Measurement of the $CP$-violating phase $\phi_\mathrm{s}$ in the B$^0_\mathrm{s}$ $\to$ J$/\psi\, \phi$(1020) $\to$ $\mu^+\mu^-$K$^+$K$^-$ channel in proton-proton collisions at $\sqrt{s} =$ 13 TeV

The $CP$-violating weak phase $\phi_\mathrm{s}$ and the decay width difference $\Delta\Gamma_\mathrm{s}$ between the light and heavy B$^0_\mathrm{s}$ mass eigenstates are measured with the CMS detector at the LHC in a sample of 48 500 reconstructed B$^0_\mathrm{s}$ $\to$ J$/\psi\, \phi$(1020) $\to$ $\mu^+\mu^-$K$^+$K$^-$ events. The measurement is based on a data sample corresponding to an integrated luminosity of 96.4 fb$^{-1}$, collected in proton-proton collisions at $\sqrt{s} =$ 13 TeV in 2017-2018. To extract the values of $\phi_\mathrm{s}$ and $\Delta\Gamma_\mathrm{s}$, a time-dependent and flavor-tagged angular analysis of the $\mu^+\mu^-$K$^+$K$^-$ final state is performed. The analysis employs a dedicated tagging trigger and a novel opposite-side muon flavor tagger based on machine learning techniques. The measurement yields $\phi_\mathrm{s} = -$11 $\pm$ 50 (stat) $\pm$ 10 (syst) mrad and $\Delta\Gamma_\mathrm{s} =$ 0.114 $\pm$ 0.014 (stat) $\pm$ 0.007 (syst) ps$^{-1}$, in agreement with the standard model predictions. When combined with the previous CMS measurement at $\sqrt{s} =$ 8 TeV, the following values are obtained: $\phi_\mathrm{s} = -$21 $\pm$ 44 (stat) $\pm$ 10 (syst) mrad, $\Delta\Gamma_\mathrm{s} =$ 0.1032 $\pm$ 0.0095 (stat) $\pm$ 0.0048 (syst) ps$^{-1}$, a significant improvement over the 8 TeV result.