${\eta }^{\prime }$ transition form factor from space- and timelike experimental data

The ${\eta }^{\prime }$ transition form factor is reanalyzed in view of the recent first observation by BESIII of the Dalitz decay ${\eta }^{\prime }\to \gamma e^{+}{e}^{-}$ in both space- and timelike regions at low and intermediate energies using the Padé approximants method. The present analysis provides a suitable parametrization for reproducing the measured form factor in the whole energy region and allows one to extract the corresponding low-energy parameters together with a prediction of their values at the origin, related to ${\mathrm{\Gamma }}_{{\eta }^{\prime }\to \gamma \gamma }$, and the asymptotic limit. The $\eta –{\eta }^{\prime }$ mixing is reassessed within a mixing scheme compatible with the large-$N_c$ chiral perturbation theory at next-to-leading order, with particular attention to the Okubo-Zweig-Iizuka-rule-violating parameters. The $J/\psi$, $Z\to {\eta }^{(\prime )}\gamma$ decays are also considered and predictions are reported.

Figure 1

Our prediction for the ${\eta }^{\prime }$ transition form factor in the timelike region obtained from the $P_1^6(\sqrt{s})$ fit to space-like data performed in Ref. [2]. Experimental points are from the BESIII measurement in Ref. [25].

Figure 2

${\eta }^{\prime }$ TFF in the SL (left) and TL (right) regions after a joint fit to the SL and TL sets of experimental data. The dotted, dashed, and solid black lines represent the fits using $P_1^1(Q^2)$, $P_1^6(Q^2)$, and $P_1^7(Q^2)$, respectively, in the left panel, while the red solid line represents ${\tilde{P}}_1^7(\sqrt{s})$ in the right panel.

Figure 3

Convergence pattern of the $P_1^L$ sequence for $F_{{\eta }^{\prime }\gamma \gamma }(0)$, $b_{{\eta }^{\prime }}$, $c_{{\eta }^{\prime }}$, and $d_{{\eta }^{\prime }}$ as obtained from fitting experimental SL and TL data together with $F_{{\eta }^{\prime }\gamma \gamma }(0)$ from the PDG [24].

Figure 4

Convergence pattern as in Fig. 3 without including information on the ${\mathrm{\Gamma }}_{{\eta }^{\prime }\to \gamma \gamma }$.

Figure 5

$\eta –{\eta }^{\prime }$ mixing parameters in the octet-singlet basis from L [48], FKS [51], BDO [52], EF [46], EMS(14) [2], and EMS(15) [11].