+1 and in 3+1 dimensional Minkowski space. The divergent loop integrals in 3+1 dimensions are regularized using dimensional regularization. We find that the CST dispersion relations exhibit, in addition to the usual right-hand branch cut, also a left-hand cut. The origin of this "spectator" left-hand cut can be understood in the context of scattering for a scalar ϕ^2 χ^2-type theory. If the in…
We present the results of the covariant spectator quark model applied to the nucleon structure function $f(x)$ measured in unpolarized deep inelastic scattering, and the structure functions $g_1(x)$ and $g_2(x)$ measured in deep inelastic scattering using polarized beams and targets ($x$ is the Bjorken scaling variable). The nucleon is modeled by a valence quark-diquark structure with $S,P$ and $D$ components. The shape of the wave functions and the relative strength of each component are fixed by making fits to the deep inelastic scattering data for the structure functions $f(x)$ and $g_1(x)$…
We present estimates from the covariant spectator quark model for the electromagnetic transition form factors for the resonances $N(1440)\frac{1}{2}^+$, $N(1535)\frac{1}{2}^-$, $N(1520)\frac{3}{2}^-$ and $Δ(1232)\frac{3}{2}^+$, at intermediate and large square momentum transfer ($Q^2$). The calculations associated to the $N(1440)\frac{1}{2}^+$, $N(1535)\frac{1}{2}^-$ and $N(1520)\frac{3}{2}^-$ states are based exclusively on the parametrizations derived for the nucleon. In the case of the $Δ(1232)\frac{3}{2}^+$ (isospin 3/2), we use lattice QCD data to estimate the radial structure, and take i…
We suggest to study the $B_{s}$ and its excitations $B_{sJ}$ in the $B_c$ decays. We calculate the $B_c\to B_{sJ}$ and $B_c\to B_{J}$ form factors within the covariant light-front quark model, where the $B_{sJ}$ and $B_{J}$ denote an s-wave or p-wave $\bar bs$ and $\bar bd$ meson, respectively. The form factors at $q^2=0$ are directly computed while their $q^2$-distributions are obtained by the extrapolation. The derived form factors are then used to study semileptonic $B_c\to (B_{sJ},B_{J})\bar\ellν$ decays, and nonleptonic $B_c\to B_{sJ}π$. Branching fractions and polarizations are predicted…
The covariant spectator quark model is applied to the $γ^\ast N \to N^\ast(1520)$ reaction. The valence quark contributions to the electromagnetic form factors and helicity transition amplitudes are estimated from the covariant structure of the $N^\ast(1520)$ wave function calibrated by the experimental data for large squared momentum transfer $Q^2$. The difference between the model results and the experimental data is then used to parameterize the low $Q^2$ behavior, where meson cloud effects are assumed to dominate. This parametrization can be very useful for future studies of the reaction, …
We formulate the covariant equations for quark-antiquark bound states in Minkowski space in the framework of the Covariant Spectator Theory. The quark propagators are dressed with the same kernel that describes the interaction between different quarks. We show that these equations are charge-conjugation invariant, and that in the chiral limit of vanishing bare quark mass, a massless pseudoscalar bound state is produced in a Nambu-Jona-Lasinio (NJL) mechanism, which is associated with the Goldstone boson of spontaneous chiral symmetry breaking. In this introductory paper, we test the formalism …
We present a model for the quark-antiquark interaction formulated in Minkowski space using the Covariant Spectator Theory. The quark propagators are dressed with the same kernel that describes the interaction between different quarks. By applying the axial-vector Ward-Takahashi identity we show that our model satisfies the Adler-zero constraint imposed by chiral symmetry. For this model, our Minkowski-space results of the dressed quark mass function are compared to lattice QCD data obtained in Euclidean space. The mass function is then used in the calculation of the electromagnetic pion form f…
The conformal algebra provides powerful constraints, which guarantee that renormalized conformally covariant operators exist in the hypothetical conformal limit of the theory, where the $β$-function vanishes. Thus, in this limit also the conformally covariant operator product expansion on the light cone holds true. This operator product expansion has predictive power for two-photon processes in the generalized Bjorken region. Only the Wilson coefficients and the anomalous dimensions that are known from deep inelastic scattering are required for the prediction of all other two-photon processes …
A covariant spectator quark model is applied to study the gamma N -> N*(1535) reaction in the large Q2 region. Starting from the relation between the nucleon and N*(1535) systems, the N*(1535) valence quark wave function is determined without the addition of any parameters. The model is then used to calculate the gamma N -> N*(1535) transition form factors. A very interesting, useful relation between the A12 and S12 helicity amplitudes for Q2 > GeV^2, is also derived.
A covariant spectator quark model is applied to estimate the valence quark contributions to the$ F1*(Q2) and F2*(Q2) transition form factors for the gamma N -> P11(1440) reaction. The Roper resonance, P11(1440), is assumed to be the first radial excitation of the nucleon. The model requires no extra parameters except for those already fixed by the previous studies for the nucleon. The results are consistent with the experimental data in the high Q2 region, and those from the lattice QCD. Finally the model is also applied to estimate the meson cloud contributions from the CLAS and MAID analysis…
In this work we compare mass spectra and decay constants obtained from two recent, independent, and fully relativistic approaches to the quarkonium bound-state problem: the Hamiltonian basis light-front quantization (BLFQ) approach, where light-front wave functions are naturally formulated; and, the covariant spectator theory (CST), based on a reorganization of the Bethe-Salpeter equation. Even though conceptually different, both solutions are obtained in Minkowski space. Comparisons of decay constants for more than ten states of charmonium and bottomonium show favorable agreement between the …
We present a study of the dressed quark mass function and the pion structure in Minkowski space using the Covariant Spectator Theory. The quark propagators are dressed with the same kernel that describes the interaction between different quarks. We use an interaction kernel in momentum space that is a relativistic generalization of the linear confining q-qbar potential and a constant potential shift that defines the energy scale. The confining interaction has a Lorentz scalar part that is not chirally invariant by itself but decouples from the equations in the chiral limit and therefore allows…
Using a covariant spectator quark model we estimate valence quark contributions to the F1*(Q2) and F2*(Q2) transition form factors for the gamma N -> P11(1440) reaction. The Roper resonance, P11(1440), is assumed to be the first radial excitation of the nucleon. The present model requires no extra parameters except for those already fixed by the previous studies for the nucleon. Our results are consistent with the experimental data in the high Q2 region, and those from lattice QCD. We also estimate the meson cloud contributions, focusing on the low Q2 region, where they are expected to be domi…
Several years ago we have proposed a manifestly covariant $\tilde{U}(12)_{SF}$-classification scheme of hadrons, which maintains the successful $SU(6)_{SF}\otimes O(3)_{L}$ framework in Non-Relativistic Quark Model and is also reconcilable with the mechanism of Spontaneously-Broken Chiral Symmetry in Relativistic Field Theoretical Models. The essential point here is to notice the overlooked freedom of the $SU(2)_ρ$-spin for (confined) light quarks, which leads to existence of new type of ``exotic'' hadrons, called ``chiralons'' and to a selection rule, $ρ_{3}$-line rule, on the spectator quark…
We study low-energy nucleon Compton scattering in the framework of baryon chiral perturbation theory (B$χ$PT) with pion, nucleon, and $Δ$(1232) degrees of freedom, up to and including the next-to-next-to-leading order (NNLO). We include the effects of order $p^2$, $p^3$ and $p^4/\varDelta$, with $\varDelta\approx 300$ MeV the $Δ$-resonance excitation energy. These are all "predictive" powers in the sense that no unknown low-energy constants enter until at least one order higher (i.e, $p^4$). Estimating the theoretical uncertainty on the basis of natural size for $p^4$ effects, we find that unc…
