Baryon transition form factors at the pole

Electromagnetic resonance properties are uniquely defined at the pole and do not depend on the separation of the resonance from background or the decay channel. Photon-nucleon branching ratios are nowadays often quoted at the pole, and we generalize the considerations to the case of virtual photons. We derive and compare relations for nucleon to baryon transition form factors both for the Breit-Wigner and the pole positions. Using the MAID2007 and SAID SM08 partial wave analyses of pion electroproduction data, we compare the $G_M, G_E$, and $G_C$ form factors for the $\mathrm{\Delta }\left(1232\right)$ resonance excitation at the Breit-Wigner resonance and pole positions up to $Q^2=5{\mathrm{GeV}}^2$. We also explore the $E/M$ and $S/M$ ratios as functions of $Q^2$. For pole and residue extraction, we apply the Laurent + Pietarinen method.

Figure 1

Figures showing the quality of the fit. From top to bottom we show all three multipoles at three different photon virtualities $Q^2=0$, 1, and $5{\mathrm{GeV}}^2$ for MAID2007 and SAID SM08 models. Black circles and brown squares are real and imaginary part of multipoles respectively. Blue solid lines are real parts and red dashed lines are imaginary parts of the L+P fit to the given model. Panels (a)–(c) show $E_{1+}^{3/2},M_{1+}^{3/2},S_{1+}^{3/2}$ of the MAID solution and (d)–(f) the same for the SAID solution.

Figure 2

Helicity transition form factors $A_{1/2}, A_{3/2}$, and $S_{1/2}$ compared at the BW and poles position. The black solid curves show the real BW results and the blue dashed and red dash-dotted lines show real and imaginary parts of the complex pole form factors. The left column shows the results with the Mainz-MAID analysis and the right column with the GWU-SAID analysis.

Figure 3

Magnetic transition form factor in the notation of Ash divided by the dipole form factor $G_M/G_D$ and $E/M$ and $S/M$ ratios compared at the BW and poles position. The black solid curves show the real BW results and the blue dashed and red dash-dotted lines show real and imaginary parts of the complex pole form factors. The left column shows the results with the Mainz-MAID analysis and the right column with the GWU-SAID analysis. The data points for $G_M$ are from Refs. [58] (black circle), [59] (black squares), [60] (blue circles), [61] (green triangles) and [62] (red diamonds); for $E/M$: [58] (black circle), [63] (black diamonds), [64] (black triangle), [65, 66] (black square), [67] (blue circle), [61] (green triangles) and [62] (red diamonds); and for $S/M$: [63] (black diamonds), [68] (black circle), [69] (black triangle), [65, 66] (black square), [67] (blue circle), [61] (green triangles) and [62] (red diamonds).

Figure 4

Magnetic, electric and charge transition form factors compared with the heavy baryon chiral effective field theory (HBChEFT) of Gail and Hemmert [7] at low $Q^2$. The blue short-dashed and red dash-dotted lines show real and imaginary parts of the complex pole form factors obtained from MAID and SAID. The blue long-dashed and the red dotted lines are the real and imaginary parts of the HBChEFT calculations.