Limits on the Observability of Direct CP Violation in the Decay $B_d^0 \rightarrow J/\psi \;K_s^0$ for ATLAS

The effects of CP violation in the decay $B_d^0 \rightarrow J/\psi\; K_s^0$ for low luminosity running at LHC for the ATLAS experiment were simulated using a simple Monte Carlo technique. A 95\% confidence limit of $-0.043<\mathcal{A}^{dir}_{CP}<0.029$ was found for the Standard Model case, where $\mathcal{A}^{dir}_{CP}=0$. The $5\sigma $ discovery limit for direct CP violation was found to be $\mathcal{A}^{dir}_{CP}=0.097\pm 0.008$. This was improved to $\mathcal{A}^{dir}_{CP}=0.078\pm 0.007$ by fixing $\sin 2\beta $ to the input value when fitting the generated asymmetries. Fixing the value of $\mathcal{A}^{dir}_{CP}$ when fitting had no effect on the mean fitted values of $\mathcal{A}^{mix-ind}_{CP}$ and reduced the errors by only about 15\%. The fitted value of $\mathcal{A}^{mix-ind}_{CP}$ and its errors were found not to vary significantly with the input value of $\mathcal{A}^{dir}_{CP}$. Using a smaller value of $\mathcal{A}^{mix-ind}_{CP}$ rather than a higher value, consistent with the Standard Model, did not produce significantly different results.