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

We present an analysis of the process $e^+e^-\to V^*\toππγ$, where $V=γ$ 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 $V^*\to ππγ$ can be described by the convolution of the hard scattering coefficient $V\to q \bar{q} γ$ and the general distribution amplitude of two pions $q\bar{q}\to ππ$. In the case of neutral pion production, $V=γ$ is the dominant process, which can therefore be used to access the GPAs of the pion, especially their $C$-even parts. The $γZ$ interference term provides an alternative approach to extract the weak mixing angle $\sin θ_W$ through measuring the helicity asymmetry in the process $e^+e^-\to π^0π^0γ$. In the case of charged pion pair production, the Bremsstrahlung process dominates and its interference with $e^+e^-\to γ^ \star \to π^+π^-γ$ can be applied to study the process $γ^ \star \to ππγ$ at the amplitude level.