# A U-spin prediction for the CP-forbidden transition $e^+ e^- \to D^0\bar{D}^0 \to (K^+ K^-)_D (\pi^+ \pi^-)_D$

## TL;DR

This paper predicts a very rare CP-forbidden transition rate in $e^+ e^- o D^0 ar{D}^0$ decays based on U-spin symmetry, linking recent CP violation observations to potential experimental detection at future colliders.

## Contribution

It provides a theoretical estimate of the CP-forbidden transition rate using U-spin symmetry, connecting recent CP violation measurements to experimental prospects.

## Key findings

- Predicted CP asymmetries in $D^0$ decays due to U-spin symmetry.
- Estimated transition rate of order $10^{-10}$ for the CP-forbidden process.
- Potential observation of the transition at high-luminosity colliders with sufficient data.

## Abstract

The LHCb Collaboration has recently reported the discovery of direct CP violation in combined $D^0 \to K^+ K^-$ and $D^0 \to \pi^+ \pi^-$ decay modes at the $5.3 \sigma$ level. Assuming U-spin symmetry (i.e., $d \leftrightarrow s$ interchange symmetry) for the strong-interaction parts of these two channels, we find that their corresponding direct CP-violating asymmetries are ${\cal A}^{}_{\rm CP} (K^+ K^-) \simeq \left(-7.7 \pm 1.5\right) \times 10^{-4}$ and ${\cal A}^{}_{\rm CP} (\pi^+ \pi^-) \simeq \left(7.7 \pm 1.5\right) \times 10^{-4}$. The CP-forbidden transition $e^+ e^- \to D^0\bar{D}^0 \to \left(K^+ K^-\right)^{}_D \left(\pi^+ \pi^-\right)^{}_D$ on the $\psi(3770)$ resonance is therefore expected to have a rate of ${\cal O}(10^{-10})$ or smaller under U-spin symmetry, and it can be observed at a high-luminosity super-$\tau$-charm factory if at least $10^{10}$ pairs of coherent $D^0$ and $\bar{D}^0$ events are accumulated.