CP VIOLATION PARAMETERS IN ${{\mathit B}^{+}}$ $\rightarrow$ ${{\mathit D}}{{\mathit K}^{+}}$ AND SIMILAR DECAYS

The parameters ${{\mathit r}}_{{{\mathit B}^{+}}}$ and $\delta _{{{\mathit B}^{+}}}$ are the magnitude ratio and strong phase difference between the amplitudes of A( ${{\mathit B}^{+}}$ $\rightarrow$ ${{\overline{\mathit D}}^{(*)0}}{{\mathit K}^{(*)+}}$ ) and A( ${{\mathit B}^{-}}$ $\rightarrow$ ${{\mathit D}^{(*)0}}{{\mathit K}^{(*)-}}$ ). The measured observables are defined as ${{\mathit x}}_{\pm{}}$ = ${{\mathit r}}_{{{\mathit B}^{+}}}$ cos($\delta _{{{\mathit B}^{+}}}$ $\pm{}{{\mathit \gamma}}$) and ${{\mathit y}}_{\pm{}}$ = ${{\mathit r}}_{{{\mathit B}^{+}}}$ sin($\delta _{{{\mathit B}^{+}}}$ $\pm{}$ $\gamma $), and can be used to measure the CKM angle $\gamma $.
"OUR EVALUATION" is provided by the Heavy Flavor Averaging Group (HFLAV). It is derived from combinations of their results on ${{\mathit B}^{+}}$ $\rightarrow$ ${{\mathit D}}{{\mathit K}^{+}}$ and related processes.

$\delta _{B}$( ${{\mathit B}^{+}}$ $\rightarrow$ ${{\mathit D}^{0}}{{\mathit K}^{+}}$ )

INSPIRE   PDGID:
S041DRX
"OUR EVALUATION" is provided by the Heavy Flavor Averaging Group (HFLAV).
VALUE ($^\circ{}$) DOCUMENT ID TECN  COMMENT
$\bf{ 127.7 {}^{+3.6}_{-3.9}}$ OUR EVALUATION
• • We do not use the following data for averages, fits, limits, etc. • •
$124.8$ $\pm12.9$ $\pm1.8$ 1, 2
ABUDINEN
2022
BELL ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$118.3$ ${}^{+5.5}_{-5.6}$ 3
AAIJ
2021L
 LHCB ${{\mathit p}}{{\mathit p}}$ at 7, 8, 13 TeV
$83.4$ ${}^{+18.3}_{-16.6}$ $\pm5.1$ 4
RESMI
2019
BELL ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$101$ $\pm11$ 5
AAIJ
2018AD
 LHCB Repl. by AAIJ 2021L
$134$ ${}^{+14}_{-15}$ 1
AAIJ
2014BA
 LHCB Repl. by AAIJ 2021L
$115$ ${}^{+41}_{-51}$ 6
AAIJ
2014BE
 LHCB Repl. by AAIJ 2014BA
$105$ ${}^{+16}_{-17}$ 7
LEES
2013B
 BABR ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$137$ ${}^{+35}_{-46}$ 8, 9
AAIJ
2012AQ
 LHCB ${{\mathit p}}{{\mathit p}}$ at 7 TeV
$129.9$ $\pm15.0$ $\pm6.0$ 9, 10
AIHARA
2012
BELL ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$119$ ${}^{+19}_{-20}$ $\pm4$ 11
DEL-AMO-SANCH..
2010F
 BABR Repl. by LEES 2013B
$136.7$ ${}^{+13.0}_{-15.8}$ $\pm23.2$ 12
POLUEKTOV
2010
BELL ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \Upsilon}{(4S)}}$
$109$ ${}^{+27}_{-30}$ $\pm8$ 13
AUBERT
2008AL
 BABR Repl. by DEL-AMO-SANCHEZ 2010F
$145.7$ ${}^{+19.0}_{-19.7}$ $\pm23.1$ 14
POLUEKTOV
2006
BELL Repl. by POLUEKTOV 2010
$104$ $\pm45$ ${}^{+23}_{-32}$ 15
AUBERT,B
2005Y
 BABR Repl. by AUBERT 2008AL
1  Uses binned Dalitz plot analysis of ${{\mathit B}^{+}}$ $\rightarrow$ ${{\mathit D}}{{\mathit K}^{+}}$ decays, with ${{\mathit D}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ and ${{\mathit D}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit K}^{+}}{{\mathit K}^{-}}$ . Strong phase measurements from CLEO-c (LIBBY 2010 ) of the ${{\mathit D}}$ decay over the Dalitz plot are used as input.
2  Supersedes AIHARA 2012 .
3  Uses binned Dalitz plot analysis of ${{\mathit D}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ and ${{\mathit K}_S^0}$ ${{\mathit K}^{+}}{{\mathit K}^{-}}$ from ${{\mathit B}^{\pm}}$ $\rightarrow$ ${{\mathit D}}{{\mathit K}^{\pm}}$ modes. Strong phase measurements from CLEO-c and BES-III of the ${{\mathit D}}$ decay over the Dalitz plot are used as input. Value is modulo 180$^\circ{}$.
4  Uses binned analysis of ${{\mathit D}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$ from ${{\mathit B}^{\pm}}$ $\rightarrow$ ${{\mathit D}}{{\mathit K}^{\pm}}$ modes over the phase space. Strong phase measurements from RESMI 2018 analysis of CLEO-c data of the ${{\mathit D}}$ decay over the phase space binning are used as input.
5  Uses binned Dalitz plot analysis of ${{\mathit D}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ and ${{\mathit K}_S^0}$ ${{\mathit K}^{+}}{{\mathit K}^{-}}$ from ${{\mathit B}^{\pm}}$ $\rightarrow$ ${{\mathit D}}{{\mathit K}^{\pm}}$ modes. Strong phase measurements from CLEO-c of the ${{\mathit D}}$ decay over the Dalitz plot are used as input.
6  AAIJ 2014BE uses model-dependent analysis of ${{\mathit D}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ amplitudes. The model is the same as in DEL-AMO-SANCHEZ 2010F.
7  Reports combination of published measurements using GGSZ, GLW, and ADS methods.
8  Reports combined statistical and systematic uncertainties.
9  Uses binned Dalitz plot of ${{\overline{\mathit D}}^{0}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ decays from ${{\mathit B}^{+}}$ $\rightarrow$ ${{\overline{\mathit D}}^{0}}{{\mathit K}^{+}}$ . Measurement of strong phases in ${{\overline{\mathit D}}^{0}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ Dalitz plot from LIBBY 2010 is used as input.
10  We combined the systematics in quadrature. The authors report separately the contribution to the systematic uncertainty due to the uncertainty on the bin-averaged strong phase difference between ${{\mathit D}^{0}}$ and ${{\overline{\mathit D}}^{0}}$ amplitudes. Superseded by ABUDINEN 2022 .
11  Uses Dalitz plot analysis of ${{\overline{\mathit D}}^{0}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ , ${{\mathit K}_S^0}$ ${{\mathit K}^{+}}{{\mathit K}^{-}}$ decays from ${{\mathit B}^{+}}$ $\rightarrow$ ${{\mathit D}^{(*)}}{{\mathit K}^{(*)+}}$ modes. The corresponding two standard deviation interval is 75$^\circ{}<{{\mathit \delta}_{{B}}}<157^\circ{}$.
12  Uses Dalitz plot analysis of ${{\overline{\mathit D}}^{0}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ decays from ${{\mathit B}^{+}}$ $\rightarrow$ ${{\overline{\mathit D}}^{0}}{{\mathit K}^{+}}$ modes. The corresponding two standard deviation interval is 102.2$^\circ{}<{{\mathit \delta}_{{B}}}<$ 162.3$^\circ{}$.
13  Uses Dalitz plot analysis of ${{\overline{\mathit D}}^{0}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ and ${{\overline{\mathit D}}^{0}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit K}^{+}}{{\mathit K}^{-}}$ decays coming from ${{\mathit B}^{\pm}}$ $\rightarrow$ ${{\mathit D}^{(*)}}{{\mathit K}^{(*)\pm}}$ modes.
14  Uses a Dalitz plot analysis of the ${{\overline{\mathit D}}^{0}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ decays; Combines the ${{\mathit D}}{{\mathit K}^{+}}$ , ${{\mathit D}^{*}}{{\mathit K}^{+}}$ and ${{\mathit D}}{{\mathit K}^{*+}}$ modes.
15  Uses a Dalitz analysis of neutral ${{\mathit D}}$ decays to ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ in the processes ${{\mathit B}^{\pm}}$ $\rightarrow$ ${{\mathit D}^{{(*)}}}{{\mathit K}^{\pm}}$ , ${{\mathit D}^{*}}$ $\rightarrow$ ${{\mathit D}}{{\mathit \pi}^{0}}$ , ${{\mathit D}}{{\mathit \gamma}}$ .
Conservation Laws:
$\mathit CP$ VIOLATION OBSERVED
References:
ABUDINEN 2022
JHEP 2202 063 Combined analysis of Belle and Belle II data to determine the CKM angle $ \phi_{3} $ using $B^+ \to D(K_{S}^0 h^- h^+) h^+$ decays
AAIJ 2021L
JHEP 2102 169 Measurement of the CKM angle $\gamma$ in $B^\pm\to D K^\pm$ and $B^\pm \to D \pi^\pm$ decays with $D \to K_\mathrm S^0 h^+ h^-$
RESMI 2019
JHEP 1910 178 First measurement of the CKM angle $\phi_3$ with $B^{\pm}\to D(K_{\rm S}^0\pi^+\pi^-\pi^0)K^{\pm}$ decays
AAIJ 2018AD
JHEP 1808 176 Measurement of the CKM angle $\gamma$ using $B^\pm\to DK^\pm$ with $D\to K_\text{S}^0\pi^+\pi^-$, $K_\text{S}^0K^+K^-$ decays
Also
JHEP 1810 107 (errat.) Measurement of the CKM angle $\gamma$ using $B^\pm\to DK^\pm$ with $D\to K_\text{S}^0\pi^+\pi^-$, $K_\text{S}^0K^+K^-$ decays
AAIJ 2014BE
NP B888 169 Measurement of $\mathit CP$ Violation and Constraints on the $\mathit CKM$ Angle ${{\mathit \gamma}}$ in ${{\mathit B}^{\pm}}$ $\rightarrow$ ${{\mathit D}}{{\mathit K}^{\pm}}$ with ${{\mathit D}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ Decays
AAIJ 2014BA
JHEP 1410 097 Measurement of the $\mathit CKM$ Angle ${{\mathit \gamma}}$ using ${{\mathit B}^{\pm}}$ $\rightarrow$ ${{\mathit D}}{{\mathit K}^{\pm}}$ with ${{\mathit D}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ , ${{\mathit K}_S^0}$ ${{\mathit K}^{+}}{{\mathit K}^{-}}$ Decays
LEES 2013B
PR D87 052015 Observation of Direct $\mathit CP$ Violation in the Measurement of the Cabibbo-Kobayashi-Maskawa Angle $\gamma $ with ${{\mathit B}^{\pm}}$ $\rightarrow$ ${{\mathit D}^{(*)}}{{\mathit K}^{(*)\pm}}$
AAIJ 2012AQ
PL B718 43 A Model-Independent Dalitz Plot Analysis of ${{\mathit B}^{\pm}}$ $\rightarrow$ ${{\mathit D}}{{\mathit K}^{\pm}}$ with ${{\mathit D}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit h}^{+}}{{\mathit h}^{-}}$ (${{\mathit h}}={{\mathit \pi}},{{\mathit K}}$) Decays and Constraints on the CKM Angle ${{\mathit \gamma}}$
AIHARA 2012
PR D85 112014 First Measurement of ${{\mathit \phi}_{{3}}}$ with a Model-Independent Dalitz Plot Analysis of ${{\mathit B}^{\pm}}$ $\rightarrow$ ${{\mathit D}}{{\mathit K}^{\pm}}$ , ${{\mathit D}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ Decay
DEL-AMO-SANCHEZ 2010F
PRL 105 121801 Evidence for Direct $\mathit CP$ Violation in the Measurement of the Cabbibo-Kobayashi-Maskawa Angle ${{\mathit \gamma}}$ with ${{\mathit B}^{\mp}}$ $\rightarrow$ ${{\mathit D}^{{(*)}}}{{\mathit K}^{{(*)}\mp}}$ Decays
POLUEKTOV 2010
PR D81 112002 Evidence for Direct $\mathit CP$ Violation in the Decay ${{\mathit B}^{\pm}}$ $\rightarrow$ ${{\mathit D}^{{(*)}}}{{\mathit K}^{\pm}}$ , ${{\mathit D}}$ $\rightarrow$ ${{\mathit K}_{{s}}^{0}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ and Measurement of the CKM Phase $\phi _{3}$
AUBERT 2008AL
PR D78 034023 Improved Measurement of the CKM Angle $\gamma $ in ${{\mathit B}^{\mp}}$ $\rightarrow$ ${{\mathit D}^{(*)}}{{\mathit K}^{(*)\mp}}$ Decays with a Dalitz Plot Analysis of ${{\mathit D}}$ Decays to ${{\mathit K}_S^0}$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ and ${{\mathit K}_S^0}$ ${{\mathit K}^{+}}{{\mathit K}^{-}}$
POLUEKTOV 2006
PR D73 112009 Measurement of ${{\mathit \phi}_{{3}}}$ with a Dalitz Plot Analysis of ${{\mathit B}^{+}}$ $\rightarrow$ ${{\mathit D}^{{(*)}}}{{\mathit K}^{{(*)}+}}$ Decay
AUBERT,B 2005Y
PRL 95 121802 Measurement of the Cabibbo-Kobayashi-Maskawa Angle $\gamma $ in ${{\mathit B}^{\mp}}$ ${}^{(*)}$ ${{\mathit K}^{\mp}}$ Decays with a Dalitz Analysis of ${{\mathit D}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit \pi}^{-}}{{\mathit \pi}^{+}}$