#### ${{\mathit B}^{0}}-{{\overline{\mathit B}}^{0}}$ MIXING PARAMETERS

For a discussion of ${{\mathit B}^{0}}-{{\overline{\mathit B}}^{0}}$ mixing see the note on “${{\mathit B}^{0}}-{{\overline{\mathit B}}^{0}}$ Mixing” in the ${{\mathit B}^{0}}$ Particle Listings above.
${{\mathit \chi}_{{{d}}}}$ is a measure of the time-integrated ${{\mathit B}^{0}}-{{\overline{\mathit B}}^{0}}$ mixing probability that a produced ${{\mathit B}^{0}}({{\overline{\mathit B}}^{0}}$) decays as a ${{\overline{\mathit B}}^{0}}({{\mathit B}^{0}}$). Mixing violates $\Delta \mathit B{}\not=$2 rule.
${{\mathit \chi}_{{{d}}}}$ = ${\mathit x{}^{2}_{\mathit d}\over 2(1+\mathit x{}^{2}_{\mathit d})}$

$\mathit x_{\mathit d}$ = ${\Delta {\mathit m}_{{{\mathit B}^{0}}}\over \Gamma _{{{\mathit B}^{0}}}}$ = (${\mathit m}_{{{\mathit B}_{{{H}}}^{0}}}$ $-$ ${\mathit m}_{{{\mathit B}_{{{L}}}^{0}}}){\mathit \tau}_{{{\mathit B}^{0}}}$ ,
where $\mathit H$, $\mathit L$ stand for heavy and light states of two ${{\mathit B}^{0}}$ $\mathit CP$ eigenstates and ${\mathit \tau}_{{{\mathit B}^{0}}}$ = ${1\over 0.5 (\Gamma _{{{\mathit B}_{{{H}}}^{0}}}+\Gamma _{{{\mathit B}_{{{L}}}^{0}}})}$.

#### $\chi _{\mathit d}$

This ${{\mathit B}^{0}}-{{\overline{\mathit B}}^{0}}$ mixing parameter is the probability (integrated over time) that a produced ${{\mathit B}^{0}}$ (or ${{\overline{\mathit B}}^{0}}$) decays as a ${{\overline{\mathit B}}^{0}}$ (or ${{\mathit B}^{0}}$), e.g. for inclusive lepton decays
 $\chi _{\mathit d}$ = $\Gamma\mathrm {( {{\mathit B}^{0}} \rightarrow {{\mathit \ell}^{-}} X (via {{\overline{\mathit B}}^{0}}))}/\Gamma\mathrm {( {{\mathit B}^{0}} \rightarrow {{\mathit \ell}^{\pm}} X)}$
 = $\Gamma\mathrm {( {{\overline{\mathit B}}^{0}} \rightarrow {{\mathit \ell}^{+}} X (via {{\mathit B}^{0}}))}/\Gamma\mathrm {( {{\overline{\mathit B}}^{0}} \rightarrow {{\mathit \ell}^{\pm}} X)}$
Where experiments have measured the parameter $\mathit r$ = $\chi /(1−\chi$), we have converted to $\chi$. Mixing violates the $\Delta \mathit B{}\not=$2 rule.

Note that the measurement of $\chi$ at energies higher than the ${{\mathit \Upsilon}{(4S)}}$ have not separated $\chi _{\mathit d}$ from $\chi _{\mathit s}$ where the subscripts indicate ${{\mathit B}^{0}}({\mathit {\overline{\mathit b}}}{\mathit {\mathit d}}$) or ${{\mathit B}_{{{s}}}^{0}}({\mathit {\overline{\mathit b}}}{\mathit {\mathit s}}$). They are listed in the ${{\mathit B}^{\pm}}/{{\mathit B}^{0}}/{{\mathit B}_{{{s}}}^{0}}/{\mathit {\mathit b}}$-baryon ADMIXTURE section.

The experiments at ${{\mathit \Upsilon}{(4S)}}$ make an assumption about the ${{\mathit B}^{0}}{{\overline{\mathit B}}^{0}}$ fraction and about the ratio of the ${{\mathit B}^{\pm}}$ and ${{\mathit B}^{0}}$ semileptonic branching ratios (usually that it equals one).

“OUR EVALUATION” is an average using rescaled values of the data listed below. The averaging/rescaling procedure takes into account correlations between the measurements, includes ${{\mathit \chi}_{{{d}}}}$ calculated from $\Delta {\mathit m}_{{{\mathit B}^{0}}}$ and $\tau _{{{\mathit B}^{0}}}$.

VALUE CL% DOCUMENT ID TECN  COMMENT
 $\bf{ 0.1860 \pm0.0011}$ OUR EVALUATION  $~~$(Produced by HFLAV)
 $\bf{ 0.182 \pm0.015}$ OUR AVERAGE
$0.198$ $\pm0.013$ $\pm0.014$ 1
 2000 B
CLE2 ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$0.16$ $\pm0.04$ $\pm0.04$ 2
 1994
ARG ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$0.149$ $\pm0.023$ $\pm0.022$ 3
 1993
CLE2 ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$0.171$ $\pm0.048$ 4
 1992 L
ARG ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
• • We do not use the following data for averages, fits, limits, etc. • •
$0.20$ $\pm0.13$ $\pm0.12$ 5
 1996 D
ARG ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$0.19$ $\pm0.07$ $\pm0.09$ 6
 1996 D
ARG ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$0.24$ $\pm0.12$ 7
 1990
JADE ${{\mathit e}^{+}}{{\mathit e}^{-}}$ $35 - 44$ GeV
$0.158$ ${}^{+0.052}_{-0.059}$
 1989
CLEO ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$0.17$ $\pm0.05$ 8
 1987 I
ARG ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$<0.19$ 90 9
 1987 B
CLEO ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$<0.27$ 90 10
 1984
CLEO ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
 1 BEHRENS 2000B uses high-momentum lepton tags and partially reconstructed ${{\overline{\mathit B}}^{0}}$ $\rightarrow$ ${{\mathit D}^{*+}}{{\mathit \pi}^{-}}$, ${{\mathit \rho}^{-}}$ decays to determine the flavor of the ${{\mathit B}}~$meson.
 2 ALBRECHT 1994 reports $\mathit r=0.194$ $\pm0.062$ $\pm0.054$. We convert to $\chi$ for comparison. Uses tagged events (lepton + pion from ${{\mathit D}^{*}}$).
 3 BARTELT 1993 analysis performed using tagged events (lepton+pion from ${{\mathit D}^{*}}$). Using dilepton events they obtain $0.157$ $\pm0.016$ ${}^{+0.033}_{-0.028}$.
 4 ALBRECHT 1992L is a combined measurement employing several lepton-based techniques. It uses all previous ARGUS data in addition to new data and therefore supersedes ALBRECHT 1987I. A value of $\mathit r$ = $20.6$ $\pm7.0\%$ is directly measured. The value can be used to measure $\mathit x$ = $\Delta \mathit M/\Gamma$ = $0.72$ $\pm0.15$ for the ${{\mathit B}_{{{d}}}}$ meson. Assumes $\mathit f_{+−}/\mathit f_{0}$ = $1.0$ $\pm0.05$ and uses ${\mathit \tau}_{{{\mathit B}^{\pm}}}/{\mathit \tau}_{{{\mathit B}^{0}}}$ = ($0.95$ $\pm0.14$) ($\mathit f_{+−}/\mathit f_{0}$).
 5 Uses ${{\mathit D}^{*+}}{{\mathit K}^{\pm}}$ correlations.
 6 Uses (${{\mathit D}^{*+}}{{\mathit \ell}^{-}}$) ${{\mathit K}^{\pm}}$ correlations.
 7 These experiments see a combination of ${{\mathit B}_{{{s}}}}$ and ${{\mathit B}_{{{d}}}}$ mesons.
 8 ALBRECHT 1987I is inclusive measurement with like-sign dileptons, with tagged ${{\mathit B}}$ decays plus leptons, and one fully reconstructed event. Measures $\mathit r=0.21$ $\pm0.08$. We convert to $\chi$ for comparison. Superseded by ALBRECHT 1992L.
 9 BEAN 1987B measured $\mathit r$ $<$ $0.24$; we converted to $\chi$.
 10 Same-sign dilepton events. Limit assumes semileptonic BR for ${{\mathit B}^{+}}$ and ${{\mathit B}^{0}}$ equal. If ${{\mathit B}^{0}}/{{\mathit B}^{\pm}}$ ratio $<$0.58, no limit exists. The limit was corrected in BEAN 1987B from $\mathit r$ $<$ $0.30$ to $\mathit r$ $<$ $0.37$. We converted this limit to $\chi$.
Conservation Laws:
$\Delta \mathit B$ = 2 VIA MIXING
References