Regge phenomenology of the $N^{*}$ and ${\mathrm{\Delta }}^{*}$ poles

We use Regge phenomenology to study the structure of the poles of the $N^{*}$ and ${\mathrm{\Delta }}^{*}$ spectrum. We employ the available pole extractions from partial wave analysis of meson scattering and photoproduction data. We assess the importance of the imaginary part of the poles (widths) to obtain a consistent determination of the parameters of the Regge trajectory. We compare the several pole extractions and show how Regge phenomenology can be used to gain insight into the internal structure of baryons. We find that the majority of the states in the parent Regge trajectories are compatible with a mostly compact three-quark state picture.

Figure 1

Chew-Frautschi plots for the leading $N^{*}$ and ${\mathrm{\Delta }}^{*}$ Regge trajectories in Tables 1–4. Solid black (blue) curves are linear fits to the displayed positive (negative) signature data points (see Sec. 2 for details.) All the curves share the same slope as required by MacDowell symmetry [51]. We do not show a fit for the ${\frac{3}{2}}_{(+)}^{+}$ states because the $\mathrm{\Delta }9/2^{-}$ pole is unreliable as will be discussed in Sec. 4b4. In order to make the plots readable, the poles are slightly displaced from the correct $\Re [J]=J_p$ value.

Figure 2

($\Im [s_p]$, $\Re [J]=J_p$) plots introduced in [6] for the leading $N^{*}$ and ${\mathrm{\Delta }}^{*}$ Regge trajectories in Tables 1–4. Solid black (blue) curves are square-root fits to the displayed positive (negative) signature data points (see Sec. 2 for details.) We do not show a fit for the ${\frac{3}{2}}_{(+)}^{+}$ states because the $\mathrm{\Delta }9/2^{-}$ pole is unreliable as will be discussed in Sec. 4b4. The different pole sets are labeled as in Fig. 1. In order to make the plots readable, the poles are slightly displaced from the correct $\Re [J]=J_p$ value as in Fig. 1. SAID(ED) $\mathrm{\Delta }9/2^{-}$ pole in the unnatural parity trajectory has a very large $\Im [s_p]$ value and it is not shown in plot (b).

Figure 3

Consistency checks (see Sec. 4a) for $I_{(\tau )}^{\eta }={\frac{1}{2}}_{(+)}^{+}$ poles from CMB, JüBo, BnGa, and SAID(ED) extractions. The left plot shows $\Re [\alpha (s_p)]$ (see Table 1 and Sec. 4b1 for their definition), computed at the poles of the resonances ($s_p$) for models 0 (black), I (red) and II (blue). The result should be equal to the corresponding angular momentum $\Re [J]=J_p$ (vertical axis) for a given resonance. The right plots depict the same calculation for $\Im [\alpha (s_p)]$, which should be equal to $\Im [J]=\Im [J_p]=0$. In this latter case, we do not show model 0 because $\Im [\alpha (s_p)]=0$ by definition. The yellow (green) bands represent up to 0.1 (from 0.1 to 0.3) deviation from the label in the vertical axis. The white band represents from 0.3 to 0.5 deviation.

Figure 4

Consistency checks for ${\frac{3}{2}}_{(-)}^{-}$ poles from CMB, SAID(ED), KH80 and KA84 extractions. Notation as in Fig. 3. See Sec. 4b4 for trajectory definition.