Hyperon electromagnetic form factors in the timelike region

Electromagnetic form factors of hyperons ($\mathrm{\Lambda }$, $\mathrm{\Sigma }$, $\mathrm{\Xi }$) in the timelike region, accessible in the reaction $e^{+}{e}^{-}\to \overline{Y}Y$, are studied. The focus is on energies close to the reaction thresholds, where the properties of these form factors are significantly influenced by the interaction in the final $\overline{Y}Y$ system. This interaction is taken into account in the calculation, utilizing $\overline{Y}Y$ potential models that have been constructed by the Jülich group for the analysis of data from the reaction $\overline{p}p\to \overline{Y}Y$ in the past. The enhancement of the effective form factor for energies close to the threshold, seen in experiments of $e^{+}{e}^{-}\to \overline{\mathrm{\Lambda }}\mathrm{\Lambda }$ and $e^{+}{e}^{-}\to {\overline{\mathrm{\Sigma }}}^0\mathrm{\Lambda }$, is reproduced. With regard to the reactions $e^{+}{e}^{-}\to {\overline{\mathrm{\Sigma }}}^{-}{\mathrm{\Sigma }}^{+}$, ${\overline{\mathrm{\Sigma }}}^0{\mathrm{\Sigma }}^0$, ${\overline{\mathrm{\Sigma }}}^{+}{\mathrm{\Sigma }}^{-}$ a delicate interplay between the three channels is observed in the results at low energies, caused by the $\overline{\mathrm{\Sigma }}\mathrm{\Sigma }$ interaction. Predictions for the electromagnetic form factors $G_M$ and $G_E$ in the timelike region are presented for the $\mathrm{\Lambda }$, $\mathrm{\Sigma }$, and $\mathrm{\Xi }$ hyperons.

Figure 1

Results for $\overline{p}p\to \overline{\mathrm{\Sigma }}\mathrm{\Sigma }$ based on the $\overline{Y}Y$ model presented in Ref. [55]. Data are taken from Refs. [49, 50, 51]. The vertical dash-dotted lines indicate the ${\overline{\mathrm{\Sigma }}}^0{\mathrm{\Sigma }}^0$ and ${\overline{\mathrm{\Sigma }}}^{+}{\mathrm{\Sigma }}^{-}$ thresholds.

Figure 2

$e^{+}{e}^{-}\to \overline{\mathrm{\Lambda }}\mathrm{\Lambda }$: Cross section (left) and effective form factor $|G_{\mathrm{eff}}|$ (right) as a function of the total cms energy. The solid, dashed, and dash-dotted lines correspond to the $\overline{\mathrm{\Lambda }}\mathrm{\Lambda }$ models I, II, and III from Ref. [52], the dash-double-dotted and dotted lines to the models K and Q described in Ref. [54]. Data are from the Refs. [11] (DM2), [12] (BABAR), and [19, 20] (BESIII). The phase space is shown by the thin (black) dashed line. The vertical dash-dotted line indicates the isospin-averaged $\overline{\mathrm{\Sigma }}\mathrm{\Sigma }$ threshold.

Figure 3

$e^{+}{e}^{-}\to \overline{\mathrm{\Lambda }}\mathrm{\Lambda }$: The ratio $|G_E/G_M|$ (left) and phase $\varphi =\arg (G_E/G_M)$ (right) as a function of the total cms energy. Same description of curves as in Fig. 2. Data are from Refs. [12] (BABAR) and [20] (BESIII).

Figure 4

$e^{+}{e}^{-}\to {\overline{\mathrm{\Sigma }}}^0\mathrm{\Lambda }$: Cross section (left) and effective form factor $|G_{\mathrm{eff}}|$ (right). The solid, dash-double-dotted and dotted lines correspond to the models I, K and Q described in Ref. [55]. Data points are taken from the DM2 [11] and BABAR [12] Collaborations. The phase space is shown by the thin (black) dashed line. The vertical dash-dotted line indicates the $\overline{\mathrm{\Sigma }}\mathrm{\Sigma }$ threshold.

Figure 5

$e^{+}{e}^{-}\to {\overline{\mathrm{\Sigma }}}^0\mathrm{\Lambda }$: Ratio $|G_E/G_M|$ (left) and phase $\varphi =\arg (G_E/G_M)$ (right). Same description of curves as in Fig. 4.

Figure 6

$e^{+}{e}^{-}\to \overline{\mathrm{\Sigma }}\mathrm{\Sigma }$: Illustrative curves/results for the ${\overline{\mathrm{\Sigma }}}^{-}{\mathrm{\Sigma }}^{+}$ and ${\overline{\mathrm{\Sigma }}}^{+}{\mathrm{\Sigma }}^{-}$ cross sections. The dashed lines represent the phase space behavior. The solid line is the result of an exploratory calculation that uses only the ${\overline{\mathrm{\Sigma }}}^{-}{\mathrm{\Sigma }}^{+}\to {\overline{\mathrm{\Sigma }}}^{-}{\mathrm{\Sigma }}^{+}$ amplitude for the FSI. The dash-dotted line represents the pQCD motivated parametrization Eq. (3) in [21]. Data are taken from Ref. [21]. The vertical lines indicate the ${\overline{\mathrm{\Sigma }}}^0{\mathrm{\Sigma }}^0$ and ${\overline{\mathrm{\Sigma }}}^{+}{\mathrm{\Sigma }}^{-}$ thresholds.

Figure 7

$e^{+}{e}^{-}\to \overline{\mathrm{\Sigma }}\mathrm{\Sigma }$: Cross section (left) and effective form factor $|G_{\mathrm{eff}}|$ (right). Results are shown based on the $\overline{\mathrm{\Sigma }}\mathrm{\Sigma }$ interaction from Ref. [55], where the dark (red) bands correspond to ${\overline{\mathrm{\Sigma }}}^{-}{\mathrm{\Sigma }}^{+}$, medium (blue) bands to ${\overline{\mathrm{\Sigma }}}^0{\mathrm{\Sigma }}^0$, and light (green) bands to ${\overline{\mathrm{\Sigma }}}^{+}{\mathrm{\Sigma }}^{-}$. For the meaning of the bands see text. Data for the ${\overline{\mathrm{\Sigma }}}^{-}{\mathrm{\Sigma }}^{+}$ and ${\overline{\mathrm{\Sigma }}}^{+}{\mathrm{\Sigma }}^{-}$ channels are from BESIII [21], those for ${\overline{\mathrm{\Sigma }}}^0{\mathrm{\Sigma }}^0$ from BABAR [12]. The vertical dash-dotted lines indicate the ${\overline{\mathrm{\Sigma }}}^0{\mathrm{\Sigma }}^0$ and ${\overline{\mathrm{\Sigma }}}^{+}{\mathrm{\Sigma }}^{-}$ thresholds.

Figure 8

$e^{+}{e}^{-}\to \overline{\mathrm{\Sigma }}\mathrm{\Sigma }$: Ratio $|G_E/G_M|$ (left) and phase $\varphi =\arg (G_E/G_M)$ (right). Same description of curves as in Fig. 7. Data for ${\overline{\mathrm{\Sigma }}}^{-}{\mathrm{\Sigma }}^{+}$ are from the BESIII Collaboraton [21].

Figure 9

$e^{+}{e}^{-}\to \overline{\mathrm{\Xi }}\mathrm{\Xi }$: Cross section (left) and effective form factor $|G_{\mathrm{eff}}|$ (right). Results for isospin $I=0$ (solid line) and $I=1$ (dashed line) are shown based on the $\overline{\mathrm{\Xi }}\mathrm{\Xi }$ interaction from Ref. [56]. Data for the ${\overline{\mathrm{\Xi }}}^{+}{\mathrm{\Xi }}^{-}$ channel are from the BESIII Collaboration [23].

Figure 10

$e^{+}{e}^{-}\to \overline{\mathrm{\Xi }}\mathrm{\Xi }$: Ratio $|G_E/G_M|$ (left) and phase $\varphi =\arg (G_E/G_M)$ (right). Same description of curves as in Fig. 9.