Constraints on the light pseudoscalar meson distribution amplitudes from their meson-photon transition form factors

The meson-photon transition form factors $\gamma {\gamma }^{*}\to P$ ($P$ stands for $\pi$, $\eta$ and ${\eta }^{\prime }$) provide strong constraints on the distribution amplitudes of the pseudoscalar mesons. In this paper, these transition form factors are calculated under the light-cone perturbative QCD approach, in which both the valence and nonvalence quarks’ contributions have been taken into consideration. To be consistent, a unified wave function model is adopted to analyze these form factors. It is shown that with the proper charm component $f_{{\eta }^{\prime }}^c\sim -30\mathrm{MeV}$ and a moderate DA with $B\sim 0.30$, the experimental data on $Q^2{F}_{\eta \gamma }(Q^2)$ and $Q^2{F}_{{\eta }^{\prime }\gamma }(Q^2)$ in the whole $Q^2$ region can be explained simultaneously. Furthermore, a detailed discussion on the form factors’ uncertainties caused by the constituent quark masses $m_q$ and $m_s$, the parameter $B$, the mixing angle $\varphi$, and $f_{{\eta }^{\prime }}^c$ are presented. It is found that, by adjusting these parameters within their reasonable regions, one can improve the form factor to a certain degree but cannot solve the puzzle for $Q^2{F}_{\pi \gamma }(Q^2)$, especially to explain the behavior of the $\pi -\gamma$ form factor within the whole $Q^2$ region consistently. We hope further experimental data on these form factors in the large $Q^2$ region can clarify the present situation.

Figure 1

DA model [Eq. (8)] for ${\varphi }_{{\eta }_q}(x)$ (left) and ${\varphi }_{{\eta }_s}(x)$ (right), with ${\mu }_0=1\mathrm{GeV}$, where $B=0.00$, 0.30, and 0.60, respectively.

Figure 2

$\pi -\gamma$ transition form factor $Q^2{F}_{\pi \gamma }(Q^2)$ with varying $B$. The solid, dotted, and dashed lines are for $B=0.00$, 0.30, and 0.60, respectively. The experimental data are taken from Refs. 12, 13, 14.

Figure 3

$\eta -\gamma$ (left) and ${\eta }^{\prime }-\gamma$ (right) transition form factors $Q^2{F}_{\eta \gamma }(Q^2)$ and $Q^2{F}_{{\eta }^{\prime }\gamma }(Q^2)$. The solid, dotted, and dashed lines are for $B=0.00$, 0.30, and 0.60, respectively. The experimental data are taken from Refs. 13, 33, 36.

Figure 4

$\pi -\gamma$ transition form factor $Q^2{F}_{\pi \gamma }(Q^2)$ with fixed $B=0.3$. The solid [red] line, dotted [blue] line, and dashed black line are for $m_q=0.25\mathrm{GeV}$, 0.30 GeV, and 0.35 GeV, respectively. The experimental data are taken from Refs. 12, 13, 14.

Figure 5

$\eta -\gamma$ (left) and ${\eta }^{\prime }-\gamma$ (right) transition form factors $Q^2{F}_{\eta \gamma }(Q^2)$ and $Q^2{F}_{{\eta }^{\prime }\gamma }(Q^2)$ with fixed $B=0.3$ and $\varphi =39.5°$. The solid [red] line, the dotted [blue] line, and the dashed black line are for [$m_q=0.25\mathrm{GeV}$ and $m_s=0.40\mathrm{GeV}$], [$m_q=0.30\mathrm{GeV}$ and $m_s=0.45\mathrm{GeV}$], and [$m_q=0.35\mathrm{GeV}$ and $m_s=0.50\mathrm{GeV}$], respectively. The experimental data are taken from Refs. 13, 33, 36.

Figure 6

$\eta -\gamma$ (left) and ${\eta }^{\prime }-\gamma$ (right) transition form factors $Q^2{F}_{\eta \gamma }(Q^2)$ and $Q^2{F}_{{\eta }^{\prime }\gamma }(Q^2)$ with fixed $B=0.3$, and with $m_q=0.30\mathrm{GeV}$ and $m_s=0.45\mathrm{GeV}$. The experimental data are taken from Refs. 13, 33, 36.

Figure 7

$\eta -\gamma$ (left) and ${\eta }^{\prime }-\gamma$ (right) transition form factors $Q^2{F}_{\eta \gamma }(Q^2)$ and $Q^2{F}_{{\eta }^{\prime }\gamma }(Q^2)$ with varying $f_{{\eta }^{\prime }}^c$, where we take $B=0.30$, $m_q=0.30\mathrm{GeV}$, $m_s=0.45\mathrm{GeV}$, and $m_c=1.50\mathrm{GeV}$. The experimental data are taken from Refs. 13, 33, 36.

Figure 8

$\eta -\gamma$ (left) and ${\eta }^{\prime }-\gamma$ (right) transition form factors $Q^2{F}_{\eta \gamma }(Q^2)$ and $Q^2{F}_{{\eta }^{\prime }\gamma }(Q^2)$ with varying $B$, where $f_{{\eta }^{\prime }}^c=-30\mathrm{MeV}$, $m_q=0.30\mathrm{GeV}$, $m_s=0.45\mathrm{GeV}$, and $m_c=1.50\mathrm{GeV}$. The experimental data are taken from Refs. 13, 33, 36.