CP-Asymmetry in \(\Upsilon \) Decays: Contributions of C-odd and C-even States of the \({{B}^{0}}{{\bar {B}}^{0}}\)

Abstract

While \(B\bar {B}\) pairs from \(\Upsilon \) decays are predominantly in a C-odd state, soft photon emission introduces a C-even admixture, modifying the time-dependent CP-asymmetry. Using the two-particle wave function formalism, we derive the CP-asymmetry for pure C-even, C-odd and mixed C-parity states and highlight the characteristic \({\text{|}}\Delta t{\text{|}}\) dependence of the C-even contribution. We analyze the magnitude of the effect for \(\Upsilon (4S)\) and \(\Upsilon (5S)\) resonances. At \(\Upsilon (5S)\) the admixture is large (\(\chi \simeq 0.4\)), while at \(\Upsilon (4S)\) it is negligible on the peak but grows dramatically with a small shift of the center-of-mass energy.