$-2\beta _{s}$ is the weak phase difference between ${{\mathit B}^{0}_{s}}$ mixing amplitude and the ${{\mathit B}^{0}_{s}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decay amplitude driven by the ${{\mathit b}}$ $\rightarrow$ ${{\mathit c}}{{\overline{\mathit c}}}{{\mathit s}}$ transition (such as ${{\mathit B}_{{{s}}}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$, ${{\mathit J / \psi}}{{\mathit K}^{+}}{{\mathit K}^{-}}$, ${{\mathit J / \psi}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$, and ${{\mathit D}_{{{s}}}^{+}}{{\mathit D}_{{{s}}}^{-}}$). The Standard Model value of $\beta _{s}$ is arg($−$ ${V_{ts} V{}^{*}_{tb}\over V_{cs} V{}^{*}_{cb} }~$) if penguin contributions are neglected.

VALUE ($ 10^{-2} $ rad) DOCUMENT ID TECN  COMMENT $\bf{ 2.0 \pm0.8}$ OUR EVALUATION  $~~$(Produced by HFLAV) $\bf{ 1.9 \pm0.8}$ OUR AVERAGE $4.3$ $\pm5.3$ $\pm1.4$ 1 AAIJ 2025 A LHCB ${{\mathit p}}{{\mathit p}}$ at 13 TeV $1.95$ $\pm1.1$ $\pm0.3$ 2 AAIJ 2024 A LHCB ${{\mathit p}}{{\mathit p}}$ at 13 TeV $4.05$ $\pm2.05$ $\pm1.1$ 3, 4 AAD 2021 AE ATLS ${{\mathit p}}{{\mathit p}}$ at 13 TeV $0$ $\pm14$ $\pm4$ 5 AAIJ 2021 AN LHCB ${{\mathit p}}{{\mathit p}}$ at 7, 8 TeV $0.55$ $\pm2.5$ $\pm0.5$ 6, 7 SIRUNYAN 2021 E CMS ${{\mathit p}}{{\mathit p}}$ at 13 TeV $2.85$ $\pm3.0$ $\pm0.55$ 8, 9 AAIJ 2019 AF LHCB ${{\mathit p}}{{\mathit p}}$ at 13 TeV $-5.95$ $\pm5.35$ $\pm1.7$ 10 AAIJ 2017 V LHCB ${{\mathit p}}{{\mathit p}}$ at 7, 8 TeV $5.05$ $\pm4.10$ $\pm2.10$ 11, 12 AAD 2016 AP ATLS ${{\mathit p}}{{\mathit p}}$ at 8 TeV $-11.5$ ${}^{+14}_{-14.5}$ $\pm1$ 13 AAIJ 2016 AK LHCB ${{\mathit p}}{{\mathit p}}$ at 7, 8 TeV $3.75$ $\pm4.85$ $\pm1.55$ 14 KHACHATRYAN 2016 S CMS ${{\mathit p}}{{\mathit p}}$ at 8 TeV $2.9$ $\pm2.5$ $\pm0.3$ 15 AAIJ 2015 I LHCB ${{\mathit p}}{{\mathit p}}$ at 7, 8 TeV $-6$ $\pm13$ $\pm3$ 16 AAD 2014 U ATLS ${{\mathit p}}{{\mathit p}}$ at 7 TeV $-1$ $\pm9$ $\pm1$ 17 AAIJ 2014 AY LHCB ${{\mathit p}}{{\mathit p}}$ at 7, 8 TeV $-3.5$ $\pm3.4$ $\pm0.4$ 18 AAIJ 2014 S LHCB ${{\mathit p}}{{\mathit p}}$ at 7, 8 TeV 19 AALTONEN 2012 AJ CDF ${{\mathit p}}{{\overline{\mathit p}}}$ at 1.96 TeV $27.5$ ${}^{+18}_{-19}$ 20 ABAZOV 2012 D D0 ${{\mathit p}}{{\overline{\mathit p}}}$ at 1.96 TeV • • We do not use the following data for averages, fits, limits, etc. • • $4.15$ $\pm2.05$ $\pm0.3$ 21 AAIJ 2019 Q LHCB Repl. by AAIJ 2024A $5$ $\pm6.5$ $\pm7$ 22 AAIJ 2018 S LHCB ${{\mathit p}}{{\mathit p}}$ at 7, 8 TeV $6$ ${}^{+8}_{-7}$ 23, 24 AAIJ 2015 K LHCB ${{\mathit p}}{{\mathit p}}$ at 7, 8 TeV $-0.5$ $\pm3.5$ $\pm0.5$ 25 AAIJ 2013 AR LHCB Repl. by AAIJ 2015I $-11$ $\pm20.5$ $\pm5$ 26 AAD 2012 CV ATLS Repl. by AAD 2014U $22$ $\pm22$ $\pm1$ 27 AAIJ 2012 B LHCB Repl. by AAIJ 2012Q $-7.5$ $\pm9$ $\pm3$ 28 AAIJ 2012 D LHCB Repl. by AAIJ 2013AR $0.95$ ${}^{+8.7}_{-8.65}$ ${}^{+0.15}_{-0.2}$ 29 AAIJ 2012 Q LHCB Repl. by AAIJ 2013AR 30 AALTONEN 2012 D CDF Repl. by AALTONEN 2012AJ 31 AALTONEN 2008 G CDF Repl. by AALTONEN 2012D $28$ ${}^{+12}_{-15}$ ${}^{+4}_{-1}$ 20, 32 ABAZOV 2008 AM D0 Repl. by ABAZOV 2012D $39.5$ $\pm28.0$ ${}^{+0.5}_{-7.0}$ 33, 34 ABAZOV 2007 D0 Repl. by ABAZOV 2007N $35$ ${}^{+20}_{-24}$ 34, 35 ABAZOV 2007 N D0 Repl. by ABAZOV 2008AM 1  AAIJ 2025A reports ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.086$ $\pm0.106$ $\pm0.028$ rad. in a time-dependent fit to ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit D}_{{{s}}}^{+}}{{\mathit D}_{{{s}}}^{-}}$, while allowing $\mathit CP$ violation in decay. 2  AAIJ 2019Q reports ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.039$ $\pm0.022$ $\pm0.006$ rad. measured using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit K}^{+}}{{\mathit K}^{-}}$ decays. 3  Reports a combination of $0.0435$ $\pm0.0180$ $\pm0.0105$ with AAD 2016AP. 4  AAD 2021AE measured ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.087$ $\pm0.036$ $\pm0.021$ rad. using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays. 5  AAIJ 2021AN measured ${{\mathit \phi}_{{{s}}}}$ = $-2{{\mathit \beta}_{{{s}}}}$ = $0.00$ $\pm0.28$ $\pm0.07$ rad, using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays with ${{\mathit J / \psi}}$ $\rightarrow$ ${{\mathit e}^{+}}{{\mathit e}^{-}}$. 6  Reports a combination of $0.0105$ $\pm0.0220$ $\pm0.0050$ with KHACHATRYAN 2016S. 7  SIRUNYAN 2021E measured $\phi _{s}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.021$ $\pm0.044$ $\pm0.010$ rad. using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays. 8  Reports a combination of $-0.001$ $\pm0.022$ $\pm0.006$ with AAIJ 2014S. 9  AAIJ 2019AF reports ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $0.002$ $\pm0.044$ $\pm0.012$ rad. and $\vert \lambda \vert $= $0.949$ $\pm0.036$ $\pm0.019$, when direct $\mathit CP$ violation is allowed. Measured using a time-dependent fit to ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ decays. 10  Measured ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $0.119$ $\pm0.107$ $\pm0.034$ rad using time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit K}^{+}}{{\mathit K}^{-}}$ in the region m(${{\mathit K}}{{\mathit K}}$) $>$ 1.05 GeV. 11  Reports a combination of $0.0435$ $\pm0.0180$ $\pm0.0105$ with AAD 2014U. 12  AAD 2016AP reports $\phi _{s}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.090$ $\pm0.078$ $\pm0.041$ rad. that was measured using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays. 13  AAIJ 2016AK reports ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $0.23$ ${}^{+0.29}_{-0.28}$ $\pm0.02$ rad. that was measured using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit \psi}{(2S)}}{{\mathit \phi}}$ decays. 14  KHACHATRYAN 2016S reports $\phi _{s}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.075$ $\pm0.097$ $\pm0.031$ rad. that was measured using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays. 15  AAIJ 2015I reports $\phi _{s}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.058$ $\pm0.049$ $\pm0.006$ rad. that was measured using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit K}^{+}}{{\mathit K}^{-}}$ decays. It also combines this result with that of AAIJ 2014S and quotes $\phi _{s}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.010$ $\pm0.039$ rad. 16  AAD 2014U reports $\phi _{s}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $0.12$ $\pm0.25$ $\pm0.05$ rad. that was measured using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays. 17  AAIJ 2014AY reports ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $0.02$ $\pm0.17$ $\pm0.02$ rad. in a time-dependent fit to ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit D}_{{{s}}}^{+}}{{\mathit D}_{{{s}}}^{-}}$, while allowing $\mathit CP$ violation in decay. 18  AAIJ 2014S reports ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $0.070$ $\pm0.068$ $\pm0.008$ rad. and $\vert \lambda \vert $= $0.89$ $\pm0.05$ $\pm0.01$, when direct $\mathit CP$ violation is allowed. Measured using a time-dependent fit to ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ decays. 19  AALTONEN 2012AJ reports $−{{\mathit \pi}}$/2 $<{{\mathit \beta}_{{{s}}}}<-1.51$ or $-0.06<{{\mathit \beta}_{{{s}}}}<$ 0.30, or 1.26 $<{{\mathit \beta}_{{{s}}}}<{{\mathit \pi}}$/2 rad. at 68$\%$ CL. Measured using the time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays. 20  ABAZOV 2012D reports ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.55$ ${}^{+0.38}_{-0.36}$ rad. that was measured using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays. A single error includes both statistical and systematic uncertainties. 21  AAIJ 2019Q reports ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.083$ $\pm0.041$ $\pm0.006$ rad. that was measured using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit K}^{+}}{{\mathit K}^{-}}$ decays. 22  AAIJ 2018S reports ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.10$ $\pm0.13$ $\pm0.14$ rad measured in ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ( ${{\mathit K}^{+}}{{\mathit \pi}^{-}}$) ( ${{\mathit K}^{-}}{{\mathit \pi}^{+}}$) in the region 0.75 $<$ m(${{\mathit K}^{\pm}}{{\mathit \pi}^{\mp}}$) $<$ 1.6 GeV. This is a ${{\mathit b}}$ $\rightarrow$ ${{\mathit d}}{{\overline{\mathit d}}}{{\mathit s}}$ transition with a decay amplitude phase different from that of ${{\mathit b}}$ $\rightarrow$ ${{\mathit c}}{{\overline{\mathit c}}}{{\mathit s}}$ transition. 23  AAIJ 2015K reports $-2{{\mathit \beta}_{{{s}}}}$ = $-0.12$ ${}^{+0.14}_{-0.16}$ rad. The value was obtained by measuring time-dependent $\mathit CP$ asymmetry in ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit K}^{+}}{{\mathit K}^{-}}$ and using a U-spin relation between ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit K}^{+}}{{\mathit K}^{-}}$ and ${{\mathit B}^{0}}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$. 24  Results are also presented using additional inputs on ${{\mathit B}^{0}}$ $\rightarrow$ ${{\mathit \pi}^{0}}{{\mathit \pi}^{0}}$ and ${{\mathit B}^{+}}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{0}}$ decays from other experiments and isospin symmetry assumptions. The dependence of the results on the maximum allowed amount of U-spin breaking up to 50$\%$ is also included. 25  AAIJ 2013AR reports ${{\mathit \phi}_{{{s}}}}$ = $-2{{\mathit \beta}_{{{s}}}}$ = $0.01$ $\pm0.07$ $\pm0.01$ rad. obtained from combined fit to ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit K}^{+}}{{\mathit K}^{-}}$ and ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ data sets. Also reports separate results of ${{\mathit \phi}_{{{s}}}}$ = $0.07$ $\pm0.09$ $\pm0.01$ rad. from ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit K}^{+}}{{\mathit K}^{-}}$ decays and ${{\mathit \phi}_{{{s}}}}$ = $-0.14$ ${}^{+0.17}_{-0.16}$ $\pm0.01$ rad. from ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ decays. 26  AAD 2012CV reports $\phi _{s}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $0.22$ $\pm0.41$ $\pm0.10$ rad. that was measured using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays. 27  Reports ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.44$ $\pm0.44$ $\pm0.02$ rad. that was measured using a time-dependent fit to ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit f}_{{{0}}}{(980)}}$ decays. 28  Reports ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $0.15$ $\pm0.18$ $\pm0.06$ rad. that was measured using a time-dependent angular analysis of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays. 29  Reports ${{\mathit \phi}_{{{s}}}}$ = $-2$ ${{\mathit \beta}_{{{s}}}}$ = $-0.019$ ${}^{+0.173}_{-0.174}{}^{+0.004}_{-0.003}$ rad. which was measured using a time-dependent fit to ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ decays, with the ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ mass within $775 - 1550$ MeV. Searches for, but finds no evidence, for direct $\mathit CP$ violation in ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \pi}}{{\mathit \pi}}$ decays. 30  Reports 0.02 $<$ ${{\mathit \phi}_{{{s}}}}<$ 0.52 or 1.08 $<$ ${{\mathit \phi}_{{{s}}}}<$ 1.55 rad. at 68$\%$ C.L. confidence regions in the two-dimensional space of ${{\mathit \phi}_{{{s}}}}$ and $\Delta {\Gamma}_{{\mathit B}_{{{s}}}^{0}}$ from ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays. 31  Reports 0.32 $<$ 2${{\mathit \beta}_{{{s}}}}<$ 2.82 rad. at 68$\%$ C.L. and confidence regions in the two-dimensional space of 2${{\mathit \beta}_{{{s}}}}$ and $\Delta \Gamma $ from the first measurement of ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays using flavor tagging. The probability of a deviation from SM prediction as large as the level of observed data is 15$\%$. 32  Reports $\phi _{s}$ = $-2$ $\beta _{s}$ and obtains 90$\%$ CL interval $-0.03$ $<$ $\beta _{s}$ $<$ 0.60 rad. 33  The first direct measurement of the $\mathit CP$-violating mixing phase is reported from the time-dependent analysis of flavor untagged ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ decays. 34  Reports ${{\mathit \phi}_{{{s}}}}$ which equals to $-2{{\mathit \beta}_{{{s}}}}$. 35  Combines D0 collaboration measurements of time-dependent angular distributions in ${{\mathit B}_{{{s}}}^{0}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit \phi}}$ and charge asymmetry in semileptonic decays. There is a 4-fold ambiguity in the solution.

Conservation Laws:

$\mathit CP$ INVARIANCE

References