Flavor analysis of nucleon, $\Delta$, and hyperon electromagnetic form factors

TL;DR

This paper analyzes nucleon, $$, and hyperon electromagnetic form factors through experimental data and tests a relativistic quark model, finding good agreement up to moderate momentum transfers and comparing results with lattice QCD.

Contribution

It evaluates the flavor structure of baryons using a relativistic quark model and experimental data, extending analysis to $$ and hyperons and comparing with lattice QCD results.

Findings

Good agreement between model and experiment up to $Q^2 \,\sim\, 4$ GeV$^2$

Model explains flavor content with three-quark configurations

Sample results for $$ and hyperons align with lattice QCD data

Abstract

By the analysis of the world data base of elastic electron scattering on the proton and the neutron (for the latter, in fact, on $^2H$ and $^3He$) important experimental insights have recently been gained into the flavor compositions of nucleon electromagnetic form factors. We report on testing the Graz Goldstone-boson-exchange relativistic constituent-quark model in comparison to the flavor contents in low-energy nucleons, as revealed from electron-scattering phenomenology. It is found that a satisfactory agreement is achieved between theory and experiment for momentum transfers up to $Q^2\sim$ 4 GeV$^2$, relying on three-quark configurations only. Analogous studies have been extended to the $\Delta$ and the hyperon electromagnetic form factors. For them we here show only some sample results in comparison to data from lattice quantum chromodynamics.

Figures

Full text

††journal: Few-Body Systems

Evidently, electromagnetic (e.m.) form factors provide stringent tests on any model for hadrons. The Goldstone-boson-exchange (GBE) relativistic constituent-quark model (RCQM) for baryons constructed by the Graz group [1] had been tested with respect to covariant predictions for the elastic e.m. $N$ form factors long ago [2; 3]. An unprecedented overall agreement with experimental data up to momentum transfers of $Q^{2}\sim$ 4 GeV2 had then been achieved in a calculation along point-form relativistic quantum mechanics. After the appearance of phenomenological flavor analyses of elastic e.m. $N$ form factors [4; 5; 6] it appeared more than interesting to check the performance of the GBE RCQM also in these respects. Recently we have performed such studies. Below we show pertinent results of selective quantities for the $N$ and from extensions of this kind of investigations to the $\Delta$ and to the hyperons with various $u$, $d$, and $s$ quark contents.

The theory and the calculations are exactly the same as explained for the point-form approach in our previous papers [2; 3; 7; 8]. The predictions for the elastic e.m. form factors fulfill Poincaré invariance as well as time-reversal invariance and current conservation [7; 8]. Accurate three-quark baryon wave functions were obtained solving a relativistically invariant mass operator along the stochastic variational method exploiting all possible symmetries in configuration, spin, and flavor spaces. For the rest frames they are depicted in Ref. [9] for singlet, octet and decuplet baryon states. In the calculation of the e.m. form factors the necessary Lorentz boosts can be executed rigorously when evaluating the matrix elements of the e.m. current operator in the framework of the point form.

In Fig. 1 we first show the e.m. form factors of both the proton ($p$) and neutron ($n$) together with their $u$- and $d$-flavor components $G^{u}_{E}$, $G^{d}_{E}$, $G^{u}_{M}$, and $G^{d}_{M}$. It is seen that not only the global predictions by the GBE RCQM agree remarkably well with experimental data but also the individual flavor contributions are quite congruent with the phenomenological analysis by Cates et al. [4]. For the particular value of $Q^{2}$=0.227$\pm$0.002 GeV2 there is also a result available from lattice quantum chromodynamics (QCD) [10], which we quote too in Fig. 1.

Sometimes the $N$ flavor components are represented also by flavor contributions to the Dirac and Pauli form factors $F_{1}(Q^{2})$ and $F_{2}(Q^{2})$, respectively. For the sake of comparison with other studies in the literature we add in Fig. 2 the $u$- and $d$-flavor components $F_{1}^{u}$, $F_{1}^{d}$, $F_{2}^{u}$, and $F_{2}^{d}$ again in comparison to the phenomenological data by Cates et al. It is seen that the GBE RCQM relying on $\{QQQ\}$ configurations only can well produce the magnitudes and shapes of all of these form-factor components. The slight deviations from the phenomenological data at $Q^{2}\geq$ 3 GeV2 in our opinion do not allow to draw conclusions of diquark clustering or higher quark Fock components in the low-energy $N$, as is sometimes advocated in the literature.

Of course, the GBE RCQM could be fine-tuned to produce an even better description of the $N$ e.m. form factors as well as electric radii and magnetic moments (cf. Refs. [2; 3; 11]). We emphasize here again that beyond the definition of the GBE RCQM no further parameters, such as, e.g., anomalous magnetic moments of constituent quarks or similar, have been introduced in the calculation of the e.m. $N$ structures. All results presented before in Refs. [2; 3] and discussed here are pure predictions by the GBE RCQM.

Next we take a look at the $\Delta$’s. There is not yet any phenomenological insight into the momentum dependences of the e.m. form factors. Experimental data exist only for the $\Delta^{+}$ and $\Delta^{++}$ magnetic moments. The GBE RCQM predictions for $\Delta$ and hyperon electric radii and magnetic moments were presented in Ref. [11]. However, in the case of the $\Delta^{+}$ we can compare the e.m. form factors with regard to their momentum dependences to lattice QCD results by Alexandrou et al. [12] and at the point $Q^{2}$=0.230$\pm$0.001 GeV2 also to results by Boinepalli et al. [13]. We find a reasonable agreement of the covariant predictions by the GBE RCQM with the lattice QCD results for the global form factors and at the single point also for the flavor components in $G_{E}^{\Delta^{+}}$; there might be a discrepancy from Ref. [13] for the flavor components in $G_{M}^{\Delta^{+}}$. However, for these lattice QCD results the theoretical errors appear to be relatively big (cf. the right panel in Fig. 3).

For an example of hyperon e.m. form factors we here address the $\Lambda^{0}$ (octet) ground state, where also an $s$ quark is involved. Fig. 4 shows the results for the total form factors and their flavor components. Like in the case of the $n$, the electric $\Lambda^{0}$ form factor is almost zero but not quite. The main reason for this behaviour is the small but relevant mixed-symmetry component in the spatial part of the octet wave function. Regarding the flavor components, $G^{u}_{E}$ is biggest, $G^{d}_{E}\sim G^{s}_{E}$, and both of the latter together almost cancel with the former producing the small values of $G^{\Lambda}_{E}$. The situation is completely different with regard to the magnetic form factor $G^{\Lambda}_{M}$. Here, both $G^{u}_{M}$ and $G^{d}_{M}$ are extremely small and in addition they are of opposite signs. As a consequence they have a negligible contribution to the total magnetic form factor, which is practically only furnished by the $s$ quark yielding $G^{\Lambda}_{M}\sim G^{s}_{M}$, a very remarkable result. Again, as in the cases of $n$ and $\Delta^{+}$ the lattice QCD results by Boinepalli et al. [10] for the magnetic form factor deviate to some extent from the GBE RCQM predictions.

All $\Delta$ and hyperon e.m. form factors will be reported and discussed in a forthcoming paper. In addition further details on the flavor decomposition of the $N$ e.m. form factors will be given therein.

Here, we summarize only by stating that the $N$, $\Delta$, and hyperon e.m. structures are remarkably well predicted by the GBE RCQM up to momentum transfers of $Q^{2}\sim$ 4 GeV2. We emphasize again that this RCQM relies on pure $\{QQQ\}$ configurations only, but implements rigorously Lorentz symmetry together with all other symmetry requirements of the Poincaré group as well as time reversal invariance and current conservation. Obviously, also the essential properties of low-energy QCD are grabbed well through the employed $Q$-$Q$ interaction, which is based on a realistic (linear) confinement and a hyperfine potential that is deduced from spontaneous breaking of chiral symmetry [1; 14].