Radiative return meets GVMD

Abstract

We improve the description of pion-photon interactions in the radiative return process $e^+e^-\to π^+π^-γ$ at the next-to-leading order by including the pion form factor in the Feynman rules. We present a general calculation of the new amplitudes, and provide an implementation easy to interface with any Monte Carlo generator. We incorporate this framework into the event generator $\texttt{Phokhara}$ and study several experimental configurations. Overall, we find percent-level effects appearing in angular differential cross section distributions at colliders whose centre-of-mass energies lie near the peak of the pion form factor. By contrast, total cross sections and distributions in charge-even variables show effects only at the permille level, or no visible differences at all. Finally, we compare the new predictions with KLOE measurements of the forward-backward asymmetry in order to assess the predictive power of the modifications.

Figures (5)

• Figure 1: Representative diagrams for the two groups, TPE and FSC, changing under GVMD. The groups are further split into ISR and FSR contributions, corresponding from left to right to $\text{TPE}_\text{ISR}$, $\text{TPE}_\text{FSR}$, $\text{FSC}_\text{FSR}$ and $\text{FSC}_\text{ISR}$.
• Figure 2: Differential cross section versus pion--pair invariant mass (left) and $\theta_{\text{avg}}$ (right) under $F_\pi\!\times\!\mathrm{sQED}$, the full GVMD modifications and only the ISR diagrams modified with GVMD. The bottom panel of each plot shows, the ratio between the modifications and $F_\pi\!\times\!\mathrm{sQED}$. Colour bands correspond to statistical uncertainties. Note the difference between y-axis scales.
• Figure 3: Differential cross section versus pion--pair invariant mass (left) and $\theta_{\text{avg}}$ (right) under $F_\pi\!\times\!\mathrm{sQED}$, the full GVMD modifications and only the ISR diagrams modified with GVMD, under the KLOE-like small angle scenario. The ratio between the GVMD modifications and $F_\pi\!\times\!\mathrm{sQED}$ can be seen in the bottom of the plot.
• Figure 4: BESIII scenario $|\cos(\theta_+)|$ (left) and B scenario $\theta_\gamma$ (right) differential cross section with the full GVMD modifications, $F_\pi\!\times\!\mathrm{sQED}$ and only the ISR diagrams modified with GVMD. The ratio between the approaches is displayed in the bottom panel.
• Figure 5: In the top panels we show the forward--backward asymmetry for a relaxed LA KLOE selection at $\sqrt{s}=1~\text{GeV}$ (left) and $\sqrt{s}=1.02~\text{GeV}$ (right). The KLOE experimental data is shown in grey, while the coloured points correspond to Phokhara predictions including different contributions (see text for details). The middle panels display the absolute difference between data and each prediction, $\Delta^{\text{KLOE}}_{X}=\text{KLOE}-X$. The bottom panels compare the MC approaches: for the left plot we take $F_\pi\!\times\!\mathrm{sQED}$ as the reference, while for the right plot we use $F_\pi\!\times\!\mathrm{sQED}$+ ISC$_\phi$.