${\gamma }^{*}{\gamma }^{*}\to {\eta }_c(1S,2S)$ transition form factors for spacelike photons

We derive the light-front wave function (LFWF) representation of the ${\gamma }^{*}{\gamma }^{*}\to {\eta }_c(1S),{\eta }_c(2S)$ transition form factor $F(Q_1^2,Q_2^2)$ for two virtual photons in the initial state. For the LFWF, we use different models obtained from the solution of the Schrödinger equation for a variety of $c\overline{c}$ potentials. We compare our results to the BABAR experimental data for the ${\eta }_c(1S)$ transition form factor, for one real and one virtual photon. We observe that the onset of the asymptotic behavior is strongly delayed and discuss applicability of the collinear and/or massless limit. We present some examples of two-dimensional distributions for $F(Q_1^2,Q_2^2)$. A factorization breaking measure is proposed and factorization breaking effects are quantified and shown to be almost model independent. Factorization is shown to be strongly broken, and a scaling of the form factor as a function of ${\overline{Q}}^2=(Q_1^2+Q_2^2)/2$ is obtained.

Figure 1

The ${\eta }_c$ production by the interaction of two virtual photons in $e^{+}{e}^{-}$ collisions.

Figure 2

Momentum space wave function $u(p)$ for ${\eta }_c(1S)$ (top) and ${\eta }_c(2S)$ (bottom) for different potentials.

Figure 3

The light-front radial wave function $\psi (z,\mathit{k})$ of Eq. (2.21) in the ($z,k_T$) space for Buchmüller-Tye potential for ${\eta }_c(1S)$ and ${\eta }_c(2S)$, respectively. Here $k_T=|\mathit{k}|$.

Figure 4

The dependence of the normalized transition form factor $\tilde{F}(Q^2,0)$ on spacelike photon virtuality for ${\eta }_c(1S)$ and ${\eta }_c(2S)$. For the case of the ${\eta }_c(1S)$, the BABAR experimental data [20] are shown for comparison.

Figure 5

$Q^{2}F(Q^2)$ as a function of photon virtuality for ${\eta }_c(1S)$ (top) and ${\eta }_c(2S)$ (bottom) for different potentials used in the present paper.

Figure 6

Distribution amplitudes for different wave functions for ${\eta }_c(1S)$ (top) and for ${\eta }_c(2S)$ (bottom).

Figure 7

$Q^{2}F(Q^2)$ for ${\eta }_c(1S)$ (top) and ${\eta }_c(2S)$ (bottom) as a function of photon virtuality. The horizontal line is the limit for $Q^2\to \infty$, calculated for the Buchmüller-Tye potential. The red dashed line (via LFWF) was obtained according to Eq. (2.38). The black dashed-dotted curve (via DA) and the blue solid curve (via DA, evolution) were calculated with Eq. (3.4) respectively with and without evolution of Gegenbauer coefficients $a_n$.

Figure 8

The ${\gamma }^{*}{\gamma }^{*}\to {\eta }_c(1S)$ (top panels) and ${\gamma }^{*}{\gamma }^{*}\to {\eta }_c(2S)$ (bottom panels) form factor as a function of ($Q_1^2,Q_2^2$) and ($\omega ,\overline{Q^2}$) for the Buchmüller-Tye potential for illustration; other potentials discussed in the present paper behave similarly.

Figure 9

Deviations from the factorization breaking ($R$) as a function of ($Q_1^2,Q_2^2$) for Buchmüller-Tye potential. Left: ${\eta }_c(1S)$; right: ${\eta }_c(2S)$.