#### ${{\mathit B}^{0}}-{{\overline{\mathit B}}^{0}}$ 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 average and rescaling were performed by the Heavy Flavor Averaging Group (HFLAV) and are described at https://hflav.web.cern.ch/. 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
 $\bf{ 0.1858 \pm0.0011}$ OUR EVALUATION
 $\bf{ 0.182 \pm0.015}$ 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:
 BEHRENS 2000B
PL B490 36 Precise Measurement of ${{\mathit B}^{0}}$ $\leftrightarrow$ ${{\overline{\mathit B}}^{0}}$ Mixing Parameters at the ${{\mathit \Upsilon}{(4S)}}$
 ALBRECHT 1996D
PL B374 256 Two Measurements of ${{\mathit B}^{0}}{{\overline{\mathit B}}^{0}}$ Mixing using Kaon Tagging
 ALBRECHT 1994
PL B324 249 A Study of ${{\overline{\mathit B}}^{0}}$ $\rightarrow$ ${{\mathit D}^{*+}}{{\mathit \ell}^{-}}{{\overline{\mathit \nu}}}$ and ${{\mathit B}^{0}}{{\overline{\mathit B}}^{0}}$ Mixing using Partial ${{\mathit D}^{*+}}$ Reconstruction
 BARTELT 1993
PRL 71 1680 Two Measurements of ${{\mathit B}^{0}}{{\overline{\mathit B}}^{0}}$ Mixing
 ALBRECHT 1992L
ZPHY C55 357 A New Determination of the ${{\mathit B}^{0}}$ $\leftrightarrow$ ${{\overline{\mathit B}}^{0}}$ Oscillation Strength
 ELSEN 1990
ZPHY C46 349 A Measurement of the Weak Axial Couplings of the ${\mathit {\mathit b}}$ and ${\mathit {\mathit c}}$ Quarks
 ARTUSO 1989
PRL 62 2233 ${{\mathit B}^{0}}{{\overline{\mathit B}}^{0}}$ Mixing at the ${{\mathit \Upsilon}{(4S)}}$
 ALBRECHT 1987I
PL B192 245 Observation of ${{\mathit B}^{0}}$ $−$ ${{\overline{\mathit B}}^{0}}$ Mixing
 BEAN 1987B
PRL 58 183 Limits on ${{\mathit B}^{0}}{{\overline{\mathit B}}^{0}}$ Mixing and ${\mathit \tau}_{{{\mathit B}^{0}}}/{\mathit \tau}_{{{\mathit B}^{+}}}$
 AVERY 1984
PRL 53 1309 Upper Limit on Flavour Changing Neutral Current Decays of the ${\mathit {\mathit b}}$ Quark