#  12th Workshop on the CKM Unitarity Triangle
Santiago de Compostela, 18-22 September 2023
Summary of Working Group 4: Mixing and mixing-related $CP$ violation in the B
system: $\Delta m$, $\Delta\Gamma$, $\phi_{s}$, $\phi_{1}/\alpha$,
$\phi_{2}/\beta$, $\phi_{3}/\gamma$

Agnieszka Dziurda  Institute of Nuclear Physics Polish Academy of Science,
Poland  Felix Erben  CERN, Theoretical Physics Department, Geneva, Switzerland
Marc Quentin Führing  TU Dortmund, Germany  Thibaud Humair  DESY, Germany
Thomas Latham  University of Warwick, Coventry, United Kingdom  Eleftheria
Malami  University of Siegen, Germany  Pascal Reeck  Karlsruhe Institute of
Technology  Vladyslav Shtabovenko  University of Siegen, Germany  Melissa Cruz
Torres  National Autonomous University of Honduras, Honduras  Yuma Uematsu
High Energy Accelerator Research Organization (KEK), Tsukuba, Japan  K. Keri
Vos  Gravitational Waves and Fundamental Physics (GWFP), Maastricht
University, Duboisdomein 30, NL-6229 GT Maastricht, the Netherlands

###### Abstract

This summary reviews contributions to the CKM 2023 workshop in Working Group
4: mixing and mixing-related $CP$ violation in B system. The theoretical and
experimental progress is discussed.

###### Contents

  1. 1 Introduction
  2. 2 Standard Model predictions of meson-mixing parameters
  3. 3 Update on SU(3)-breaking ratios and bag parameters for $B_{(s)}$ mesons
  4. 4 Mixing angles
    1. 4.1 Determinations of the $\phi_{s}$ angle
    2. 4.2 $\sin 2\beta$ determination at LHCb
    3. 4.3  $\beta$ and related measurements at Belle II
    4. 4.4 Time-dependent measurements of the CKM $\gamma$ angle
    5. 4.5 New physics in $B_{q}-\bar{B_{q}}$ in connection with the CKM $\gamma$ angle.
    6. 4.6 Determination of $\gamma$ from $B_{(s)}\to hh$ decays
  5. 5 Determination of the $\Delta\Gamma$ width difference
    1. 5.1 New results from LHCb
    2. 5.2 NNLO QCD corrections
      1. 5.2.1 Calculation overview
      2. 5.2.2 Results
  6. 6 Conclusion

##  1 Introduction

The study of neutral $B$-meson oscillations provides many insights into quark
flavour dynamics. The oscillation frequencies can be calculated precisely
within the Standard Model (SM), and any experimentally measured deviation
could hint at contributions from New Physics (NP). In addition, the study of
mixing-induced $CP$ violation gives access to angles of the $B^{0}_{s}$ and
$B^{0}_{d}$ Unitary Triangles (UT) which, in combination with independent
measurement, provide a stringent test of the flavour structure of the SM.

These proceedings make a summary of contributions from Working Group 4 at the
CKM workshop in 2023. In particular, updates on SU(3)-breaking ratio and bag
parameters, improvements in determination of $\Delta\Gamma$, and the Cabbibo-
Kobayashi-Maskawa (CKM) angle $\phi_{3}/\gamma$ as well as measurements of
$\phi_{s}$ and $\sin 2\beta$ parameters are discussed.

##  2 Standard Model predictions of meson-mixing parameters

In the SM, neutral mesons mixing arises from a mismatch between the mass and
flavour eigenstates describing the meson-antimeson system. In the
$B^{0}_{q}-\bar{B}^{0}_{q}$ system, where $q=d,s$, the mixing process is
described by a box-diagram with internal up, charm and top quarks as shown in
Fig. 1.

Figure 1: SM diagrams for the $B_{q}^{0}-\bar{B}_{q}^{0}$ mixing

The contribution from internal on-shell and off-shell particles are denoted as
$\Gamma^{q}_{12}$ and $M_{12}^{q}$ parameters. These parameters are related to
the mass ($\Delta M_{q}$) and width ($\Delta\Gamma_{q}$) difference as
follows:

$\Delta M_{q}\approx 2|M_{12}^{q}|,\quad\Delta\Gamma_{q}\approx
2|\Gamma^{q}_{12}|\cos{\phi^{q}_{12}},\quad
a^{q}_{sl}\approx\Bigg{|}\frac{\Gamma^{q}_{12}}{M_{12}^{q}}\Bigg{|}\sin{\phi^{q}_{12}}$

with $\phi^{q}_{12}=\arg(-M_{12}^{q}/\Gamma^{q}_{12})$ and the parameter
$a^{q}_{sl}$ being the semi-leptonic asymmetry. The
$B^{0}_{d,s}-\bar{B}^{0}_{d,s}$ mixing is hence described by the three
physical quantities $\Delta M_{q}$, $\Delta\Gamma^{q}_{12}$ $a^{q}_{sl}$, that
are equally well accessible to experimental measurements and theoretical
calculations. These observables are important for obtaining a better handle on
understanding of the SM and searching for NP effects.

The decay-width difference $\Delta\Gamma_{d,s}$ would deviate from its
predicted value if New Physics enters via light particles with masses below
the electroweak scale. On the other hand, heavy (multi-TeV) degrees of freedom
would induce a shift in the measured value of $\Delta m_{d,s}$. The most up-
to-date experimental measurements for these two quantities yield [1]

$\displaystyle\Delta m^{\rm exp}_{d}$ $\displaystyle=(0.5065\pm
0.0019)\;\text{ps${}^{-1}$},$ (2.1) $\displaystyle\Delta\Gamma_{d}^{\rm
exp}/\Gamma_{d}^{\rm exp}$ $\displaystyle=0.001\pm 0.010$ (2.2)
$\displaystyle\Delta m^{\rm exp}_{s}$ $\displaystyle=(17.765\pm
0.006)\;\text{ps${}^{-1}$},$ (2.3) $\displaystyle\Delta\Gamma^{\rm exp}_{s}$
$\displaystyle=(0.083\pm 0.005)\;\text{ps}^{-1}.$ (2.4)

The calculation of neutral meson mixing parameters can be addressed in the
framework of an effective theory obtained by integrating out heavy degrees of
freedom above the $W$ mass scale. The resulting effective Hamiltonian consists
of Wilson coefficients (calculable perturbatively) multiplying nonperturbative
matrix elements that can be obtained from the lattice. In particular, this
applies to the dimension-6 matrix elements describing $\Delta B=2$ transitions
that enter theory predictions for $\Delta\Gamma_{d,s}$. The transition to the
bag parameters is done by normalising those matrix elements by their vacuum
saturation values.

The bag parameter $\hat{B}^{(1)}_{B_{q}}$ related to the matrix element
$O_{1}=\bar{b}^{i}_{L}\gamma^{\mu}q^{i}_{L}\bar{b}^{j}_{L}\gamma^{\mu}q^{j}_{L}$
is also necessary for the theory prediction of $\Delta m_{d,s}$ that can be
written as

$\Delta
m_{q}=\left|{V_{tb}V_{tq}^{*}}\right|^{2}\,\mathcal{K}\,M_{B_{q}}\,f^{2}_{B_{q}}\hat{B}^{(1)}_{B_{q}}\,,$
(2.5)

where $f_{B_{q}}$ is the nonperturbative decay constant, while $\mathcal{K}$
can be computed in perturbation theory.

##  3 Update on SU(3)-breaking ratios and bag parameters for $B_{(s)}$ mesons

Lattice QCD calculations of $B$-meson mixing parameters are available from
ETMC [2], Fermilab/MILC [3], RBC/UKQCD [4] and HPQCD [5]. These computations
are summarised in the FLAG 21 review [6] as well as a recent review on
$B$-meson physics on the lattice [7]. There are some small tensions between
lattice results, with HPQCD confirming the experimental average of oscillation
frequencies $\Delta m_{d,s}$ and Fermilab/MILC confirming it only at the
$2\sigma$ level. The situation is summarised as part of sum-rules calculations
of the same quantities [8, 9]. Generally, all available theory predictions of
$\Delta m_{d,s}$ are one or two orders of magnitude less precise than the
experimental measurements. A current calculation that can help to shed light
into these tensions is a joint effort by the RBC/UKQCD and the JLQCD
collaborations [10], aiming to compute $B_{d,s}$ mixing parameters from chiral
domain-wall fermions using lattice QCD on a large set of lattice ensembles.

Using lattice QCD, the dominant short-distance contribution to
$B_{d,s}$-mixing can be computed nonperturbatively. This $\Delta B=2$
transition is typically parameterised via an operator-product expansion

$\displaystyle\langle\bar{B}^{0}_{q}|\mathcal{H}^{\Delta
B=2}_{\mathrm{eff}}|B^{0}_{q}\rangle=\sum_{i=1}^{5}C_{i}\langle\bar{B}^{0}_{q}|\mathcal{O}_{i}|B^{0}_{q}\rangle,$
(3.1)

where $|B^{0}_{q}\rangle$ is a state interpolating a $B^{0}_{d}$ meson ($q=d$)
or a $B^{0}_{s}$ meson ($q=s$), $\mathcal{H}^{\Delta B=2}_{\mathrm{eff}}$ is
the effective Hamiltonian, $\mathcal{O}_{i}$ are the 5 parity-even,
dimension-6 $\Delta B=2$ operators and $C_{i}$ are the corresponding Wilson
coefficients. By computing lattice-QCD two-point functions, masses $M_{B_{q}}$
and decay constants $f_{B_{q}}$ can be extracted via

$\displaystyle\langle
B_{q}(t)B_{q}^{\dagger}(0)\rangle_{L,a,m_{l},m_{h}}\Rightarrow
M_{B_{q}}(L,a,m_{l},m_{h}),f_{B_{q}}(L,a,m_{l},m_{h})\,,$ (3.2)

all at a given volume $L$, lattice spacing $a$, simulated light-quark mass
$m_{l}$ and heavy-quark mass $m_{h}$. From three-point functions

$\displaystyle\langle B_{q}(\Delta
T)\mathcal{O}_{i}(t)B_{q}^{\dagger}(0)\rangle_{L,a,m_{l},m_{h}}\Rightarrow\langle\bar{B}^{0}_{q}|\mathcal{O}_{i}|B^{0}_{q}\rangle(L,a,m_{l},m_{h})\,,$
(3.3)

one can extract the desired matrix elements, or alternatively bag parameters

$\displaystyle\mathcal{B}_{B_{q}}^{[i]}=\frac{\langle\bar{B}^{0}_{q}|\mathcal{O}_{i}|B^{0}_{q}\rangle}{\langle\bar{B}^{0}_{q}|\mathcal{O}_{i}|B^{0}_{q}\rangle_{\mathrm{VSA}}}\,,$
(3.4)

which are the ratios of the matrix elements by their vacuum saturation
approximation. These extracted quantities on the lattice can then be combined
and by taking the relevant limits, e.g.

$\displaystyle\Delta m_{q}=|V_{td}V_{tq}^{*}|^{2}\kappa\lim_{a\to
0}\lim_{L\to\infty}\lim_{m_{l}\to m_{l}^{P}}\lim_{m_{h}\to
m_{h}^{P}}(M_{B_{q}}f_{B_{q}}^{2}\mathcal{B}^{[1]}_{B_{q}})(L,a,m_{l},m_{h})\,,$
(3.5)

where $\kappa$ is some known perturbative factor and the CKM matrix elements
$V_{tq}$ can be taken from experiment to predict $\Delta m_{q}$ or the other
way round.

It is currently unfeasible to simulate both at the physical light-quark mass
$m_{l}^{P}$ and physical heavy-quark mass $m_{h}^{P}=m_{b}$. In the project
currenlty pursued by the RBC/UKQCD and the JLQCD collaborations [10], 15
different ensembles at 6 lattice spacings between $0.044\mathrm{fm}$ and
$0.11\mathrm{fm}$ are employed, two of which are at the physical pion mass.
The heavy-quarks are simulated fully relativistically, using the domain-wall
fermion action also employed for the light quarks. Thanks to the ensembles
with the finest lattice spacings, heavy-qaurk masses almost reaching $m_{b}$
can be simulated.

The bare bag parameters from Eq. (3.4) or the bare matrix elements
$\langle\bar{B}^{0}_{q}|\mathcal{O}_{i}|B^{0}_{q}\rangle$ obtained from
lattice QCD need to be renormalised and matched to a continuum scheme to be
compared with experiment. This project uses a nonperturbative renomalisation
in the so-called RI-SMOM scheme [11]. These renormalisation constants cancel
in certain ratios such as

$\displaystyle\xi=\frac{f_{B_{s}}\sqrt{\mathcal{B}^{[1]}_{B_{s}}}}{f_{B_{d}}\sqrt{\mathcal{B}^{[1]}_{B_{d}}}}\,.$
(3.6)

Figure 2 shows preliminary results of the RBC/UKQCD lattice QCD project both
on $\xi$ as well as renormalised bag parameters $\mathcal{B}^{[1]}_{B_{d}}$ in
the RI-SMOM scheme for all ensembles and heavy-quark masses employed in this
work. In particular, the very fine JLQCD ensembles lead to an increased reach
in the simulated heavy-quark masses towards the physical $b$-quark mass, the
extrapolation of which was one of the dominating sources of uncertainty in the
chiral-continuum fit of the RBC/UKQCD ensembles only [4].

Figure 2: Top panel: The $SU(3)$-breaking ratio $\xi$. Bottom panel:
Renormalised bag parameters $\mathcal{B}^{[1]}_{B_{d}}$ in the RI-SMOM scheme,
not yet matched to a continuum scheme. In both panels, multiple heavy-quark
masses are shown per ensemble plotted via the inverse heavy-strange meson mass
$1/m_{sh}$. Ensemble names indicate the lattice spacing in GeV and pion mass
in MeV - e.g. ensemble “a1.7m14” has an inverse lattice spacing of $a^{-1}\sim
1.7$GeV and a pion mass of $m_{\pi}\sim 140$MeV. The vertical green lines
indicate the inverse physical $B_{s}$ mass (left) and $D_{s}$ mass (right).
The RBC/UKQCD ensembles already used in [4] have square symbols and the set of
JLQCD ensembles has triangle symbols.

##  4 Mixing angles

Neutral $B$-meson decays with flavour-changing neutral current are excellent
to probe New Physics contributions, potentially occurring via loop (penguin)
processes.

The unitarity triangle with the largest area is defined in the complex plane
as

$V^{\ast}_{ud}V_{ub}+V^{\ast}_{cd}V_{cb}+V^{\ast}_{td}V_{tb}=0.$ (4.1)

Division of this equation by $V_{cb}^{\ast}V_{cd}$ fixes the base of the
triangle to have unit length, while its side lengths are given by

$R_{u}=\left|\frac{V_{ub}^{\ast}V_{ud}}{V_{cb}^{\ast}V_{cd}}\right|,\quad
R_{t}=\left|\frac{V_{tb}^{\ast}V_{td}}{V_{cb}^{\ast}V_{cd}}\right|,$ (4.2)

as shown in Fig. 3.

$\bar{\rho}$$\bar{\eta}$$\gamma=\varphi_{3}$$R_{u}$$\beta=\varphi_{1}$$\alpha=\varphi_{2}$$R_{t}$
Figure 3: CKM unitarity triangle in the $\bar{\rho}-\bar{\eta}$ plane.

Accordingly, the angles of the triangle are defined as:

$\displaystyle\beta$
$\displaystyle=\varphi_{1}=\arg\left(-\frac{{V^{\ast}_{cb}V_{cd}}}{{V^{\ast}_{tb}V_{td}}}\right),$
(4.3) $\displaystyle\alpha$
$\displaystyle=\varphi_{2}=\arg\left(-\frac{{V^{\ast}_{tb}V_{td}}}{{V^{\ast}_{ub}V_{ud}}}\right),$
(4.4) $\displaystyle\gamma$
$\displaystyle=\varphi_{3}=\arg\left(-\frac{{V^{\ast}_{ub}V_{ud}}}{{V^{\ast}_{cb}V_{cd}}}\right).$
(4.5)

The $\alpha$, $\beta$, $\gamma$ naming convention is used hereafter. In the
context of $B^{0}_{s}-\bar{B}^{0}_{s}$ mixing it is also useful to consider
the rather flat triangle

$V^{\ast}_{us}V_{ub}+V^{\ast}_{cs}V_{cb}+V^{\ast}_{ts}V_{tb}=0,$ (4.6)

with

$\beta_{s}=\arg\left(-\frac{V_{ts}V_{tb}^{\ast}}{V_{cs}V_{cb}^{\ast}}\right).$
(4.7)

Experimentally, the angles $\beta$ and $\beta_{s}$ are determined from the
measurements of time-dependent CP asymmetries in the $B^{0}_{q}$ decays to a
$CP$ eigenstate $f$. The asymmetry $\mathcal{A}^{C\\!P}$ between the number of
initially-produced $B^{0}_{q}$ and of initially-produced $\bar{B}^{0}_{q}$
mesons is function of the decay-time $t$ and reads111All equations related to
the determination of the mixing phases here neglect CP violation in the
mixing.

$\mathcal{A}^{C\\!P}_{f}(t)=\frac{S_{f}\sin{(\Delta m_{q}t)}-C_{f}\cos(\Delta
m_{q}t)}{\cosh(\frac{\Delta\Gamma_{q}}{2}t)+\mathcal{A}^{\Delta\Gamma}\sinh(\frac{\Delta\Gamma_{q}}{2}t)}.$
(4.8)

The parameter $S_{f}$ is related to the mixing phase as $\phi^{f}_{q}$
$S=sin(2\phi^{f}_{q})$. Within the SM, and neglecting penguin pollution
effects, $\phi_{s}^{f}=-2\beta_{s}$ and $\phi^{f}=2\beta$. The parameter
$C_{f}$ is $C_{f}=\frac{1-|\lambda^{f}|^{2}}{1+|\lambda^{f}|^{2}}$, where
$|\lambda^{f}|$ is the ratio between the $\bar{B}^{0}_{q}\to f$ and
$B^{0}_{q}\to f$ decay amplitudes:
$|\lambda^{f}|=\left|\frac{\bar{A}_{f}}{A_{f}}\right|$.

###  4.1 Determinations of the $\phi_{s}$ angle

Figure 4: Average of experimental results for $\phi^{c\bar{c}s}_{s}$ vs
$\Delta\Gamma_{s}$ parameters [1].

The ($\phi^{c\bar{c}s}_{s}$,$\Delta\Gamma_{s}$) plane with the individual 68%
confidence-level contours of ATLAS, CMS, CDF, D0 and LHCb results for 2021
average is shown in Fig. 4.

The golden channel, $B_{s}^{0}\to J/\psi K^{-}K^{+}$, has been used in the
improved determination of the $\phi^{c\bar{c}s}_{s}$ parameter based on the 6
fb-1 of data collected by the LHCb detector between 2015 and 2018 [12]. This
measurement supersedes the previous LHCb analysis [13]. The results are

$\phi^{c\bar{c}s}_{s}=-0.039\pm 0.022\pm
0.006\quad\rm{rad},\quad|\lambda^{c\bar{c}s}|=1.001\pm 0.011\pm 0.005,$

finding a good agreement with the SM predictions. No evidence for $CP$
violation is found. Further combination with all LHCb $\phi^{c\bar{c}s}_{s}$
measurements [14, 15, 16, 17, 18, 19] (from $B_{s}^{0}\to J/\psi K^{-}K^{+}$
above $\phi(1020)$ resonance, $B_{s}^{0}\to D_{s}^{-}D_{s}^{+}$, $B_{s}^{0}\to
J/\psi\pi^{-}\pi^{+}$, $B_{s}^{0}\to\psi(2S)K^{-}K^{+}$ decays), gives
$\phi^{c\bar{c}s}_{s}=-0.031\pm 0.018$. In addition, the analysis provides
determination of $\Delta\Gamma_{s}$ and $\Gamma_{s}-\Gamma_{d}$ finding:

$\displaystyle\Delta\Gamma_{s}$ $\displaystyle=0.0845\pm 0.0044\pm
0.0024\quad\rm{ps}^{-1},$ $\displaystyle\Gamma_{s}-\Gamma_{d}$
$\displaystyle=-0.0056^{+0.0013}_{-0.0015}\pm 0.0014\quad\rm{ps}^{-1},$

where the first uncertainty is statistical and second is systematic. The
obtained values are the most precise up to date and agrees well with SM
predictions.

The CMS experiment reported the measured values of $\phi_{s}$ and
$\Delta\Gamma_{s}$ to be [20]

$\phi^{c\bar{c}s}_{s}=-11\pm 50\pm
0.10\quad\rm{mrad},\quad\Delta\Gamma_{s}=0.114\pm 0.014\pm
0.007\quad\rm{ps}^{-1},$

using a data sample corresponding to an integrated luminosity of 96.4 fb-1
collected in proton-proton collisions at $\sqrt{s}$ = 13 TeV in 2017-2018.

On the other hand, the latest results reported by the ATLAS collaboration [21]
are the following

$\displaystyle\phi^{c\bar{c}s}_{s}=-0.087\pm 0..036\pm 0.021\quad\rm{rad},$
$\displaystyle\Delta\Gamma_{s}=0.065\pm 0.0043\pm 0.0037\quad\rm{ps}^{-1},$
$\displaystyle\Gamma_{s}=0.6703\pm 0.0014\pm 0.0018\quad\rm{ps}^{-1},$

Obtained by combining the results of 13 TeV and 7 and 8 TeV of data taking.

While $B_{s}^{0}\to J/\psi K^{-}K^{+}$ decays are dominated by the tree-level
transition, $B_{s}^{0}\to\phi\phi$ modes allow to measure mixing angle via
$b\to s\bar{s}s$ processes. Within LHCb experiment this channel is a benchmark
mode to study $CP$ violation in flavour-changing neutral currents. In addition
to the $\phi^{s\bar{s}s}_{s}$ mixing angle, the parameter $|\lambda|$ is
measured. Due to a cancellation of the mixing and decay weak phase the former
is expected to be zero, while for the latter absence of direct $CP$ asymmetry
leads to expected value of unity. Any deviations from these values would be
signs of New Physics in these decays. In addition, $B_{s}^{0}\to\phi\phi$
decays probe possible polarisation dependent effects in NP thanks to three
linear polarisation states in the $\phi\phi$ system. The updated measurement
[22] uses the data collected in 2015-2018 which corresponds to 6 fb-1
collected by the LHCb detector at the centre-of-mass energy 13 TeV and
supersedes previous measurement presented in [23] The results are:

$\phi^{s\bar{s}s}_{s}=-0.042\pm 0.075\pm
0.009\quad\rm{rad},\quad|\lambda|=1.004\pm 0.030\pm 0.009,$

where the first uncertainty is statistical and second systematic. Further
combination with the LHCb result from 2011-2012 data [24] yields to
$\phi^{s\bar{s}s}_{s}=-0.074\pm 0.069$ rad, $|\lambda|=1.009\pm 0.030$. The
comparison with other LHCb analyses is shown in Fig. 5. The measurement agrees
with SM predictions. In addition, for the first time three polarisations are
measured finding a good agreement among them. It is the most precise
determination of time dependent $CP$ asymmetry in $B_{s}^{0}\to\phi\phi$ modes
and in any penguin-dominated beauty-meson decay.

Figure 5: Comparison of $\phi^{s\bar{s}s}_{s}$ measurements from this and
previous analyses by the LHCb collaboration. The vertical band indicates the
SM prediction. Run1 stands for 2011-2012 period, while Run2 denotes 2015-2018
years [22].

###  4.2 $\sin 2\beta$ determination at LHCb

The CKM angle $\beta$ can be determined from the measurements of time-
dependent $C\\!P$ asymmetries in the decays of neutral $B$ mesons to final
states containing charmonium mesons, namely $B^{0}\to\psi K^{0}_{S}$ with
$\psi=[J/\psi,\psi(2S)]$, which are dominated by $b\to c\bar{c}s$ tree-level
transitions. As the decay-width difference is small in the $B^{0}$ system, the
denominator of Eq. (4.8) is equal to unity, leaving $C_{\psi K_{S}}$ and
$S_{\psi K_{S}}$ as experimentally measurable observables. In the SM, $C_{\psi
K_{S}}=0$ and $S_{\psi K_{S}}=\sin 2\beta$, with corrections due to penguin
pollution expected to be smaller than the currently achievable experimental
precision. Historically, these measurements have been performed by the BaBar,
Belle and LHCb collaborations, resulting in a world-average of
$\sin(2\beta)=0.699\pm 0.017$[1].

Figure 6: Time-dependent $C\\!P$ asymmetry of $B^{0}\to J/\psi K^{0}_{S}$
decays [25].

The LHCb collaboration has performed new measurements of $C\\!P$-violating
parameters from $B^{0}\to\psi K^{0}_{S}$ decays [25] using the full Run2
(2015-2018) data set, corresponding to the integrated luminosity of 6 fb-1
collected by the LHCb detector at the centre-of-mass energy $\sqrt{s}=13$ TeV.
Three decay modes are used: $B^{0}\to J/\psi(\to\mu^{+}\mu^{-})K^{0}_{S}$,
$B^{0}\to\psi(2S)(\to\mu^{+}\mu^{-})K^{0}_{S}$ and $B^{0}\to J/\psi(\to
e^{+}e^{-})K_{S}^{0}$. In all cases, the $K^{0}_{S}$ candidates are
reconstructed in the decay $K^{0}_{S}\to\pi^{+}\pi^{-}$. Due to the long
lifetime of the $K^{0}_{S}$, it is necessary to consider a number of different
reconstruction categories, depending on whether the pions left hits in the
various parts of the LHCb tracking system. Two such categories are included
for the first time in a $C\\!P$-violation measurement: that where only one of
the pions leaves hits in the vertex locator, and that where one of the pions
leaves hits only upstream of the magnet. Together, these boost the signal
yields by approximately 13%. The time-dependent asymmetries are shown in Fig.
6, while the $C\\!P$-violating observables are measured to be:

$\displaystyle S_{\psi K^{0}_{S}}^{\text{Run2}}$ $\displaystyle=0.717\pm
0.013(\text{stat})\pm 0.008(\text{syst}),$ $\displaystyle C_{\psi
K^{0}_{S}}^{\text{Run2}}$ $\displaystyle=0.008\pm 0.012(\text{stat})\pm
0.003(\text{syst}),$

where the first uncertainty is statistical and second is systematic, and a
correlation coefficient of 0.441 is found between the two observables. These
are the most precise measurement of these quantities and are more precise than
the current world average values [1]. The combination with the previous
measurement using Run 1 data is performed, resulting in:

$\displaystyle S_{\psi K^{0}_{S}}^{\text{Run1+2}}$ $\displaystyle=0.724\pm
0.014,$ $\displaystyle C_{\psi K^{0}_{S}}^{\text{Run1+2}}$
$\displaystyle=0.010\pm 0.012,$

where statistical and systematic uncertainties are combined and the
correlation between observables is 0.40. The results agree with the world
average [1] and global fits by CKMFitter [26] and UTFit [27] collaborations.

###  4.3  $\beta$ and related measurements at Belle II

The Belle II collaboration has recently performed several measurements of the
mixing-induced $CP$-violating parameters in $B^{0}\to\eta^{\prime}K_{S}^{0}$,
$B^{0}\to\phi K^{0}_{S}$, $B^{0}\to K^{0}_{S}\pi^{0}$ and $S^{\rm
Belle}_{K^{0}_{S}K^{0}_{S}K^{0}_{S}}$ decays. These decays are mediated by a
$b\to s$ loop-suppressed transition in the SM, making them sensitive to NP.
These transitions carry approximately the same weak phase as the tree-level
$b\to\bar{c}\bar{c}s$ transition. Hence $S\approx\sin 2\beta$ in these decays,
up to small corrections related to sub-leading amplitudes.

Using a dataset of $362\;\text{fb}^{-1}$ collected between 2019 and 2021,
Belle II obtained the following results:

$\displaystyle S^{\rm Belle\ II}_{\eta^{\prime}K^{0}_{S}}$
$\displaystyle=\phantom{-}0.67\pm 0.10\pm
0.04~{}\text{\cite[cite]{[\@@bibref{}{Belle-II:2024xzm}{}{}]}},$
$\displaystyle S^{\rm Belle\ II}_{K^{0}_{S}\pi^{0}}$
$\displaystyle=\phantom{-}0.75^{+0.20}_{-0.23}\pm
0.04~{}\text{\cite[cite]{[\@@bibref{}{Belle-II:2023grc}{}{}]}},$
$\displaystyle S^{\rm Belle\ II}_{\phi K^{0}_{S}}$
$\displaystyle=\phantom{-}0.54\pm
0.26^{+0.06}_{-0.08}~{}\text{\cite[cite]{[\@@bibref{}{Belle-
II:2023uql}{}{}]}},$ $\displaystyle S^{\rm Belle\
II}_{K^{0}_{S}K^{0}_{S}K^{0}_{S}}$ $\displaystyle=-1.37^{+0.35}_{-0.45}\pm
0.03~{}\text{\cite[cite]{[\@@bibref{}{Belle-II:2024tqu}{}{}]}},$

where the first uncertainty is statistical and the second is systematic. Using
the same dataset, Belle II also performed a measurement of the $CP$-violation
parameters in $B^{0}\to K^{*0}\gamma$ decays. In those decays, also mediated
by a one-loop transition in the SM, mixing-induced $CP$ violation is
suppressed by the polarisation of the photon and $S$ is expected to be close
to $0$. Belle II finds

$\displaystyle S^{\rm Belle\ II}_{K^{*0}\gamma}$
$\displaystyle=\phantom{-}0.00^{+0.27}_{-0.26}\pm 0.03.$

All the above results are compatible with the SM expectation and with previous
Belle and BaBar measurements [1]. The results for the $B^{0}\to
K^{0}_{S}\pi^{0}$ decay and the $B^{0}\to K^{*0}(\to K^{0}_{S}\pi^{0})\gamma$
decay are more precise than those of Belle[32, 33], in spite of the smaller
dataset used. This is mainly achieved thanks to an improved $K^{0}_{S}$ and
$\pi^{0}$ selection based on machine-learning algorithms. The decay-time
distributions for these two decays are shown in Fig. 7.

In addition, Belle II also introduced a new flavour-tagging algorithm based on
a graph-neural-network which improves the effective tagging efficiency by 18%
with respect to the previous algorithm [34]. The $C\\!P$-violating parameters
in $B^{0}\to J/\psi K^{0}_{S}$ decays are measured with this new algorithm to
be

$\displaystyle S^{\rm Belle~{}II}_{J/\psi K^{0}_{S}}$
$\displaystyle=\phantom{-}0.724\pm 0.035\pm 0.014,$ $\displaystyle C^{\rm
Belle~{}II}_{J/\psi K^{0}_{S}}$ $\displaystyle=-0.035\pm 0.026\pm 0.013,$

where the first uncertainty is statistical and the second is systematic. The
results are consistent with the world average and the new LHCb result.

(a) $B^{0}\to K^{0}_{S}\pi^{0}$ decay (b) $B^{0}\to K^{0}_{S}\pi^{0}\gamma$
decay

Figure 7:  Decay-time difference $\Delta t$ distribution in $B^{0}\to
K^{0}_{S}\pi^{0}$ decay [29] and $B^{0}\to K^{0}_{S}\pi^{0}\gamma$ decay. The
latter decay is separately analyzed depending on $K^{0}_{S}\pi^{0}$ invariant
mass: $m_{K^{0}_{S}\pi^{0}}\in(0.8,1.0)\,{\rm GeV}\\!/c^{2}$, corresponding to
$K^{*0}\to K^{0}_{S}\pi^{0}$ decays; and
$m_{K^{0}_{S}\pi^{0}}\in(0.6,0.8)\cup(1.0,1.8)\,{\rm GeV}\\!/c^{2}$,
corresponding to the non-resonant decays. $\Delta t$ distribution is shown
separately for a flavor of the accompanying $B$ mesons from $\Upsilon(4S)$
decays, $B_{\rm tag}$. The signal-subtracted distribution is shown for
$B^{0}\to K^{0}_{S}\pi^{0}$ decay. The asymmetry $[N_{\rm sig}(B^{0}_{\rm
tag})-N_{\rm sig}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{\rm
tag})]/[N_{\rm sig}(B^{0}_{\rm tag})+N_{\rm sig}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{\rm tag})]$ is shown in the bottom
panels.

###  4.4 Time-dependent measurements of the CKM $\gamma$ angle

The CKM angle $\gamma$ is defined in Eq. (4.5). The world average is dominated
by LHCb measurements of $B^{+}$, $B^{0}$ and $B_{s}^{0}$ decays, leading to
the value of $\gamma=(63.8^{+3,5}_{-3.7})^{\circ}$ [35]. The time-integrated
measurements are covered in the summary of WG5 [36]. A unique way to measure
the CKM angle $\gamma$ is via time-dependent measurements of $B_{s}^{0}\to
D_{s}^{\mp}K^{\pm}$-like decays. Due to the interference between mixing and
decay amplitudes, the $CP$-violating parameters in these decays are a
combination of the CKM angle $\gamma$ and the relevant mixing phase, namely
$\gamma-2\beta_{s}$, where $\beta_{s}$ is defined in Eq. (4.7). Therefore any
measurement from $B_{s}^{0}\to D_{s}^{\mp}K^{\pm}$ decays can be interpreted
in terms of either the CKM angle $\gamma$ or $-2\beta_{s}=\phi_{s}$ when
taking the other parameters from an independent source.

The LHCb collaboration presented an updated measurement of the CKM angle
$\gamma$ using $B_{s}^{0}\to D_{s}^{\mp}K^{\pm}$ decays reconstructed in the
Run 2 data-set representing $6\,\mathrm{fb}^{-1}$ of data [37]. The time-
dependent asymmetries are defined in Eq. (4.8). As the decay-width difference
significantly differs from zero in the $B_{s}$-$\bar{B}_{s}$ system, this
channel gives access to the hyperbolic terms, allowing for measuring five
$CP$-violating observables: $C$, $S$, $\bar{S}$, $\mathcal{A}^{\Delta\Gamma}$
and $\mathcal{\bar{A}}^{\Delta\Gamma}$, where $C=-\bar{C}$. These parameters
are related to two final states, either $f=D_{s}^{-}K^{+}$ or
$\bar{f}=D_{s}^{+}K^{-}$.

The values of $CP$-violating parameters are measured to be:

$\displaystyle C_{D_{s}K}^{\text{Run\,2}}$ $\displaystyle=\phantom{-}0.791\pm
0.061\pm 0.022,$
$\displaystyle\mathcal{A}^{\Delta\Gamma,\,\text{Run\,2}}_{D_{s}K}$
$\displaystyle=-0.051\pm 0.134\pm 0.037$
$\displaystyle\mathcal{\bar{A}}^{\Delta\Gamma,\,\text{Run\,2}}_{D_{s}K}$
$\displaystyle=-0.303\pm 0.125\pm 0.036,$ $\displaystyle
S_{D_{s}K}^{\text{Run\,2}}$ $\displaystyle=-0.571\pm 0.084\pm 0.023,$
$\displaystyle\bar{S}_{D_{s}K}^{\text{Run\,2}}$ $\displaystyle=-0.503\pm
0.084\pm 0.025,$

where the first uncertainty is statistical and the second is systematic. The
$CP$ violation in the interference of mixing and decay, i.e. $S\neq\bar{S}$ is
obtained at the level of 8.8$\sigma$. The interpretation in terms of the
physical observables with neglecting penguin-loop diagrams or processes beyond
the SM gives:

$\displaystyle\gamma^{\text{Run\,2}}$ $\displaystyle=(74\pm 11)^{\circ},$
$\displaystyle\delta_{D_{s}K}^{\text{Run\,2}}$ $\displaystyle=(346.9\pm
6.6)^{\circ},$ $\displaystyle r_{D_{s}K}^{\text{Run\,2}}$
$\displaystyle=0.327\pm 0.038,$

where uncertainties are combined and $\delta$ denotes the strong phase in this
process, while $r_{D_{s}K}$ is an amplitude ratio defined as
$r_{D_{s}K}=|A(\bar{B}_{s}^{0}\to D_{s}^{-}K^{+})|/|A(B_{s}^{0}\to
D_{s}^{-}K^{+})|$. The time-dependent asymmetry is shown in Fig. 8.

Figure 8: $CP$ asymmetry for the (blue) $D_{s}^{-}K^{+}$ and the (red)
$D_{s}^{+}K^{-}$ final states, folded into one mixing period, $2\pi/\Delta
m_{s}$[37].

The presented measurement is the most precise time-dependent measurement of
the CKM angle $\gamma$ to-date and agrees with the world average [1] and
global fits by the CKMFitter [26] and UTFit [27] collaborations. The overall
agreement of this new measurement with the previous measurement of
$B_{s}^{0}\to D_{s}^{\mp}K^{\pm}$ using an independent data set of
$3\,\mathrm{fb}^{-1}$ recorded by the LHCb experiment at centre-of-mass
energies of 6 and 7 TeV [38] is at the level of $1.3\,\sigma$. This small
deviation is driven by the high value of
$\gamma^{\text{Run\,1}}=(128^{+17}_{-22})^{\circ}$ obtained in the previous
measurement.

In the past, two alternative decay modes have been used to extract the CKM
angle $\gamma$ using a time-dependent analysis. The analysis of $B_{s}^{0}\to
D_{s}^{\mp}K^{\pm}\pi^{+}\pi^{-}$ decays based on the $9\,\mathrm{fb}^{-1}$
LHCb data set[39] requires precise understanding of the phase space and
possible intermediate resonances. The angular analysis yields an angle of
$\gamma_{D_{s}K\pi\pi}=(44\pm 12)^{\circ}$, while a phase-space integrated
approach gives $\gamma_{D_{s}K\pi\pi}=(44^{+20}_{-13})^{\circ}$. Time-
dependent measurements using $B^{0}\to D^{\mp}\pi^{\pm}$ decays suffer from
low sensitivity due to the small amplitude ratio $r_{D\pi}\approx 0.02$. The
corresponding measurement using $3\,\mathrm{fb}^{-1}$ of data recorded by the
LHCb experiment at centre-of-mass energies of 6 and 7 TeV [40] is used to set
limits of
$\gamma_{D\pi}\in[5^{\circ},\,86^{\circ}]\cup[185^{\circ},\,266^{\circ}]$ at
68% CL by using external input for the angle $\beta$.

Currently, the LHCb collaboration takes action to combine the two time-
dependent measurements of $\gamma$ based on $B_{s}^{0}\to D_{s}^{\mp}K^{\pm}$
decays. The result will replace the previous stand-alone result using only the
$3\,\mathrm{fb}^{-1}$ data set in upcoming LHCb-wide combinations of the CKM
angle $\gamma$.

###  4.5 New physics in $B_{q}-\bar{B_{q}}$ in connection with the CKM
$\gamma$ angle.

In Ref. [41], it is explored how much space is left for NP through the current
data, performing a careful analysis of the Unitarity Triangle. The choice for
the input measurements, in particular on the determination of the apex of the
UT, plays a central role in the NP searches. Tensions arise between inclusive
and exclusive determinations of the CKM matrix elements $|V_{ub}|$ and
$|V_{cb}|$. Separate analyses for these two cases is important to be
performed. A third possibility, a hybrid scenario combining the exclusive
$|V_{ub}|$ with the inclusive $|V_{cb}|$ value is also studied [41]. Sizeable
differences are observed between the three cases, indicating that it is
essential for the puzzles between the different determinations of the CKM
parameters to finally be resolved.

In the method that Ref. [41] chooses to follow for the extraction of the UT
apex in the SM determination, only two observables are used: the UT side
$R_{b}$ and the UT angle $\gamma$. The two parameters are experimentally
measured using tree-level decays, and hence possible NP contamination is
expected to be minimal. The side $R_{b}$ is defined as

$R_{b}\equiv\left(1-\frac{\lambda^{2}}{2}\right)\frac{1}{\lambda}\left|\frac{V_{ub}}{V_{cb}}\right|=\sqrt{\bar{\rho}\,^{2}+\bar{\eta}\,^{2}}\>,$
(4.9)

and hence it strongly depends on the determination of the CKM matrix elements
$|V_{cb}|$ and $|V_{ub}|$.

In addition of utilising purely tree $B\to DK$ decays, the CKM angle $\gamma$
can also be determined using $B\to\pi\pi$, $\rho\pi$, $\rho\rho$ modes and
isospin relations. These modes are usually interpreted in terms of the UT
angle $\alpha$, from which $\gamma$ can be extracted using the relation
$\gamma=\pi-\alpha-\beta$. Despite the very different dynamics between the
corresponding decays, excellent agreement has been found between the two
determinations. Although such an agreement is expected in the SM,
discrepancies may arise due to NP contributions to the decay amplitudes.

To quantify the impact of different NP scenarios in
$B_{q}^{0}-\bar{B_{q}}^{0}$ mixing, the generalised expressions of the mixing
parameters is utilised [42]:

$\displaystyle\phi_{q}$
$\displaystyle=\phi_{q}^{\text{SM}}+\phi_{q}^{\text{NP}}=\phi_{q}^{\text{SM}}+\arg\left(1+\kappa_{q}e^{i\sigma_{q}}\right)\>,$
(4.10) $\displaystyle\Delta m_{q}$ $\displaystyle=\Delta
m_{q}^{\text{SM}}\left|1+\kappa_{q}e^{i\sigma_{q}}\right|\>.$ (4.11)

Here, $\kappa_{q}$ describes the size of the NP effects and $\sigma_{q}$ is a
complex phase accounting for additional $CP$-violating effects. This
parametrisation is model independent.

We study two NP scenarios. Scenario I is the most general one and uses as
input $R_{b}$ and $\gamma$, assuming that there is no NP contribution in these
input parameters. The extraction of the parameters $\kappa_{q}$ and
$\sigma_{q}$, is performed separately for the $B_{d}^{0}$ and the $B_{s}^{0}$
system. In Scenario II, we use a Flavour Universal New Physics (FUNP)
assumption. Here, the assumption is that
$(\kappa_{d},\sigma_{d})=(\kappa_{s},\sigma_{s})$, which means that the NP
contributions are equal in the $B_{d}^{0}$ and the $B_{s}^{0}$ system. The
determination of the UT apex does not rely on $\gamma$ now but in the sides
$R_{b}$ and $R_{t}$ instead.

Comparing Scenarios I and II provides a test of whether the data are
compatible with the FUNP assumption, and characterizes how the FUMP assumption
constrains the NP parameter space. For each scenario, three different
determinations of $R_{b}$ are compared using exclusive, inclusive or hybrid
inputs for $V_{cb}$ and $V_{ub}$. Fig. 9 illustrates the correlation between
$\kappa_{d}$ and $\kappa_{s}$. All three $R_{b}$ solutions are compatible with
the FUNP assumption $\kappa_{d}=\kappa_{s}$.

Figure 9:  Fit for the NP parameters $\kappa_{d}$ and $\kappa_{s}$, comparing
the inclusive, exclusive and hybrid scenarios. The black diagonal line denotes
the case $\kappa_{d}=\kappa_{s}$, which is the FUNP assumption. The plot is
taken from Ref. [41]

These NP results have also interesting applications in the analysis of
leptonic rare decays $B_{q}^{0}\to\mu^{+}\mu^{-}$. Key observable is the
corresponding SM branching ratio. The branching ratio requires information on
$|V_{ts}|$, which is determined through the $|V_{cb}|$ matrix element. It is
shown that the dependence on $|V_{cb}|$ can be canceled in the SM by creating
the following ratio with the $B_{s}$ mass difference $\Delta m_{s}$ [43, 44,
45]:

$\mathcal{R}_{s\mu}\equiv{\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})}/{\Delta
m_{s}}\>,$ (4.12)

where possible NP contributions to $B^{0}_{q}$-$\bar{B}^{0}_{q}$ mixing can be
taken into account, following the strategy proposed in [41].

This NP analysis is extrapolated to the high-precision era and future
projections are made [41]. In the $B_{d}^{0}$-$\bar{B}_{d}^{0}$ system, the UT
apex is a limiting factor. This is however not the case for the
$B_{s}^{0}$-$\bar{B}_{s}^{0}$ system, making it a good candidate for searching
for new sources of NP.

New opportunities are expected in NP searches, both related to the
determination of the angle $\gamma$ and of the $B^{0}_{q}$-$\bar{B}^{0}_{q}$
mixing parameters. Future improvements of the precision on $\gamma$ are
especially important, given that an improved precision might reveal
significant deviations between the different determinations of $\gamma$ due to
NP contributions.

###  4.6 Determination of $\gamma$ from $B_{(s)}\to hh$ decays

Charmless two-body $B_{(s)}\to hh$ decays, where $h=\pi,K$, are important
probes of $CP$ violation. Among these decays, the penguin-dominated
$B_{s}^{0}\to K^{-}K^{+}$ decay can also be used to determine the angle
$\gamma$ and the $B_{s}^{0}$–$\bar{B}_{s}^{0}$ mixing phase $\phi_{s}$ [46,
47, 48, 49, 51, 52, 50, 53]. The LHCb collaboration recently observed $CP$
violation in this decay for the first time [54], allowing for the
determination of either $\gamma$ or $\phi_{s}$ from a loop-topology dominated
decay [53]. These penguin modes are very sensitive to effects of new
particles. Therefore it is interesting to compare $\gamma$ or $\phi_{s}$
extracted from these decays with determinations from tree-level dominated
decays.

Looking at the differences between the direct $CP$ asymmetries of several
newly measured $B_{(s)}\to hh$ modes by the LHCb collaboration also reveals an
interesting pattern. Comparing the modes that only differ by a spectator
quark, leads to [53]

$\displaystyle\mathcal{A}_{\rm CP}^{\rm dir}(B_{s}^{0}\to
K^{-}K^{+})-\mathcal{A}_{\rm CP}^{\rm dir}(B_{d}^{0}\to\pi^{-}K^{+})$
$\displaystyle=0.089\pm 0.031\ ,$ $\displaystyle\mathcal{A}_{\rm CP}^{\rm
dir}(B_{d}^{0}\to\pi^{-}\pi^{+})-\mathcal{A}_{\rm CP}^{\rm dir}(B_{s}^{0}\to
K^{-}\pi^{+})$ $\displaystyle=-0.095\pm 0.040.$ (4.13)

which is quite striking as the difference from zero is at the $(2-3)\sigma$
level. This difference is quite difficult to explain through new-physics
effects, because the above decays only differ through their spectator quarks.
On the other hand, the $B_{s}^{0}\to K^{-}K^{+}$ and
$B_{d}^{0}\to\pi^{-}\pi^{+}$ modes also have contributions from Exchange ($E$)
and penguin-annihilation ($PA$) topologies while the
$B_{d}^{0}\to\pi^{-}K^{+}$ and $B_{s}^{0}\to K^{-}\pi^{+}$ decays do not. This
raises the question if the differences in Eq. (4.13) can be accommodated by
reasonable $E$ and $PA$ contributions. In Ref. [53], both the determination of
$\gamma$ and $\phi_{s}$ from these decays and a new strategy to determine the
size of the $E$ and $PA$ contributions was presented. In this Section, these
findings are reviewed.

Extracting $\gamma$ using flavour symmetries The $B_{s}^{0}\to K^{-}K^{+}$ and
$B_{d}^{0}\to\pi^{-}\pi^{+}$ decays form a $U$-spin system, with tree ($T$),
QCD penguin ($P$) and $E$ and $PA$ contributions. Their amplitudes are
parametrised in terms of the penguin parameters $d^{(^{\prime})}$ and their
$CP$-conserving phase $\theta^{(^{\prime})}$, where the prime indicates the
$b\to s$ transition through [51]

$\displaystyle A(B_{s}^{0}\to K^{+}K^{-})$
$\displaystyle=\sqrt{\epsilon}e^{i\gamma}\mathcal{C}^{\prime}\left[1+\frac{1}{\epsilon}d^{\prime}e^{i\theta^{\prime}}e^{-i\gamma}\right]\
,$ (4.14) $\displaystyle A(B_{d}^{0}\to\pi^{+}\pi^{-})$
$\displaystyle=e^{i\gamma}\mathcal{C}(1+de^{i\theta}e^{-i\gamma})\ ,$ (4.15)

where $\epsilon\equiv\lambda^{2}/(1-\lambda^{2})$, with $\lambda$ being the
Wolfenstein parameter of the CKM matrix. The penguin parameter is

$de^{i\theta}\equiv\frac{1}{R_{b}}\left[\frac{P^{(ct)}+PA^{(ct)}}{T+E+P^{(ut)}+PA^{(ut)}}\right]\
,$ (4.16)

with a similar expression for $d^{\prime}$. In addition, $P^{(qt)}\equiv
P^{(q)}-P^{(t)}$, and in analogy for the $PA^{(qt)}$ contribution and
$\mathcal{C}$ parametrises the $T+E+P^{(ut)}+PA^{(ut)}$ amplitudes. In the
$U$-spin limit, the penguin parameters are related via
$de^{i\theta}=d^{\prime}e^{i\theta^{\prime}}$ [46].

The direct and mixing-induced $CP$ asymmetries of the $U$-spin system, depend
only on $d^{(^{\prime})},\theta^{(^{\prime})}$ and $\gamma$, $\phi_{s}$ and
$\phi_{d}$. Assuming $U$-spin symmetry and taking $\phi_{s}$ and $\phi_{d}$,
the penguin parameter $d$ and $\gamma$ simultaneously using only the $CP$
asymmetries and by taking $\phi_{s}$ and $\phi_{d}$ as inputs. The current
data in [54] give the contours in the $d$-$\gamma$ plane given in Fig. 10. The
intersection of the two curves gives

$d=d^{\prime}=0.52_{-0.09}^{+0.13}\ $ (4.17)

and

$\gamma=(65_{-5}^{+7})^{\circ}\ .$ (4.18)

Comparing with the CKM angle $\gamma$ determinations from tree-level decays
described in Sec. 4.4, especially not considering the determinations from
$B_{s}^{0}$ decays leads to impressive agreement.

Figure 10: Dependence of the $CP$ asymmetries of the $B_{d}\to\pi^{-}\pi^{+}$
and $B_{s}\to K^{-}K^{+}$ modes on the penguin parameter $d$ and the CKM angle
$\gamma$ using data from [54]. The overlap region gives the determination of
$\gamma$. Plot taken from [53].

Probing the Exchange and Penguin annihilation contributions The new LHCb data
revealed a large difference between the direct $CP$ asymmetries of
$B_{d}^{0}\to\pi^{-}\pi^{+},B_{s}^{0}\to K^{-}K^{+}$ system and the
$B_{d}^{0}\to\pi^{-}K^{+},B_{s}^{0}\to K^{-}\pi^{+}$ system in Eq. (4.13).
Within the SM, this difference can only be explained through (sizeable)
exchange ($E$) and penguin-annihilation ($PA$) contributions, which contribute
to the first $U$-spin system but not to the second. These contributions are
denoted by the hadronic parameters $x^{(\prime)}$ and $r_{PA}^{(\prime)}$,
respectively. The mismatch between the two $U$-spin systems is given by

$\zeta^{\prime}\equiv|\zeta^{\prime}|e^{i\omega^{\prime}}\equiv\frac{1+x^{\prime}}{1+r_{PA}^{\prime}}$
(4.19)

with

$x^{(\prime)}\equiv|x^{(\prime)}|e^{i\sigma^{(\prime)}}\equiv\frac{E^{(\prime)}+PA^{(ut)(\prime)}}{T^{(\prime)}+P^{(ut)(\prime)}},\quad
r_{PA}^{(\prime)}\equiv|r_{PA}^{(\prime)}|e^{i\theta_{PA}^{(\prime)}}\equiv\frac{PA^{(ct)(\prime)}}{P^{(ct)(\prime)}}.$
(4.20)

Figure 2 in Ref. [53], describes in detail a new strategy to determine these
parameters from the experimental data. An brief outline of this new strategy
is presented below.

$\bullet$ Step 1: Using instead $\gamma$ as an input, allows to determine the
hadronic parameters defined in Eq. (4.16) by assuming the $U$-spin symmetry in
the $B_{d}^{0}\to\pi^{-}\pi^{+},B_{s}^{0}\to K^{-}K^{+}$ system. This gives

$d=d^{\prime}=0.39\pm 0.05,\qquad\theta=\theta^{\prime}=(140\pm 5)^{\circ}\ ,$
(4.21)

where compared to the above $d$ is shifted down by $1\,\sigma$ level.

$\bullet$ Step 2: Similarly, the analogous penguin parameter $\tilde{d}$ can
be obtained from the $CP$ asymmetries of the $B_{s}\to K^{-}\pi^{+}$ and
$B_{d}s\to\pi^{-}K^{+}$decays. Comparing gives

$|\zeta|\equiv\tilde{d}/d=1.3\pm 0.2\
,\quad\omega\equiv\tilde{\theta}-\theta=(16\pm 5)^{\circ}\ ,$ (4.22)

which measure contribution of the $E^{(\prime)}$ and $PA^{(\prime)}$
topologies in the $B_{d}^{0}\to\pi^{-}\pi^{+}$ ($B_{s}^{0}\to K^{-}K^{+}$)
decay with respect to the tree and penguin amplitudes. From the values in Eq.
(4.22), it is concluded in [53] that the difference in the direct $CP$
asymmetries in Eq. (4.13) can be accommodated by exchange and penguin
annihilation effects at the level of (20–30)%. This size lies in the range of
what is expected theoretically for such contributions and does not signal any
anomalously enhanced rescattering or long-distance effects.

$\bullet$ Step 3: To determine $x$ and $r_{PA}$ from $\zeta$ requires
additional information. This can be provided by the
$B_{s}^{0}\to\pi^{-}\pi^{+},B_{d}^{0}\to K^{-}K^{+}$ $U$-spin system, which
receives only contributions from $E$ and $PA$ topologies. At the moment, only
the branching ratios of these decays are measured. In Ref. [53] the
constraints Fig. 11 were obtained. In the future, possible measurements of the
$CP$ asymmetries in the $B_{s}^{0}\to\pi^{-}\pi^{+},B_{d}^{0}\to K^{-}K^{+}$
system would allow pinning down these contributions even further.

Figure 11: Extracted $1\,\sigma$ allowed space for the exchange ($x,\sigma$)
and penguin-annihilation ($r_{PA},\theta_{PA}$) parameters. Figure taken from
[53].

Extracting $\phi_{s}$ from penguin decays Finally, the penguin-dominated
$B_{s}^{0}\to K^{-}K^{+}$ decay mode can also be used to determine the
$B_{s}^{0}$–$\bar{B}_{s}^{0}$ mixing phase $\phi_{s}$. Doing so requires
taking $\gamma$ as an input.

The $CP$ asymmetries of $B_{s}^{0}\to K^{-}K^{+}$ allow the extraction of the
effective phase

$\sin\phi_{s}^{\rm eff}=\frac{\mathcal{A}_{\rm CP}^{\rm mix}(B_{s}^{0}\to
K^{-}K^{+})}{\sqrt{1-\left[\mathcal{A}_{\rm CP}^{\rm dir}(B_{s}^{0}\to
K^{-}K^{+})\right]^{2}}}\ .$ (4.23)

Using the LHCb data for the $CP$ asymmetries from [54], gives $\phi_{s}^{\rm
eff}=-(8.1\pm 1.9)^{\circ}$ which is defined as

$\phi_{s}^{\rm eff}\equiv\phi_{s}+\Delta\phi_{KK}\ ,$ (4.24)

where the parameter $\Delta\phi_{KK}$ is a hadronic phase shift

$\tan\Delta\phi_{KK}=2\epsilon\sin\gamma\left[\frac{d^{\prime}\cos\theta^{\prime}+\epsilon\cos\gamma}{d^{\prime
2}+2\epsilon d^{\prime}\cos\theta^{\prime}\cos\gamma+\epsilon^{2}\cos
2\gamma}\right]\ ,$ (4.25)

which depends on the hadronic parameters $d^{\prime},\theta^{\prime}$
discussed before. In [51] and [53], two strategies to determine
$\Delta\phi_{KK}$ using semileptonic decays were presented:

Strategy I: This strategy uses (double) ratios of non-leptonic and
semileptonic $B_{(s)}$ decay rates [51]:

$R_{\pi}\equiv\frac{\Gamma(B_{d}^{0}\to\pi^{-}\pi^{+})}{|d\Gamma(B^{0}_{d}\rightarrow\pi^{-}\ell^{+}\nu_{\ell})/dq^{2}|_{q^{2}=m_{\pi}^{2}}}\
,\quad\quad R_{K}\equiv\frac{\Gamma(B_{s}^{0}\to K^{-}K^{+})_{\rm
theo}}{|d\Gamma(B^{0}_{s}\rightarrow
K^{-}\ell^{+}\nu_{\ell})/dq^{2}|_{q^{2}=m_{K}^{2}}}\ .$ (4.26)

In terms of the hadronic parameters, $R_{\pi}$ is proportional to

$r_{\pi}\equiv 1+d^{2}-2d\cos\theta\cos\gamma\ ,$ (4.27)

while $R_{K}$ is proportional to

$r_{K}\equiv
1+\left(\frac{d^{\prime}}{\epsilon}\right)^{2}+2\frac{d^{\prime}}{\epsilon}\cos\theta^{\prime}\cos\gamma.$
(4.28)

Taking then the ratio of $R_{\pi}$ and ${R}_{K}$ yields

${r}_{K}=\frac{{R}_{K}}{R_{\pi}}\left(\frac{|V_{ud}|f_{\pi}}{|V_{us}|f_{K}}\right)^{2}\frac{X_{\pi}}{{X}_{K}}\left({\xi}_{\text{NF}}^{a}\right)^{2}r_{\pi}\
,$ (4.29)

where also the decay constants $f_{\pi,K}$ and the ratio of $B_{s}\to K$ and
$B\to\pi$ form factors enter through $X_{K}$ and $X_{\pi}$, respectively (see
(56) in Ref. [53]). These inputs are precisely determined, and the remaining
theoretical uncertainty comes from the ratio

${\xi}_{\text{NF}}^{a}\equiv\left|\frac{1+r_{P}}{1+{r}^{\prime}_{P}}\right|\left|\frac{1+x}{1+x^{\prime}}\right|\left|\frac{a_{\rm
NF}^{T}}{{a}_{\rm NF}^{T^{\prime}}}\right|,$ (4.30)

which parametrises the non-factorisable $U$-spin-breaking contributions. Here
$r_{P}=P^{(ut)}/T$ and $a_{\rm NF}^{T}$ describes the non-factorisable
contributions to the colour-allowed tree topology, which can be obtained using
QCD factorisation. In the exact $U$-spin symmetry, $\xi_{\rm NF}^{a}=1$, while
potential $U$-spin-breaking corrections only enter through the double-ratio in
$\xi$.

In Ref. [53], the uncertainty on $\xi_{\rm NF}^{a}$ was re-evaluated, using
the results of Fig. 11. This also allowed to check the assumptions made in
Ref. [51]. This led to:

$\xi_{\rm NF}^{a}=1.00\pm 0.07\ ,$ (4.31)

which results in a theoretical uncertainty of only $0.8^{\circ}$ on
$\Delta\phi_{KK}$. Recently this decay has been observed for the first time by
the LHCb collaboration [55]. However, for this strategy the differential
semileptonic $B_{s}^{0}\to K^{-}\ell^{+}\nu_{\ell}$ decay rate at
$q^{2}=m_{K}^{2}$ is required as input, which is currently not available.
Therefore, this strategy cannot be fully implemented at this moment.

Strategy II: An alternative strategy is to use only the nonleptonic
$B_{s}^{0}\to K^{-}K^{+}$ and $B_{d}^{0}\to\pi^{-}\pi^{+}$ rates and the
ratio:

$\displaystyle K$
$\displaystyle\equiv\frac{1}{\epsilon}\left[\frac{m_{B_{s}}}{m_{B_{d}}}\frac{\Phi(m_{\pi}/m_{B_{d}},m_{\pi}/m_{B_{d}})}{\Phi(m_{K}/m_{B_{s}},m_{K}/m_{B_{s}})}\frac{\tau_{B_{d}}}{\tau_{B_{s}}}\right]\frac{\mathcal{B}(B_{s}^{0}\to
K^{-}K^{+})_{\rm theo}}{\mathcal{B}(B_{d}^{0}\to\pi^{-}\pi^{+})}$
$\displaystyle=\left|\frac{\mathcal{C}^{\prime}}{\mathcal{C}}\right|^{2}\frac{1+2(d^{\prime}/\epsilon)\cos\theta^{\prime}\cos\gamma+(d^{\prime}/\epsilon)^{2}}{1-2d\cos\theta\cos\gamma+d^{2}},$
(4.32)

where $\Phi$ is a phase space function. The experimental data from LHCb [54]
gives $K=105.3\pm 9.6$. To use this value to extract the penguin parameters
requires a theoretical determination of $\mathcal{C}/\mathcal{C}^{\prime}$,
which can be estimated through

$\left|\frac{\mathcal{C}}{\mathcal{C}^{\prime}}\right|=\frac{f_{\pi}}{f_{K}}\left[\frac{m_{B_{d}}^{2}-m_{\pi}^{2}}{m_{B_{s}}^{2}-m_{K}^{2}}\right]\left[\frac{F_{0}^{B_{d}\pi}(m_{\pi}^{2})}{F_{0}^{B_{s}K}(m_{K}^{2})}\right]\xi^{a}_{\rm
NF}\ ,$ (4.33)

where the form factors are known from [56]. Finally, using also $\gamma$ as
input and $\xi_{\rm NF}^{a}$ from Eq. (4.31), gives

$\Delta\phi_{KK}=-(4.5\pm 5.3)^{\circ}\ ,\quad\quad\phi_{s}=-(3.6\pm
5.7)^{\circ}\ .$ (4.34)

The uncertainty on $\phi_{s}$ is dominated by the experimental uncertainties
on $B_{s}^{0}\to K^{-}K^{+}$ $CP$ asymmetries and those on the form factors.

This new determination is in good agreement with the $\phi_{s}$ obtained from
$B_{s}^{0}\to J/\psi\phi$ and agrees with the SM predictions. However, more
interestingly, it also leaves significant room for new physics.

In the future, with more precise $CP$ asymmetries, it will be interesting to
see if the value of $\phi_{s}$ from the penguin-modes and that from
$B_{s}^{0}\to J/\psi\phi$ analyses remain in agreement. If a discrepancy would
arise, this signifies new $CP$-violating physics, to which especially the
penguin-dominated determination discussed above is sensitive.

##  5 Determination of the $\Delta\Gamma$ width difference

In the $B^{0}_{s}$ system estimation the decay-width difference
$\Delta\Gamma_{s}$ varies in the range (7.6-9.2)$\times$10-2 ps-1 depending on
the renormalisation choice [57, 58, 59, 60]. This quantity is measured in
channels such as $B_{s}^{0}\to J/\psi\phi$ [21, 20, 12]. The experimental
results are in tension as shown in Fig. 12. Therefore new studies from
experimental and theoretical sides are needed.

Figure 12: Experimental results of $\Gamma_{s}$ vs $\Delta\Gamma_{s}$ from
$B_{s}^{0}\to J/\psi KK$ decays averaged by HFLAV group [1].

###  5.1 New results from LHCb

The LHCb experiment presented for the first time measurement on the
$\Delta\Gamma_{s}$ parameter exploiting new decays: $B_{s}^{0}\to
J/\psi\eta^{\prime}$ and $B_{s}^{0}\to J/\psi\pi^{+}\pi^{-}$ [61]. The
analysis uses the full LHCb Run 1 and Run 2 data, corresponding to 9 fb-1 of
integrated luminosity. The width difference, $\Delta\Gamma_{s}$, is obtained
from the decay-width difference between a $CP$-odd and a $CP$-even $B_{s}^{0}$
mode. It is possible thanks to the small value of $\phi_{s}$, which implies
that with a good approximation $CP$-even modes determine the light mass
eigenstate lifetime ($\tau_{L}=1/\Gamma_{L}$), while $CP$-odd corresponds to
the heavy mass eigenstate lifetime $\tau_{H}=1/\Gamma_{H}$. The measurement is
performed in bins of decay time using the ratio of yields in given intervals
($t_{1}$,$t_{2}$):

$R_{i}=\frac{N_{L}}{N_{H}}\propto\frac{[e^{-\Gamma_{s}t(1+y)}]^{t1}_{t2}}{[e^{-\Gamma_{s}t(1-y)}]^{t1}_{t2}}\times\frac{1-y}{1+y}$

where $y=\Delta\Gamma_{s}/2\Gamma_{s}$ and $N_{L}$ ($N_{H}$) denotes the yield
of $CP$-even ($CP$-odd) mode in given interval. The results are obtained for
periods of data-taking as shown in Fig. 13, the combined value of
$\Delta\Gamma_{s}$ yields:

$\Delta\Gamma_{s}=0.087\pm 0.012\pm 0.009\quad\rm{ps}^{-1},$

where the first uncertainty is statistical and second is systematic. The value
agrees with both HLFAV averages [1] obtained from the time dependent
measurements of $B_{s}^{0}\to J/\psi\phi$ decays, as well as including
constraints from other untagged effective lifetime measurements.

Figure 13: Measurement of $\Delta\Gamma_{s}$ for the four datasets (2011-2012,
2015-2016, 2017 and 2018) and their weighted average. The orange band
corresponds to 1 standard deviation. Taken from [61].

###  5.2 NNLO QCD corrections

####  5.2.1 Calculation overview

The latest update from on the theoretical calculation of $\Delta\Gamma$ from
Refs. [62, 63] focuses on reducing the perturbative uncertainties in the
leading $\mathcal{O}((\Lambda_{\text{QCD}}/m_{b})^{0})$ terms. This is
achieved through a matching calculation of a $\lvert\Delta B\rvert=2$ matrix
element calculated within effective $\lvert\Delta B\rvert=1$ and $\lvert\Delta
B\rvert=2$ theories, where the high-energy and low-energy effects factorise
into the matching coefficients and the operator matrix elements respectively.
The Wilson coefficients of the first Operator Product Expansion, i.e. for the
$\lvert\Delta B\rvert=1$ Hamiltonian, have been calculated to three-loop order
in previous works [65, 66, 67]. To obtain only the leading terms in
$\Lambda_{\text{QCD}}/m_{b}$, the Heavy Quark Expansion (HQE) is used for the
transition operator on the $\lvert\Delta B\rvert=2$ side, which allows us to
expand the operators in $\Lambda_{\text{QCD}}/m_{b}$ [68, 69, 70, 71, 72, 73,
74, 75, 76, 77]. The matching calculation is done methodically by first
calculating the imaginary part of the $B_{s}\rightarrow\overline{B}_{s}$
mixing amplitude in the two effective field theories, renormalising the
results and then matching the coefficients of the $\lvert\Delta B\rvert=2$
operators to the result from the $\lvert\Delta B\rvert=1$ calculation.

The definition of the Hamiltonian for the $\lvert\Delta B\rvert=1$ theory in
the Chetyrkin-Misiak-Münz basis can be found in [64]. It is worth noting that
in the application this basis is particularly useful for automated
calculations as it circumvents all of the complications related to
$\gamma_{5}$ in dimensional regularisation. Another issue related to
dimensional regularisation with $d=4-2\epsilon$ dimensions is the appearance
of so-called evanescent operators, which are of order $\epsilon$ and vanish in
four dimensions due to Fierz identities. However, the evanescent operators mix
with physical operators and need to be taken into consideration when
renormalising bare amplitudes.

To calculate the width difference $\Delta\Gamma$, the absorptive part of the
scattering matrix element needs to be evaluated, which decomposes into a sum
of terms with different CKM factors. This prompts to decompose the
$\lvert\Delta B\rvert=2$ matching coefficients in an analogous fashion. From
these considerations, the off-diagonal matrix element of the decay width in
the $\lvert\Delta B\rvert=1$ theory can be written as

$\Gamma_{12}=\frac{1}{M_{B}}\sum_{\alpha,\beta}\lambda_{\alpha}\lambda_{\beta}\,\text{Im}(\mathcal{M}_{\alpha\beta}),$
(5.1)

and in the $\lvert\Delta B\rvert=2$ theory as

$\Gamma_{12}=-\frac{G_{F}^{2}m_{b}^{2}}{24\pi^{2}M_{B}}\sum_{\alpha,\beta}\lambda_{\alpha}\lambda_{\beta}\left[H^{\alpha\beta}\langle
B\rvert Q\lvert\bar{B}\rangle+\tilde{H}_{S}^{\alpha\beta}\langle
B\rvert\tilde{Q}_{S}\lvert\bar{B}\rangle\right]+\mathcal{O}\left(\frac{\Lambda_{\text{QCD}}}{m_{b}}\right),$
(5.2)

where $M_{B}$ is the mass of the $B$ meson. In the context of the HQE, the
matching coefficients $H^{\alpha\beta}$ and $\tilde{H}_{S}^{\alpha\beta}$ are
calculated as expansions in $z\equiv m_{c}^{2}/m_{b}^{2}$. $Q$ and
$\tilde{Q}_{S}$ are the physical operators of the $\lvert\Delta B\rvert=2$
transition operator defined in Ref. [62].

As alluded to previously, the low-energy and high-energy physics factorise
with the matching coefficients $H^{\alpha\beta}$ and
$\tilde{H}_{S}^{\alpha\beta}$ containing the perturbative high-energy physics
that is the main goal of the theoretical calculation described here. The low-
energy behaviour captured in the operator matrix elements of the physical
operators needs to be extracted from either QCD sum rules [78, 79, 80, 81, 82,
83, 9, 84] or lattice QCD calculations [85, 5] and is used as an input in the
prediction of $\Delta\Gamma$. For more details on the calculation, see Ref.
[86].

####  5.2.2 Results

The update on the theoretical calculation in Refs. [62, 63] provides new
results for the current-current operator contributions at NNLO by combining
novel calculations with known results for diagrams with closed charm loops.
Contributions from penguin operators are also updated to higher accuracy than
in previous calculations. All calculations are done as a naive expansion in
$m_{c}/m_{b}$ and only the leading term in $z\equiv m_{c}^{2}/m_{b}^{2}$ is
determined, see Tab. 1 for a summary.

Contribution | Previous results | Refs.[62, 63]
---|---|---
$P_{1,2}\times P_{3-6}$ | 2 loops, $z$-exact, $n_{f}$-part only [89, 90] | 2 loops, $\mathcal{O}(z)$, full
$P_{1,2}\times P_{8}$ | 2 loops, $z$-exact, $n_{f}$-part only [89, 90] | 2 loops, $\mathcal{O}(z)$, full
$P_{3-6}\times P_{3-6}$ | 1 loop, $z$-exact, full [91] | 2 loops, $\mathcal{O}(z)$, full
$P_{3-6}\times P_{8}$ | 1 loop, $z$-exact, $n_{f}$-part only [89, 90] | 2 loops, $\mathcal{O}(z)$, full
$P_{8}\times P_{8}$ | 1 loop, $z$-exact, $n_{f}$-part only [89, 90] | 2 loops, $\mathcal{O}(z)$, full
$P_{1,2}\times P_{1,2}$ | 3 loops, $\mathcal{O}(\sqrt{z})$, $n_{f}$-part only [89, 90] | 3 loops, $\mathcal{O}(z)$, full
Table 1: Updated contributions to the theoretical value of $\Delta\Gamma$.
“Full” contributions refers to the fact that both the fermionic and non-
fermionic pieces have been calculated.

To improve the numerical accuracy of the result, it is helpful to consider the
ratio $\Delta\Gamma/\Delta M$. In this ratio, the dependence on $\lvert
V_{cb}\rvert$ drops out and most of the dependence on the bag parameters is
also removed. The NNLO corrections from QCD to $\Delta M$ are already known
and have been published in Ref. [87]. Finally, the theoretical uncertainty
also depends on the choice of renormalisation scheme of the mass $m_{b}^{2}$
in Eq. (5.2) multiplying $\Delta\Gamma$. In a first step, one usually employs
a pole mass but subsequently trades it for an MS mass or the potential-
subtracted (PS) mass [88], which have better infrared properties. The results
for the resulting ratio in the different mass schemes of the overall
$m_{b}^{2}$ factor are given by

$\displaystyle\left.\frac{\Delta\Gamma_{s}}{\Delta
M_{s}}\right\rvert_{\overline{\text{MS}}}$
$\displaystyle=\left(4.33\,\begin{subarray}{c}+0.23\\\
-0.44\end{subarray}_{\text{scale}}\,\begin{subarray}{c}+0.09\\\
-0.19\end{subarray}_{\text{scale},1/m_{b}}\pm 0.12_{B\tilde{B}_{S}}\pm
0.78_{1/m_{b}}\pm 0.05_{\text{input}}\right)\times 10^{-3},$ (5.3)
$\displaystyle\left.\frac{\Delta\Gamma_{s}}{\Delta
M_{s}}\right\rvert_{\text{PS}}$
$\displaystyle=\left(4.20\,\begin{subarray}{c}+0.36\\\
-0.39\end{subarray}_{\text{scale}}\,\begin{subarray}{c}+0.09\\\
-0.19\end{subarray}_{\text{scale},1/m_{b}}\pm 0.12_{B\tilde{B}_{S}}\pm
0.78_{1/m_{b}}\pm 0.05_{\text{input}}\right)\times 10^{-3},$ (5.4)

where the subscripts on the uncertainties indicate their origin [62].

Using the experimental value for $\Delta M_{s}$ [1],

$\Delta M_{s}^{\text{exp}}=(17.7656\pm 0.0057)\;\text{ps}^{-1},$ (5.5)

the theoretical prediction for $\Delta\Gamma_{s}$ is updated to be

$\Delta\Gamma_{s}^{\text{th}}=(0.076\pm 0.017)\;\text{ps}^{-1}.$ (5.6)

With this update, the theoretical uncertainty is about three times as large as
the experimental uncertainty given in Eq. (2.4). Calculations to further
improve upon this are already underway and are aiming to improve the accuracy
by including higher-order terms in $z$ as well as the penguin operator
contributions at NNLO.

##  6 Conclusion

New advancements in the accuracy of mixing and mixing-related $CP$-observables
were presented at the 12th Workshop on the CKM Unitarity Triangle. They
provide important insights into the principles governing nature, both in
experimental findings and theoretical developments.

Acknowledgements. Agnieszka, Thibaud and Vladyslav would like to thank all
speakers of the session for their excellent talks. The proceedings authors are
also grateful to the organisers for preparing the CKM 2023 workshop so well.
Agnieszka Dziurda expresses her gratitude to the Ministry of Science and
Higher Education in Poland, for financial support under the contract no
2022/WK/03. Felix Erben has received funding from the European Union’s Horizon
Europe research and innovation programme under the Marie Skłodowska-Curie
grant agreement No. 101106913. Thomas Latham is supported by the Science and
Technology Facilities Council (UK). Pascal Reeck was supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under grant 396021762
- TRR 257 “Particle Physics Phenomenology after the Higgs Discovery”.

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