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

TL;DR

The paper analyzes how a small $C$-even admixture in the $B^0\bar{B}^0$ wave function, induced by soft-photon emission in $\Upsilon(4S)$ decays, alters the time-dependent CP asymmetry. Using a two-particle wave function formalism, it derives analytic expressions for CP asymmetry in pure $C$-odd, pure $C$-even, and mixed parity states, highlighting that the $C$-even component makes the asymmetry depend on the absolute time difference $|\Delta t|$ and shifts the zero-crossing. Quantitatively, the correction is negligible on-shell with ${\rm Br}(\Upsilon(4S)\to B^0\bar{B}^0\gamma)\sim10^{-9}$, but off-resonance by about 25 MeV the radiative width can grow to ${\rm Br}\sim10^{-3}$, yielding corrections up to ${\cal O}(10^{-4})$–${\cal O}(10^{-3})$ that can rival penguin pollution. The authors compare these contributions with penguin effects, noting that near the resonance the penguin term dominates, while the $C$-even admixture becomes significant off-resonance, and they discuss additional mechanisms such as intermediate $\chi_b$ states. The work provides a framework for incorporating this subtle admixture into high-precision CP-violation analyses and emphasizes its energy dependence in experimental planning.

Abstract

$BB$ pairs from $Υ(4S)\to BB$ 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.

Figures (2)

• Figure 1: Time-dependent $CP$-asymmetry in the case of $C$-odd (red dashed) and $C$-even (green dot-dashed) states and their mixture (blue solid) with an admixture of $\chi = 0.4$ in the case of $\eta_S = -1$.
• Figure 2: The decay width of $\Upsilon(4S) \to B^0 \bar{B}^0 \gamma$ as a function of $\sqrt{s}$.