Extraction and interpretation of $\gamma N\to \Delta$ form factors within a dynamical model

Within the dynamical model of Refs. [Phys. Rev. C 54, 2660 (1996); 63, 055201 (2001)], we perform an analysis of recent data of pion electroproduction reactions at energies near the $\Delta (1232)$ resonance. We discuss possible interpretations of the extracted bare and dressed $\gamma N\to \Delta$ form factors in terms of relativistic constituent quark models and lattice quantum chromodynamics calculations. Possible future developments are discussed.

Figure 1

Phase shifts of $\pi N$ elastic scattering up to $T_L=250$ MeV. Solid and dotted stand for model SL and SL2, respectively. Data, $L_{2T,2J}$, are from the energy-independent SAID [25] analysis plus 8 points from their energy dependent solution for the $P_{13}$ and $P_{31}$ partial waves at lower energies.

Figure 2

Graphic representation of the dressed $\gamma N\to \Delta$ vertex defined by Eqs. (2.8, 2.9, 2.10).

Figure 3

Magnetic dipole transition form factor $G_M^{*}$ for ${\gamma }^{*}N\to \Delta (1232)$, normalized to the proton dipole form factor $G_D(Q^2)=1/[1+Q^2/{\Lambda }^2]{}^2$ with ${\Lambda }^2=0.71$ (GeV/$c$)${}^2$. Experimental points are analyses of inclusive data (○) from pre-1990 experiments at DESY and SLAC [30] and recent exclusive $p(e,e^{\text{'}}p){\pi }^o$ data (▪) from BATES [4], MAMI [6, 7], and JLAB [2, 3, 14, 27, 31]. Solid curve is from the dressed calculation of this work using the parametrization of Eqs. (3.15) and (3.16). The dotted curve is obtained when the meson cloud effect, defined by Eq. (2.9) is turned off.

Figure 4

The ratios ${\overline{R}}_{\mathrm{EM}}$ and ${\overline{R}}_{\mathrm{EM}}$ [defined by Eqs. (3.17) and (3.18)] calculated from the dynamical model using the parametrization Eqs. (3.15) and (3.16) (solid line). The data are from recent experiments at BATES: □ [4], MAMI: ▪ [5], ○ [6], ▵ [7], ◊ [32], JLAB/CLAS: ▾ [3], ▴ [2] and JLAB/Hall A: • [27]. CLAS points from Ref. [3] were obtained from the average of UIM and SL model estimates in Table .

Figure 5

The dressed $\gamma N\to \Delta$ form factors (solid curves) calculated using the parametrization Eqs. (3.15) and (3.16) are compared with experiment. The data are from BATES: □ [4], MAMI: ▪ [5], ○ [6], ▵ [7] and JLAB/CLAS: ▾ [3], ▴ [2].

Figure 6

Fits to experimental $p(e,e^{\text{'}}p){\pi }^0$ structure functions. Solid lines are from the fits with the bare form factors $G_M(Q^2),G_E(Q^2)$ and $G_C(Q^2)$ adjusted at each $Q^2$. The dashed curves are from the calculations using the parametrization Eqs. (3.15) and (3.16). Data are from MAMI [6] at $Q^2=0.06$ GeV${}^2$, BATES [4] at $Q^2=0.127$ GeV${}^2$, CLAS [3] ($W=1220$ MeV) and MAMI [7] ($W=1221$ MeV) at $Q^2=0.2$ GeV${}^2$ and CLAS [2] at $Q^2=0.9,1.45$ GeV${}^2$.

Figure 7

Bare form factors for the $\gamma N\to \Delta$ transition as a function of $Q^2$. The points have been obtained by performing individual fits for each $Q^2$ value to the corresponding pion electroproduction data (BATES: □ [4], MAMI: ▪ [5], ○ [6], ▵ [7] and JLAB/CLAS: ▾ [3], ▴ [2].) The dashed curves are from the front form quark model calculations of Refs. [15, 34]. The dotted curves are from the instant form quark model calculations of Ref. [16].

Figure 8

The extracted $\gamma N\to \Delta$ form factors. Dark squares(triangles) are the dressed (bare) values. Open squares are the lattice QCD calculation of Ref. [18].