Implication on the pion distribution amplitude from the pion-photon transition form factor with the new BABAR data

The new BABAR data on the pion-photon transition form factor arouses people’s interest for the determination of the pion distribution amplitude. To explain the data, we take both the leading valence quark state’s and the nonvalence quark state’s contributions into consideration, where the valence quark part up to next-to-leading order is presented and the nonvalence quark part is estimated by a phenomenological model based on its limiting behavior at both $Q^2\to 0$ and $Q^2\to \infty$. Our results show that to be consistent with the new BABAR data at the large $Q^2$ region, a broader amplitude other than the asymptoticlike pion distribution amplitude should be adopted. The broadness of the pion distribution amplitude is controlled by a parameter $B$. It has been found that the new BABAR data at low and high energy regions can be explained simultaneously by setting $B$ to be around 0.60, in which the pion distribution amplitude is closed to the Chernyak-Zhitnitsky form.

Figure 1

Typical diagrams that contribute to the pion-photon transition form factor $F_{\pi \gamma }(Q^2)$, where $x^{\prime }=(1-x)$. The rightmost shaded oval with a slant pattern stands for the strong interactions.

Figure 2

Comparison of the pion DA model defined in Eq. (8) with the asymptotic-form DA and the CZ-form DA, where $B=0.00$, 0.30, and 0.60, respectively.

Figure 3

Comparison of the pion DA model defined in Eq. (8) with Brodsky and Teramond’s (BT) holographic model, where $B=0.00$, 0.05, 0.10, and 0.15, respectively.

Figure 4

$Q^2{F}_{\pi \gamma }(Q^2)$ with the model wave function (3) by taking $m_q=0.30\mathrm{GeV}$ and by varying $B$ within the region of [0.00, 0.60]. The dash-dotted, the dotted, and the dashed lines are for $B=0.00$, $B=0.30$, and $B=0.60$, respectively. The solid line is the fitted curve (1) derived by BABAR [3].

Figure 5

$Q^2{F}_{\pi \gamma }(Q^2)$ with the model wave function (3) by taking $m_q=0.30\mathrm{GeV}$ and $B=0.60$. The solid, the dotted, and the dashed lines are for the total contribution, the leading valence quark contribution, and the nonvalence quark contribution to the form factor, respectively.

Figure 6

$Q^2{F}_{\pi \gamma }(Q^2)$ with the model wave function (3) by fixing $B=0.60$ and by varying $m_q$ within the region [0.30, 0.50] GeV. The solid line is for $m_q=0.40\mathrm{GeV}$, and the shaded band shows its uncertainty.