  We investigate radial excitation of the quark-antiquark pair in the $\pi^0$
meson and its effects on the $\gamma \gamma^* \rightarrow \pi^0$ transition
form factor in the framework of light-cone perturbative QCD. The existing
constraints on the light-cone wave function of the lowest Fock state $|q { \bar
q} \rangle$ in the $\pi^0$ meson allow a sizeable radial excitation of the
quark-antiquark pair. We construct the light-cone wave function for the
quark-antiquark pair in the first radially excited state (the 2S state) using a
simple harmonic oscillator potential. The distribution amplitude obtained for
the 2S state has two nodes in $x$ at low scale of $Q$ and thereby has a much
strong scale dependence than the 1S state. Contributions from this radial
excitation to the $\gamma \gamma^* \rightarrow \pi^0$ transition form factor
exhibit different $Q^2$-dependence behavior from the ground state and thus can
modify the prediction for the transition form factor in the medium-large region
of $Q^2$.
