${\eta }_c$ elastic and transition form factors: Contact interaction and algebraic model

For the flavor-singlet heavy-quark system of charmonia in the pseudoscalar [${\eta }_c(1S)$] channel, we calculate the elastic (EFF) and transition form factors (TFFs) [${\eta }_c(1S)\to \gamma {\gamma }^{*}$] for a wide range of photon momentum transfer squared ($Q^2$). The framework for this analysis is provided by a symmetry-preserving Schwinger-Dyson equation and Bethe-Salpeter equation treatment of a $\text{vector}\times \text{vector}$ contact interaction. We also employ an algebraic model, developed earlier to describe the light-quark systems. It correctly correlates infrared and ultraviolet dynamics of quantum chromodynamics (QCD). The contact interaction results agree with the lattice data for low $Q^2$. For $Q^2\ge Q_0^2$, the results start deviating from the lattice results by more than 20%. $Q_0^2\approx 2.5{\mathrm{GeV}}^2$ for the EFF, and $\approx 25{\mathrm{GeV}}^2$ for the TFF. We also present the results for the EFF, TFF, and ${\eta }_c(1S)$ parton distribution amplitude for the algebraic model. Wherever the comparison is possible, these results are in excellent agreement with the lattice, perturbative QCD, results obtained through a Schwinger-Dyson equation–Bethe-Salpeter equation study, employing refined truncations, and the experimental findings of the BABAR experiment.

Figure 1

CI results for the ${\eta }_c$ elastic form factor with and without the dressing of the quark-photon vertex. The lattice QCD curve is from Ref. [37], and the VMD monopole result is defined by the mass scale $m_V=3.096\mathrm{GeV}$. We also include the result obtained with the AM; see Eq. (24).

Figure 2

CI results for the ${\eta }_c$ and $\pi$ elastic form factors. The parameter set used for the calculation of the ${\eta }_c$ form factor is that used to produce Tables (1 and 2) and Fig. 1, while the one used to compute the $\pi$ form factor is that given in Ref. [21].

Figure 3

CI for the transition ${\gamma }^{*}\gamma \to {\eta }_c$ form factor. The lattice QCD curve is a fit [37] to data of the form $G_{{\gamma }^{*}\gamma {\eta }_c}^{\text{Lattice}}(Q^2)=\frac{{\mu }^2}{{\mu }^2+Q^2}$ with $\mu =3.43\mathrm{GeV}$, while the BABAR data is a fit [39] of the form $G_{{\gamma }^{*}\gamma {\eta }_c}^{\text{BABAR}}(Q^2)=\frac{1}{1+Q^2/\mathrm{\Lambda }}$ with $\mathrm{\Lambda }=8.5{\mathrm{GeV}}^2$. The perturbative QCD (pQCD) limit is due to Feldmann and Kroll [75]. We also include the plot obtained with the algebraic model; see Eq. (24).

Figure 4

Numerical results for $Q^2{G}_{{\gamma }^{*}\gamma {\eta }_c}$. See the caption for Fig. 3.

Figure 5

Numerical results for the ${\eta }_c$ PDA, $\varphi (x)$, obtained with the AM. We have also plotted the resulting PDA evolved to $50\mathrm{G}\mathrm{e}{\mathrm{V}}^2$, using the leading-order QCD ERBL evolution equation (see text). For the sake of comparison, we have also included the perturbative QCD (pQCD) results of Ref. [75], the asymptotic QCD expression $6x(1-x)$ (Conformal QCD), and the recent SDE prediction (SDE-DB) of Ref. [77].