Correction to $CP$-asymmetry in $\Upsilon(4S)$ decays due to the admixture of $B^0 \bar B^0$ in a $C$-even state

TL;DR

This paper analyzes how a small $C$-even admixture, caused by soft photon emission, affects the $CP$-asymmetry in $S$ decays, providing analytical formulas and emphasizing energy dependence.

Contribution

It introduces a formalism to quantify the impact of $C$-even admixture on $CP$-asymmetry in $S$ decays, accounting for soft photon effects.

Findings

Derived analytical expressions for $CP$-asymmetry in mixed $C$-parity states.

Showed the $C$-even admixture depends strongly on energy.

Quantified the correction to $CP$-asymmetry due to soft photon emission.

Abstract

$BB$ pairs from $\Upsilon(4S)\to BB$ decays are in $C$-odd state. However, there is a small admixture of the $C$-even state and it modifies the time dependent $CP$-asymmetry. The $C$-even component appears due to soft photon emission, breaking the pure $C$-odd nature of the initial state. Using the two-particle wave function formalism, we derive analytical expressions for $CP$-asymmetry in both pure and mixed $C$-parity states. We emphasize the strong energy dependence of the admixture magnitude.