Weak transition form factors of heavy-light pseudoscalar mesons for space- and timelike momentum transfers

Weak $B^-rightarrow D^0, pi^0$ and $D^-rightarrow {K}^0, pi^0$ transition form factors are described in both the space- and time-like momentum transfer regions, within a constituent-quark model. Neutrino-meson scattering and semileptonic weak decays are formulated within the point form of relativistic quantum mechanics to end up with relativistic invariant process amplitudes from which meson transition currents and form factors are extracted in an unambiguous way. For space-like momentum transfers, form factors depend on the frame in which the $W M M^prime$ vertex is considered. Such a frame dependence is expected from a pure valence-quark picture, since a complete, frame independent description of form factors is supposed to include non-valence contributions. The most important of such contributions are the $Z$-graphs, which are, however, suppressed in the infinite-momentum frame ($q^2<0$). On the other hand, they can play a significant role in the Breit frame ($q^2<0$) and in the direct decay calculation ($q^2>0$), as a comparison with the infinite-momentum-frame form factors (analytically continued to $q^2>0$) reveals. Numerical results for the analytically continued infinite-momentum-frame form factors agree very well with lattice data in the time-like momentum transfer region and the experimental value for the slope of the $F^+_{Brightarrow D}$ transition form factor at zero recoil is reproduced satisfactorily. These predictions satisfy heavy-quark-symmetry constraints and their $q^2$ dependence is well approximated by a pole fit, reminiscent of a vector-meson-dominance-like decay mechanism. We discuss how such a decay mechanism can be accommodated within an extension of our constituent-quark model, by allowing for a non-valence component in the meson wave functions. We also address the question of wrong cluster properties inherent in the Bakamjian-Thomas formulation.

Polynomial Solutions of Generalized Quartic Anharmonic Oscillators

We introduce a nilpotent group to write a generalized quartic anharmonic oscillator Hamiltonian as a polynomial in the generators of the group. Energy eigenvalues are then seen to depend on the values of the two Casimir operators of the group. This dependence exhibits a scaling law which follows from the scaling properties of the group generators. Demanding that the potential give rise to polynomial solutions in a particular Lie algebra element puts constraints on the four potential parameters, leaving only two of them free. For potentials satisfying such constraints at least one of the energy eigenvalues and the corresponding eigenfunctions can be obtained in closed analytic form; examples, beyond those available in the literature, are given. Finally, we find solutions for particles in external electromagnetic fields in terms of these quartic polynomial solutions.

Pion-cloud contribution to the $Nrightarrow Delta$ transition form factors

We examine the contribution of the pion cloud to the electromagnetic $N rightarrow Delta$ transition form factors within a relativistic hybrid constituent-quark model. In this model baryons consist not only of the $3q$ valence component, but contain, in addition, a $3 q pi$ non-valence component. We start with constituent quarks which are subject to a scalar, isoscalar confining force. This leads to an $SU(6)$ spin-flavor symmetric spectrum with degenerate nucleon and Delta masses. Mass splitting is caused by pions which are assumed to couple directly to the quarks. The point-form of relativistic quantum mechanics is employed to achieve a relativistically invariant description of this system. The $N rightarrow Delta$ transition current is then determined from the one-photon exchange contribution to the $Delta$ electroproduction amplitude. We will give predictions for the ratios $R_{EM}$ and $R_{SM}$ of electric to magnetic and Coulomb to magnetic form factors, which are supposed to be most sensitive to pion-cloud effects.

The problem of cluster separability in relativistic few-body systems

An appropriate framework for dealing with hadron structure and hadronic physics in the few-GeV energy range is relativistic quantum mechanics. The Bakamjian-Thomas construction provides a systematic procedure for implementing interactions in a relativistic invariant way. It leads, however, to problems with cluster separability. It has been known for some time, due to Sokolovs pioneering work, that mass operators with correct cluster properties can be obtained through a series of unitary transformations making use of so-called packing operators. In the present contribution we sketch an explicit construction of packing operators for three-particle systems consisting of distinguishable, spinless particles.

Constituent-quark model with pionic contributions: electromagnetic $NrightarrowDelta$ transition

We report on ongoing work to determine the pion-cloud contribution to the electromagnetic $NrightarrowDelta$ transition form factors. The starting point is an $SU(6)$ spin-flavor symmetric constituent-quark model with instantaneous confinement that is augmented by dynamical pions which couple directly to the quarks. This system is treated in a relativistically invariant way within the framework of point-form quantum mechanics using a multichannel formulation. The first step is to determine the electromagnetic form factors of the bare particles that consist only of three quarks. These form factors are basic ingredients for calculating the pion-cloud contributions. Already without the pion cloud, electromagnetic nucleon and $Nrightarrow Delta$ transition form factors compare reasonably well with the data. By inclusion of the pion-cloud contribution coming from the $pi$-$N$ intermediate state the reproduction of the data is further improved.

$rho$-Meson Form Factors in the Point Form

We present a calculation of the electromagnetic form factors of the $rho^+$ meson. Our formalism is based on the point-form of relativistic quantum mechanics. Electron-$rho$-meson scattering is formulated as a coupled-channel problem for a Bakamjian-Thomas mass operator, such that the dynamics of the exchanged photon is taken explicitly into account. The $rho$-meson current is extracted from on-shell matrix elements of the optical potential of the scattering process. As a consequence of the violation of cluster separability in the Bakamjian-Thomas framework, our current includes additional, unphysical contributions, which can be separated from the physical ones uniquely. Our results for the form factors are in good agreement with other approaches.

On the microscopic structure of $pi NN$, $pi NDelta$ and $piDeltaDelta$ vertices

We use a hybrid constituent-quark model for the microscopic description of $pi N N$, $pi N Delta$ and $pi Delta Delta$ vertices. In this model quarks are confined by an instantaneous potential and are allowed to emit and absorb a pion, which is also treated as dynamical degree of freedom. The point form of relativistic quantum mechanics is employed to achieve a relativistically invariant description of this system. Starting with an $SU(6)$ spin-flavor symmetric wave function for $N_0$ and $Delta_0$, i.e. the eigenstates of the pure confinement problem, we calculate the strength of the $pi N_0 N_0$, $pi N_0 Delta_0$ and $pi Delta_0 Delta_0$ couplings and the corresponding vertex form factors. Interestingly the ratios of the resulting couplings resemble strongly those needed in purely hadronic coupled-channel models, but deviate significantly from the ratios following from SU(6) spin-flavor symmetry in the non-relativistic constituent-quark model.

The microscopic structure of $pi NN$, $pi NDelta$ and $piDeltaDelta$ vertices in a hybrid constituent quark model

We present a microscopic description of the strong $pi NN$, $pi NDelta$ and $piDeltaDelta$ vertices. Our starting point is a constituent-quark model supplemented by an additional $3qpi$ non-valence component. In the spirit of chiral constituent-quark models, quarks are allowed to emit and reabsorb a pion. This multichannel system is treated in a relativistically invariant way within the framework of point-form quantum mechanics. Starting with a common $SU(6)$ spin-flavor-symmetric wave function for $N$ and $Delta$, we calculate the strength of the $pi NN$, $pi NDelta$ and $piDeltaDelta$ couplings and the corresponding vertex form factors. Our results are in accordance with phenomenological fits of these quantities that have been obtained within purely hadronic multichannel models for baryon resonances.

The Pion-Cloud Contribution to the Electromagnetic Nucleon Form Factors

We study the electromagnetic structure of the nucleon within a hybrid constituent-quark model that comprises, in addition to the $3q$ valence component, also a $3q$+$pi$ non-valence component. To this aim we employ a Poincare-invariant multichannel formulation based on the point-form of relativistic quantum mechanics. With a simple 3-quark wave function for the bare nucleon, i.e. the $3q$-component, we obtain reasonable results for the nucleon form factors and predict the meson-cloud contribution to be significant only below $Q^2lesssim 0.5$,GeV$^2$ amounting to about 10% for $Q^2rightarrow 0$, in accordance with the findings of other authors.

$ pi^- p rightarrow D^- Lambda_{c}^{+} $ within the Generalized Parton Picture

We investigate the reaction $ pi^- p rightarrow D^- Lambda_{c}^{+} $ within the generalized parton picture. The process is described by a handbag-type mechanism with the charm-quark mass acting as the hard scale. As in the case of preceding work on $bar{p} p rightarrow bar{Lambda}^-_c Lambda_{c}^{+} $ we argue that the process amplitude factorizes into one for the perturbatively calculable partonic subprocess $bar{u} urightarrow bar{c} c$ and hadronic matrix elements that can be parameterized in terms of generalized parton distributions. Modeling the generalized parton distributions by overlaps of (valence-quark) light-cone wave functions for the hadrons involved, we obtain numerical results for unpolarized differential and integrated cross sections as well as spin observables. Our approach works well above the production threshold ($s gtrsim 20 $GeV$^2$) in the forward hemisphere and predicts unpolarized cross sections of the order of nb, a finding that could be of interest in view of plans to measure $ pi^- p rightarrow D^- Lambda_{c}^{+} $ at J-PARC.