Relativistic effects and the constituent quark model of heavy quarkonia

Charmonium \(c\bar c\) and bottomonium \(b\bar b\) are investigated in the framework of a constituent quark model. A scalar confining and a one-gluon exchange (OGE) potential are used in a nonrelativistic reduction to order (p/m)². Therefore the model includes spin dependent as well as spin independent terms. Their influence on the meson mass spectra and decay widths is analysed. We find that the experimental spectra can be reproduced by using a full model as well as by using a reduced version neglecting the spin independent terms. For both versions we calculate leptonic and radiative decay widths including relativistic corrections for the current operators. We find that for leptonic decays inclusion of all terms of the OGE potential gives better results than the non-relativistic formulas. For radiative transitions relativistic corrections are important.     

Pseudoscalars in the charmonium model

We take the recently found charmonium state at 2.976 GeV to be the η _c and show that it can be included in a charmonium model with relativistic corrections which reproduces the s-wave spectrum, the leptonic widths Γ(V→e ⁺ e^?) and the p-wave splittings. The upsilon spectrum is discussed as are the effects of radial and pseudoscalar mixing.     

Heavy quarkonia from classical SU(3) Yang-Mills configurations

A generalized Cho-Faddeev-Niemi ansatz for SU(3) Yang-Mills is investigated. The corresponding classical field equations are solved for its simplest parametrization. From these solutions it is possible to define a confining non-relativistic central potential used to study heavy quarkonia. The associated spectra reproduces the experimental spectra with an error of less than 3% for charmonium and 1% for bottomonium. Moreover, the recently discovered new charmonium states can be accomodate in the spectra, keeping the same level of precision. The leptonic widths show good agreement with the recent measurements. The charmonium and bottomonium E1 electromagnetic transitions widths are computed and compared with the experimental values.     

Revisiting radiative decays of $$1^{+-}$$ heavy quarkonia in the covariant light-front approach

We revisit the calculation of the width for the radiative decay of a \(1^{+-}\) heavy \(Q \bar{Q}\) meson via the channel \(1^{+-} \rightarrow 0^{-+} +\gamma \) in the covariant light-front quark model. We carry out the reduction of the light-front amplitude in the non-relativistic limit, explicitly computing the leading and next-to-leading order relativistic corrections. This shows the consistency of the light-front approach with the non-relativistic formula for this electric dipole transition. Furthermore, the theoretical uncertainty in the predicted width is studied as a function of the inputs for the heavy-quark mass and wave function structure parameter. We analyze the specific decays \(h_{\mathrm{c}}(1P) \rightarrow \eta _{\mathrm{c}}(1S) + \gamma \) and \(h_{\mathrm{b}}(1P) \rightarrow \eta _{\mathrm{b}}(1S) + \gamma \). We compare our results with experimental data and with other theoretical predictions from calculations based on non-relativistic models and their extensions to include relativistic effects, finding reasonable agreement.     

Constraints on the decay rates of quarkonium

We derive inequalities between the leptonic decay rates of 1S and 2S states of quarkonium, when the binding potential is an increasing concave (convex) function of the inter-quark distance in a framework where some relativistic corrections have been made to the Van Royen-Weisskopf formula for these rates. Experimental decay rates of the γ and γ′ rule out the convex increasing potential.     

A relativistic correction to semiclassical charmonium

It is shown that the relativistic linear potentials, introduced by the author within the particle à la Wheeler-Feynman direct-interaction (AAD) theory, applied to the semiclassically quantized charmonium, yield energy spectrum comparable to that of some known models. Using the expansion of the relativistic linear AAD potentials in powers ofc ^–1, the charmonium spectrum, given as a rule by Bohr-Sommerfeld quantization of circular orbits, is extended up to the second order of relativistic corrections.     

Gluonic corrections to the decay rate of heavy quarkonium states into lepton pairs

We calculate one-gluon corrections to the decay rate of heavy quarkonium states into lepton pairs. We find that the correction due to one-gluon exchange is not very significant for charmonium, and for practical purposes only the relativistic kinematical effects need to be taken into account. For theY family the gluonic corrections are, relatively, of greater importance.     

Momentum-dependent effects in the decays of charmonium and upsilon

Decay widths of quarkonia are determined in a calculation in which the momentum distribution resulting from the localization of the quarks is taken into account. The e⁺e⁻ decay modes of charmonium and upsilon are considered with the momentum-dependent corrections to the well-known van Royen-Weisskopf formula for leptonic widths. The momentum-dependent effects are found to be significant, and amount to approximately a 20% correction. The γη(1440) decay mode is also examined to investigate a decay involving a propagating quark in the bound state. Here the momentum-dependent corrections to the static approximation are very large and suppress the decay widths by well over a factor of 3. Therefore one must exercise caution in using a static approximation.     

Bethe-salpeter equation with instantaneous harmonic oscillator exchange

The Bethe-Salpeter equation in the form due to Cung et al. (Ann. Phys. (N.Y.)98 (1976), 516) is investigated for the special case of instantaneous harmonic oseillator exchange, An exact reduction to a pair of coupled ordinary differential equations for the radial excitations of the ³(J ± 1)_J modes is achieved. The equations in the mass zero case are brought to a form which is quite close to Whittaker's equation. This similarity to Whittaker's equation is exploited in a computer study of the level structure as a function of the quark mass. This study covers the region from a highly relativistic spectrum depending only upon J to the nonrelativistic regime where the spectrum depends only upon L. An expression for the leptonic width of a ³S₁ state in terms of the Bethe-Salpeter wave function is derived and applied to the ψ-family. The effect of relativistic corrections is to reduce the predicted value of the leptonic width compared to the value calculated by assuming nonrelativistic kinematics. It is also shown that the relativistic treatment allows a ³D₁ state to couple directly to a virtual photon.     

Search for higher laying charmonium states and charmed hybrids in experiments using antiproton beam with momentum ranging from 1 GeV/<Emphasis Type="Italic">c</Emphasis> to 15 GeV/<Emphasis Type="Italic">c</Emphasis>

The elaborated analysis of spectrum of scalar and vector charmonium states in the mass region above D[`(D)]D\bar D — threshold is given. The combined approach based on the potential model and relativistic spherical symmetric top model for decay products has been proposed. Ten radial excited states of charmonium in the mass region above D[`(D)]D\bar D — threshold are anticipated to exist in the framework of the combined approach. The experimental data from different collaborations were analyzed. Especial attention was given to the new states with the hidden charm discovered recently. Eight of these states may be interpreted as higher laying radial excited charmonium states. But much more data on different decay modes are needed for deeper analysis. These data can be derived directly from the experiments using high quality antiproton beam with the momentum ranging from 1 GeV/c to 15 GeV/c (PANDA experiment at FAIR).     

LOCV calculation of nuclear matter with relativistic Hamiltonian

The equation of state of symmetric nuclear matter is calculated using the relativistic Hamiltonian (H_R) with potentials which have been fitted with the N -N scattering data using the relativistic two-body Hamiltonian ( [(v)\tilde]₁₄ \tilde{{v}}_{{14}}^{} and the non-relativistic two-body Hamiltonian, i.e. the Argonne V₁₄ interaction. The boost interaction corrections as well as the relativistic one-body and two-body kinetic energy corrections in cluster expansion energy within the lowest-order-constrained variational method are calculated. It is shown that the relativistic corrections reduce the binding energy by 1.5MeV for [(v)\tilde]₁₄ \tilde{{v}}_{{14}}^{} and AV₁₄ interactions. The symmetric nuclear-matter saturation energy is about -16.43 MeV at r \rho = 0.253 (fm^-3) with [(v)\tilde]₁₄ \tilde{{v}}_{{14}}^{} interaction plus relativistic corrections. Finally, various properties of the symmetric nuclear matter are given and a comparison is made with the other many-body calculations.     

On the decay of θ mesons in charmonium models

We derive a new formula for the decay rate of the ψ mesons by properly treating the dependence of lowest order electromagnetic annihilation on the quark-antiquark bound state wave function. Results are presented for currently used non-relativistic potentials models for charmonium. In all cases substantial corrections to previous estimates from commonly used formulas occur. Our results are of relevance also for hadronic decay via gluons in QCD and weak decay of pseudoscalar mesons.     

Neutrinoless double beta decay and a new limit on the lepton number violation

We have calculated the neutrinoless double beta decay rate of ⁷⁶Ge. We take into account for the first time a relativistic correction to the nuclear current including weak magnetism. Its effect is to cancel a considerable part of the decay amplitude and we obtain less stringent upper limits on the neutrino Majorana mass and the right-handed weak leptonic current compared with previous calculations.     

Saturation of confining forces in a relativistic quark model for the baryons

A relativistic quark model for the baryons with saturating three-particle forces is investigated. The properties of relativistic three-fermion amplitudes are analyzed with respect to Lorentz transformations and permutations. Six different classes of possible structures in spin space are found. They serve as an appropriate basis for the classification and calculation of spin-dependent interactions. The quantum numbers and amplitudes for the orbital part are determined for Euclidean relative vectors with help of the irreducible representations of the groups SU(4) SO(4) SO(3). These kinematical results together with the Green's function techniques of relativistic quantum field theory are applied to a Bethe-Salpeter model for the binding of three heavy quarks inside a baryon. We give as an example a confining saturating interaction which yields baryon quantum numbers similar to those of the non-relativistic harmonic oscillator model. However, the spin structure of the amplitudes obtained in this way differs from the boosted non-relativistic ones. This feature is important, since the phenomenological discussion of photoproduction and strong decays of the baryon resonances shows that at least sizable corrections to the non-relativistic amplitudes are necessary.     

A Partial Width Calculation of OZI-Allowed Charmonium Decays in a Coupled Channel Framework

Okubo–Zweig–Iizuka-allowed partial decay widths, masses, and total decay width of charmonium states are studied in a nonrelativistic coupled-channel framework based on microscopic effective quark interactions. With the help of the complex scale transformation, the coupled channel equation is easily solved under the proper boundary condition for resonances. The obtained result as a whole is very successful and encouraging for the traditional charmonium states including ψ(4040) whose features of mass and partial decay widths have been argued historically. The coupling mechanisms of these states are investigated by reducing artificially the channel coupling strengths little by little and finally turning the coupling off. The situations turn out to be quite different from what we would have naively supposed. Other solutions than the traditional charmonium states were obatined at the same time. Some of them are discussed in relation with new particles observed recently.     

Ab initio all-electron fully relativistic Dirac—Fock self-consistent field calculations for UCI6

Ab initio all-electron fully relativistic Dirac-Fock and non-relativistic Hartree-Fock self-consistent field (SCF) calculations are reported at four UCl bond distances, assuming octahedral UCl₆. The results are fitted to a polynomial, obtaining thereby the optimized values of the bond distance and the corresponding total electronic energy for the UCl₆. The nonrelativistic Hartree-Fock (HF) and Dirac-Fock (DF) SCF calculations predict UCl₆ to be bound, with a predicted dissociation (atomization) energy D _e of 11.88 eV and 17.89 eV, respectively. Relativistic effects lead to ~51% increment in the predicted atomization energy of UCl₆. The UCl bond lengths predicted for UCl₆ with the relativistic DF and non-relativistic HF wave-functions are 2.46 Å and 2.58 Å, respectively. Complete neglect in the SCF step of the two-electron [SS|SS] integrals involving the small components of the spinors (NOSS) in the DF SCF calculation for UCl₆ predicts a D _e of 18.25 eV and essentially the same bond length (2.48 Å) as that predicted with the full SCF procedure. Thus the small components contribute an antibinding relativistic effect of ~0.4 eV to the D _e of UCl₆ and have a negligible effect on the bond length. The calculations show that relativistic effects are significant for the bonding and the dissociation (atomization) energy of UCl₆, and that these may be treated accurately using Dirac's fully relativistic equation for an electron.