Systematic studies on the $B_s$ proper time measurement for the determination of the mixing phase $\phi_s$ in the channel $B^0_s \to J/ \psi \phi$

The decay $B^0_s \to J/ \psi \phi$ is one of the most promising channels that allows the measurement of the $B_s$ mixing phase $\phi_s$. In this document we introduce the formalism necessary to extract this CP-violating phase $\phi_s$ from a full time-dependent angular analysis. We estimate a sensitivity on $\phi_s$ equal to 0.03 if $\phi_s$ is equal to the expected Standard Model value -0.0368 and to 0.05 if $\phi_s$ is equal to a new Physics value chosen arbitrarily as -0.736. We report studies on systematic effects on the value of $\phi_s$ that can arise by using different proper time resolution models as well as in the case of a wrong proper time measurement. We find that up to 2fb$^{-1}$ the determination of $\phi_s$ is rather robust respect the time resolution model approximation or systematic effects.