EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2021-183 LHCb-PAPER-2021-033 21 Dec 2021

Simultaneous determination

of CKM angle $\gamma$ and

charm mixing parameters

LHCb collaboration†††Authors are listed at the end of this paper.

A combination of measurements sensitive to the $C\\!P$ violation angle
$\gamma$ of the Cabibbo–Kobayashi–Maskawa unitarity triangle and to the charm
mixing parameters that describe oscillations between ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons is performed. Results from
the charm and beauty sectors, based on data collected with the LHCb detector
at CERN’s Large Hadron Collider, are combined for the first time. This method
provides an improvement on the precision of the charm mixing parameter $y$ by
a factor of two with respect to the current world average. The charm mixing
parameters are determined to be $x=(0.400^{\,+0.052}_{\,-0.053})\%$ and
$y=(0.630^{\,+0.033}_{\,-0.030})\%$. The angle $\gamma$ is found to be
${\gamma}=(65.4^{\,+3.8}_{\,-4.2})^{\circ}$ and is the most precise
determination from a single experiment.

Published in JHEP 12 (2021) 141

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

##  1 Introduction

Precise measurements of the Cabibbo–Kobayashi–Maskawa (CKM) unitarity triangle
provide a strict test of the Standard Model (SM) and allow for indirect new
physics searches in the quark sector up to very high mass scales. The $C\\!P$
violating phase $\gamma\equiv\arg[-V_{ud}V^{*}_{ub}/V_{cd}V^{*}_{cb}]$, where
$V_{qq^{\prime}}$ is the relevant CKM matrix element, is the only angle of the
unitarity triangle that can be determined using solely measurements of tree-
level $B$-meson decays [1, 2, 3, 4, 5, 6, 7, 8] with negligible theoretical
uncertainty [9], assuming no sizeable new physics effects are present at tree
level [10]. Deviations between direct measurements of $\gamma$ and the value
derived from global CKM fits, which assume validity of the SM and hence
unitarity of the CKM matrix, would be a clear indication of physics beyond the
SM. Furthermore, comparisons between the value of $\gamma$ measured using
decays of different $B$-meson species provide sensitivity to possible new
physics effects at tree level given the different decay topologies involved.
The world average for direct measurements of
${\gamma}=(66.2^{\,+3.4}_{\,-3.6})^{\circ}$ [11] is dominated by LHCb results.
The experimental uncertainty on $\gamma$ is larger than that obtained from
global CKM fits, ${\gamma}=(65.6^{+0.9}_{-2.7})^{\circ}$ [12] using a
frequentist framework, and ${\gamma}=(65.8\pm 2.2)^{\circ}$[13] with a
Bayesian approach. Closing this sensitivity gap is a key physics goal of the
LHCb experiment and the comparison between the direct and indirect
determinations of $\gamma$ is an important test of the SM.

The CKM angle $\gamma$ is measured in decays which are sensitive to
interference between favoured ${b}\rightarrow{c}$ and suppressed
${b}\rightarrow{u}$ quark transition amplitudes that are proportional to
$V_{{c}{b}}^{\phantom{\ast}}$ and $V_{{u}{b}}^{\phantom{\ast}}$,
respectively.111Charge conjugation is implied throughout unless stated
otherwise. The ratio of these two amplitudes is given by
$\mathcal{A}_{\text{sup}}/\mathcal{A}_{\text{fav}}=r_{B}e^{i\delta_{B}\pm{\gamma}}$,
where the $+$ or $-$ sign indicates whether the initial state contains a
$\overline{b}$\- or $b$-quark, $r_{B}$ is the ratio of the amplitude
magnitudes, and $\delta_{B}$ their $C\\!P$-conserving strong-phase difference.
This interference effect is typically measured in $B$-meson decays such as
${{B}^{\pm}}\rightarrow{D}h^{\pm}$, where $D$ is an admixture of the ${D}^{0}$
and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ flavour states, and
$h^{\pm}$ is either a charged kaon or pion. Figure 1 shows the leading-order
Feynman diagrams for the favoured and suppressed processes. Interference
effects, providing sensitivity to $\gamma$, only occur when the $D$ meson
decays to a final state, $f$, accessible to both ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons.

Figure 1: Leading-order Feynman diagrams for ${{{B}^{-}}}\rightarrow Dh^{-}$
decays with a (left) favoured ${b}\rightarrow{c}$ and (right) suppressed
${b}\rightarrow{u}$ quark transition.

Neglecting mixing in the neutral charm system, the decay rate for a
${{B}^{\pm}}\rightarrow{D}h^{\pm}$ decay is given by

$\Gamma({{B}^{\pm}}\rightarrow{D}h^{\pm})\propto|r_{D}e^{-i\delta_{D}}+r_{B}e^{i(\delta_{B}\pm{\gamma})}|^{2}\Rightarrow
r_{D}^{2}+r_{B}^{2}+2\kappa_{D}\kappa_{B}r_{D}r_{B}\cos(\delta_{B}+\delta_{D}\pm{\gamma}),$
(1)

where $r_{D}$ and $\delta_{D}$ are the magnitude ratio and strong-phase
difference between the ${{D}^{0}}\rightarrow f$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rightarrow f$ amplitudes. For $D$
decays to $C\\!P$ eigenstates, e.g. $D\rightarrow{{K}^{+}}{{K}^{-}}$, these
values are $r_{D}=1$ and $\delta_{D}=0$. The coherence factors of $B$ and $D$
decays, $\kappa_{B}$ and $\kappa_{D}$, are equal to unity for two-body decays,
and account for a dilution of the interference term due to incoherence (strong
phase variation) between contributing intermediate resonances in multibody
decays. The hadronic parameters, $r_{B}$, $\delta_{B}$, $r_{D}$, $\delta_{D}$,
are specific to each $B$ decay and subsequent $D$ decay, respectively.
However, the $C\\!P$-violating weak phase difference between ${{B}^{+}}$ and
${{B}^{-}}$ amplitudes, $\gamma$, is shared by all such decays.

Equation (1) has at least five unknown parameters, even more if the coherence
factors are not set to unity, hence they cannot be determined using a single
pair of ${B}^{\pm}$ decay rates. This is overcome by combining the results
from many different $D$-decay modes to overconstrain the parameters of the
$B$-meson decay, provided that the corresponding $r_{D}$, $\delta_{D}$ and
$\kappa_{D}$ parameters are constrained by other measurements. In past
combinations these parameters have been taken as external inputs using
dedicated charm-meson measurements. The large $B$-meson samples now constrain
$\gamma$ and $\delta_{B}$ so precisely that $\delta_{D}^{K\pi}$, the strong
phase difference between ${D}^{0}$ $\rightarrow$ ${K}^{-}$ ${\pi}^{+}$ and
${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ $\rightarrow$ ${K}^{-}$
${\pi}^{+}$ decays, can be measured with similar precision as $\gamma$ and
$\delta_{B}$, a factor of about two better than the previous world average
[11]. This improved precision on $\delta_{D}^{K\pi}$ can then be used to
improve knowledge of charm mixing as described below.

The mass eigenstates of the neutral charm mesons can be written as
$|D_{1,2}\rangle\equiv p|{{D}^{0}}\rangle\pm q|{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rangle$, where $p$ and $q$ are
complex parameters such that $|p|^{2}+|q|^{2}=1$. The $D_{1}$ ($D_{2}$) state
corresponds to the $+$ ($-$) sign and is approximately $C\\!P$ even (odd) in
the chosen convention. The mixing of charm flavour states can be described by
two dimensionless parameters, $x\equiv(m_{1}-m_{2})/\Gamma$ and
$y\equiv(\Gamma_{1}-\Gamma_{2})/2\Gamma$, where $m_{i}$ ($\Gamma_{i}$) is the
mass (width) of the appropriate $D$ mass state, and $\Gamma$ their average
decay width.222Natural units, with $c=\hbar=1$, are used throughout. Effects
of $C\\!P$ violation in ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ decays to a common final state,
$f$, can be seen in mixing if $|q/p|\neq 1$, or in the interference between
mixing and decay if $\phi\equiv\arg(q/p)\neq 0,\,\pi$.333The Wolfenstein
parametrisation and the convention that ${C\\!P}|{{D}^{0}}\rangle=|{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rangle$ is used. Study of the
charm mixing parameters is of high interest in its own right, because the
flavour-changing neutral currents responsible for the mixing transition do not
occur at tree-level in the SM, and thus can be significantly affected by
contributions from new heavy particles. The world averages for
$x=(4.09^{\,+0.48}_{\,-0.49})\times 10^{-3}$ and
$y=(6.15^{\,+0.56}_{\,-0.55})\times 10^{-3}$ [11] are dominated by LHCb
results.

The mixing parameters, $x$ and $y$, can be determined using the ratio of
wrong-sign (WS), ${D}^{0}$ $\rightarrow$ ${K}^{+}$ ${\pi}^{-}$, and right-sign
(RS), ${D}^{0}$ $\rightarrow$ ${K}^{-}$ ${\pi}^{+}$, time-dependent decay
rates. This ratio is

$\displaystyle R^{\pm}(t)\approx
R^{\pm}+\sqrt{R^{\pm}}y^{\prime\pm}\left(\frac{t}{\tau}\right)+\frac{(x^{\prime\pm})^{2}+(y^{\prime\pm})^{2}}{4}\left(\frac{t}{\tau}\right)^{2},$
(2)

up to second order in the mixing parameters, where $t$ is the decay time,
$\tau$ is the ${D}^{0}$ meson lifetime, and the $+$ ($-$) signs correspond to
the decay-rate ratio for a flavour-tagged ${D}^{0}$ (${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$) initial state.444 It should be
noted that there are multiple conventions in the literature for the strong
phase $\delta_{D}^{K\pi}$, depending on whether the discussion involves the
CKM angle $\gamma$ or charm mixing. The convention in which
$\delta_{D}\rightarrow\pi$ in the SU(3) limit is used, which is shifted by
$\pi$ with respect to the convention employed by the HFLAV Charm group. The
parameter $R^{\pm}=r_{D}^{2}(1\pm A_{D})$ is the ratio of suppressed-to-
favoured decay rates, modulated by the direct $C\\!P$ asymmetry, $A_{D}$,
between ${D}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ WS
decays. The parameters $x^{\prime\pm}\equiv-\left|q/p\right|^{\pm
1}\left[x\cos(\delta_{D}^{K\pi}\pm\phi)+y\sin(\delta_{D}^{K\pi}\pm\phi)\right]$
and $y^{\prime\pm}\equiv-\left|q/p\right|^{\pm
1}\left[y\cos(\delta_{D}^{K\pi}\pm\phi)-x\sin(\delta_{D}^{K\pi}\pm\phi)\right]$
encode the mixing. Since $\delta_{D}^{K\pi}$ is close to $\pi$ and $\phi$ is
almost zero [11], it follows that $R^{\pm}(t)$ is mostly sensitive to the
parameter $y$ through the term linear in decay time and mixing parameters, and
currently the precision on $y$ is limited by the precision with which
$\delta_{D}^{K\pi}$ is known. Consequently, a simultaneous combination using
both beauty and charm observables from LHCb is performed for the first time,
improving the precision on $y$ ($x$) by about 50% (2%).

A further motivation for the simultaneous combination of both beauty and charm
measurements is that non-negligible effects due to charm-meson mixing give
rise to additional terms in Eq. (1). Incorporating the effect of $D$-meson
mixing, up to first order in $x$ and $y$, means the decay rate of Eq. (1)
becomes [14]

$\displaystyle\Gamma({{B}^{\pm}}\rightarrow{D}h^{\pm})\propto$
$\displaystyle\phantom{+}r_{D}^{2}+r_{B}^{2}+2\kappa_{D}\kappa_{B}r_{D}r_{B}\cos(\delta_{B}+\delta_{D}\pm{\gamma})$
$\displaystyle-\alpha\left[(1+r_{B}^{2})\kappa_{D}r_{D}\cos(\delta_{D})+(1+r_{D}^{2})\kappa_{B}r_{B}\cos(\delta_{B}\pm{\gamma})\right]y$
$\displaystyle+\alpha\left[(1-r_{B}^{2})\kappa_{D}r_{D}\,\sin(\delta_{D})-(1-r_{D}^{2})\kappa_{B}r_{B}\sin(\delta_{B}\pm{\gamma})\right]x,$
(3)

where the $\alpha$ coefficient accounts for the non-uniform decay-time
acceptance of the LHCb detector. For cases where $r_{B}\gg x,\,y$, such as the
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ decay, the effect of $D$ mixing is
small. However, for decays like ${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$,
where $r_{B}\sim x,\,y$, the effect is significant [14]. Studies of this
combination, which includes both ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ modes, suggest that not accounting
for the effect of $D$-meson mixing results in a bias on $\gamma$ of
approximately $1.8^{\circ}$, and an ever larger bias for the hadronic
parameters, $r_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}}$ and
$\delta_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}}$, of the
${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ system. Thus an unbiased
determination of $\gamma$, $x$ and $y$ requires the simultaneous combination
produced in this article.

This article presents results for the weak phase $\gamma$ and charm mixing and
$C\\!P$ violation parameters $x$, $y$, $|q/p|$ and $\phi$, as well as for
several additional amplitude ratios and strong phases, using data collected at
the LHCb experiment during the first two runs of the LHC. The statistical
procedure is identical to that described in Ref. [15] and follows a
frequentist treatment which is described in detail in Ref. [16] and briefly
recapped in Sec. 3. The results have additionally been cross-checked using
Bayesian inference, which finds very similar values. The results presented
here supersede previous LHCb combinations [16, 15, 17, 18].

The full list of LHCb measurements that are used as inputs to the combination
is provided in Table 1. In the beauty sector this includes decay-rate ratios
and charge asymmetries of ${{B}^{\pm}}\rightarrow{D}h^{\pm}$,
${{B}^{\pm}}\rightarrow{{D}^{*}}h^{\pm}$,
${{B}^{\pm}}\rightarrow{D}{{K}^{*\pm}}$,
${{B}^{\pm}}\rightarrow{D}h^{\pm}{{\pi}^{+}}{{\pi}^{-}}$,
${{B}^{0}}\rightarrow{D}{{K}^{*0}}$,
${{B}^{0}}\rightarrow{{D}^{\mp}}{{\pi}^{\pm}}$,
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ and
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}$
decays, where ${{D}^{*}}$ is an admixture of ${{D}^{*0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{*0}}$ flavour states. In the charm
sector this includes time-dependent measurements of ${{D}^{0}}\rightarrow
h^{+}h^{-}$, ${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$,
${{D}^{0}}\rightarrow{{K}^{\pm}}{{\pi}^{\mp}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ decays.
There are seven new or updated measurements from beauty-meson decays since the
last combination, including LHCb Run 2 updates from the highly sensitive
${{B}^{\pm}}\rightarrow{D}h^{\pm}$ with
$D\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ [19] and $D\rightarrow
h^{+}h^{-}$ [20] decays. The eight inputs from LHCb charm analyses are
included in the combination for the first time.

Additional external constraints are summarised in Table 2; these are used
predominantly to provide auxiliary information on the hadronic parameters and
coherence factors in multibody $B$ and $D$ decays. In the case of
quasi-$C\\!P$-eigenstate decays, such as
$D\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, the coherence
factor is determined by the fraction of $C\\!P$-even content in the final-
state amplitude, $F^{+}=(\kappa_{D}+1)/2$. In the case of the
${{B}^{0}}\rightarrow{{D}^{\mp}}{{\pi}^{\pm}}$,
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ and
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}$
modes, the weak phases measured through the time-dependent $C\\!P$ asymmetry
are $({\gamma}+2\beta)$ and $({\gamma}-2\beta_{s})$, induced via interference
between $B^{0}_{(s)}$ mixing and decay. Therefore, in order to obtain
sensitivity to $\gamma$, external constraints from the world averages of
$\beta\equiv\arg[-V_{cd}V_{cb}^{*}/V_{td}V_{tb}^{*}]$ and
$\phi_{s}\approx-2\beta_{s}\equiv-2\arg[-V_{ts}V_{tb}^{*}/V_{cs}V_{cb}^{*}]$
[11] are included.

Table 1: Measurements used in the combination. Inputs from the charm system
appear in the lower part of the table. Those that are new, or that have
changed, since the previous combination [17] are highlighted in bold.
Measurements denoted by (*) include only a fraction of the Run 2 sample,
corresponding to data taken in 2015 and 2016. Where multiple references are
cited, measured values are taken from the most recent results, which include
information from the others.

${B}$ decay |  ${D}$ decay | Ref. | Dataset | Status since
---|---|---|---|---
|  |  |  | Ref. [17]
${{B}^{\pm}}\rightarrow{D}h^{\pm}$ | $D\rightarrow h^{+}h^{-}$ | [20] | Run 1&2 | Updated
${{B}^{\pm}}\rightarrow{D}h^{\pm}$ | $D\rightarrow h^{+}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ | [21] | Run 1 | As before
${{B}^{\pm}}\rightarrow{D}h^{\pm}$ | $D\rightarrow h^{+}h^{-}{{\pi}^{0}}$ | [22] | Run 1 | As before
${{B}^{\pm}}\rightarrow{D}h^{\pm}$ | $D\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ | [19] | Run 1&2 | Updated
${{B}^{\pm}}\rightarrow{D}h^{\pm}$ | $D\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{\pm}}{{\pi}^{\mp}}$ | [23] | Run 1&2 | Updated
${{B}^{\pm}}\rightarrow{{D}^{*}}h^{\pm}$ | $D\rightarrow h^{+}h^{-}$ | [20] | Run 1&2 | Updated
${{B}^{\pm}}\rightarrow{D}{{K}^{*\pm}}$ | $D\rightarrow h^{+}h^{-}$ | [24] | Run 1&2(*) | As before
${{B}^{\pm}}\rightarrow{D}{{K}^{*\pm}}$ | $D\rightarrow h^{+}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ | [24] | Run 1&2(*) | As before
${{B}^{\pm}}\rightarrow{D}h^{\pm}{{\pi}^{+}}{{\pi}^{-}}$ | $D\rightarrow h^{+}h^{-}$ | [25] | Run 1 | As before
${{B}^{0}}\rightarrow{D}{{K}^{*0}}$ | $D\rightarrow h^{+}h^{-}$ | [26] | Run 1&2(*) | Updated
${{B}^{0}}\rightarrow{D}{{K}^{*0}}$ | $D\rightarrow h^{+}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ | [26] | Run 1&2(*) | New
${{B}^{0}}\rightarrow{D}{{K}^{*0}}$ | $D\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ | [27] | Run 1 | As before
${{B}^{0}}\rightarrow{{D}^{\mp}}{{\pi}^{\pm}}$ | ${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ | [28] | Run 1 | As before
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ | ${{D}^{+}_{s}}\rightarrow h^{+}h^{-}{{\pi}^{+}}$ | [29] | Run 1 | As before
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}$ | ${{D}^{+}_{s}}\rightarrow h^{+}h^{-}{{\pi}^{+}}$ | [30] | Run 1&2 | New
${D}$ decay | Observable(s) | Ref. | Dataset | Status since
|  |  |  | Ref. [17]
${{D}^{0}}\rightarrow h^{+}h^{-}$ | $\Delta A_{{C\\!P}}$ | [31, 32, 33] | Run 1&2 | New
${{D}^{0}}\rightarrow h^{+}h^{-}$ | $y_{{C\\!P}}$ | [34] | Run 1 | New
${{D}^{0}}\rightarrow h^{+}h^{-}$ | $\Delta Y$ | [35, 36, 37, 38] | Run 1&2 | New
${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ (Single Tag) |  $R^{\pm}$, $(x^{\prime\pm})^{2}$, $y^{\prime\pm}$ | [39] | Run 1 | New
${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ (Double Tag) |  $R^{\pm}$, $(x^{\prime\pm})^{2}$, $y^{\prime\pm}$ | [40] | Run 1&2(*) | New
${{D}^{0}}\rightarrow{{K}^{\pm}}{{\pi}^{\mp}}{{\pi}^{+}}{{\pi}^{-}}$ | $(x^{2}+y^{2})/4$ | [41] | Run 1 | New
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ |  $x$, $y$ | [42] | Run 1 | New
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ |  $x_{{C\\!P}}$, $y_{{C\\!P}}$, $\Delta x$, $\Delta y$ | [43] | Run 1 | New
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ |  $x_{{C\\!P}}$, $y_{{C\\!P}}$, $\Delta x$, $\Delta y$ | [44] | Run 2 | New

Table 2: Auxiliary inputs used in the combination. Those highlighted in bold
have changed since the previous combination [17].

Decay | Parameters | Source | Ref. | Status since
---|---|---|---|---
|  |  |  | Ref. [17]
${{B}^{\pm}}\rightarrow{D}{{K}^{*\pm}}$ | $\kappa_{{{B}^{\pm}}}^{D{{K}^{*\pm}}}$ | LHCb | [24] | As before
${{B}^{0}}\rightarrow{D}{{K}^{*0}}$ | $\kappa_{{{B}^{0}}}^{D{{K}^{*0}}}$ | LHCb | [45] | As before
${{B}^{0}}\rightarrow{{D}^{\mp}}{{\pi}^{\pm}}$ | $\beta$ | HFLAV | [11] | Updated
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ $(\pi\pi)$ | $\phi_{{s}}$ | HFLAV | [11] | Updated
$D\rightarrow h^{+}h^{-}{{\pi}^{0}}$ |  $F^{+}_{\pi\pi{{\pi}^{0}}}$, $F^{+}_{K\pi{{\pi}^{0}}}$ | CLEO-c | [46] | As before
$D\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ | $F^{+}_{4\pi}$ | CLEO-c | [46] | As before
$D\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ |  $r_{{D}}^{K\pi{{\pi}^{0}}}$, $\delta_{{D}}^{K\pi{{\pi}^{0}}}$, $\kappa_{{D}}^{K\pi{{\pi}^{0}}}$ | CLEO-c+LHCb+BESIII | [47, 48, 49] | Updated
$D\rightarrow{{K}^{\pm}}{{\pi}^{\mp}}{{\pi}^{+}}{{\pi}^{-}}$ |  $r_{{D}}^{K3\pi}$, $\delta_{{D}}^{K3\pi}$, $\kappa_{{D}}^{K3\pi}$ | CLEO-c+LHCb+BESIII | [47, 48, 41, 49] | Updated
$D\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{\pm}}{{\pi}^{\mp}}$ |  $r_{{D}}^{{{K}^{0}_{\mathrm{S}}}K\pi}$, $\delta_{{D}}^{{{K}^{0}_{\mathrm{S}}}K\pi}$, $\kappa_{{D}}^{{{K}^{0}_{\mathrm{S}}}K\pi}$ | CLEO | [50] | As before
$D\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{\pm}}{{\pi}^{\mp}}$ | $r_{{D}}^{{{K}^{0}_{\mathrm{S}}}K\pi}$ | LHCb | [51] | As before

##  2 Assumptions

The mathematical formulae relating the input observables to the parameters of
interest, via Eq. (3), contain a few assumptions. These are detailed below and
their impact on the results has been checked to be negligible at the current
precision. In the future, as the precision on $\gamma$ approaches one degree,
many of them will need to be reassessed.

### Neutral kaon mixing

The extraction of $\gamma$ from decays where the final state of the $D$ meson
decay contains a neutral kaon is affected by $C\\!P$ violation in
${K}^{0}$–${\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{0}$ mixing and
decay and by regeneration [52]. For the
$D\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ final state, a relative shift
of approximately $\Delta\gamma/\gamma\approx\mathcal{O}(10^{-3})$ is expected;
this has been studied in detail in Ref. [52]. Furthermore, the result of the
relevant input analysis includes a small systematic uncertainty to account for
this [19], so these effects are not considered further in this combination.
The size of the effect in
$D\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{\pm}}{{\pi}^{\mp}}$ final states is
larger, $\Delta\gamma/\gamma\approx\mathcal{O}({\varepsilon_{K}}/r_{B})$,
where ${\varepsilon_{K}}=(2.10^{+0.27}_{-0.20})\times 10^{-3}$ quantifies
$C\\!P$ violation in neutral kaon mixing [12]. However, the impact on $\gamma$
is negligible at present because the sensitivity of the input measurement is
relatively low [23].

###  $C\\!P$ violation in $D$-meson decays

The effect of $C\\!P$ violation in the direct decay of
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ is not considered
because it is negligible in the SM for the Cabibbo-favoured (CF) and doubly
Cabibbo-suppressed (DCS) amplitudes contributing to that process. However, the
effect of $C\\!P$ violation in the direct decay of
${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ is allowed for in the charm part of
the combination, and is denoted by $A_{D}$. A non-zero value of $A_{D}$ would
cause a small shift in the charge asymmetries measured for
${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ final states in the beauty system,
which is not accounted for in this combination. The impact on the
determination of $\gamma$ is found to be smaller than $0.2^{\circ}$. The
difference between the size of direct $C\\!P$ violation in
${{D}^{0}}\rightarrow{{K}^{+}}{{K}^{-}}$ and
${{D}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays is included as an input in
the charm part of the fit [31]. In the beauty system, this value is used to
account for direct $C\\!P$ violation in ${{D}^{0}}\rightarrow h^{+}h^{-}$
decays for the most sensitive analyses, where the $D$ meson is produced in
${{B}^{\pm}}\rightarrow{D}h^{\pm}$ or ${{B}^{\pm}}\rightarrow{{D}^{*}}h^{\pm}$
decays, under the hypothesis of $U$-spin symmetry,
$A_{{C\\!P}}(KK)=-A_{{C\\!P}}(\pi\pi)=\Delta A_{{C\\!P}}/2$, where
$A_{{C\\!P}}(f)$ is the $C\\!P$ asymmetry of the $D$ meson decay to the final
state $f$. Any $U$-spin breaking effects are negligible given that ignoring
any direct $C\\!P$ violation in $D\rightarrow h^{+}h^{-}$ decays only has a
small impact, below $0.3^{\circ}$, on the determination of $\gamma$. Time-
dependent $C\\!P$ violation in charm mixing, which would add additional terms
to Eq. (3), is also neglected in the beauty system, since its impact on the
determination of $\gamma$ is smaller than $0.1^{\circ}$ [14].

### Strong phases in $D\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ decays

The input measurements containing
$D\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ final states [19, 27, 43, 44],
both in the charm and beauty systems, require external knowledge of the
strong-phase difference between the
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$
amplitudes across the phase space of the $D$ decay. These values are taken
from a combination of CLEO-c and BES-III measurements [53, 54, 55, 56] and
their uncertainties propagated to the uncertainties of the input measurements
listed in Table 1. In this combination, each set of input measurements is
treated as statistically independent and thus a small part of the already sub-
dominant systematic uncertainties of these measurements is being counted twice
in the $D\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ system i.e.
the appropriate correlation is not accounted for. This correlation is non-
trivial to compute owing to the different binning schemes employed by the
different input analyses. In any case, the effect on this combination is small
since the uncertainty on the strong phases accounts for approximately
$0.5^{\circ}$ of the uncertainty on $\gamma$, 40% of the uncertainty on $x$
and 1% of the uncertainty on $y$, and these parameters are nearly
uncorrelated. Studies suggest that incorporation of this correlation will
become important only with a three times larger data sample.

### Correlations of systematic uncertainties between input measurements

In addition to the effect of strong phases in
$D\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ decays, there are various other
potential systematic correlations that are not accounted for. Whilst the
individual input analyses provide both statistical and systematic covariance
matrices between the sets of observables they measure, there are in principle
sub-leading systematic correlations between input analyses which are not
accounted for. For example, systematic uncertainties originating from
production and detection asymmetries will be correlated for most time-
integrated measurements and those originating from knowledge of decay-time
acceptance and resolution will be correlated for time-dependent measurements.
The impact of ignoring these small correlations is a marginal underestimation
of the uncertainties (assuming the correlation is positive), but given that
the combination is still statistically dominated ($3.3^{\circ}$ out of
$3.6^{\circ}$) the effect is expected to be negligible.

##  3 Statistical treatment

The results are obtained using a frequentist treatment, with a likelihood
function built from the product of the probability density functions, $f_{i}$,
of experimental observables $\vec{A}_{i}$, defined as

$\mathcal{L}(\vec{\alpha})=\prod_{i}f_{i}(\vec{A}_{i}^{\rm
obs}|\vec{\alpha}).$ (4)

Here, $\vec{A}_{i}^{\rm obs}$ denotes the measured observables from analysis
$i$, and $\vec{\alpha}$ is the set of underlying physics parameters on which
they depend. The observables of each input are assumed to follow a multi-
dimensional Gaussian distribution

$f_{i}(\vec{A}_{i}^{\rm
obs}|\vec{\alpha})\propto\exp\left(-\frac{1}{2}(\vec{A}_{i}(\vec{\alpha})-\vec{A}_{i}^{\rm
obs})^{T}\,V_{i}^{-1}\,(\vec{A}_{i}(\vec{\alpha})-\vec{A}_{i}^{\rm
obs})\right)\,,$ (5)

where $V_{i}$ is the experimental covariance matrix, including both
statistical and systematic uncertainties and their correlations.

A $\chi^{2}$-function is defined as
$\chi^{2}(\vec{\alpha})=-2\ln\mathcal{L}(\vec{\alpha})$, with the best-fit
point given by the global minimum of the $\chi^{2}$ function,
$\chi^{2}(\vec{\alpha}_{\min})$. The confidence level (CL) for a parameter at
a given value, denoted $\alpha_{0}$, is determined in the following way.
First, for every fixed $\alpha_{0}$, a new minimum of $\alpha^{\prime}$ is
found, $\chi^{2}(\vec{\alpha}^{\,\prime}_{\min})$, and the deviation from the
global minimum,
$\Delta\chi^{2}=\chi^{2}(\vec{\alpha}^{\,\prime}_{\min})-\chi^{2}(\vec{\alpha}_{\min})$,
is computed. Second, an ensemble of pseudoexperiments, $\vec{A}^{\,\rm
MC}_{j}$, is generated according to the probability distribution of Eq. (5),
with parameters $\vec{\alpha}=\vec{\alpha}^{\,\prime}_{\min}$. Finally, for
each pseudoexperiment the $\chi^{2}$-function is minimised once with the
parameter of interest free to vary and once with it at a fixed value
$\alpha_{0}$, to obtain the difference, $(\Delta\chi^{2})^{\rm MC}$, from
$\vec{A}^{\rm\,MC}_{j}$, in the same way as $\Delta\chi^{2}$ was computed from
$\vec{A}^{\rm obs}_{i}$. The $p$-value, or $1-{\rm CL}$, is then defined as
the fraction of pseudoexperiments with $(\Delta\chi^{2})^{\rm
MC}>\Delta\chi^{2}$. This method is often referred to as the $\hat{\mu}$ or
Plugin method; see Ref. [57] for details. Its coverage is not guaranteed [57]
for the full parameter space, but can be evaluated at various points across
the phase space. The coverage of the intervals quoted in this combination has
been computed at several points across the phase space, including at the
global minimum, by generating large samples of pseudoexperiments and computing
the fraction which contains the generated value within a given confidence
level. The coverage of the quoted 68.3% interval for $\gamma$ is $(67.3\pm
1.5)\%$, for $x$ is $(68.2\pm 1.5)\%$, for $y$ is $(67.6\pm 1.5)\%$, for
$|q/p|$ is $(66.6\pm 1.5)\%$, and for $\phi$ is $(67.7\pm 1.5)\%$. Similar
coverage is seen for the $95.4\%$ intervals and no correction to the quoted
intervals is applied.

##  4 Results

The combination uses a total of 151 input observables to determine 52 free
parameters, and the goodness of fit is found to be $84\%$, evaluated using the
best-fit $\chi^{2}$ and cross-checked with pseudoexperiments. The resulting
confidence intervals for each parameter of interest, except for externally
constrained nuisance parameters, are provided in Table 3. The correlation
matrix of the parameters in Table 3 is given in Appendix A, Tables 7, 8 and 9.
The $p$-value (or $1-{\rm CL}$) distribution as a function of $\gamma$ is
shown in Fig. 2 for the total combination and for subsets in which the input
observables are split by the species of the initial $B$ meson. The
corresponding confidence intervals are provided in Table 4. Significant
differences between initial state $B$ mesons could be an indication of new
physics entering at tree-level, as the decay topologies for charged and
neutral initial states are different. Figure 2 shows a moderate tension, 2.2
standard deviations ($\sigma$), between the charged and neutral $B$ states.
The uncertainties in the ${B}^{0}$ and ${B}^{0}_{s}$ modes are considerably
larger than in the dominant ${{B}^{+}}$ modes. The sensitivity of the
${B}^{0}$ and ${B}^{0}_{s}$ modes is expected to improve by approximately a
factor of $2$ with the analysis of
${{B}^{0}}\rightarrow{D}{{K}^{+}}{{\pi}^{-}}$ with
$D\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ and
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ decays using the full
LHCb data sample. Table 5 presents the confidence intervals for $\gamma$ as
determined from inputs of time-dependent methods and time-integrated methods
only. Two-dimensional profile likelihood contours in the $(x,y)$ and
$(|q/p|,\phi)$ planes are shown in Fig. 3. The significant improvement, of a
factor of two, in the precision to $y$ demonstrates the advantage of this
combination over the current world average in the charm system.

Table 3: Confidence intervals and central values for each of the parameters of
interest. Entries marked with an asterisk show where the scan has hit a
physical boundary at the lower limit.

Quantity | Value | 68.3% CL | 95.4% CL
---|---|---|---
Uncertainty | Interval | Uncertainty | Interval
${\gamma}\,[^{\circ}]$ | $65.4$ | ${}^{+3.8}_{-4.2}$ | $[61.2,69.2]$ | ${}^{+7.5}_{-8.7}$ | $[56.7,72.9]$
$r_{{{B}^{\pm}}}^{D{{K}^{\pm}}}$ | $0.0984$ | ${}^{+0.0027}_{-0.0026}$ | $[0.0958,0.1011]$ | ${}^{+0.0056}_{-0.0052}$ | $[0.0932,0.1040]$
$\delta_{{{B}^{\pm}}}^{D{{K}^{\pm}}}\,[^{\circ}]$ | $127.6$ | ${}^{+4.0}_{-4.2}$ | $[123.4,131.6]$ | ${}^{+7.8}_{-9.2}$ | $[118.4,135.4]$
$r_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}}$ | $0.00480$ | ${}^{+0.00070}_{-0.00056}$ | $[0.00424,0.00550]$ | ${}^{+0.0017}_{-0.0011}$ | $[0.0037,0.0065]$
$\delta_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}}\,[^{\circ}]$ | $288$ | ${}^{+14}_{-15}$ | $[273,302]$ | ${}^{+26}_{-31}$ | $[257,314]$
$r_{{{B}^{\pm}}}^{{{D}^{*}}{{K}^{\pm}}}$ | $0.099$ | ${}^{+0.016}_{-0.019}$ | $[0.080,0.115]$ | ${}^{+0.030}_{-0.038}$ | $[0.061,0.129]$
$\delta_{{{B}^{\pm}}}^{{{D}^{*}}{{K}^{\pm}}}\,[^{\circ}]$ | $310$ | ${}^{+12}_{-23}$ | $[287,322]$ | ${}^{+20}_{-71}$ | $[239,330]$
$r_{{{B}^{\pm}}}^{{{D}^{*}}{{\pi}^{\pm}}}$ | $0.0095$ | ${}^{+0.0085}_{-0.0061}$ | $[0.0034,0.0180]$ | ${}^{+0.017}_{-0.0089}$ | $[0.0006,0.026]$
$\delta_{{{B}^{\pm}}}^{{{D}^{*}}{{\pi}^{\pm}}}\,[^{\circ}]$ | $139$ | ${}^{+22}_{-86}$ | $[53,161]$ | ${}^{+32}_{-129}$ | $[10,171]$
$r_{{{B}^{\pm}}}^{D{{K}^{*\pm}}}$ | $0.106$ | ${}^{+0.017}_{-0.019}$ | $[0.087,0.123]$ | ${}^{+0.031}_{-0.040}$ | $[0.066,0.137]$
$\delta_{{{B}^{\pm}}}^{D{{K}^{*\pm}}}\,[^{\circ}]$ | $35$ | ${}^{+20}_{-15}$ | $[20,55]$ | ${}^{+57}_{-28}$ | $[7,92]$
$r_{{{B}^{0}}}^{D{{K}^{*0}}}$ | $0.250$ | ${}^{+0.023}_{-0.024}$ | $[0.226,0.273]$ | ${}^{+0.044}_{-0.052}$ | $[0.198,0.294]$
$\delta_{{{B}^{0}}}^{D{{K}^{*0}}}\,[^{\circ}]$ | $197$ | ${}^{+10}_{-9.3}$ | $[187.7,207]$ | ${}^{+24}_{-18}$ | $[179,221]$
$r_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}}$ | $0.310$ | ${}^{+0.098}_{-0.092}$ | $[0.218,0.408]$ | ${}^{+0.20}_{-0.21}$ | $[0.10,0.51]$
$\delta_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}}\,[^{\circ}]$ | $356$ | ${}^{+19}_{-18}$ | $[338,375]$ | ${}^{+39}_{-39}$ | $[317,395]$
$r_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $0.460$ | ${}^{+0.081}_{-0.084}$ | $[0.376,0.541]$ | ${}^{+0.16}_{-0.17}$ | $[0.29,0.62]$
$\delta_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}\,[^{\circ}]$ | $345$ | ${}^{+13}_{-12}$ | $[333,358]$ | ${}^{+26}_{-25}$ | $[320,371]$
$r_{{{B}^{0}}}^{{{D}^{\mp}}{{\pi}^{\pm}}}$ | $0.030$ | ${}^{+0.014}_{-0.012}$ | $[0.018,0.044]$ | ${}^{+0.036}_{-0.028}$ | $[0.002,0.066]$
$\delta_{{{B}^{0}}}^{{{D}^{\mp}}{{\pi}^{\pm}}}\,[^{\circ}]$ | $30$ | ${}^{+26}_{-37}$ | $[-7,56]$ | ${}^{+45}_{-81}$ | $[-51,75]$
$r_{{{B}^{\pm}}}^{D{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $0.079$ | ${}^{+0.028}_{-0.034}$ | $[0.045,0.107]$ | ${}^{+0.050}_{-0.079}$ | $[0.000,0.129]^{*}$
$r_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $0.067$ | ${}^{+0.025}_{-0.029}$ | $[0.038,0.092]$ | ${}^{+0.040}_{-0.067}$ | $[0.000,0.107]^{*}$
$x\,[\%]$ | $0.400$ | ${}^{+0.052}_{-0.053}$ | $[0.347,0.452]$ | ${}^{+0.10}_{-0.11}$ | $[0.29,0.50]$
$y\,[\%]$ | $0.630$ | ${}^{+0.033}_{-0.030}$ | $[0.600,0.663]$ | ${}^{+0.069}_{-0.058}$ | $[0.572,0.699]$
$r_{D}^{K\pi}$ | $0.05867$ | ${}^{+0.00015}_{-0.00015}$ | $[0.05852,0.05882]$ | ${}^{+0.00031}_{-0.00030}$ | $[0.05837,0.05898]$
$\delta_{D}^{K\pi}\,[^{\circ}]$ | $190.0$ | ${}^{+4.2}_{-4.1}$ | $[185.9,194.2]$ | ${}^{+8.6}_{-8.3}$ | $[181.7,198.6]$
$r_{{D}}^{{{K}^{0}_{\mathrm{S}}}K\pi}$ | $0.6150$ | ${}^{+0.0056}_{-0.0054}$ | $[0.6096,0.6206]$ | ${}^{+0.011}_{-0.011}$ | $[0.604,0.626]$
$\delta_{{D}}^{{{K}^{0}_{\mathrm{S}}}K\pi}[^{\circ}]$ | $17$ | ${}^{+13}_{-14}$ | $[3,30]$ | ${}^{+26}_{-30}$ | $[-13,43]$
$\kappa_{{D}}^{{{K}^{0}_{\mathrm{S}}}K\pi}$ | $0.82$ | $\pm 0.10$ | $[0.72,0.92]$ | $\pm 0.20$ | $[0.62,1.02]$
$r_{{D}}^{K3\pi}$ | $0.05560$ | ${}^{+0.00059}_{-0.00058}$ | $[0.05502,0.05619]$ | ${}^{+0.0012}_{-0.0012}$ | $[0.0544,0.0568]$
$\delta_{{D}}^{K3\pi}[^{\circ}]$ | $154$ | ${}^{+15}_{-16}$ | $[138,169]$ | ${}^{+29}_{-33}$ | $[121,183]$
$\kappa_{{D}}^{K3\pi}$ | $0.480$ | $\pm 0.064$ | $[0.416,0.544]$ | $\pm 0.13$ | $[0.35,0.60]$
$|q/p|$ | $0.997$ | $\pm 0.016$ | $[0.981,1.013]$ | $\pm 0.033$ | $[0.964,1.030]$
$\phi\,[^{\circ}]$ | $-2.4$ | $\pm 1.2$ | $[-3.6,-1.2]$ | $\pm 2.5$ | $[-4.9,0.1]$
$\Delta a_{{C\\!P}}^{\text{dir}}$ | $-0.00152$ | $\pm 0.00029$ | $[-0.00181,-0.00123]$ | $\pm 0.00058$ | $[-0.00210,-0.00094]$

Figure 2: One dimensional $1-{\rm CL}$ profiles for $\gamma$ from the combination using inputs from ${B}^{0}_{s}$ (light blue), ${B}^{0}$ (orange), ${{B}^{+}}$ mesons (red) and all species together (dark blue). Table 4: Confidence intervals and best-fit values for $\gamma$ when splitting the combination inputs by initial $B$ meson species. Species | Value $[^{\circ}]$ | 68.3% CL | 95.4% CL
---|---|---|---
Uncertainty | Interval | Uncertainty | Interval
${{B}^{+}}$ | $61.7$ | ${}^{+4.4}_{-4.8}$ | $[56.9,66.1]$ | ${}^{+8.6}_{-9.5}$ | $[52.2,70.3]$
${B}^{0}$ | $82.0$ | ${}^{+8.1}_{-8.8}$ | $[73.2,90.1]$ | ${}^{+17}_{-18}$ | $[64,99]$
${B}^{0}_{s}$ | $79$ | ${}^{+21}_{-24}$ | $[55,100]$ | ${}^{+51}_{-47}$ | $[32,130]$
Table 5: Confidence intervals and best-fit values for $\gamma$ when splitting the combination inputs by time-dependent and time-integrated methods. Method | Value $[^{\circ}]$ | 68.3% CL | 95.4% CL
---|---|---|---
Uncertainty | Interval | Uncertainty | Interval
Time-dependent | $79$ | ${}^{+21}_{-24}$ | $[55,100]$ | ${}^{+51}_{-47}$ | $[32,130]$
Time-integrated | $64.9$ | ${}^{+3.9}_{-4.5}$ | $[60.4,68.8]$ | ${}^{+7.8}_{-9.6}$ | $[55.3,72.7]$

Figure 3: Two-dimensional profile likelihood contours for (left) the charm
mixing parameters $x$ and $y$, and (right) the $\phi$ and $|q/p|$ parameters.
The blue contours show the current charm world average from Ref. [11]; the
brown contours show the result of this combination. Contours are drawn out
from 1 ($68.3\%$) to 5 standard deviations.

Breakdowns of the contributing components in the combination are shown in
Figs. 4 and 5. These highlight the complementary nature of the input
measurements to constrain both $\gamma$ and the charm mixing parameters. In
Fig. 5 (top left) the dark orange band shows external constraints from CLEO-c
[58] and BES-III [59]. These are required to constrain $\delta_{D}^{K\pi}$
when obtaining the “All Charm Modes” contours, but are not used in the full
combination. In the top right and bottom plots the orange bands show the
constraints from ${{D}^{0}}\rightarrow h^{+}h^{-}$ modes, but these cannot
provide bands in $(x,y)$ or $(|q/p|,\phi)$ without other constraints [60].
Consequently, when these orange bands are produced in the top right plot
$(|q/p|,\phi,r_{D}^{K\pi},\delta_{D}^{K\pi})$ are fixed to their best fit
values from Table 3, while in the bottom plot
$(x,y,r_{D}^{K\pi},\delta_{D}^{K\pi})$ are fixed to their best fit values. In
the bottom figure the red contour is mostly hidden behind the blue; this is
because no significant additional sensitivity to $C\\!P$ violation in the
charm system is provided by the inclusion of the beauty observables in the
simultaneous fit.

Figure 4: Profile likelihood contours for the beauty decay parameters versus
$\gamma$, showing the breakdown of sensitivity amongst different sub-
combinations of modes. The contours indicate the 68.3% and 95.4% confidence
region.

Figure 5: Profile likelihood contours for the charm decay parameters, showing
the breakdown of sensitivity amongst different sub-combinations of modes. The
contours indicate the 68.3% and 95.4% confidence region.

The value of ${\gamma}=(65.4^{\,+3.8}_{\,-4.2})^{\circ}$ determined from this
combination is compatible with, but lower than that of the previous LHCb
combination ${\gamma}=(74^{\,+5.0}_{\,-5.8})^{\circ}$ [17]. This change is
driven by improved treatments of background sources in the major inputs
described in Refs. [19, 20]. An assessment of the compatibility between this
and the previous combination, which considers the full parameter space and the
correlation between the current set of inputs and the previous set of inputs,
finds they are compatible at the level of $2.1\sigma$. The new result is in
excellent agreement with the global CKM fit results [12, 13].

The charm mixing parameters, $x$ and $y$, are determined simultaneously with
$\gamma$ in this combination for the first time. The precision on $x$ is
driven by the recent measurement described in Ref. [44]. The result
$y=(0.630^{\,+0.033}_{\,-0.030})\%$ is more precise than the world average,
$y=(0.603^{\,+0.057}_{\,-0.056})\%$ [11], by approximately a factor of two,
driven entirely by the improved measurement of $\delta_{D}^{K\pi}$ from the
beauty system and the simultaneous averaging methodology employed in this
article. The correlation between $\delta_{D}^{K\pi}$ and
$\delta_{{{B}^{\pm}}}^{D{{K}^{\pm}}}$ is $-57\%$, highlighting
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ decays as the source of this
improvement.

The beauty part of the combination is cross-checked with an independent
framework using a Bayesian statistical treatment. A flat prior is used for
$\gamma$ and the relevant hadronic parameters and results in a value of
${\gamma}=(65.6^{\,+3.7}_{\,-3.8})^{\circ}$, in agreement with the default
frequentist results. Good agreement between the frequentist and Bayesian
interpretations is also seen for the other hadronic parameters. A second
cross-check using an independent fitting framework with frequentist
interpretation gives consistent results to better than $1\%$ precision.
Finally, the charm sector of the combination was validated by accurately
reproducing the HFLAV results [11].

The relative impact of systematic uncertainties on the input observables is
studied, and found to contribute approximately $1.2^{\circ}$ to the result for
$\gamma$, demonstrating that the uncertainty of this combination is still
dominated by the data sample size.

In previous combinations, the experimental input from
${{B}^{0}}\rightarrow{{D}^{\mp}}{{\pi}^{\pm}}$ decays was included with an
external theoretical prediction of
$r_{{{B}^{0}}}^{{{D}^{\mp}}{{\pi}^{\pm}}}=0.0182\pm 0.0038$ [28]. This
prediction assumes SU(3) symmetry, and was the only input from theory. This
external input is no longer used, and the combination gives an experimental
determination of
$r_{{{B}^{0}}}^{{{D}^{\mp}}{{\pi}^{\pm}}}=0.030^{\,+0.014}_{\,-0.012}$. This
is in agreement with the theory-based prediction and provides confidence that
the assumption of SU(3) symmetry is valid within the current precision. This
change has a negligible impact on the determination of other parameters.

##  5 Conclusion

A simultaneous combination of LHCb measurements sensitive to the CKM angle
$\gamma$ and charm mixing parameters, along with auxiliary information from
other experiments, is performed for the first time. This includes seven new
and updated inputs from $B$-meson decays and eight inputs from $D$-meson
decays. The result,

${\gamma}=(65.4^{\,+3.8}_{\,-4.2})^{\circ}\,,$

provides the most precise measurement from a single experiment. The charm
mixing parameters are found to be

$\displaystyle x$ $\displaystyle=(0.400^{\,+0.052}_{\,-0.053})\%\,,$
$\displaystyle y$ $\displaystyle=(0.630^{\,+0.033}_{\,-0.030})\%\,,$

which are the most precise determinations to date. In particular, the
uncertainty on $y$ is reduced by a factor of two by using the new procedure
described in this paper.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MICINN (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and NERSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from ARC and ARDC (Australia); AvH Foundation
(Germany); EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union);
A*MIDEX, ANR, Labex P2IO and OCEVU, and Région Auvergne-Rhône-Alpes (France);
Key Research Program of Frontier Sciences of CAS, CAS PIFI, CAS CCEPP,
Fundamental Research Funds for the Central Universities, and Sci. & Tech.
Program of Guangzhou (China); RFBR, RSF and Yandex LLC (Russia); GVA, XuntaGal
and GENCAT (Spain); the Leverhulme Trust, the Royal Society and UKRI (United
Kingdom).

## Appendices

##  Appendix A Correlation matrix

The global fit correlation matrix between each of the parameters presented in
Table 3 is provided in Tables 7, 8 and 9. A subset of this matrix, including
only the parameters of greatest interest, is given in Table 6. The correlation
coefficients for $\beta$ and $\phi_{s}$ are not included as they are almost
all smaller than $0.001$, an exception is $\rho(\gamma,\phi_{s})=-0.009$.

Table 6: Reduced correlation matrix for the parameters of greater interest.
Values smaller than $0.001$ are replaced with a - symbol.

| $\gamma$ | $A_{D}$ | $r_{D}^{K\pi}$ | $\delta_{D}^{K\pi}$ | $x$ | $y$ | $|q/p|$ | $\phi$
---|---|---|---|---|---|---|---|---
$\gamma$ | $1.000$ | - | $-0.003$ | $0.003$ | $-0.002$ | $0.003$ | - | -
$A_{D}$ |  | $1.000$ | $-0.016$ | $-0.083$ | $-0.003$ | $-0.083$ | $-0.316$ | $0.295$
$r_{D}^{K\pi}$ |  |  | $1.000$ | $0.295$ | $0.282$ | $-0.095$ | $-0.070$ | $0.015$
$\delta_{D}^{K\pi}$ |  |  |  | $1.000$ | $0.029$ | $0.891$ | $-0.020$ | $0.055$
$x$ |  |  |  |  | $1.000$ | $0.013$ | $-0.129$ | $0.083$
$y$ |  |  |  |  |  | $1.000$ | $-0.018$ | $0.040$
$|q/p|$ |  |  |  |  |  |  | $1.000$ | $0.554$
$\phi$ |  |  |  |  |  |  |  | $1.000$

Table 7: Correlation matrix of the fit result, part 1 of 3. Values smaller
than $0.001$ are replaced with a - symbol.

| $\gamma$ | $r_{{{B}^{\pm}}}^{D{{K}^{\pm}}}$ | $\delta_{{{B}^{\pm}}}^{D{{K}^{\pm}}}$ | $r_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}}$ | $\delta_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}}$ | $r_{{{B}^{\pm}}}^{{{D}^{*}}{{K}^{\pm}}}$ | $\delta_{{{B}^{\pm}}}^{{{D}^{*}}{{K}^{\pm}}}$ | $r_{{{B}^{\pm}}}^{{{D}^{*}}{{\pi}^{\pm}}}$ | $\delta_{{{B}^{\pm}}}^{{{D}^{*}}{{\pi}^{\pm}}}$ | $r_{{{B}^{\pm}}}^{D{{K}^{*\pm}}}$ | $\delta_{{{B}^{\pm}}}^{D{{K}^{*\pm}}}$ | $r_{{{B}^{0}}}^{D{{K}^{*0}}}$ | $\delta_{{{B}^{0}}}^{D{{K}^{*0}}}$ | $r_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\gamma$ | $1.000$ | $0.490$ | $0.613$ | $-0.051$ | $0.229$ | $0.138$ | $0.331$ | $0.116$ | $0.149$ | $-0.158$ | $-0.012$ | $0.206$ | $-0.111$ | $-0.003$
$r_{{{B}^{\pm}}}^{D{{K}^{\pm}}}$ |  | $1.000$ | $0.442$ | $-0.048$ | $0.113$ | $0.058$ | $0.174$ | $0.070$ | $0.100$ | $-0.079$ | $0.026$ | $0.098$ | $-0.047$ | $-0.001$
$\delta_{{{B}^{\pm}}}^{D{{K}^{\pm}}}$ |  |  | $1.000$ | $-0.055$ | $0.236$ | $0.062$ | $0.231$ | $0.102$ | $0.156$ | $-0.094$ | $0.054$ | $0.120$ | $-0.052$ | $-0.002$
$r_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}}$ |  |  |  | $1.000$ | $0.629$ | $-0.006$ | $-0.018$ | $-0.008$ | $-0.011$ | $0.006$ | $-0.003$ | $-0.009$ | $0.004$ | -
$\delta_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}}$ |  |  |  |  | $1.000$ | $0.025$ | $0.085$ | $0.034$ | $0.052$ | $-0.035$ | $0.011$ | $0.046$ | $-0.022$ | -
$r_{{{B}^{\pm}}}^{{{D}^{*}}{{K}^{\pm}}}$ |  |  |  |  |  | $1.000$ | $0.764$ | $-0.211$ | $-0.213$ | $-0.022$ | $-0.006$ | $0.029$ | $-0.016$ | -
$\delta_{{{B}^{\pm}}}^{{{D}^{*}}{{K}^{\pm}}}$ |  |  |  |  |  |  | $1.000$ | $-0.172$ | $-0.164$ | $-0.052$ | $0.001$ | $0.067$ | $-0.035$ | -
$r_{{{B}^{\pm}}}^{{{D}^{*}}{{\pi}^{\pm}}}$ |  |  |  |  |  |  |  | $1.000$ | $0.895$ | $-0.018$ | $0.004$ | $0.023$ | $-0.011$ | -
$\delta_{{{B}^{\pm}}}^{{{D}^{*}}{{\pi}^{\pm}}}$ |  |  |  |  |  |  |  |  | $1.000$ | $-0.023$ | $0.010$ | $0.029$ | $-0.013$ | -
$r_{{{B}^{\pm}}}^{D{{K}^{*\pm}}}$ |  |  |  |  |  |  |  |  |  | $1.000$ | $-0.323$ | $-0.033$ | $0.017$ | -
$\delta_{{{B}^{\pm}}}^{D{{K}^{*\pm}}}$ |  |  |  |  |  |  |  |  |  |  | $1.000$ | $-0.004$ | $0.006$ | -
$r_{{{B}^{0}}}^{D{{K}^{*0}}}$ |  |  |  |  |  |  |  |  |  |  |  | $1.000$ | $-0.139$ | -
$\delta_{{{B}^{0}}}^{D{{K}^{*0}}}$ |  |  |  |  |  |  |  |  |  |  |  |  | $1.000$ | -
$r_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}}$ |  |  |  |  |  |  |  |  |  |  |  |  |  | $1.000$

Table 8: Correlation matrix of the fit result, part 2 of 3. Values smaller
than $0.001$ are replaced with a - symbol.

| $\delta_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}}$ | $r_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $\delta_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $r_{{{B}^{0}}}^{{{D}^{\mp}}{{\pi}^{\pm}}}$ | $\delta_{{{B}^{0}}}^{{{D}^{\mp}}{{\pi}^{\pm}}}$ | $r_{{{B}^{\pm}}}^{D{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $r_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $r_{D}^{K\pi}$ | $\delta_{D}^{K\pi}$ | $x$ | $y$ | $|q/p|$ | $\phi$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\gamma$ | $-0.019$ | $-0.027$ | $-0.144$ | $-0.065$ | $-0.168$ | $0.024$ | $-0.181$ | $-0.003$ | $0.003$ | $-0.002$ | $0.003$ | - | -
$r_{{{B}^{\pm}}}^{D{{K}^{\pm}}}$ | $-0.009$ | $-0.013$ | $-0.071$ | $-0.032$ | $-0.082$ | $-0.006$ | $0.038$ | $-0.079$ | $-0.236$ | $0.011$ | $-0.206$ | $0.003$ | $-0.010$
$\delta_{{{B}^{\pm}}}^{D{{K}^{\pm}}}$ | $-0.012$ | $-0.017$ | $-0.088$ | $-0.040$ | $-0.103$ | $-0.028$ | $0.196$ | $-0.161$ | $-0.573$ | $0.020$ | $-0.510$ | $0.007$ | $-0.029$
$r_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}}$ | - | $0.001$ | $0.007$ | $0.003$ | $0.009$ | - | $-0.003$ | $0.084$ | $0.022$ | $-0.005$ | $-0.004$ | $-0.005$ | $-0.004$
$\delta_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}}$ | $-0.004$ | $-0.006$ | $-0.033$ | $-0.015$ | $-0.038$ | $-0.007$ | $0.048$ | $0.048$ | $-0.169$ | - | $-0.180$ | $-0.004$ | $-0.016$
$r_{{{B}^{\pm}}}^{{{D}^{*}}{{K}^{\pm}}}$ | $-0.003$ | $-0.004$ | $-0.020$ | $-0.009$ | $-0.023$ | $0.006$ | $-0.046$ | $0.013$ | $0.040$ | $-0.002$ | $0.035$ | - | $0.002$
$\delta_{{{B}^{\pm}}}^{{{D}^{*}}{{K}^{\pm}}}$ | $-0.006$ | $-0.009$ | $-0.048$ | $-0.022$ | $-0.055$ | $0.004$ | $-0.034$ | $-0.012$ | $-0.048$ | - | $-0.044$ | - | $-0.003$
$r_{{{B}^{\pm}}}^{{{D}^{*}}{{\pi}^{\pm}}}$ | $-0.002$ | $-0.003$ | $-0.017$ | $-0.008$ | $-0.019$ | $-0.001$ | $0.008$ | $-0.024$ | $-0.053$ | $0.002$ | $-0.045$ | $0.001$ | $-0.002$
$\delta_{{{B}^{\pm}}}^{{{D}^{*}}{{\pi}^{\pm}}}$ | $-0.003$ | $-0.004$ | $-0.021$ | $-0.010$ | $-0.025$ | $-0.005$ | $0.033$ | $-0.039$ | $-0.112$ | $0.004$ | $-0.097$ | $0.002$ | $-0.005$
$r_{{{B}^{\pm}}}^{D{{K}^{*\pm}}}$ | $0.003$ | $0.004$ | $0.023$ | $0.010$ | $0.026$ | $-0.004$ | $0.032$ | $-0.003$ | -0.006 | $-0.002$ | $-0.005$ | - | -
$\delta_{{{B}^{\pm}}}^{D{{K}^{*\pm}}}$ | - | - | $0.002$ | - | $0.002$ | $-0.008$ | $0.057$ | $-0.027$ | $-0.102$ | $0.011$ | $-0.091$ | - | $-0.004$
$r_{{{B}^{0}}}^{D{{K}^{*0}}}$ | $-0.004$ | $-0.006$ | $-0.030$ | $-0.013$ | $-0.034$ | $0.006$ | $-0.043$ | $0.002$ | $0.011$ | $-0.002$ | $0.010$ | - | -
$\delta_{{{B}^{0}}}^{D{{K}^{*0}}}$ | $0.002$ | $0.003$ | $0.016$ | $0.007$ | $0.019$ | $-0.005$ | $0.035$ | $-0.007$ | $-0.028$ | $0.002$ | $-0.025$ | - | $-0.001$
$r_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}}$ | $-0.034$ | - | - | - | - | - | - | - | - | - | - | - | -

Table 9: Correlation matrix of the fit result, part 3 of 3. Values smaller
than $0.001$ are replaced with a - symbol.

| $\delta_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}}$ | $r_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $\delta_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $r_{{{B}^{0}}}^{{{D}^{\mp}}{{\pi}^{\pm}}}$ | $\delta_{{{B}^{0}}}^{{{D}^{\mp}}{{\pi}^{\pm}}}$ | $r_{{{B}^{\pm}}}^{D{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $r_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ | $r_{D}^{K\pi}$ | $\delta_{D}^{K\pi}$ | $x$ | $y$ | $|q/p|$ | $\phi$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\delta_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}}$ | $1.000$ | - | $0.003$ | $0.001$ | $0.003$ | - | $0.003$ | - | - | - | - | - | -
$r_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ |  | $1.000$ | $0.006$ | $0.002$ | $0.005$ | - | $0.005$ | - | - | - | - | - | -
$\delta_{{{B}^{0}_{s}}}^{{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ |  |  | $1.000$ | $0.009$ | $0.024$ | -0.003 | $0.026$ | - | - | - | - | - | -
$r_{{{B}^{0}}}^{{{D}^{\mp}}{{\pi}^{\pm}}}$ |  |  |  | $1.000$ | $0.485$ | $-0.002$ | $0.012$ | - | - | - | - | - | -
$\delta_{{{B}^{0}}}^{{{D}^{\mp}}{{\pi}^{\pm}}}$ |  |  |  |  | $1.000$ | $-0.004$ | $0.030$ | - | - | - | - | - | -
$r_{{{B}^{\pm}}}^{D{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ |  |  |  |  |  | $1.000$ | $-0.044$ | $0.020$ | $0.074$ | $-0.003$ | $0.066$ | - | $0.004$
$r_{{{B}^{\pm}}}^{D{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}}$ |  |  |  |  |  |  | $1.000$ | $-0.148$ | $-0.531$ | $0.025$ | $-0.473$ | 0.005 | $-0.026$
$r_{D}^{K\pi}$ |  |  |  |  |  |  |  | $1.000$ | $0.295$ | $0.282$ | $-0.095$ | -0.070 | $0.015$
$\delta_{D}^{K\pi}$ |  |  |  |  |  |  |  |  | $1.000$ | $0.029$ | $0.892$ | $-0.020$ | $0.055$
$x$ |  |  |  |  |  |  |  |  |  | $1.000$ | $0.013$ | $-0.129$ | $0.083$
$y$ |  |  |  |  |  |  |  |  |  |  | $1.000$ | $-0.018$ | $0.040$
$|q/p|$ |  |  |  |  |  |  |  |  |  |  |  | $1.000$ | $0.554$
$\phi$ |  |  |  |  |  |  |  |  |  |  |  |  | $1.000$

##  Appendix B Contribution of each input measurement to the global $\chi^{2}$

The contribution of each input measurement to the global $\chi^{2}$ is shown
in Table 10.

Table 10: Contributions to the total $\chi^{2}$ and the number of observables of each input measurement. | Measurement | $\chi^{2}$ | No. of obs.
---|---|---|---
Beauty sector | $B^{\pm}\rightarrow Dh^{\pm},D\rightarrow h^{\pm}h^{\prime\mp}$ | $2.71$ | 8
$B^{\pm}\rightarrow Dh^{\pm},D\rightarrow h^{\pm}\pi^{\mp}\pi^{+}\pi^{-}$ | $7.36$ | 8
$B^{\pm}\rightarrow Dh^{\pm},D\rightarrow h^{\pm}h^{\prime\mp}\pi^{0}$ | $7.14$ | 11
$B^{\pm}\rightarrow Dh^{\pm},D\rightarrow K_{S}^{0}h^{+}h^{-}$ | $4.67$ | 6
$B^{\pm}\rightarrow Dh^{\pm},D\rightarrow K_{S}^{0}K^{\pm}\pi^{\mp}$ | $7.57$ | 7
$B^{\pm}\rightarrow D^{*}h^{\pm},D\rightarrow h^{\pm}h^{\prime\mp}$ | $7.31$ | 16
$B^{\pm}\rightarrow DK^{*\pm},D\rightarrow h^{\pm}h^{\prime\mp}(\pi^{+}\pi^{-})$ | $3.71$ | 12
$B^{0}\rightarrow DK^{*0},D\rightarrow h^{\pm}h^{\prime\mp}(\pi^{+}\pi^{-})$ | $9.45$ | 12
$B^{0}\rightarrow DK^{*0},D\rightarrow K_{S}^{0}h^{+}h^{-}$ | $3.26$ | 4
$B^{\pm}\rightarrow Dh^{\pm}\pi^{+}\pi^{-},D\rightarrow h^{\pm}h^{\prime\mp}$ | $1.34$ | 11
$B^{0}_{s}\rightarrow D^{\mp}_{s}K^{\pm}$ | $5.71$ | 5
$B^{0}_{s}\rightarrow D^{\mp}_{s}K^{\pm}\pi^{+}\pi^{-}$ | $2.88$ | 5
$B^{0}\rightarrow D^{\mp}\pi^{\pm}$ | $0.00$ | 2
Charm sector | $D\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\;2011$ | $5.38$ | 2
$D\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\;\mathrm{Run\;1}$ | $0.77$ | 4
$D\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\;\mathrm{Run\;2}$ | $1.37$ | 4
$D\rightarrow K^{\pm}\pi^{\mp}\;\mathrm{Run\;1}$ | $1.29$ | 6
$D\rightarrow h^{+}h^{-}\;\Delta A_{CP}$ | $0.00$ | 2
$D\rightarrow K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}$ | $3.59$ | 1
$D\rightarrow h^{+}h^{-}\;y_{CP}$ | $0.40$ | 1
$D\rightarrow h^{+}h^{-}\;\Delta Y$ | $0.15$ | 1
$D\rightarrow K^{\pm}\pi^{\mp}\;\mathrm{Run\;2}$ | $2.23$ | 6
External constraints | $D\rightarrow K^{\pm}\pi^{\mp}\pi^{0},\;D\rightarrow K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}$ | $0.79$ | 6
$D\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ | $0.03$ | 1
$D\rightarrow h^{+}h^{-}\pi^{0}$ | $0.01$ | 2
$D\rightarrow K_{S}^{0}K^{\pm}\pi^{\mp}\;\mathrm{WS}$ | $0.60$ | 1
$D\rightarrow K_{S}^{0}K^{\pm}\pi^{\mp}$ | $3.79$ | 3
$B^{\pm}\rightarrow DK^{*\pm}$ | $0.02$ | 1
$B^{0}\rightarrow DK^{*0}$ | $0.01$ | 1
$\phi_{s}$ | $0.00$ | 1
$\beta$ | $0.00$ | 1
| Total | 83.53 | 151

##  Appendix C Pull distribution of each input observable

The pull of each input observable with respect to the global best fit point is
shown in Figs. 6–8. The pull is defined as $(A_{\rm exp}-A_{\rm
fit})/\sigma(A_{\rm exp})$, where $A_{\rm exp}$ and $A_{\rm fit}$ are the
experimental input value and value at the best-fit point, respectively, and
$\sigma(A_{\rm exp})$ is the experimental uncertainty.

Figure 6: Pulls of the input observables, part 1 of 3.

Figure 7: Pulls of the input observables, part 2 of 3.

Figure 8: Pulls of the input observables, part 3 of 3.

##  Appendix D Additional figures

Figure 9 shows the $p$-value distribution as a function of $\gamma$ for the
global fit. A summary of LHCb $\gamma$ combination results as a function of
time is given in Fig. 10.

Figure 9: One dimensional $1-{\rm CL}$ profile for $\gamma$ from all inputs
used in the combination. Figure 10: Evolution of the LHCb combination result
for $\gamma$, with the central values and $1\sigma$ uncertainties in black.
This result is the 2021 data point, the value and uncertainty are highlighted
by the dashed blue line and band, respectively.

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LHCb collaboration

R. Aaij32, A.S.W. Abdelmotteleb56, C. Abellán Beteta50, F. Abudinén56, T.
Ackernley60, B. Adeva46, M. Adinolfi54, H. Afsharnia9, C. Agapopoulou13, C.A.
Aidala87, S. Aiola25, Z. Ajaltouni9, S. Akar65, J. Albrecht15, F. Alessio48,
M. Alexander59, A. Alfonso Albero45, Z. Aliouche62, G. Alkhazov38, P. Alvarez
Cartelle55, S. Amato2, J.L. Amey54, Y. Amhis11, L. An48, L. Anderlini22, A.
Andreianov38, M. Andreotti21, F. Archilli17, A. Artamonov44, M. Artuso68, K.
Arzymatov42, E. Aslanides10, M. Atzeni50, B. Audurier12, S. Bachmann17, M.
Bachmayer49, J.J. Back56, P. Baladron Rodriguez46, V. Balagura12, W.
Baldini21, J. Baptista Leite1, M. Barbetti22,g, R.J. Barlow62, S. Barsuk11, W.
Barter61, M. Bartolini24,h, F. Baryshnikov83, J.M. Basels14, S. Bashir34, G.
Bassi29, B. Batsukh68, A. Battig15, A. Bay49, A. Beck56, M. Becker15, F.
Bedeschi29, I. Bediaga1, A. Beiter68, V. Belavin42, S. Belin27, V. Bellee50,
K. Belous44, I. Belov40, I. Belyaev41, G. Bencivenni23, E. Ben-Haim13, A.
Berezhnoy40, R. Bernet50, D. Berninghoff17, H.C. Bernstein68, C. Bertella48,
A. Bertolin28, C. Betancourt50, F. Betti48, Ia. Bezshyiko50, S. Bhasin54, J.
Bhom35, L. Bian73, M.S. Bieker15, S. Bifani53, P. Billoir13, M. Birch61,
F.C.R. Bishop55, A. Bitadze62, A. Bizzeti22,k, M. Bjørn63, M.P. Blago48, T.
Blake56, F. Blanc49, S. Blusk68, D. Bobulska59, J.A. Boelhauve15, O. Boente
Garcia46, T. Boettcher65, A. Boldyrev82, A. Bondar43, N. Bondar38,48, S.
Borghi62, M. Borisyak42, M. Borsato17, J.T. Borsuk35, S.A. Bouchiba49, T.J.V.
Bowcock60, A. Boyer48, C. Bozzi21, M.J. Bradley61, S. Braun66, A. Brea
Rodriguez46, J. Brodzicka35, A. Brossa Gonzalo56, D. Brundu27, A. Buonaura50,
L. Buonincontri28, A.T. Burke62, C. Burr48, A. Bursche72, A. Butkevich39, J.S.
Butter32, J. Buytaert48, W. Byczynski48, S. Cadeddu27, H. Cai73, R.
Calabrese21,f, L. Calefice15,13, L. Calero Diaz23, S. Cali23, R. Calladine53,
M. Calvi26,j, M. Calvo Gomez85, P. Camargo Magalhaes54, P. Campana23, A.F.
Campoverde Quezada6, S. Capelli26,j, L. Capriotti20,d, A. Carbone20,d, G.
Carboni31, R. Cardinale24,h, A. Cardini27, I. Carli4, P. Carniti26,j, L.
Carus14, K. Carvalho Akiba32, A. Casais Vidal46, G. Casse60, M. Cattaneo48, G.
Cavallero48, S. Celani49, J. Cerasoli10, D. Cervenkov63, A.J. Chadwick60, M.G.
Chapman54, M. Charles13, Ph. Charpentier48, G. Chatzikonstantinidis53, C.A.
Chavez Barajas60, M. Chefdeville8, C. Chen3, S. Chen4, A. Chernov35, V.
Chobanova46, S. Cholak49, M. Chrzaszcz35, A. Chubykin38, V. Chulikov38, P.
Ciambrone23, M.F. Cicala56, X. Cid Vidal46, G. Ciezarek48, P.E.L. Clarke58, M.
Clemencic48, H.V. Cliff55, J. Closier48, J.L. Cobbledick62, V. Coco48, J.A.B.
Coelho11, J. Cogan10, E. Cogneras9, L. Cojocariu37, P. Collins48, T.
Colombo48, L. Congedo19,c, A. Contu27, N. Cooke53, G. Coombs59, I. Corredoira
46, G. Corti48, C.M. Costa Sobral56, B. Couturier48, D.C. Craik64, J.
Crkovská67, M. Cruz Torres1, R. Currie58, C.L. Da Silva67, S. Dadabaev83, L.
Dai71, E. Dall’Occo15, J. Dalseno46, C. D’Ambrosio48, A. Danilina41, P.
d’Argent48, J.E. Davies62, A. Davis62, O. De Aguiar Francisco62, K. De
Bruyn79, S. De Capua62, M. De Cian49, J.M. De Miranda1, L. De Paula2, M. De
Serio19,c, D. De Simone50, P. De Simone23, F. De Vellis15, J.A. de Vries80,
C.T. Dean67, F. Debernardis19,c, D. Decamp8, V. Dedu10, L. Del Buono13, B.
Delaney55, H.-P. Dembinski15, A. Dendek34, V. Denysenko50, D. Derkach82, O.
Deschamps9, F. Desse11, F. Dettori27,e, B. Dey77, A. Di Cicco23, P. Di
Nezza23, S. Didenko83, L. Dieste Maronas46, H. Dijkstra48, V. Dobishuk52, C.
Dong3, A.M. Donohoe18, F. Dordei27, A.C. dos Reis1, L. Douglas59, A.
Dovbnya51, A.G. Downes8, M.W. Dudek35, L. Dufour48, V. Duk78, P. Durante48,
J.M. Durham67, D. Dutta62, A. Dziurda35, A. Dzyuba38, S. Easo57, U. Egede69,
V. Egorychev41, S. Eidelman43,u,†, S. Eisenhardt58, S. Ek-In49, L.
Eklund59,86, S. Ely68, A. Ene37, E. Epple67, S. Escher14, J. Eschle50, S.
Esen13, T. Evans48, A. Falabella20, J. Fan3, Y. Fan6, B. Fang73, S. Farry60,
D. Fazzini26,j, M. Féo48, A. Fernandez Prieto46, A.D. Fernez66, F.
Ferrari20,d, L. Ferreira Lopes49, F. Ferreira Rodrigues2, S. Ferreres Sole32,
M. Ferrillo50, M. Ferro-Luzzi48, S. Filippov39, R.A. Fini19, M. Fiorini21,f,
M. Firlej34, K.M. Fischer63, D.S. Fitzgerald87, C. Fitzpatrick62, T.
Fiutowski34, A. Fkiaras48, F. Fleuret12, M. Fontana13, F. Fontanelli24,h, R.
Forty48, D. Foulds-Holt55, V. Franco Lima60, M. Franco Sevilla66, M. Frank48,
E. Franzoso21, G. Frau17, C. Frei48, D.A. Friday59, J. Fu6, Q. Fuehring15, E.
Gabriel32, G. Galati19,c, A. Gallas Torreira46, D. Galli20,d, S.
Gambetta58,48, Y. Gan3, M. Gandelman2, P. Gandini25, Y. Gao5, M. Garau27, L.M.
Garcia Martin56, P. Garcia Moreno45, J. García Pardiñas26,j, B. Garcia
Plana46, F.A. Garcia Rosales12, L. Garrido45, C. Gaspar48, R.E. Geertsema32,
D. Gerick17, L.L. Gerken15, E. Gersabeck62, M. Gersabeck62, T. Gershon56, D.
Gerstel10, L. Giambastiani28, V. Gibson55, H.K. Giemza36, A.L. Gilman63, M.
Giovannetti23,p, A. Gioventù46, P. Gironella Gironell45, L. Giubega37, C.
Giugliano21,f,48, K. Gizdov58, E.L. Gkougkousis48, V.V. Gligorov13, C.
Göbel70, E. Golobardes85, D. Golubkov41, A. Golutvin61,83, A. Gomes1,a, S.
Gomez Fernandez45, F. Goncalves Abrantes63, M. Goncerz35, G. Gong3, P.
Gorbounov41, I.V. Gorelov40, C. Gotti26, E. Govorkova48, J.P. Grabowski17, T.
Grammatico13, L.A. Granado Cardoso48, E. Graugés45, E. Graverini49, G.
Graziani22, A. Grecu37, L.M. Greeven32, N.A. Grieser4, L. Grillo62, S.
Gromov83, B.R. Gruberg Cazon63, C. Gu3, M. Guarise21, M. Guittiere11, P. A.
Günther17, E. Gushchin39, A. Guth14, Y. Guz44, T. Gys48, T. Hadavizadeh69, G.
Haefeli49, C. Haen48, J. Haimberger48, T. Halewood-leagas60, P.M. Hamilton66,
J.P. Hammerich60, Q. Han7, X. Han17, T.H. Hancock63, E.B. Hansen62, S.
Hansmann-Menzemer17, N. Harnew63, T. Harrison60, C. Hasse48, M. Hatch48, J.
He6,b, M. Hecker61, K. Heijhoff32, K. Heinicke15, A.M. Hennequin48, K.
Hennessy60, L. Henry48, J. Heuel14, A. Hicheur2, D. Hill49, M. Hilton62, S.E.
Hollitt15, R. Hou7, Y. Hou8, J. Hu17, J. Hu72, W. Hu7, X. Hu3, W. Huang6, X.
Huang73, W. Hulsbergen32, R.J. Hunter56, M. Hushchyn82, D. Hutchcroft60, D.
Hynds32, P. Ibis15, M. Idzik34, D. Ilin38, P. Ilten65, A. Inglessi38, A.
Ishteev83, K. Ivshin38, R. Jacobsson48, H. Jage14, S. Jakobsen48, E. Jans32,
B.K. Jashal47, A. Jawahery66, V. Jevtic15, F. Jiang3, M. John63, D. Johnson48,
C.R. Jones55, T.P. Jones56, B. Jost48, N. Jurik48, S.H. Kalavan Kadavath34, S.
Kandybei51, Y. Kang3, M. Karacson48, M. Karpov82, F. Keizer48, D.M. Keller68,
M. Kenzie56, T. Ketel33, B. Khanji15, A. Kharisova84, S. Kholodenko44, T.
Kirn14, V.S. Kirsebom49, O. Kitouni64, S. Klaver32, N. Kleijne29, K.
Klimaszewski36, M.R. Kmiec36, S. Koliiev52, A. Kondybayeva83, A.
Konoplyannikov41, P. Kopciewicz34, R. Kopecna17, P. Koppenburg32, M.
Korolev40, I. Kostiuk32,52, O. Kot52, S. Kotriakhova21,38, P. Kravchenko38, L.
Kravchuk39, R.D. Krawczyk48, M. Kreps56, F. Kress61, S. Kretzschmar14, P.
Krokovny43,u, W. Krupa34, W. Krzemien36, M. Kucharczyk35, V. Kudryavtsev43,u,
H.S. Kuindersma32,33, G.J. Kunde67, T. Kvaratskheliya41, D. Lacarrere48, G.
Lafferty62, A. Lai27, A. Lampis27, D. Lancierini50, J.J. Lane62, R. Lane54, G.
Lanfranchi23, C. Langenbruch14, J. Langer15, O. Lantwin83, T. Latham56, F.
Lazzari29,q, R. Le Gac10, S.H. Lee87, R. Lefèvre9, A. Leflat40, S. Legotin83,
O. Leroy10, T. Lesiak35, B. Leverington17, H. Li72, P. Li17, S. Li7, Y. Li4,
Y. Li4, Z. Li68, X. Liang68, T. Lin61, R. Lindner48, V. Lisovskyi15, R.
Litvinov27, G. Liu72, H. Liu6, Q. Liu6, S. Liu4, A. Lobo Salvia45, A. Loi27,
J. Lomba Castro46, I. Longstaff59, J.H. Lopes2, S. Lopez Solino46, G.H.
Lovell55, Y. Lu4, C. Lucarelli22,g, D. Lucchesi28,l, S. Luchuk39, M. Lucio
Martinez32, V. Lukashenko32,52, Y. Luo3, A. Lupato62, E. Luppi21,f, O.
Lupton56, A. Lusiani29,m, X. Lyu6, L. Ma4, R. Ma6, S. Maccolini20,d, F.
Machefert11, F. Maciuc37, V. Macko49, P. Mackowiak15, S. Maddrell-Mander54, O.
Madejczyk34, L.R. Madhan Mohan54, O. Maev38, A. Maevskiy82, D. Maisuzenko38,
M.W. Majewski34, J.J. Malczewski35, S. Malde63, B. Malecki48, A. Malinin81, T.
Maltsev43,u, H. Malygina17, G. Manca27,e, G. Mancinelli10, D. Manuzzi20,d, D.
Marangotto25,i, J. Maratas9,s, J.F. Marchand8, U. Marconi20, S. Mariani22,g,
C. Marin Benito48, M. Marinangeli49, J. Marks17, A.M. Marshall54, P.J.
Marshall60, G. Martelli78, G. Martellotti30, L. Martinazzoli48,j, M.
Martinelli26,j, D. Martinez Santos46, F. Martinez Vidal47, A. Massafferri1, M.
Materok14, R. Matev48, A. Mathad50, V. Matiunin41, C. Matteuzzi26, K.R.
Mattioli87, A. Mauri32, E. Maurice12, J. Mauricio45, M. Mazurek48, M.
McCann61, L. Mcconnell18, T.H. Mcgrath62, N.T. Mchugh59, A. McNab62, R.
McNulty18, J.V. Mead60, B. Meadows65, G. Meier15, N. Meinert76, D.
Melnychuk36, S. Meloni26,j, M. Merk32,80, A. Merli25,i, L. Meyer Garcia2, M.
Mikhasenko48, D.A. Milanes74, E. Millard56, M. Milovanovic48, M.-N. Minard8,
A. Minotti26,j, L. Minzoni21,f, S.E. Mitchell58, B. Mitreska62, D.S. Mitzel15,
A. Mödden 15, R.A. Mohammed63, R.D. Moise61, S. Mokhnenko82, T. Mombächer46,
I.A. Monroy74, S. Monteil9, M. Morandin28, G. Morello23, M.J. Morello29,m, J.
Moron34, A.B. Morris75, A.G. Morris56, R. Mountain68, H. Mu3, F. Muheim58,48,
M. Mulder48, D. Müller48, K. Müller50, C.H. Murphy63, D. Murray62, P.
Muzzetto27,48, P. Naik54, T. Nakada49, R. Nandakumar57, T. Nanut49, I.
Nasteva2, M. Needham58, I. Neri21, N. Neri25,i, S. Neubert75, N. Neufeld48, R.
Newcombe61, E.M. Niel11, S. Nieswand14, N. Nikitin40, N.S. Nolte64, C.
Normand8, C. Nunez87, A. Oblakowska-Mucha34, V. Obraztsov44, T. Oeser14, D.P.
O’Hanlon54, S. Okamura21, R. Oldeman27,e, F. Oliva58, M.E. Olivares68, C.J.G.
Onderwater79, R.H. O’Neil58, J.M. Otalora Goicochea2, T. Ovsiannikova41, P.
Owen50, A. Oyanguren47, K.O. Padeken75, B. Pagare56, P.R. Pais48, T. Pajero63,
A. Palano19, M. Palutan23, Y. Pan62, G. Panshin84, A. Papanestis57, M.
Pappagallo19,c, L.L. Pappalardo21,f, C. Pappenheimer65, W. Parker66, C.
Parkes62, B. Passalacqua21, G. Passaleva22, A. Pastore19, M. Patel61, C.
Patrignani20,d, C.J. Pawley80, A. Pearce48, A. Pellegrino32, M. Pepe
Altarelli48, S. Perazzini20, D. Pereima41, A. Pereiro Castro46, P. Perret9, M.
Petric59,48, K. Petridis54, A. Petrolini24,h, A. Petrov81, S. Petrucci58, M.
Petruzzo25, T.T.H. Pham68, A. Philippov42, L. Pica29,m, M. Piccini78, B.
Pietrzyk8, G. Pietrzyk49, M. Pili63, D. Pinci30, F. Pisani48, M.
Pizzichemi26,48,j, Resmi P.K10, V. Placinta37, J. Plews53, M. Plo Casasus46,
F. Polci13, M. Poli Lener23, M. Poliakova68, A. Poluektov10, N. Polukhina83,t,
I. Polyakov68, E. Polycarpo2, S. Ponce48, D. Popov6,48, S. Popov42, S.
Poslavskii44, K. Prasanth35, L. Promberger48, C. Prouve46, V. Pugatch52, V.
Puill11, H. Pullen63, G. Punzi29,n, H. Qi3, W. Qian6, J. Qin6, N. Qin3, R.
Quagliani49, B. Quintana8, N.V. Raab18, R.I. Rabadan Trejo6, B. Rachwal34,
J.H. Rademacker54, M. Rama29, M. Ramos Pernas56, M.S. Rangel2, F.
Ratnikov42,82, G. Raven33, M. Reboud8, F. Redi49, F. Reiss62, C. Remon
Alepuz47, Z. Ren3, V. Renaudin63, R. Ribatti29, S. Ricciardi57, K. Rinnert60,
P. Robbe11, G. Robertson58, A.B. Rodrigues49, E. Rodrigues60, J.A. Rodriguez
Lopez74, E.R.R. Rodriguez Rodriguez46, A. Rollings63, P. Roloff48, V.
Romanovskiy44, M. Romero Lamas46, A. Romero Vidal46, J.D. Roth87, M.
Rotondo23, M.S. Rudolph68, T. Ruf48, R.A. Ruiz Fernandez46, J. Ruiz Vidal47,
A. Ryzhikov82, J. Ryzka34, J.J. Saborido Silva46, N. Sagidova38, N. Sahoo56,
B. Saitta27,e, M. Salomoni48, C. Sanchez Gras32, R. Santacesaria30, C.
Santamarina Rios46, M. Santimaria23, E. Santovetti31,p, D. Saranin83, G.
Sarpis14, M. Sarpis75, A. Sarti30, C. Satriano30,o, A. Satta31, M. Saur15, D.
Savrina41,40, H. Sazak9, L.G. Scantlebury Smead63, A. Scarabotto13, S.
Schael14, S. Scherl60, M. Schiller59, H. Schindler48, M. Schmelling16, B.
Schmidt48, S. Schmitt14, O. Schneider49, A. Schopper48, M. Schubiger32, S.
Schulte49, M.H. Schune11, R. Schwemmer48, B. Sciascia23,48, S. Sellam46, A.
Semennikov41, M. Senghi Soares33, A. Sergi24,h, N. Serra50, L. Sestini28, A.
Seuthe15, Y. Shang5, D.M. Shangase87, M. Shapkin44, I. Shchemerov83, L.
Shchutska49, T. Shears60, L. Shekhtman43,u, Z. Shen5, V. Shevchenko81, E.B.
Shields26,j, Y. Shimizu11, E. Shmanin83, J.D. Shupperd68, B.G. Siddi21, R.
Silva Coutinho50, G. Simi28, S. Simone19,c, N. Skidmore62, T. Skwarnicki68,
M.W. Slater53, I. Slazyk21,f, J.C. Smallwood63, J.G. Smeaton55, A. Smetkina41,
E. Smith50, M. Smith61, A. Snoch32, M. Soares20, L. Soares Lavra9, M.D.
Sokoloff65, F.J.P. Soler59, A. Solovev38, I. Solovyev38, F.L. Souza De
Almeida2, B. Souza De Paula2, B. Spaan15, E. Spadaro Norella25,i, P.
Spradlin59, F. Stagni48, M. Stahl65, S. Stahl48, S. Stanislaus63, O.
Steinkamp50,83, O. Stenyakin44, H. Stevens15, S. Stone68, M. Straticiuc37, D.
Strekalina83, F. Suljik63, J. Sun27, L. Sun73, Y. Sun66, P. Svihra62, P.N.
Swallow53, K. Swientek34, A. Szabelski36, T. Szumlak34, M. Szymanski48, S.
Taneja62, A.R. Tanner54, M.D. Tat63, A. Terentev83, F. Teubert48, E. Thomas48,
D.J.D. Thompson53, K.A. Thomson60, V. Tisserand9, S. T’Jampens8, M. Tobin4, L.
Tomassetti21,f, X. Tong5, D. Torres Machado1, D.Y. Tou13, E. Trifonova83, C.
Trippl49, G. Tuci6, A. Tully49, N. Tuning32,48, A. Ukleja36, D.J. Unverzagt17,
E. Ursov83, A. Usachov32, A. Ustyuzhanin42,82, U. Uwer17, A. Vagner84, V.
Vagnoni20, A. Valassi48, G. Valenti20, N. Valls Canudas85, M. van Beuzekom32,
M. Van Dijk49, H. Van Hecke67, E. van Herwijnen83, C.B. Van Hulse18, M. van
Veghel79, R. Vazquez Gomez45, P. Vazquez Regueiro46, C. Vázquez Sierra48, S.
Vecchi21, J.J. Velthuis54, M. Veltri22,r, A. Venkateswaran68, M. Veronesi32,
M. Vesterinen56, D. Vieira65, M. Vieites Diaz49, H. Viemann76, X. Vilasis-
Cardona85, E. Vilella Figueras60, A. Villa20, P. Vincent13, F.C. Volle11, D.
Vom Bruch10, A. Vorobyev38, V. Vorobyev43,u, N. Voropaev38, K. Vos80, R.
Waldi17, J. Walsh29, C. Wang17, J. Wang5, J. Wang4, J. Wang3, J. Wang73, M.
Wang3, R. Wang54, Y. Wang7, Z. Wang50, Z. Wang3, Z. Wang6, J.A. Ward56, N.K.
Watson53, S.G. Weber13, D. Websdale61, C. Weisser64, B.D.C. Westhenry54, D.J.
White62, M. Whitehead54, A.R. Wiederhold56, D. Wiedner15, G. Wilkinson63, M.
Wilkinson68, I. Williams55, M. Williams64, M.R.J. Williams58, F.F. Wilson57,
W. Wislicki36, M. Witek35, L. Witola17, G. Wormser11, S.A. Wotton55, H. Wu68,
K. Wyllie48, Z. Xiang6, D. Xiao7, Y. Xie7, A. Xu5, J. Xu6, L. Xu3, M. Xu7, Q.
Xu6, Z. Xu5, Z. Xu6, D. Yang3, S. Yang6, Y. Yang6, Z. Yang5, Z. Yang66, Y.
Yao68, L.E. Yeomans60, H. Yin7, J. Yu71, X. Yuan68, O. Yushchenko44, E.
Zaffaroni49, M. Zavertyaev16,t, M. Zdybal35, O. Zenaiev48, M. Zeng3, D.
Zhang7, L. Zhang3, S. Zhang71, S. Zhang5, Y. Zhang5, Y. Zhang63, A.
Zharkova83, A. Zhelezov17, Y. Zheng6, T. Zhou5, X. Zhou6, Y. Zhou6, V.
Zhovkovska11, X. Zhu3, X. Zhu7, Z. Zhu6, V. Zhukov14,40, J.B. Zonneveld58, Q.
Zou4, S. Zucchelli20,d, D. Zuliani28, G. Zunica62.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4Institute Of High Energy Physics (IHEP), Beijing, China
5School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
6University of Chinese Academy of Sciences, Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy, France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
12Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut
Polytechnique de Paris, Palaiseau, France
13LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
14I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
15Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
16Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
17Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
18School of Physics, University College Dublin, Dublin, Ireland
19INFN Sezione di Bari, Bari, Italy
20INFN Sezione di Bologna, Bologna, Italy
21INFN Sezione di Ferrara, Ferrara, Italy
22INFN Sezione di Firenze, Firenze, Italy
23INFN Laboratori Nazionali di Frascati, Frascati, Italy
24INFN Sezione di Genova, Genova, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Milano-Bicocca, Milano, Italy
27INFN Sezione di Cagliari, Monserrato, Italy
28Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
29INFN Sezione di Pisa, Pisa, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31INFN Sezione di Roma Tor Vergata, Roma, Italy
32Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
33Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
36National Center for Nuclear Research (NCBJ), Warsaw, Poland
37Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
38Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
39Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
40Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
41Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia
42Yandex School of Data Analysis, Moscow, Russia
43Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
44Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
45ICCUB, Universitat de Barcelona, Barcelona, Spain
46Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
47Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
48European Organization for Nuclear Research (CERN), Geneva, Switzerland
49Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
50Physik-Institut, Universität Zürich, Zürich, Switzerland
51NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
52Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
53University of Birmingham, Birmingham, United Kingdom
54H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
55Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
56Department of Physics, University of Warwick, Coventry, United Kingdom
57STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
58School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
59School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
60Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
61Imperial College London, London, United Kingdom
62Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
63Department of Physics, University of Oxford, Oxford, United Kingdom
64Massachusetts Institute of Technology, Cambridge, MA, United States
65University of Cincinnati, Cincinnati, OH, United States
66University of Maryland, College Park, MD, United States
67Los Alamos National Laboratory (LANL), Los Alamos, United States
68Syracuse University, Syracuse, NY, United States
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 56
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71Physics and Micro Electronic College, Hunan University, Changsha City,
China, associated to 7
72Guangdong Provincial Key Laboratory of Nuclear Science, Guangdong-Hong Kong
Joint Laboratory of Quantum Matter, Institute of Quantum Matter, South China
Normal University, Guangzhou, China, associated to 3
73School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
74Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 13
75Universität Bonn - Helmholtz-Institut für Strahlen und Kernphysik, Bonn,
Germany, associated to 17
76Institut für Physik, Universität Rostock, Rostock, Germany, associated to 17
77Eotvos Lorand University, Budapest, Hungary, associated to 48
78INFN Sezione di Perugia, Perugia, Italy, associated to 21
79Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 32
80Universiteit Maastricht, Maastricht, Netherlands, associated to 32
81National Research Centre Kurchatov Institute, Moscow, Russia, associated to
41
82National Research University Higher School of Economics, Moscow, Russia,
associated to 42
83National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 41
84National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
41
85DS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain, associated to 45
86Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden,
associated to 59
87University of Michigan, Ann Arbor, United States, associated to 68

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bHangzhou Institute for Advanced Study, UCAS, Hangzhou, China
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Genova, Genova, Italy
iUniversità degli Studi di Milano, Milano, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Modena e Reggio Emilia, Modena, Italy
lUniversità di Padova, Padova, Italy
mScuola Normale Superiore, Pisa, Italy
nUniversità di Pisa, Pisa, Italy
oUniversità della Basilicata, Potenza, Italy
pUniversità di Roma Tor Vergata, Roma, Italy
qUniversità di Siena, Siena, Italy
rUniversità di Urbino, Urbino, Italy
sMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
tP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
uNovosibirsk State University, Novosibirsk, Russia
†Deceased