Pion pair production in $e^+e^-$ annihilation

We present an analysis of the process $e^+e^-\to V^*\to\pi\pi\gamma$, where $V=\gamma$ or $V=Z$ boson, in the kinematical region where $\sqrt{s}$, the c.m. energy of the $e^+e^-$ pair, is large but much below the $Z$-pole. The subprocess $V^*\to \pi\pi \gamma$ can be described by the convolution of the hard scattering coefficient $V\to q \bar{q} \gamma$ and the general distribution amplitude of two pions $q\bar{q}\to \pi \pi$. In the case of neutral pion production, $V=\gamma$ is the dominant process, which can therefore be used to access the GPAs of the pion, especially their $C$-even parts. The $\gamma Z$ interference term provides an alternative approach to extract the weak mixing angle $\sin \theta_W$ through measuring the helicity asymmetry in the process $e^+e^-\to \pi^0\pi^0\gamma$. In the case of charged pion pair production, the Bremsstrahlung process dominates and its interference with $e^+e^-\to \gamma^ \star \to \pi^+\pi^-\gamma$ can be applied to study the process $\gamma^ \star \to \pi\pi\gamma$ at the amplitude level.