Electric quadrupole and magnetic octupole moments of the Δ

Abstract

Using a covariant spectator constituent quark model we predict an electric quadrupole moment Q ∆ + = −0.043 efm 2 and a magnetic octupole moment O ∆ + = −0.0035 efm 3 for the ∆ + excited state of the nucleon.Although it was the first nucleon resonance to be discovered, the properties of the ∆ are almost completely unknown. Only the ∆ ++ and ∆ + magnetic moments have been measured, and these measurements have large error bars [1,2,3]. Most of the information we have about the ∆ comes from indirect information, such as the study of the γN → ∆ transition [4].The dominant ∆ elastic form factors are the electric charge G E0 and magnetic dipole G M1 . The subleading form factors are the electric quadrupole (G E2 ) and magnetic octupole (G M3 ). Those form factors measure the deviation of the charge and magnetic dipole distribution from a symmetric form [5]. At Q 2 = 0 the form factors define the magnetic dipolemoments, where e is the electric charge and M ∆ the ∆ mass.Until recently, there were essentially only theoretical predictions for µ ∆ (see Ref.[6] for details) and Q ∆ [7,8,9,10,11,12,13,14]. The exception was the pioneering work in lattice QCD [15], where all the form factors were estimated for low Q 2 , although the statistics for G E2 and G M3 were very poor.Recent lattice QCD calculations of all four form factors over a limited Q 2 range have revived interest in the ∆ moments, especially the interesting quadrupole and octupole moments [16,17]. These results are obtained only for unphysical pion masses in the range of 350-700 MeV so some extrapolation to the physical pion mass is required [18,19]. Still, in the absence of direct experimental information, lattice QCD provides the best reference for theoretical calculations. Stimulated by these new lattice results the covariant spectator quark model [6] and chiral Quark-Soliton model (χQSM) [20] have been used to estimate the ∆ form factors. Simultaneously, a lattice technique based on the background-field method [21] has been used to estimate the µ ∆ with great precision [22]. The octupole moment O ∆ has also been evaluated by Buchmann [5] using a deformed pion cloud model, and QCD sum rules (QCDSR) have been used to estimateThe size of the moments Q ∆ and O ∆ tells us if the ∆ is deformed, and in which direction. The nucleon, as a spin 1/2 particle, can have no electric quadrupole moment [24] [although the possibility remains, as pointed out by Buchmann and Henley [25], that it might be a collective state with an intrinsic quadrupole moment]. While the measurement of the quadrupole form factors for the γN → ∆ transition gives some information about the deformation of the ∆ [26], it is very important to obtain an independent estimate [17,27]. Motivated by these considerations, the Nicosia-MIT and the Adelaide groups are presently working on an evaluation of G M3 using lattice QCD [17,28]. Also Ledwig and collaborators are working in the same subject [20] using the χQSM.In this Letter we use the covariant spectator formalism [29] to evaluate Q ∆ and O ∆ . Foll...

Baryons as relativistic three-quark bound states

“…Nonrelativistically, a negative sign for the electric quadrupole moment G E2 (0) indicates an oblate charge distribution for the ∆ in the Breit frame, with a different interpretation arising in the infinite-momentum frame [398]. From the measurement of the γN → ∆ transition one can infer the value G E2 (0) = −1.87 (8) in the large-N C limit [398,604]; comparable values are predicted by a range of constituent-quark models [605] and they are in the ballpark of the lattice results. Similar numbers arise from the Dyson-Schwinger calculations in Fig.…”

“…In the covariant spectator quark model of Refs. [545,605] the magnetic octupole form factor is strongly dependent on the d-wave content of the wave function; two different parametrizations for the ∆ wave function lead to different signs although in both cases the result is negative at zero photon momentum. We note again that in the Dyson-Schwinger calculations the partial-wave content of the nucleon and ∆ amplitudes in terms of s, p, d and f waves is not restricted in any way but determined dynamically when solving the bound-state equations.…”

Relativistic quark model for theΩ−electromagnetic form factors

“…Even then one must be careful, since the lattice QCD results are obtained with unphysical pion masses (heavy quark masses) which induce extra ambiguities, as discussed in Refs. [58,59]. On the other hand, the situation for the Ω − is completely different.…”

“…In the previous work this spectator formalism was applied to the γN → ∆ transition form factors [54,55,56] as well as the nucleon [57] and ∆ [58,59] electromagnetic form factors. The flavor-spin structure of the Ω − is very similar to that of the ∆.…”

Octet to decuplet electromagnetic transition in a relativistic quark model

“…For this purpose, we rely on the covariant spectator quark model [32][33][34][35][36][37][38], since it was successful in the studies of the electromagnetic structure of nucleon [39][40][41], octet and decuplet baryons [34][35][36][37][42][43][44][45], transition form factors of the reactions à N ! Áð1232Þ, à N !…”