Current conservation, screening and the magnetic moment of the Δ\Delta resonance: 2. Formulation with quark degrees of freedom 3. Magnetic moment of the Δo\Delta^o and Δ−\Delta^- resonances

Our previous paper \cite{MFNEW} is generalized within the field theoretical formulation with the quark degrees of freedom \cite{HW,H,N,Z}, where pions and nucleons are treated as the bound systems of quarks. It is shown that relations generated by current conservation for the on shell $\pi N$ bremsstrahlung amplitude with composite nucleons and pions have the same form as in the usual quantum field theory \cite{IZ,BD} without quark degrees of freedom \cite{MFNEW}. Consequently, the model independent relations for the magnetic dipole moments of the $\Delta^+$ and $\Delta^{++}$ resonances in \cite{MFNEW} remain be the same in the quantum field theory with the quark degrees of freedom. These relations are extended for the magnetic dipole moments of the $\Delta^o$ and $\Delta^-$ resonances which are determined via the anomalous magnetic moment of the neutron $\mu_n$ as $\mu_{\Delta^o}={{M_{\Delta}}\over {m_p}} \mu_n$ and $\mu_{\Delta^{-}}={3\over 2}\mu_{\Delta^o}$.Comment: The second part of Preprint arXiv:nucl-th/0804.1322 2008; 17 pages 3 figures, 1 Tabl

Current conservation, screening and the magnetic moment of the Δ\Delta resonance. -- 1. Formulation without quark degrees of freedom

The pion-nucleon bremsstrahlung $\pi+N\Longrightarrow\gamma'+\pi'+N'$ is studied in a new form of current conservation. According to this condition, the internal and external particle radiation parts of the $\pi N$ radiation amplitude have opposite signs, i.e., they contain terms which must cancel each other. Therefore, one has a screening of the internal and external particle radiation in the $\pi N$ bremsstrahlung. In particular, it is shown that the double $\Delta$ exchange diagram with the $\Delta-\gamma' \Delta'$ vertex cancel against the appropriate longitudinal part of the external particle radiation diagrams. Consequently, a model independent relation between the magnetic dipole moments of the $\Delta^+$ and $\Delta^{++}$ resonances and the anomalous magnetic moment of the proton $\mu_p$ is obtained, where $\mu_{\Delta}$ is expressed by $\mu_p$ as $\mu_{\Delta^+}={{M_{\Delta}}\over {m_p}} \mu_p$ and $\mu_{\Delta^{++}}={3\over 2}\mu_{\Delta^+}$ in agreement with the values extracted from the fit for the experimental cross section of the $\pi^+ p\to\gamma'\pi^+ p$ reaction.Comment: 37 pages, 2 figures and 1 tabl

Magnetic dipole moment of the Δ+\Delta^+(1232) from the γp→γπ0p\gamma p \to \gamma \pi^0 p reaction

The $\gamma p \to \gamma \pi^0 p$ reaction in the $\Delta(1232)$-resonance region is investigated as a method to access the $\Delta^+(1232)$ magnetic dipole moment. The calculations are performed within the context of an effective Lagrangian model containing both the $\Delta$-resonant mechanism and a background of non-resonant contributions to the $\gamma p \to \gamma \pi^0 p$ reaction. Results are shown both for existing and forthcoming $\gamma p \to \gamma \pi^0 p$ experiments. In particular, the sensitivity of unpolarized cross sections and photon asymmetries to the $\Delta^+$ magnetic dipole moment is displayed for those forthcoming data.Comment: 25 pages, 11 figure

The reaction γp→π∘γ′p\gamma p \to \pi^\circ \gamma^\prime p and the magnetic dipole moment of the Δ+(1232)\Delta^+(1232) resonance

The reaction $\gamma p \to \pi^\circ \gamma^\prime p$ has been measured with the TAPS calorimeter at the Mainz Microtron accelerator facility MAMI for energies between $\sqrt{s}$ = 1221--1331 MeV. Cross sections differential in angle and energy have been determined for all particles in the final state in three bins of the excitation energy. This reaction channel provides access to the magnetic dipole moment of the $\Delta^{+}(1232)$ resonance and, for the first time, a value of $\mu_{\Delta^+} = (2.7_{-1.3}^{+1.0}(stat.) \pm 1.5 (syst.) \pm 3(theo.)) \mu_N$ has been extracted

Radiative π⁰ photoproduction on protons in the Δ⁺ (1232) region

The reaction γp → pπ<sup> </sup>0γ<sup>′</sup> has been measured with the Crystal Ball/TAPS detectors using the energy-tagged photon beam at the electron accelerator facility MAMI-B. Energy and angular differential cross-sections for the emitted photon γ<sup>′</sup> and angular differential cross-sections for the π<sup>0</sup> have been determined with high statistics in the energy range of the Δ<sup>+</sup> (1232)-resonance. Cross-sections and the ratio of the cross-section to the nonradiative process γp → pπ <sup>0</sup> are compared to theoretical reaction models, having the anomalous magnetic moment <i>κ<sub> Δ</sub></i> as free parameter. As the shape of the experimental distributions is not reproduced in detail by the model calculations, currently no extraction of <i>κ<sub> Δ</sub></i> is feasible

Highly multimodal structure of high topological charge extreme ultraviolet vortex beams

Les faisceaux optiques portant un Moment Angulaire Orbital (OAM) font l'objet de recherches très actives,en raison de l'intérêt de leurs applications potentielles, particulièrement à courtes longueurs d'onde. Nous nous intéressons ici à des faisceaux produits par Génération d'Harmoniques d'Ordre Elevé (HOE) dans un gaz rare, et analysons la structure du vortex de haute charge grâce à la caractérisation complète obtenue par un capteur de front d'onde de type Hartmann dans l'Extrême UltraViolet (EUV). Nous montrons que ces faisceaux HOE sont généralement constitués d'un ensemble de nombreux modes de vortex. Un miroir déformable nous permet d'étudier la sensibilité de l'éclairement et de la phase du faisceau HOE vis à vis des aberrations du laser infrarouge (IR).Optical beams carrying Orbital Angular Momentum (OAM) are a very active field of research for their prospective applications, especially at short wavelengths. We consider here such beams produced through high-harmonic generation (HHG) in a rare gas and analyze the characterization of their high charge vortex structure by an EUV (Extreme UltraViolet) Hartmann wavefront sensor. We show that such HHG beams are generally composed of a set of numerous vortex modes. The sensitivity of the intensity and phase of the HHG beam to the infrared (IR) laser aberrations is investigated using a deformable mirror.Optische Strahlen mit Orbital Angular Momentum (OAM) sind ein sehr aktives Forschungsfeld für ihre prospektiven Anwendungen, insbesondere bei kurzen Wellenlängen. Wir betrachten hier solche Strahlen, die durch hochharmonische Erzeugung (HHG) in einem seltenen Gas erzeugt werden, und analysieren die Charakterisierung ihrer Hochladungs-Wirbelstruktur durch einen EUV (Extreme UltraViolet) Hartmann-Wellenfrontsensor. Wir zeigen, dass solche HHG-Strahlen in der Regel aus einer Reihe von zahlreichen Vortex-Modi bestehen. Die Empfindlichkeit der Intensität und Phase des HHG-Strahls auf die infraroten (IR) Laseraberrationen wird mit einem deformierbaren Spiegel untersucht