12 December 2022

Heavy Flavour Physics and CP Violation at LHCb: a Ten-Year Review

Shanzhen Chen1, Yiming Li1, Wenbin Qian2, Zhihong Shen3, Yuehong Xie4,

Zhenwei Yang3, Liming Zhang5, Yanxi Zhang3

1Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, China
2University of Chinese Academy of Sciences, Beijing, China
3State Key Laboratory of Nuclear Physics and Technology & School of Physics,
Peking University, Beijing, China
4Key Laboratory of Quark and Lepton Physics of Ministry of Education &
Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
5Department of Engineering Physics & Center for High Energy Physics, Tsinghua
University, Beijing, China

Heavy flavour physics provides excellent opportunities to indirectly search
for new physics at very high energy scales and to study hadron properties for
deep understanding of the strong interaction. The LHCb experiment has been
playing a leading role in the study of heavy flavour physics since the start
of the LHC operations about ten years ago, and made a range of high-precision
measurements and unexpected discoveries, which may have far-reaching
implications on the field of particle physics. This review highlights a
selection of the most influential physics results on CP violation, rare
decays, and heavy flavour production and spectroscopy obtained by LHCb using
the data collected during the first two operation periods of the LHC. The
upgrade plan of LHCb and the physics prospects are also briefly discussed.

###### Contents

  1. 1 Introduction
  2. 2 Heavy-flavour production and spectroscopy
    1. 2.1 Production
    2. 2.2 Spectroscopy
  3. 3 Rare beauty hadron decays
    1. 3.1 Effective field theory for $b\rightarrow s$ transitions
    2. 3.2 Purely leptonic $B$ meson decays
    3. 3.3 Semileptonic $b\rightarrow s\ell^{+}\ell^{-}$ decays
    4. 3.4 Radiative $b\rightarrow s\gamma$ decays
    5. 3.5 Other rare decays of beauty hadrons
  4. 4 CP violation in beauty and CKM parameters
    1. 4.1 CKM angle $\gamma$
    2. 4.2 CKM angle $\beta$
    3. 4.3 CKM angle $\beta_{s}$
    4. 4.4 CKM elements $V_{ub}$ and $V_{cb}$
    5. 4.5 $\Delta m_{d}$ and $\Delta m_{s}$
    6. 4.6 Global fit
  5. 5 Charm mixing and CP violation
    1. 5.1 Neutral $D$ meson mixing
    2. 5.2 CP violation
  6. 6 Prospects and summary
    1. 6.1 Upgrade plan of LHCb
    2. 6.2 Physics prospects
    3. 6.3 Summary

##  1 Introduction

With the discovery of the Higgs boson at the Large Hadron Collider (LHC) in
2012 [1, 2, 3], all fundamental particles expected in the Standard Model (SM)
of particle physics are found and the SM is finally completed. The SM has
achieved tremendous success in explaining experimental results in high-energy
physics. However, there are still many key questions that are not answered in
the SM, such as the mechanism to generate the matter-antimatter asymmetry in
the Universe, the origin of the three generations of fermions and their
mixing, and the nature of dark matter and dark energy. It is commonly believed
that new physics (NP) beyond the SM should exist at or above the  TeV energy
scale. Flavour physics can provide a unique approach to indirectly probe NP at
energy scales far above  TeV via precision study of charge-parity (CP)
violation and rare phenomena, complementary to the direct search for new
particles and interactions at the energy frontier. Flavour physics also serves
as a natural laboratory to test quantum chromodynamics (QCD), the theory of
the strong interaction, via measurements of hadron production and
spectroscopy. The Large Hadron Collider beauty (LHCb) experiment has been
playing a leading role in the study of heavy-flavour physics since the start
of the LHC, and has made a series of discoveries and improvements in CP
violation, rare decays, and hadron production and spectroscopy. This review
aims to present a selection of the high-impact physics results on the above
subjects from the LHCb experiment, and to briefly discuss the prospects. Due
to the limited space, not all interesting results can be covered here. For a
complete list of all LHCb physics results, please refer to the official LHCb
summary [4].

The LHCb detector [5] is optimised for the study of the decays of heavy-
flavour hadrons, i.e. hadrons containing heavy quarks ($b$ or $c$ quarks,
often collectively referred to as $Q$). In proton-proton ($pp$) collisions at
LHC energies, the ${b}{\overline{b}}$ pairs are produced dominantly through
the gluon fusion process $gg\rightarrow{{b}{\overline{b}}}$. Due to the large
Lorentz boost along the proton beam in the laboratory frame, the $b$ and
$\overline{b}$ quarks generated in a pair are highly correlated in their
momentum directions, either both in the forward region or both in the backward
region in the majority of cases. In order to take advantage of this
characteristic of the ${b}{\overline{b}}$ pair at the LHC, the LHCb detector
is designed to have a forward geometry as shown in Fig. 1 to cover the forward
region of $pp$ collisions.

Figure 1:  Layout of the LHCb detector [5].

The flavour of the $b$-hadron under study can usually be tagged by the other
$b$-hadron that is also inside the LHCb acceptance. This enhances the
potential of the LHCb experiment in the study of CP violation and mixing with
${{B}^{0}}$ and ${{B}^{0}_{s}}$ decays.

Excellent vertex and momentum resolution, particle identification etc. are key
ingredients for flavour physics measurements at hadron colliders. A silicon
vertex locator (VELO) surrounding the $pp$ collision region is used to
precisely determine the primary interaction vertices (PVs) and the displaced
secondary/tertiary vertices (SVs/TVs) formed by the decay products of heavy-
flavour hadrons. The VELO system offers decay time measurements with a typical
resolution better than $50\text{\,fs}$, thus allows the LHCb experiment to
make precision measurement of hadron lifetimes and resolve the fast
${{B}^{0}_{s}}$-${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$
oscillation, which has a period of about 350 fs. The clear separation of SVs
from PVs also allows for substantial suppression of the combinatorial
background, which is extremely high in $pp$ collisions. In addition to the
VELO, the LHCb tracking system includes also four trackers, namely the TT and
T1-T3 stations, located upstream and downstream of the dipole magnet,
respectively. Together with a magnet that has a bending power of
$4\,\mathrm{Tm}$, the tracking system provides precise measurements of the
momenta of charged particles. The momentum resolution ($\Delta p/p$) is
typically 0.5% for low momentum tracks and 1.0% for track momentum up to
$200\text{\,Ge\kern-1.00006ptV\\!/}c$. The mass resolution for ${b}$-hadrons
can be as good as $8\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, precise enough to
distinguish decays of ${{B}^{0}}$ and ${{B}^{0}_{s}}$ mesons to the same final
state.

There are two ring-imaging Cherenkov detectors (RICH1 and RICH2) used to
identify charged hadrons in the momentum range
$p\in(2,100)\text{\,Ge\kern-1.00006ptV\\!/}c$. The RICH detectors are very
powerful at suppressing misidentification background for $b$-hadron decays to
final states containing charged kaons, pions or protons. The muon system
provides excellent muon identification and is essential for reconstruction of
decays with muons in the final states. The electromagnetic calorimeter is used
for photon and electron reconstruction and identification. Together with the
hadronic calorimeter, it also provides information for event trigger.

The trigger system is crucial for the success of the LHCb experiment. The
hardware trigger at the first level reduces the data rate from
$40\text{\,MHz}$ down to $1\text{\,MHz}$, at which point the flexible
software-based trigger takes over to further reduce the rate to around
$12\text{\,kHz}$ for offline processing and analysis. The ability to sustain
such large rates enables the LHCb experiment to trigger with high efficiency
on decay processes across a wide range of final states and to provide large
data samples for study of exclusive rare decay modes as well as for inclusive
data mining. Of particular importance is the LHCb muon trigger, which allows
events containing one muon or two muons to be selected with greater than 95%
efficiency. This is ideal for studying $b$-hadron decays to
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ or ${\psi{(2S)}}$ mesons that further
decay to ${\mu^{+}}{\mu^{-}}$ pairs and semileptonic decays into muons.
Meanwhile, there are also highly efficient triggers that are purely based on
the multi-body topology of the final state hadrons. These triggers are
important for reconstructing decays of heavy-quark hadrons to final states
without muons.

The LHCb experiment collected a data sample corresponding to an integrated
luminosity of 3$\text{\,fb}^{-1}$ in $pp$ collisions at centre-of-mass
energies $\sqrt{s}=7$ and $8\text{\,Te\kern-1.00006ptV}$ from 2011 to 2012
(Run 1), and another sample of 6$\text{\,fb}^{-1}$ at
$\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$ from 2015 to 2018 (Run 2). Results
discussed in this review are based on either full Run 1 and Run 2 data samples
or a subset. Since the cross-sections for $c$\- and $b$-quark production in
$pp$ collisions at 13 TeV are about twice of the cross-sections at 7 and 8 TeV
[6, 7, 8, 9, 10, 11, 12], and the trigger scheme for Run 2 has also been
improved compared with Run 1, the number of recorded charm and beauty decays
available for physics analysis is more than four times higher in the Run 2
data than in Run 1 data. In total, more than $10^{11}$ $b$-hadrons and
$10^{12}$ $c$-hadrons have been produced within the LHCb detector acceptance.
The typical trigger and selection efficiencies are of the order of $10^{-3}$
to $10^{-2}$ for decays only to charged particles and $10^{-4}$ to $10^{-3}$
for decays to final states involving photons, ${{\pi}^{0}}$, ${\mathchar
28931\relax}^{0}$ or ${{K}^{0}_{\mathrm{S}}}$ particles. The enormous $b$\-
and $c$-hadron samples form the basis of precision measurements of CP
violation, exploration of rare decays, and searches for new hadrons.

This review is structured as follows, with each section covering a different
subject. Recent results of heavy-flavour production and spectroscopy are shown
in Section 2. For heavy-flavour production, recent results of associated
production and the studies in heavy-ion collisions are shown; for
spectroscopy, the results of conventional hadrons and exotic hadrons are
summarised. Section 3 discusses rare $B$-hadron decays. The results on purely
leptonic $B$-meson decays, semileptonic $b\rightarrow{s}{\ell^{+}}{\ell^{-}}$
decays, and radiative $b\rightarrow{s}\gamma$ decays are shown.111Charge
conjugation is implied throughout unless stated otherwise. Some puzzling
results in angular distributions and lepton flavour universality tests are
discussed in details. Section 4 presents the latest results on CP violation in
the beauty sector. Emphasis is put on the progresses that have been made for
precision test of the Cabibbo-Kobayashi-Maskawa (CKM) mechanism, such as
significant improvement in the determination of the parameters $\gamma$,
$\beta$, $\beta_{s}$, ${V_{{u}{b}}^{\phantom{\ast}}}$,
${V_{{c}{b}}^{\phantom{\ast}}}$, and $\Delta m_{s/d}$. Section 5 provides
recent results of CP violation in the charm sector, including those for charm
mixing and the observation of CP violation in ${{D}^{0}}$ decays. The final
section provides a picture of the LHCb upgrade, as well as a brief summary of
the content in this review. The main goals and modifications to the detector
in Upgrade I and Upgrade II are introduced, and the prospects of some key
measurements are presented.

##  2 Heavy-flavour production and spectroscopy

Rich information on QCD dynamics can be deciphered from measurements of heavy-
flavour production [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49] and studies of heavy-flavour spectroscopy [50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71]. The mass
of a heavy quark, which is much larger than the nonperturbative QCD scale
$\Lambda_{\mathrm{QCD}}$, provides an energy scale that allows for
perturbative calculation of heavy-quark production. The production of heavy
quark pairs, $Q\overline{Q}$, is predominantly in the initial stage of the
collision, thus can be used to probe properties of the colliding system and
the possibly created QCD medium [72, 73]. The presence of heavy quark(s) also
provides practical benefits for theoretical and experimental studies of
spectroscopy. Heavy quarks are approximately nonrelativistic in hadrons, which
makes it possible to simplify theoretical calculations. The large mass and the
weak decay of heavy-flavour hadrons offer essential features, such as decay
products with high transverse momentum, $p_{\mathrm{T}}$, and vertices
displaced from the PVs, which can be exploited to reject the huge QCD
background at hadron colliders.

As discussed in Section 1, the excellent performance of the LHCb detector,
owing to the dedicated design for heavy-flavour hadrons, enables LHCb to make
great achievements in the study of heavy-flavour production and spectroscopy,
e.g. the observation of pentaquark states and the doubly charmed baryon
${\mathchar 28932\relax}_{cc}^{++}$. In this section, relevant results from
the LHCb experiment are reviewed, with a focus on recent developments.

###  2.1 Production

A summary of LHCb production measurements for open heavy-flavour hadrons,
heavy quarkonia and pairs of heavy-flavour hadrons at LHCb [74, 7, 75, 76, 77,
78, 79, 80, 81, 82, 83, 84, 9, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96,
10, 11, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110,
111, 112, 113, 114, 115, 116, 117, 118, 119] are listed in Table 1. Inclusive
hadroproduction of open heavy-flavour hadrons ($H_{Q}$) factorises into three
components in perturbative QCD (pQCD) calculations: the parton distribution
function (PDF) in the two initial projectiles $f_{i,j}$, the parton level
cross-section $\sigma_{ij\rightarrow Q+X}$ of a heavy-quark $Q$ production,
and the heavy-quark fragmentation function $D_{Q\rightarrow H_{Q}}$ [120].
Differential cross-section for $H_{Q}$ production in $A$-$B$ collisions is
expressed as

$\mathrm{d}\sigma_{AB\rightarrow H_{Q}}=\sum_{i,j}(f_{i}^{A}\otimes
f_{j}^{B})\otimes\mathrm{d}\sigma_{ij\rightarrow Q+X}\otimes D_{Q\rightarrow
H_{Q}},$ (1)

where the indices $i,j$ run over all possible parton species, and at LHC
energies heavy-flavour production is dominated by gluons. The PDF and
fragmentation function include nonperturbative effects, and can be determined
from a global fit of available data [120, 14]. The results of open charm and
beauty production at LHCb are consistent with pQCD models, for example the
calculation based on fixed-order plus next-to-leading logs (FONLL) [121]. It
turns out that the LHCb results on charm and beauty cross-sections have a
better precision than theoretical calculations [11, 105, 100], and can be used
to reduce the uncertainties on gluon PDF, in particular in the small
Bjorken-$x$ region, $x{~{}\raise 1.49994pt\hbox{$<$}\kern-8.50006pt\lower
3.50006pt\hbox{$\sim$}~{}}10^{-5}$ [122].

Table 1: LHCb measurements of production cross-sections (or ratios) for various heavy hadrons in $pp$ and $p$Pb collisions at different centre-of-mass energies. System |  $\sqrt{s}_{(NN)}$ ( TeV)
---|---
$5\,(2.76)$ |  $7$ |  $\hskip 34.14322pt8\,(8.16)$ |  $13$
$pp$ |  ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$[84], $\mathchar 28935\relax$[88], $D$[100] |  ${\eta_{c}}$[91], ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$[7], $\chi_{c1}(3872)$[78], ${\chi_{c}}/{{J\mskip-3.0mu/\mskip-2.0mu\psi}}$[77], ${\chi_{{c}2}}/{\chi_{{c}1}}$[76, 86], ${\psi{(2S)}}$[81, 111], $\mathchar 28935\relax$[79, 97], ${\chi_{{b}2}}/{\chi_{{b}1}}$[93, 92], $D,{{\mathchar 28931\relax}^{+}_{c}}$[9], $B$[80, 85, 105, 117], ${{\mathchar 28931\relax}^{0}_{b}}$[89, 96], ${{\mathchar 28932\relax}_{b}}$[109], ${{B}_{c}^{+}}$[83], ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{J\mskip-3.0mu/\mskip-2.0mu\psi}}$[75], $\mathchar 28935\relax D$[98], ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}D,DD,D{{\kern 1.61993pt\overline{\kern-1.61993ptD}}{}^{0}}$[82] |  ${\eta_{c}}$[91], ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$[95], ${\chi_{{b}2}}/{\chi_{{b}1}}$[93, 92], $\chi_{c1}(3872)/{\psi{(2S)}}$[116], $\chi_{c1}(3872)$[119], $\mathchar 28935\relax$[95, 97], ${{B}^{0}_{s}}/{{B}^{0}}$[117], ${{\mathchar 28931\relax}^{0}_{b}}$[96], ${{\mathchar 28932\relax}_{b}}$[109], $\mathchar 28935\relax D$[98], ${{B}_{c}^{+}}$[94] |  ${\eta_{c}}$[112], ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$[10, 102], $\chi_{c1}(3872)$[119], ${\psi{(2S)}}$[111], $\mathchar 28935\relax$[106], $D$[11], ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$[114], ${{{B}^{+}}}$[105], ${{\mathchar 28932\relax}_{b}}$[109], ${{B}_{c}^{+}}$[113], ${{B}^{0}_{s}}/{{B}^{0}}$[117], ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{J\mskip-3.0mu/\mskip-2.0mu\psi}}$[101]
$p$Pb |  ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$[87], ${\psi{(2S)}}$[99], $\mathchar 28935\relax$[90], ${{D}^{0}}$[104], ${{\mathchar 28931\relax}^{+}_{c}}$[107] |  |  ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$[103], $\mathchar 28935\relax$[108], ${{D}^{0}}$[74], ${{{B}^{+}}},{{B}^{0}},{{\mathchar 28931\relax}^{0}_{b}}$[110], ${\chi_{{c}2}}/{\chi_{{c}1}}$[118], $DD,D{{\kern 1.61993pt\overline{\kern-1.61993ptD}}{}^{0}}$[115] |

Figure 2: (Left) Differential cross-section of
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ in $pp$ collisions at
$\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$, taken from Ref. [10]. (Right)
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ polarisation in $pp$ collisions at
$\sqrt{s}=7\text{\,Te\kern-1.00006ptV}$, taken from Ref. [123]. The
measurements are compared with NRQCD [124, 125, 126] and colour singlet
calculations [125].

For quarkonium production, assumptions have to be put on how heavy-quark
pairs, $Q\overline{Q}$, produced with various possible colour, spin and parity
configurations, transform into specific colourless quarkonia [127, 128, 129,
130]. Cross-section measurements favour calculations using the nonrelativistic
QCD (NRQCD) framework [131], as shown on the left of Fig. 2, for
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ production in $pp$ collisions at
$\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$. The NRQCD framework introduces long
distance matrix elements (LDMEs) as model parameters [132] to account for
transition probabilities from heavy-quark pairs to quarkonia. LDMEs are
assumed to be independent of quarkonium production environments and
kinematics, and are fixed by matching the predicted $p_{\mathrm{T}}$ spectrum
to data. The polarisation of heavy quarkonia is another observable sensitive
to the $Q\overline{Q}$ production mechanism and LDMEs. Inconsistencies are
observed between LHCb data and theoretical calculations on the $\psi$ and
$\mathchar 28935\relax$ polarisations [133, 134, 123]. Only a level of 10% or
smaller polarisation is observed in the LHCb acceptance, in contrast to a
dominant transverse polarisation predicted by NRQCD [124, 126, 125]. The
measurement of ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ polarisation is shown on
the right of Fig. 2. Even though the discrepancy can be reduced by tuning the
LDMEs, a coherent description of production cross-section and polarisation is
still a difficult theoretical problem [126, 135, 31, 26, 136, 137, 138, 139,
140]. If only the $Q\overline{Q}$ state that has the same quantum number as
the final quarkonium is considered, the NRQCD framework reduces to the colour
singlet model, which underestimates production cross-sections [125] and
disagrees with data on $\psi$ and $\mathchar 28935\relax$ polarisations [123,
134, 133].

####  2.1.1 Associated production

Recent LHCb production measurements focus on associated production of multiple
heavy flavours and quantities probing properties of QCD matter. Associated
heavy-flavour production provides an approach to study the multiple parton
interactions (MPIs). MPIs are sensitive to correlations between partons in
space, momentum, flavour, colour, spin etc. inside the colliding projectiles
[141, 142, 143, 144, 145, 146]. Usually in an MPI process these correlations
are assumed to be absent initially such that each parton-scattering is
independent from each other, and then consistency checks are performed to
verify this assumption. Under this assumption, the cross-section for
associated production of $ab$ through a double-parton-scattering (DPS) process
is related to the single inclusive production of $a$ and $b$ as [141]

$\sigma^{ab}=\kappa\frac{\sigma^{a}\sigma^{b}}{\sigma_{\mathrm{eff}}},$ (2)

where $\kappa$ is a symmetry factor with $\kappa=1$ if $a\neq b$ and the
effective cross-section $\sigma_{\mathrm{eff}}$ is assumed to be universal.
Heavy-quark fragmentations in MPIs are implied to be identical to that in
inclusive production defined in Eq. 2. In particular, the kinematics of $a$
and $b$ is uncorrelated and each of them is similar to that in single particle
inclusive production. Besides MPI, the single parton scattering (SPS) is also
able to generate associated production, but in SPS the final-state kinematics
is correlated. This difference between DPS and SPS is used to identify DPS.
Studies of DPS include measurements of the $\sigma_{\mathrm{eff}}$ parameter
and tests of its universality for different states, and investigations of
kinematic correlations between $a$ and $b$. One example of correlation
variables is the relative azimuthal angle $\Delta\phi$ between $a$ and $b$ and
to infer the correlations between colliding partons. For DPS production
$\Delta\phi$ distribution is approximately flat, while in SPS events a
concentration at $\Delta\phi\sim 0$ or $\pi$ is expected.

Measurements of associated production in $pp$ collisions are made at LHCb for
two open charm hadrons [82], a heavy quarkonium plus an open charm [98, 82],
and double ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ mesons [101]. To subtract the
SPS contribution from data, one usually relies on theoretical inputs for
cross-sections of SPS or fits to data using templates of correlation variables
built for both SPS and DPS production. Note that theoretical uncertainties are
still much larger than experimental ones for these measurements.

For some associated production, SPS is estimated or assumed to be negligible,
resulting in a sample of approximately pure DPS events. In this case the
$\sigma_{\mathrm{eff}}$ parameter measured using Eq. 2 is around
$15\text{\,mb}$ for ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}D$ and $\mathchar
28935\relax D$ production, independent of the $D$ species and collision
centre-of-mass energies [82, 98]. The results are similar to those extracted
using multi-jet production at Tevatron [147]. However, the values obtained
using same-sign $DD$ pairs are around $20\text{\,mb}$, and that for
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ pairs
is about $7\text{\,mb}$ [101]. The former are consistently higher than the
value of $15\text{\,mb}$ [82], while the latter is significantly lower. Higher
values of $\sigma_{\mathrm{eff}}$ for
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{J\mskip-3.0mu/\mskip-2.0mu\psi}}$
production are obtained if a fraction of SPS is subtracted. The SPS fraction
is estimated to be between $20\%$ and $40\%$ depending on the choice of
control variables and input templates for SPS and DPS distributions [101]. The
smaller $\sigma_{\mathrm{eff}}$ measurement for
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ pairs
confirms previous observations by D0, CMS and ATLAS experiments for quarkonium
pairs [148, 149, 150].

For correlation variables, as shown on the left of Fig. 3, the $\Delta\phi$
distributions of ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}D$ events are reasonably
flat [82], consistent with the DPS production in which
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ and $D$ kinematics is uncorrelated. This
observation is a sign of dominant or pure DPS contribution for
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}D$ samples. For same-sign $DD$ production,
the correlation variables also favour DPS dominance [82].

The $p_{\mathrm{T}}$ distribution of each hadron in the pair production is
also studied. For DPS events, it is expected to be similar to that in single
inclusive production. For $\mathchar 28935\relax D$ samples, both the
$\mathchar 28935\relax$ and the $D$ meson have a $p_{\mathrm{T}}$ distribution
similar to that in single inclusive production [98]. The same conclusion holds
for the $p_{\mathrm{T}}$ distribution of $D$ mesons in
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}D$ events. However, the $p_{\mathrm{T}}$ of
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ mesons in
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}D$ events is significantly harder than that
in inclusive production, indicated by the right of Fig. 3. For the same-sign
$DD$ production, the $p_{\mathrm{T}}$ distribution of $D$ mesons is also
significantly harder than that in single inclusive $D$ production, but are
similar to those in opposite-sign $D{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ samples [82]. This result is
inconsistent with the observation in correlation variables, which hints at a
dominant DPS (SPS) contribution in the same-sign (opposite-sign) pair
production. Note that for associated production of $DD$ pairs, the
$p_{\mathrm{T}}$ distribution of $D$ mesons is similar for different $D$
species, indicating that charm hadron fragmentations are not modified, so that
the unexpected $p_{\mathrm{T}}$ distribution is not due to the fragmentation
process. A detailed theoretical calculation on
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}D$ production was performed recently to
understand the problem, but a solid conclusion is not available yet [146].

Figure 3: (Left) Distribution of relative azimuthal angle, $\Delta\phi$,
between ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ and $D$ mesons in pair
production, showing flat behaviour; (Right) the $p_{\mathrm{T}}$ distribution
of ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ mesons in
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}D$ pair production, compared with that in
inclusive production (shown in black open circle) [7]. Figures are taken from
Ref. [82].

####  2.1.2 Production in $p$Pb collisions

Charm pair production is also studied in $p$Pb collisions of a center-of-mass
energy per nucleon pair
$\sqrt{s_{\scriptscriptstyle\text{NN}}}=8.16\text{\,Te\kern-1.00006ptV}$
[115]. The DPS cross-section in $p$Pb collisions is expected to scale with
three times of the Pb mass number ($A_{\mathrm{Pb}}=208$) with respect to that
in $pp$ data at the same $\sqrt{s_{\scriptscriptstyle\text{NN}}}$, rather than
a simple scale factor of $A_{\mathrm{Pb}}$, when nuclear matter effects are
not considered [151]. The $A$-scaling is relevant for SPS production in the
absence of nuclear matter effects [151]. In the end DPS production has a
factor of three enhancement compared with SPS. The cross-section ratio between
same-sign $DD$ and opposite-sign $D{\kern
1.79993pt\overline{\kern-1.79993ptD}}$ signals is measured to be around three
times of that in $pp$ collisions [115]. The result is in favour of the
expected factor-three enhancement. The parameter $\sigma_{\mathrm{eff}}$ is
measured with ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}D$ and same-sign $DD$
production as shown on the left of Fig. 4, for the positive rapidity region,
which corresponds to the Pb-beam direction (high Bjorken-$x$ of Pb nucleus),
and the negative rapidity region, which corresponds to the $p$-beam direction
(low Bjorken-$x$ of Pb nucleus). The measurements show that, similar to the
results in $pp$ data, the $\sigma_{\mathrm{eff}}$ parameter for
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}D$ production is about 30% smaller than
that for same-sign $DD$ production. Besides, the results in negative rapidity
hint at smaller values than those in positive rapidity for both
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}D$ and $DD$ pair production. It may be a
sign of universality violation of $\sigma_{\mathrm{eff}}$. This will be
explored with better precision in Run 3 heavy-ion collision data, where ten
times more luminosity is expected to be collected.

Figure 4: (Left) The parameter $\sigma_{\mathrm{eff}}$ measured in $p$Pb
collisions at
$\sqrt{s_{\scriptscriptstyle\text{NN}}}=8.16\text{\,Te\kern-1.00006ptV}$ using
(green) ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{D}^{0}}$ and (brown)
${{D}^{0}}{{D}^{0}}$ production for (positive rapidity) $p$ beam and (negative
rapidity) Pb-beam direction, taken from [115]. The prediction shown in shaded
area is from Ref. [152]. (Right) The ratio of cross-sections between
$\mathchar 28935\relax(3S)$ and $\mathchar 28935\relax(1S)$ over that in $pp$
data at $\sqrt{s_{\scriptscriptstyle\text{NN}}}=8\text{\,Te\kern-1.00006ptV}$,
taken from Ref. [108], is compared with the comovers model [153]. The
rapidities are defined in the rest frame of two colliding nucleons with
respect to proton-beam direction.

Besides the enhancement of DPS production, heavy nuclear collisions have many
more new phenomena compared with $pp$ collisions, collectively called nuclear
matter effects. Presence of nuclear matter effects in $p$Pb collisions could
modify the PDF, or reduce parton energies or dissociate heavy quarkonia, which
can be probed using heavy-flavour production in $p$Pb data compared with the
$A_{\mathrm{Pb}}$ scaled $pp$ cross-section [154, 155, 156]. Measurements of
${{D}^{0}}$ and ${{{B}^{+}}}$ production in $p$Pb data suggest heavy-quark
production in the $p$-beam direction is significantly suppressed compared with
the $A_{\mathrm{Pb}}$ scaling, by about 30%, while the production in Pb-beam
direction approximately scales with $A_{\mathrm{Pb}}$ [104, 110]. The results
are consistent with modifications of the gluon PDF in a Pb nucleus compared
with that in a free nucleon. The LHCb measurements are found to be able to
reduce the gluon PDF uncertainties in the Pb nucleus by about a factor of
three compared with the commonly used nuclear PDF sets [157]. A new precise
measurement of ${{D}^{0}}$ production in $p$Pb shows that the magnitude of the
${{D}^{0}}$ suppression in $p$ beam direction over that in Pb-beam direction,
i.e. the forward-backward ratio $R_{\mathrm{FB}}$, increases significantly at
high $p_{\mathrm{T}}({{D}^{0}})$ and seems to reach unity at
$p_{\mathrm{T}}({{D}^{0}})>8\text{\,Ge\kern-1.00006ptV\\!/}c$ [74]. However
according to predictions using the nuclear PDF, the $R_{\mathrm{FB}}$ is about
70%, almost independent of $p_{\mathrm{T}}({{D}^{0}})$. The observed trend of
$R_{\mathrm{FB}}$ for ${{D}^{0}}$ may be caused by the parton energy loss
effect which alters heavy-flavour kinematic distribution, whose impact is
reduced at high $p_{\mathrm{T}}$ [158], otherwise the result will require a
modification of current knowledge of the nuclear PDF.

Measurement of ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ production in $p$Pb data
shows a similar trend of suppression compared with open heavy-flavour hadrons
[87, 103], suggesting that they suffer from a common influence by nuclear
matter effects. However, the result for ${\psi{(2S)}}$ in $p$Pb data suggests
a stronger suppression compared with ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$, in
particularly in the Pb-beam direction [99]. Similarly, the $\mathchar
28935\relax(3S)$ meson is measured to be more suppressed compared with
$\mathchar 28935\relax(1S)$ [108], as shown on the right of Fig. 4. The
stronger suppression for excited quarkonium cannot be explained using the
nuclear PDF modification or the parton energy loss effect. The comover model
introducing final state interactions between a heavy quarkonium and comoving
particles is able to explain data [153]. The comovers effect is stronger in
events of higher occupancy and for particles with larger sizes, such that it
is more pronounced in Pb-beam direction and for excited states, reducing their
yields more significantly than for the ground state in the $p$-beam direction
$A_{\mathrm{Pb}}$ . The comovers mechanism also exists in $pp$ collisions, and
is probed using heavy quarkonium production. The cross-section ratio between
prompt $\chi_{c1}(3872)$ and ${\psi{(2S)}}$ mesons is measured to decrease
with the increase of the number of reconstructed tracks in the vertex detector
[116]. It suggests that the $\chi_{c1}(3872)$ state has a larger size or a
smaller binding energy compared with the ${\psi{(2S)}}$ meson, and is
consistent with a component of ${{D}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}+{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{*0}}{{D}^{0}}$ hadron molecule in the
$\chi_{c1}(3872)$ wave function [159, 160]. The same measurement in $p$Pb data
feasible in LHC Run 3 period would be very important to confirm this result.

In the near future, new data will provide enough statistics for associated
production of triple heavy flavours and heavy quarkonium pairs beyond
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{J\mskip-3.0mu/\mskip-2.0mu\psi}}$, which
help to further understand MPI and heavy quarkonium production mechanism
[141]. Concerning studies of nuclear matter effects using heavy-flavour
production, a rich program is foreseen in Run 3, including measurements
probing the DPS enhancement, the modification of nuclear PDF and heavy-quark
fragmentation and the heavy quarkonium dissociation mechanism.

###  2.2 Spectroscopy

The strong interaction confines quarks (and/or gluons) to form various colour-
singlet hadrons that are accessible experimentally. This confinement
phenomenon is nonperturbative and is not fully understood yet from the current
QCD theory. In analogy to photon spectroscopy in atomic physics, hadron
spectroscopy provides a way to understand dynamics of QCD at low energies.
Hadrons composed of a quark and an antiquark are called mesons, those of three
quarks are called baryons, and those composed of more than three quarks are
usually referred to as exotic hadrons. Existence of exotic hadrons have been
predicted since the birth of the quark model and their properties are
reexamined by refined theoretical approaches in the past decades [161, 162,
163, 164, 165, 166, 167, 168, 169]. The past years witnessed the observations
of a plethora of new conventional and exotic hadrons containing heavy quarks
and LHCb is one of the leading players in the field. Figure 5 displays the 68
new hadrons that are discovered by the LHC experiments, and most of them by
LHCb.

Figure 5: New hadrons that are discovered by the LHC experiments, taken from
[170].

####  2.2.1 Conventional hadrons

LHCb have filled many gaps in conventional heavy meson and baryon spectra.
This section will focus on more recent highlights in baryon spectroscopy and
discovery of doubly heavy hadrons. Though not discussed in detail, it should
not be ignored that multiple excited beauty [171, 172] and charm [173] mesons
are observed in high energy $pp$ collisions by LHCb. Moreover, the beauty
hadron decay is an ideal place to study excited charm hadron as $b\rightarrow
c$ is the dominant transition of the $b$ quark. Among the many new charm
mesons discovered in $B$ decay at LHCb [174, 175, 176], an interesting example
is a new excited $D_{s}^{+}$ meson, $D_{s}(2590)^{+}$, consistent with the
radial excited state $D_{s}(2^{1}S_{0})^{+}$. Its mass and width will help to
understand the excitation spectrum of $D_{s}^{+}$ mesons which are found to be
not fully consistent with the quark model predictions [177, 178, 179, 180,
181, 182].

##### Classification of heavy baryons

Following the heavy-quark symmetry, baryons with a heavy quark $Q$ are
organised into multiplets according to quantum configurations of the two light
quarks [183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196,
197, 198, 199]. The total wave function including flavour ($F$), spin
($s_{qq^{\prime}}$) and orbital angular momentum ($l_{qq^{\prime}}$) must be
symmetric for the two light quarks $qq^{\prime}$ to form an antisymmetric
state together with their antisymmetric colour configuration. Baryons with
$l_{qq^{\prime}}=s_{qq^{\prime}}=0$ (antisymmetric in spin space) have a spin-
parity of $J^{P}=1/2^{+}$, and are grouped into a multiplet of three flavour-
antisymmetric states for each heavy quark $Q$. While baryons with
$s_{qq^{\prime}}=1,l_{qq^{\prime}}=0$ (symmetric in spin space) have
$J^{P}=1/2^{+}$ or $J^{P}=3/2^{+}$, and form a multiplet of six flavour-
symmetric states for each $J^{P}$. These three different multiplets are shown
in the bottom row of Fig. 6. Orbital and radial excitation can happen inside
the two light quarks ($\rho$-mode) or between $Q$ and the $qq^{\prime}$ system
($\lambda$-mode). The parity of a baryon is determined to be
$P=(-1)^{l_{\rho}+l_{\lambda}}$, where $l_{\rho}\equiv l_{qq^{\prime}}$, and
$l_{\lambda}$ is the orbital angular momentum between the $Q$ and
$qq^{\prime}$. Beauty and charm baryons with
$l_{\lambda}=l_{qq^{\prime}}=s_{qq^{\prime}}=0$ decay weakly and have been
well established. However a chart of their excited states are far from being
complete. As a matter of fact only a few low lying states are observed, in
particularly for beauty baryons as can be seen in Fig. 6. Up to date no sign
of $\rho$-mode states have been identified experimentally, probably because
they are too wide (hundreds of  MeV) to be resolved from underlying
background.

##### Charm baryons

In the invariant mass spectrum of ${\mathchar 28932\relax}^{+}_{c}{{K}^{-}}$
hadrons shown on the left of Fig. 7, LHCb observed five states whose quark
contents are considered to be $css$: ${\mathchar 28938\relax}_{c}(3000)^{0}$,
${\mathchar 28938\relax}_{c}(3050)^{0}$, ${\mathchar
28938\relax}_{c}(3066)^{0}$, ${\mathchar 28938\relax}_{c}(3090)^{0}$ and
${\mathchar 28938\relax}_{c}(3119)^{0}$ [200]. All these states have narrow
widths, below $10\text{\,Me\kern-1.00006ptV}$, and their mass differences are
only tens of  MeV. The first four states are confirmed by Belle in
${e^{+}e^{-}}$ collisions [201] and by LHCb in the exclusive ${\mathchar
28938\relax}_{b}^{-}\rightarrow{\mathchar
28932\relax}^{+}_{c}{{K}^{-}}{{\pi}^{-}}$ decay [202]. The spin assignments of
the first four states favour $1/2,3/2,3/2,5/2$, consistent with the
expectations for $P$-wave $\lambda$ excitation [202]. A determination of their
parities will help to make firm conclusions. According to phenomenological
models [203, 204, 205, 206, 207, 208, 209], one of the five $1P$ states with
$l_{\rho}=0,l_{\lambda}=1$ in the mass region of observed states is missing,
and the ${\mathchar 28938\relax}_{c}(3119)^{0}$ state may be a $2S$ or
$D$-wave baryon. In the high mass region of LHCb data, a hint of a wide state
${\mathchar 28938\relax}_{c}(3188)^{0}$ is present, to be confirmed in future
analysis with additional statistics. It is noted that some of these states are
considered to be exotic states of quark constituents
$cssu{\overline{u}}/d{\overline{d}}$ rather than conventional $css$ baryons
[210, 211, 212, 213, 214].

Similarly, excited ${\mathchar 28932\relax}^{0}_{c}$ states are searched for
by LHCb [215] in the ${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}$ invariant
mass spectrum shown on the right of Fig. 7. A new state ${{\mathchar
28932\relax}_{c}}(2965)^{0}$ is observed, and the state ${{\mathchar
28932\relax}_{c}}(2930)^{0}$ claimed by the Belle experiment [216] now splits
into two structures, ${{\mathchar 28932\relax}_{c}}(2923)^{0}$ and
${{\mathchar 28932\relax}_{c}}(2939)^{0}$. Separation of the ${{\mathchar
28932\relax}_{c}}(2923)^{0}$ and ${{\mathchar 28932\relax}_{c}}(2939)^{0}$ is
recently confirmed in $B$ decay by LHCb [217]. The widths of these three
states are determined to be around $10\text{\,Me\kern-1.00006ptV}$. These
states and previously known ${{\mathchar 28932\relax}_{c}}(2790)^{0}$ and
${{\mathchar 28932\relax}_{c}}(2815)^{0}$ lie in the mass region of $1P$
excitation [218, 219, 220, 221, 222]. There are in total seven $1P$
${\mathchar 28932\relax}^{0}_{c}$ states of the $\lambda$-excitation. At least
two of these $1P$ states are still missing; Besides, a complete and solid
matching of these observed states to predicted spectrum is not resolved yet
[223].

Figure 6: Experimentally identified bottom baryons, grouped according to spin
and flavour symmetry of the light quarks as well as the excitation between the
light quark system and $b$ quark. The assignments of quantum numbers are based
on measured properties and relevant predictions. Note that for recently
discovered states their $J^{P}$ need confirmation and the assignments of
excited ${\mathchar 28938\relax}_{b}$ are not certain, though they likely have
$J^{P}=1^{-}$. In the figure, 1$P$ excitation of flavour symmetric bottom
baryons are gathered into a single multiplet (shown in purple). Only $\lambda$
mode radial and orbital angular momentum excitation are considered. Missing
states are marked as question marks in the figure. The notation
$s_{qq^{\prime}}$ indicates the total spin of the light quark system.

Figure 7: (Left) Invariant mass distribution of ${\mathchar
28932\relax}^{+}_{c}{{K}^{-}}$, overlaid with the fit including five narrow
${\mathchar 28938\relax}_{c}$ states, taken from Ref. [200]. (Right) Invariant
mass distribution of ${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}$, overlaid
with the fit including three narrow ${{\mathchar 28932\relax}_{c}}$ states
(red, brown and purple dashed lines) and partially reconstructed components,
taken from Ref. [215].

##### Bottom baryons

A summary of all bottom baryons with clear experiment evidences are shown in
Fig. 6, organised in multiplets of $qq^{\prime}$ flavour symmetry and
$\lambda$-mode excitation when possible. The spin-parity quantum numbers for
most of these states are not measured, so we rely on theoretical calculations
as a guidance to make the classification. Actually, there is not always a
consensus on the $J^{P}$ of each state, in particularly for ${\mathchar
28938\relax}_{b}$ baryons [224, 225, 226, 227, 228, 229, 230, 231, 232, 233,
234, 235, 236, 237, 238, 239].

In total, five states have been reported in the ${{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}}$ mass spectrum: ${{\mathchar
28931\relax}_{b}}(5912)^{0}$ and ${{\mathchar 28931\relax}_{b}}(5920)^{0}$
with widths below $1\text{\,Me\kern-1.00006ptV}$ [240], ${{\mathchar
28931\relax}_{b}}(6146)^{0}$ and ${{\mathchar 28931\relax}_{b}}(6152)^{0}$
with widths of about $2\text{\,Me\kern-1.00006ptV}$ [241], and ${{\mathchar
28931\relax}_{b}}(6072)^{0}$ with a width around
$70\text{\,Me\kern-1.00006ptV}$ [242]. Their masses match two $1P$, two $1D$
and $2S$ ${{\mathchar 28931\relax}^{0}_{b}}$ $\lambda$-mode excitation
respectively, though other assignments are also discussed [243, 244, 245, 246,
247]. It is useful to note that intermediate ${\mathchar
28934\relax}^{(*)\pm}_{b}(\rightarrow{{\mathchar
28931\relax}^{0}_{b}}\pi^{\pm})$ states are found to be present in the
${{\mathchar 28931\relax}_{b}}^{*(*)0}\rightarrow{{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}}$ decays.

The ground ${\mathchar 28934\relax}^{\pm}_{b}$ and ${\mathchar
28934\relax}^{*\pm}_{b}$ states were first detected in the ${{\mathchar
28931\relax}^{0}_{b}}\pi^{\pm}$ mass spectrum by CDF [248]. In the same final
state, two new ones, ${{\mathchar 28934\relax}_{b}}(6097)^{+}$ and
${{\mathchar 28934\relax}_{b}}(6097)^{-}$, are observed by LHCb [249], whose
widths are about $30\text{\,Me\kern-1.00006ptV}$. These two new states belong
to the $P$-wave family, and many more of them are still missing, like for
charm baryons.

In analogy, excited ${{\mathchar 28932\relax}_{b}}$ states are searched for by
the LHC experiments in the ${\mathchar 28932\relax}^{0}_{b}\pi^{-}/{\mathchar
28932\relax}^{-}_{b}\pi^{+}$ spectra. New states close to ${{\mathchar
28932\relax}_{b}}\pi$ mass thresholds are observed, which include the low
lying ${\mathchar 28932\relax}^{*0}_{b}$ baryon discovered by CMS [250] and
${\mathchar 28932\relax}^{{}^{\prime}-}_{b}$, ${\mathchar
28932\relax}^{*-}_{b}$ states discovered by LHCb [251]. The states ${\mathchar
28932\relax}^{{}^{\prime}-}_{b}$ and ${\mathchar 28934\relax}^{\pm}_{b}$
belong to the flavour symmetric multiplet with
$s_{qq^{\prime}}=1,J^{P}=1/2^{+}$, while ${\mathchar 28932\relax}^{*0}_{b}$,
${\mathchar 28932\relax}^{*-}_{b}$ and ${\mathchar 28934\relax}^{*\pm}_{b}$
belong to the flavour symmetric multiplet with
$s_{qq^{\prime}}=1,J^{P}=3/2^{+}$. Going to the higher mass region, a state
${{\mathchar 28932\relax}_{b}}(6227)^{-}$, with a width around
$20\text{\,Me\kern-1.00006ptV}$, is found in both ${{\mathchar
28932\relax}^{0}_{b}}{{\pi}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}{{K}^{-}}$ final states [252], and its flavour partner
${{\mathchar 28932\relax}_{b}}(6227)^{0}$ is found in the ${{\mathchar
28932\relax}^{-}_{b}}{{\pi}^{+}}$ mass spectrum [253]. They can be matched to
$P$-wave states or a mixture of several $P$-wave states with masses close to
$6227\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. Very recently, two new states
${{\mathchar 28932\relax}_{b}}(6327)^{0}$ and ${{\mathchar
28932\relax}_{b}}(6333)^{0}$, with widths below
$2\text{\,Me\kern-1.00006ptV}$, are found in the ${{\mathchar
28931\relax}^{0}_{b}}{{K}^{-}}{{\pi}^{+}}$ mass spectrum [254], consistent
with the $1D$ excitation of the ${{\mathchar 28932\relax}^{0}_{b}}$ baryon.
These two states may also be present in the ${{\mathchar
28932\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}}$ sample as well, demanding a
future investigation of this decay mode by LHCb. In fact, in the ${{\mathchar
28932\relax}^{-}_{b}}{{\pi}^{+}}{{\pi}^{-}}$ spectrum, a ${{\mathchar
28932\relax}_{b}}(6100)^{-}$ state is observed by CMS [255], consistent with
the $1P$ excitation of the flavour antisymmetric ${{\mathchar
28932\relax}^{-}_{b}}$ state with $J^{P}=3/2^{-}$. Apparently, the other $1P$
state with $J^{P}=1/2^{-}$ and a mass around
$6100\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ is missing. No states with higher
masses, for example flavour partners of ${{\mathchar
28932\relax}_{b}}(6327)^{0}$ and ${{\mathchar 28932\relax}_{b}}(6333)^{0}$,
are reported by CMS, which may be explained by lower production rate for these
states.

Excited ${{\mathchar 28938\relax}^{-}_{b}}$ states are searched for in the
${{\mathchar 28932\relax}^{0}_{b}}{{K}^{-}}$ mass spectrum [256]. Four narrow
(width $<5\text{\,Me\kern-1.00006ptV}$) peaking structures are identified with
two of them having significance greater than five standard deviations
($5\sigma$), named as ${\mathchar 28938\relax}_{b}(6340)^{-}$ and ${\mathchar
28938\relax}_{b}(6350)^{-}$ respectively. These states lie in the mass region
of $P$-wave excitation. More statistics in Run 3 will allow for a further
investigation of these states.

Figure 8: (Left) Invariant mass distribution of $\Delta M\equiv
m_{{{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}}-m_{{{B}_{c}^{+}}}$, overlaid with the
fit including two narrow structures, corresponding to ${B}_{c}^{*}(2S)^{+}$
and ${B}_{c}(2S)^{+}$ respectively, taken from Ref. [257]. (Right)
Distribution of the reconstructed invariant mass
$m_{\mathrm{cand}}({{\mathchar 28932\relax}^{++}_{{c}{c}}})\equiv
m({{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}})-m_{\mathrm{cand}}({{\mathchar
28931\relax}^{+}_{c}})+m_{\mathrm{PDG}}({{\mathchar 28931\relax}^{+}_{c}})$,
overlaid with the fit projections, taken from Ref. [258].

##### Doubly heavy hadrons

LHCb has observed a few new hadrons with two heavy quarks. In the final states
of ${{D}^{0}}{{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ and
${{D}^{+}}{{D}^{-}}$ mesons produced promptly, a new particle $X(3842)$, with
a width of about $3\text{\,Me\kern-1.00006ptV}$, is discovered [259]. It is
consistent with the spin-3 conventional charmonium $\psi_{3}({}^{3}D_{3})$
with $J^{PC}=3^{--}$ [260].

As shown on the left of Fig. 8, two narrow structures are detected in the
${{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ invariant mass spectrum of LHCb data
[257]. The left one corresponds to the
${B}_{c}^{*}(2^{3}S_{1})^{+}\rightarrow{B}_{c}^{*}(1^{3}S_{1})^{+}{{\pi}^{+}}{{\pi}^{-}}$
decay with the photon in decay of
${B}_{c}^{*}(1^{3}S_{1})^{+}\rightarrow{{B}_{c}^{+}}\gamma$ not detected. The
peak on the right is consistent with the
${B}_{c}(2^{1}S_{0})^{+}\rightarrow{{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decay.
Almost in parallel, these two states are independently observed by CMS [261].
As the ${{B}_{c}^{+}}$ meson is composed of two heavy quarks, its excitation
mass spectroscopy can be calculated using models similar to those applied to
heavy quarkonia [262, 263, 264, 265, 266, 267] despite their hadroproduction
mechanisms are very different [268, 269].

LHCb opens a new era in studies of doubly heavy baryons by observing the
${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ baryon in the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ mass spectrum [258], as
shown on the right of Fig. 8. This discovery decay mode is predicted to have a
relatively large branching fraction [270, 271]. The ${{\mathchar
28932\relax}^{++}_{{c}{c}}}$ state is later confirmed using the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ decay [272]. Its lifetime is measured to be
about $0.25\text{\,ps}$ [273] and its mass is precisely determined to be
$(3621.55\pm 0.38)\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ [273]. Its SU(3)
partners, ${{\mathchar 28932\relax}^{+}_{{c}{c}}}$ and ${{\mathchar
28938\relax}^{+}_{{c}{c}}}$, are also searched for by LHCb, but with no sign
of observation yet [274, 275, 276]. Theoretically, an important question is to
understand the mass spectrum of the doubly charm baryons and related systems,
which depends on the binding energy between the two heavy quarks [277, 278,
279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289]. The successful
discovery of ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ has triggered wide
theoretical work to understand the properties of baryons with more than a
heavy quark [290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302,
303, 304, 305, 306, 307, 308, 309, 310, 311, 312]. In addtion, the
${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ baryon mass is used to study the
stability of tetraquark states with $QQ$ contents [313, 314] with the
assumption that $QQ$ form a heavy diquark [315, 316, 317, 318, 319].

####  2.2.2 Exotic hadrons

Many theoretical efforts have been placed to understand how quarks are
combined to form a multi-body system [320, 321, 322, 323]. At the same time,
more and more new states are observed experimentally which cannot fit into the
conventional hadron spectra [54]. As a result, the study on exotic hadrons has
been a hot topic for the past decade. New results from LHCb are discussed
below.

##### Tetraquark states

LHCb provides essential information for the understanding of previously known
tetraquark states. For example LHCb determined the quantum number of the
$X(3872)$ state, first reported by Belle [324], to be $J^{PC}=1^{++}$ through
a full amplitude analysis [325]. LHCb also precisely measured the mass of the
$X(3872)$ state (referred to as $\chi_{c1}(3872)$ in Ref. [326]) to be
$m_{\chi_{c1}(3872)}-m_{{\psi{(2S)}}}=185.49\pm 0.06\pm
0.03\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ [327, 328]. Its Breit-Wigner (BW)
width is determined by LHCb to be
$\Gamma_{\chi_{c1}(3872)}^{\mathrm{BW}}=0.96^{+0.19}_{-0.18}\pm
0.21\text{\,Me\kern-1.00006ptV}$ [327, 328]. Evidence of its decay to
$\psi(2S)\gamma$ is found and the branching fraction relative to
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\gamma$ is measured to $2.46\pm 0.64\pm
0.29$ [329], disfavouring a pure $D\bar{D}^{*}$ molecule intepretation. The
$\rho^{0}$ and $\omega$ contributions are disentangled in its decay to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{\pi}^{+}}{{\pi}^{-}}$ and a sizeable
contribution from $\omega$ is confirmed [330]. The $\chi_{c1}(3872)$ state is
the mostly studied exotic candidate, and its exotic behaviours include a
extremely narrow width, isospin breaking decays and a mass close to the
$D^{*}D$ threshold. Despite all the available information we are still not
sure whether it is a compact tetraquark state, a ${{D}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ hadron molecule, a mixture of
${{D}^{*0}}{{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ molecule with
a ${\chi_{{c}1}}(2P)$ charmonium component or just caused by kinematic
rescattering effect [331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341,
342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356,
357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371,
372, 373].

The exotic candidate $T_{\psi 1}^{b}(4430)^{+}$ was first observed by Belle in
the ${\psi{(2S)}}{{\pi}^{-}}$ mass spectrum in
${{B}^{0}}\rightarrow{\psi{(2S)}}{{K}^{+}}{{\pi}^{-}}$ decays [374].222The
exotic hadron naming scheme is used [375]. An amplitude analysis of the
${{B}^{0}}\rightarrow{\psi{(2S)}}{{K}^{+}}{{\pi}^{-}}$ decay is performed at
LHCb, confirming the existence of the $T_{\psi 1}^{b}(4430)^{+}$ state and
determining it to be consistent with a Breit-Wigner resonance with
$J^{P}=1^{+}$ [376]. The quark contents of $T_{\psi 1}^{b}(4430)^{+}$,
$c{\overline{c}}{\overline{u}}d$, are the same as the $T_{\psi
1}^{b}(3900)^{+}$ state observed by the BESIII experiment [377]. Being
charged, they are definitely not consistent with conventional charmonia and
many phenomenological calculations are performed to explain their internal
structure and properties [378, 379, 380, 381, 382, 383, 384, 385, 386, 387,
388, 389, 390, 391, 392, 393, 394, 395].

The ${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi{{K}^{+}}$
decay is a zoo of exotic hadrons. In 2009, a narrow state $X(4140)$ was
reported by CDF in the ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi$ mass spectrum
of the ${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi{{K}^{+}}$
decay [396, 397], and is later confirmed by CMS [398]. The quark contents of
the $X(4140)$ state is likely to be ${c}{\overline{c}}{s}{\overline{s}}$,
consistent with an exotic hadron [399, 400], even though excited conventional
charmonia may have the chance to decay into
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi$ too. In the amplitude analysis by
LHCb using Run 1 data, four exotic candidates $X(4140)$, $X(4274)$, $X(4500)$
and $X(4700)$ are observed [401]. Currently, these $X$ states are considered
to be either hadron molecules or compact tetraquark states or high-mass
conventional charmonia in various calculations [402, 403, 404, 405, 406, 407,
408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422,
423, 424, 425, 426, 427, 428]. The LHCb analysis is updated recently with a
sample that has six times more statistics, in which three more $X$ states are
reported [429]. In addition, two $T_{\psi s}^{+}$ structures are observed in
the ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ mass spectrum. The mass
spectra and fit projections are shown in Fig. 9 and the properties of these
exotic candidates are summarised in Table 2. The $T_{\psi s}^{+}$ states mark
the first observation of exotic hadrons with an $s$ quark through beauty
decays. It is noted that another $T_{\psi s}^{+}$ state is reported by BESIII
in the final state of $D_{s}^{-}D^{*0}+D_{s}^{*-}{{D}^{0}}$ pairs [430], with
a mass and width different from those observed by LHCb. Very recently,
evidence of $T_{\psi s}^{0}$, isopin partner of $T_{\psi s}^{+}$, is found by
LHCb in the
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi{{K}^{0}_{\mathrm{S}}}$
decay through a combined amplitude analysis of both
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi{{K}^{0}_{\mathrm{S}}}$
and ${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi{{K}^{+}}$
decays [431]. The masses, widths of these two states and their contributions
to the $B\rightarrow T_{\psi s}K$ decay are similar, confirming that they are
isospin partners. There should be more states of
$c{\overline{c}}q{\overline{s}}$ quark contents, and their discovery will
definitely help to understand the internal structure of strange tetraquark
states [432, 433].

Figure 9: Distributions of (left) $\phi{{K}^{+}}$, (middle)${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi$ and (right) ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ invariant masses for (black data points) the ${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi{{K}^{+}}$ candidates compared with (red solid lines) the fit results using (top row) the final model and (bottom row) the model of LHCb Run 1 analysis [401]. Figure taken from Ref. [429]. Table 2: Spin-parity, significance, masses, widths, and fit fractions of exotic candidates observed in the ${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi{{K}^{+}}$ decay by LHCb [429]. Contribution | Significance [$\times\sigma$] |  $M_{0}$ [ MeV] |  $\Gamma_{0}$ [ MeV] | FF [%]
---|---|---|---|---
$X(2^{-})$ |  |  |  |
$X(4150)$ | 4.8 (8.7) |  $4146$ $\pm$ | $18\pm 33$ |  $135$ $\pm$ | $28\,_{-\,30}^{+\,59}$ |  $2.0$ $\pm$ | $0.5\,_{-\,1.0}^{+\,0.8}$
$X(1^{-})$ |  |  |  |
$X(4630)$ | 5.5 (5.7) |  $4626$ $\pm$ | $16\,_{-\,110}^{+\,\phantom{0}18}$ |  $174$ $\pm$ | $27\,_{-\,\phantom{0}73}^{+\,134}$ |  $2.6$ $\pm$ | $0.5\,_{-\,1.5}^{+\,2.9}$
All $X(0^{+})$ |  |  |  |  $20$ $\pm$ | $5\,_{-\,\phantom{0}7}^{+\,14}$
$X(4500)$ | 20 (20) |  $4474$ $\pm$ | $3\pm 3$ |  $77$ $\pm$ | $6\,_{-\,\phantom{0}8}^{+\,10}$ |  $5.6$ $\pm$ | $0.7\,_{-\,0.6}^{+\,2.4}$
$X(4700)$ | 17 (18) |  $4694$ $\pm$ | $4\,_{-\,\phantom{0}3}^{+\,16}$ |  $87$ $\pm$ | $8\,_{-\,\phantom{0}6}^{+\,16}$ |  $8.9$ $\pm$ | $1.2\,_{-\,1.4}^{+\,4.9}$
All $X(1^{+})$ |  |  |  |  $26$ $\pm$ | $3\,_{-\,10}^{+\,\phantom{0}8}$
$X(4140)$ | 13 (16) |  $4118$ $\pm$ | $11\,_{-\,36}^{+\,19}$ |  $162$ $\pm$ | $21\,_{-\,49}^{+\,24}$ |  $17$ $\pm$ | $3\,_{-\,6}^{+\,19}$
$X(4274)$ | 18 (18) |  $4294$ $\pm$ | $4\,_{-\,6}^{+\,3}$ |  $53$ $\pm$ | $5\pm 5$ |  $2.8$ $\pm$ | $0.5\,_{-\,0.4}^{+\,0.8}$
$X(4685)$ | 15 (15) |  $4684$ $\pm$ | $7\,_{-\,16}^{+\,13}$ |  $126$ $\pm$ | $15\,_{-\,41}^{+\,37}$ |  $7.2$ $\pm$ | $1.0\,_{-\,2.0}^{+\,4.0}$
All $T_{\psi s}(1^{+})$ |  |  |  |  $25$ $\pm$ | $5\,_{-\,12}^{+\,11}$
$T_{\psi s}(4000)$ | 15 (16) |  $4003$ $\pm$ | $6\,_{-\,14}^{+\,\phantom{0}4}$ |  $131$ $\pm$ | $15\pm 26$ |  $9.4$ $\pm$ | $2.1\pm 3.4$
$T_{\psi s}(4220)$ | 5.9 (8.4) |  $4216$ $\pm$ | $24\,_{-\,30}^{+\,43}$ |  $233$ $\pm$ | $52\,_{-\,73}^{+\,97}$ |  $10$ $\pm$ | $4\,_{-\,\phantom{0}7}^{+\,10}$

Exotic hadrons are also searched for in open charm final states using fully
reconstructed beauty hadron decays. A Dalitz analysis of the
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decay is performed by LHCb
[434, 435], and two exotic states, $T_{cs0}(2900)^{0}$ and
$T_{cs1}(2900)^{0}$, are required to have a good fit to the
${{D}^{-}}{{K}^{+}}$ invariant mass spectrum. The ${{D}^{-}}{{K}^{+}}$
invariant-mass distribution and the fit projections are shown in Fig. 10.
Their spin-parities are measured to be $J^{P}=0^{+}$ and $1^{-}$, and widths
to be about $50\text{\,Me\kern-1.00006ptV}$ and
$100\text{\,Me\kern-1.00006ptV}$ respectively. The quark contents of these $X$
states are ${c}{s}{\overline{u}}{\overline{d}}$. These two $X$ states
contribute up to $35\%$ of the total
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decay branching fraction,
a magnitude similar to those of conventional charmonia in the decay. A final
state rescattering effect is considered in order to explain such a large
branching fraction [436]. In a recent analysis of the
${{B}^{0}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{D}^{+}_{s}}{{\pi}^{-}}$ and
${{{B}^{+}}}\rightarrow{{D}^{-}}{{D}^{+}_{s}}{{\pi}^{+}}$ decays by LHCb, two
new states $T_{c\overline{s}0}^{a}(2900)^{++/0}$ are observed in the
${{D}^{+}_{s}}\pi^{+/-}$ systems [437, 438]. The quark contents of these two
states are ${c}{\overline{s}}{d}{\overline{u}}$ and
${c}{\overline{s}}{u}{\overline{d}}$ respectively. The invariant mass
distributions of ${{D}^{+}_{s}}{{\pi}^{-}}$ and ${{D}^{+}_{s}}{{\pi}^{+}}$ are
shown in Fig. 10 superimposed with the amplitude fit results. The masses and
width of these two states are measured to be about
$2.9\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ and
$0.15\text{\,Ge\kern-1.00006ptV}$ respectively, and they contribute to about
3% of the total $B\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{D}^{+}_{s}}{{\pi}^{-}}$ decay.
The mass of the $T_{{c}{\overline{s}}0}^{a}(2900)$ is consistent with the
previously mentioned $T_{cs0}(2900)^{0}$ discovered in the
${{D}^{-}}{{K}^{+}}$ final state, but their widths and flavour contents are
different. The observation of $X\rightarrow Dh$ ($D={{D}^{+}_{s}},D$,
$h=\pi,K$) states in the $B\rightarrow D\overline{D}h$ decay opens a new
avenue for studies of exotic hadrons composed of four different quark flavours
[439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452].
Actually there are more than two dozens of $B\rightarrow DDK(\pi)$ decays, and
also a few similar decays for $b$-baryons, and it is promising that more $X$
states will be observed in these decays.

Recently, a Dalitz analysis for the
${{{B}^{+}}}\rightarrow{{D}^{+}_{s}}{{D}^{-}_{s}}{{K}^{+}}$ decay was
performed by the LHCb collaboration [453, 454]. A near-threshold structure in
the ${{D}^{+}_{s}}{{D}^{-}_{s}}$ system is observed with a significance larger
than $12\sigma$. The ${{D}^{+}_{s}}{{D}^{-}_{s}}$ invariant-mass distribution
is shown on the left of Fig. 10. The spin-parity of the $X(3960)$ state is
determined to be $J^{PC}=0^{++}$. The $X(3960)$ state is similar to the
$\chi_{c0}(3930)$ state in Ref. [326]. If they are the same states, the
partial width ratio is measured to be

$\frac{\Gamma\left(X\rightarrow D^{+}D^{-}\right)}{\Gamma\left(X\rightarrow
D_{s}^{+}D_{s}^{-}\right)}=0.29\pm 0.09\pm 0.10\pm 0.08,$

where the first uncertainty is statistical, the second systematic, and the
third external. The ratio is smaller than unity. Since the creation of
${s}{\overline{s}}$ from vacuum is suppressed than ${u}{\overline{u}}$ or
${d}{\overline{d}}$ and the phase-space factor of $X\rightarrow D^{+}D^{-}$ is
smaller than $X\rightarrow{{D}^{+}_{s}}{{D}^{-}_{s}}$, the
$\Gamma\left(X\rightarrow D^{+}D^{-}\right)$ is expected to be larger than
$\Gamma\left(X\rightarrow D_{s}^{+}D_{s}^{-}\right)$, which is inconsistent
with the results from experiment. The inconsistency indicating the exotic
nature of the $X(3960)$ states under the assumption of the $X(3960)$ and the
$\chi_{c0}(3930)$ being the same states.

Figure 10:  (Top left) Invariant-mass distribution of ${{D}^{-}}{{K}^{+}}$ in
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decays in data and the fit
projections, taken from Ref. [435]. The yellow dot-dashed and red dotted line
correspond to $T_{cs0}(2900)^{0}$ and $T_{cs1}(2900)^{0}$, respectively, and
more details can be found in Ref. [435]. (Top right) The distribution of the
invariant mass of the ${{D}^{+}_{s}}{{D}^{-}_{s}}$ system in the
${{{B}^{+}}}\rightarrow{{D}^{+}_{s}}{{D}^{-}_{s}}{{K}^{+}}$ decay. A near-
threshold structure, denoted as $X(3960)$, is marked in red [453]. The
distribution of the invariant mass of (bottom left) the
${{D}^{+}_{s}}{{\pi}^{-}}$ pair in the ${{B}^{0}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{D}^{+}_{s}}{{\pi}^{-}}$ and
(bottom right) ${{D}^{+}_{s}}{{\pi}^{+}}$ system in the
${{{B}^{+}}}\rightarrow{{D}^{-}}{{D}^{+}_{s}}{{\pi}^{+}}$ decay [438].

Prompt production of di-charm hadrons has been suggested to search for exotic
states containing multiple charm quarks [455]. In the invariant mass spectrum
of ${{D}^{0}}{{D}^{0}}{{\pi}^{+}}$, shown in Fig. 11, a new structure is
observed close to the $D^{*+}{{D}^{0}}$ mass threshold [456, 457]. The
structure is measured to be consistent with the ground state of a $T_{cc}^{+}$
isoscalar tetraquark, with $J^{P}=1^{+}$ and quark contents
$cc{\overline{u}}{\overline{d}}$. Its Breit-Wigner mass is measured to be
$-273\pm 61\pm 5^{+11}_{-14}\text{\,ke\kern-1.00006ptV\\!/}c^{2}$ below
$m_{D^{*+}}+m_{{D}^{0}}$, and its Breit-Wigner width is
$\Gamma_{\mathrm{BW}}=410\pm 165\pm
43^{+18}_{-38}\text{\,ke\kern-1.00006ptV}$. The same state also appears in the
${{D}^{0}}{{D}^{0}}$ and ${{D}^{0}}{{D}^{+}}$ mass spectra, with a
${{\pi}^{+}}$, ${{\pi}^{0}}$ or $\gamma$ in the
$T_{cc}^{+}\rightarrow{{D}^{0}}{{D}^{0}}{{\pi}^{+}}$,
$T_{cc}^{+}\rightarrow{{D}^{0}}{{D}^{+}}{{\pi}^{0}}$ or
$T_{cc}^{+}\rightarrow{{D}^{0}}{{D}^{+}}\gamma$ decays undetected,
respectively. Dedicated studies of the $T_{cc}^{+}$ resonance lineshape are
performed using a unitarised Breit-Wigner distribution, considering
$T_{cc}^{+}$ decays in
${{D}^{0}}{{D}^{0}}{{\pi}^{+}},{{D}^{0}}{{D}^{+}}{{\pi}^{0}}$ and
${{D}^{0}}{{D}^{0}}\gamma$ final states. The pole mass of the resonance in
this advanced model is measured to be $-360\pm
4^{+4}_{-0}\text{\,ke\kern-1.00006ptV\\!/}c^{2}$ below
$m_{{D}^{*}}+m_{{D}^{0}}$, and the pole width is
$\Gamma_{\mathrm{pole}}=-48\pm 2^{+0}_{-14}\text{\,ke\kern-1.00006ptV}$. This
extremely narrow width has attracted many theoretical interests [458, 459,
460, 461, 462, 463]. The $T_{cc}^{+}$ state is the first observed tetraquark
candidates with two heavy quarks of the same flavour. Many theoretical models
have been applied to explain the existence and structure of such a tetraquark
state with a large fraction of them favouring a $D^{*+}{{D}^{0}}$ hadron
molecule interpretation [464, 465, 466, 467, 468, 469, 470, 471, 472, 473,
474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486].

Figure 11:  (Left) Distribution of ${{D}^{0}}{{D}^{0}}{{\pi}^{+}}$ invariant
mass overlaid with fit projections. The contribution of the non-${{D}^{0}}$
background has been statistically subtracted. Vertical dashed lines show the
$D^{*+}{{D}^{0}}$ and ${{D}^{*0}}{{D}^{+}}$ mass thresholds respectively for a
comparison. Figure taken from Ref. [457]. (Right) Invariant mass distribution
of the combination of two ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ mesons overlaid
with fit projections, taken from Ref. [487]. The fit model does not contain
interference between any components.

In addition to two open charm final states,
di-${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ mass spectrum of prompt production is
also studied using full LHCb data [487]. Two peaking structures are observed
in the mass range
$6.2<m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{J\mskip-3.0mu/\mskip-2.0mu\psi}}}<7.4\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$,
where fully charmed tetraquark states are predicted [488, 489, 490, 491, 492,
493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507,
508, 509, 510, 511, 512]. The first structure (referred to as the threshold
peak) covers the range between $6.2$ and
$6.6\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ close to the
di-${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ mass threshold, and the other one sits
at $6.9\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, as shown on the left of Fig. 11.
The di-${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ mass spectrum is modelled with a
combination of BW functions for the two peaking structures and empirical
smooth functions for SPS, DPS production of nonpeaking background. When no
interference between BW and SPS is applied, the threshold structure can be
described by two BW functions and the one at
$6.9\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ is well described by a BW. The
narrow structure, denoted as $T_{\psi\psi}(6900)$, is measured to have a mass
of $m_{T_{\psi\psi}(6900)}=6905\pm 11\pm
7\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and a width of
$\Gamma_{T_{\psi\psi}(6900)}=80\pm 19\pm 33\text{\,Me\kern-1.00006ptV}$.
Interpretations of the threshold peak using feeddown decays from excited
quarkonium pairs are also possible. For various fit models without any
interference, the dip around $6.8\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ can’t
be well described. Advance fit studies are performed introducing interference
between SPS and resonant structures. In one such fit, two BW functions are
considered: a broad BW interfering with SPS used to describe the threshold
structure, and a stand-alone narrow one used to model the
$6.9\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ peak. This new model could fit well
the overall spectrum, and the broad structure is now measured to have a mass
around $6.7\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ and a width of about
$0.3\text{\,Ge\kern-1.00006ptV}$, while the $T_{\psi\psi}(6900)$ structure has
a mass consistent with the no-interference fit model, but its width becomes
about twice larger. As the fit results for the broad structure is not stable
in different models, its nature is not fully resolved and more data are needed
to provide better information. The $T_{\psi\psi}(6900)$ and higher resonances
have been recently confirmed by CMS [513] and ATLAS [514] The
$T_{\psi\psi}(6900)$ state is consistent with a genuine fully charmed
tetraquark, however, in some models possible origins due to rescatterings of
multiple charmonia or dibaryon molecules etc. are also discussed [515, 516,
517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531,
532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546,
547, 548, 549, 550, 551]. Other fully heavy tetraquarks, such as
${b}{\overline{b}}{c}{\overline{c}}$ and ${b}{\overline{b}}{b}{\overline{b}}$,
can be searched for in $\mathchar
28935\relax{{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ and $\mathchar
28935\relax\mathchar 28935\relax$ or similar final states, however current
limited data are estimated to have small sensitivities for these states,
demanding the increased luminosity in LHCb upgrades.

##### Pentaquark states

Following the successful discoveries of tetraquarks with $Q\overline{Q}$
contents, pentaquark states with $Q\overline{Q}$ were predicted, in the form
of either meson-baryon hadron molecules or compact five-quark hadrons [552,
553, 554, 555]. Since 2015, several pentaquark candidates are found in LHCb,
as summarised in Table 3. The first observation of them is made in
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}p$ final states in ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}p{{K}^{-}}$
decays through an amplitude analysis of Run 1 data [556]. Two states were
reported, $P_{\psi}^{N}(4380)^{+}$ and $P_{\psi}^{N}(4450)^{+}$, and their
evidence is also found in the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}p{{\pi}^{-}}$
decays with a similar amplitude study [557]. A model-independent moment
analysis of the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}p{{K}^{-}}$
decay concludes that contributions of ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}p$
exotics are essential, since only allowing ${\mathchar
28931\relax}^{*}\rightarrow p{{K}^{-}}$ resonances in the decay are not
sufficient to describe data [558]. With full LHCb data, an amplitude analysis
of ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}p{{K}^{-}}$
decays becomes computationally very difficult. On the other hand, benefiting
from the high statistics, one dimensional
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}p$ mass spectrum is investigated to look
for narrow pentaquark states [559]. In this new analysis, the
$P_{\psi}^{N}(4450)^{+}$ structure is found to consist of two narrow
overlapping peaks $P_{\psi}^{N}(4440)^{+}$ and $P_{\psi}^{N}(4457)^{+}$, and a
new structure $P_{\psi}^{N}(4312)^{+}$ is observed. It is noted that the
$P_{\psi}^{N}(4312)^{+}$ and $P_{\psi}^{N}(4457)^{+}$ states are close to the
${\mathchar 28934\relax}^{+}_{c}{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ and ${\mathchar
28934\relax}^{+}_{c}{{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{*0}}$
mass thresholds respectively, as shown in Fig. 12 making them ideal candidates
of meson-baryon molecules. Recently, LHCb reported the evidence of a new
pentaquark state, $P_{\psi}^{N}(4337)^{+}$, in the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}p({\overline{p}})$ mass spectrum of
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}p{\overline{p}}$
decays [560]. This possible state is different from those observed in
${{\mathchar 28931\relax}^{0}_{b}}$ decays, making beauty meson decays a new
place to search for pentaquarks.

Pentaquark candidates with strangeness are predicted in the ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar
28931\relax}{{K}^{-}}$ decay [561, 562], which is an analogy of the
${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}p{{K}^{-}}$
channel by replacing the $d$ quark by the $s$ quark. An amplitude analysis of
the ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar
28931\relax}{{K}^{-}}$ decay is performed using full LHCb data, resulting in
the evidence of a new state, $P_{\psi s}^{{\mathchar 28931\relax}}(4459)^{0}$,
in the ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar 28931\relax}$ mass
spectrum, with quark contents, ${c}{\overline{c}}{u}{d}{s}$. Its mass is
measured to be $4458.8\pm
2.9^{+4.7}_{-1.1}\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and width to be
$17.3\pm 6.5^{+8.0}_{-5.7}\text{\,Me\kern-1.00006ptV}$ [563]. According to the
prediction in Ref. [561], there are two ${{\mathchar
28932\relax}_{c}}{{D}^{*}}$ hadron molecules with masses within a few
$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ around the observed structure. If the
$P_{\psi s}^{{\mathchar 28931\relax}}(4459)^{0}$ structure is found to be
composed of two nearby states with LHCb upgrade data, it will be a strong
proof of the molecular interpretation of such states. Recently, an amplitude
analysis of the
${{{B}^{-}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar
28931\relax}{\overline{p}}$ was performed [564]. A narrow structure in the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar 28931\relax}$ system, denoted as
$P_{\psi s}^{{\mathchar 28931\relax}}(4338)^{0}$, is observed with high
significance, which is consistent with pentaquark state with strangeness. The
invariant-mass distribution of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar 28931\relax}$ system is shown in
Fig. 12. The mass and width of the state are measured to be $4338.2\pm 0.7\pm
0.4\text{\,Me\kern-1.00006ptV}$ and $7.0\pm 1.2\pm
1.3\text{\,Me\kern-1.00006ptV}$, respectively, where the first uncertainty is
statistical and the second systematic. The spin-parity of the state is
determined to be $J^{P}=\frac{1}{2}^{-}$.

Figure 12:  (Left) Invariant mass distribution of
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}p$ in the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{p}{{K}^{-}}$
decay fitted with three BW functions plus polynomial background, taken from
Ref. [487]. The vertical lines mark the ${\mathchar
28934\relax}^{+}_{c}{{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ and
the ${\mathchar 28934\relax}^{+}_{c}{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{*0}}$ mass thresholds respectively.
(Right) Invariant mass distribution of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar 28931\relax}$ system in the
${{\mathchar
28932\relax}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar
28931\relax}{{K}^{-}}$ decay overlaid with fit projections, taken from Ref.
[564].

Observations of pentaquark states with heavy-quark contents have triggered
many studies from theorists, with the purpose of understanding their nature.
In general, these states are considered to either be compact tetraquarks,
hadronic molecules or simple bumps due to kinematic rescatterings [565, 566,
567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581,
582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596,
597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611,
612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626,
627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638]. Measurements of
their $J^{P}$ and production properties, and finding their flavour partners
will shed light on the problem. However, it is likely that debates on the
nature of exotic hadrons will continue before we have a complete and coherent
theory to explain all of them. In the past years, phenomenology models on the
hadron spectroscopy evolved quickly and some patterns have been revealed, for
example, dynamics close to two-hadron mass thresholds [639]. Besides, lattice
QCD simulations have improved in computational performances and application
scopes substantially over the years and will play more and more important
roles in our understanding of low-energy QCD [640, 641, 642, 643, 644, 645,
646]. Hopefully, one day we can predict low energy QCD phenomena as precise as
other parts of the SM.

Table 3: Detection decay channels, experimental significance, masses and widths of pentaquark states reported by LHCb. State | Decays | Significance [$\sigma$] | Mass [$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$] | Width [ MeV]
---|---|---|---|---
$P_{\psi}^{N}(4312)^{+}$ | ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}p$ | $7.3\sigma$ | $4311.9\pm 0.7^{+6.8}_{-0.6}$ | $\phantom{0}9.8\pm 2.7^{+\phantom{0}3.7}_{-\phantom{0}4.5}$
$P_{\psi}^{N}(4440)^{+}$ | ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}p$ | $5.4\sigma$ | $4440.3\pm 1.3^{+4.1}_{-4.7}$ | $20.6\pm 4.9^{+\phantom{0}8.7}_{-10.1}$
$P_{\psi}^{N}(4457)^{+}$ | ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}p$ | $5.4\sigma$ | $4457.3\pm 0.6^{+4.1}_{-1.7}$ | $0.53\pm 2.0^{+\phantom{0}5.7}_{-\phantom{0}1.9}$
$P_{\psi}^{N}(4337)^{+}$ | ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}p$ | $3.1\sigma$ | $\phantom{0}4337\phantom{0}^{+\phantom{0}7}_{-\phantom{0}4}\phantom{0}\pm 2$ | $\phantom{0}\phantom{0}29^{+\phantom{0}26}_{-\phantom{0}12}\pm 14$
$P_{\psi s}^{{\mathchar 28931\relax}}(4459)^{0}$ | ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar 28931\relax}$ | $3.1\sigma$ | $4458.8\pm 2.9^{+4.7}_{-1.1}$ | $17.3\pm 6.5^{+\phantom{0}8.0}_{-\phantom{0}5.7}$
$P_{\psi s}^{{\mathchar 28931\relax}}(4338)^{0}$ | ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar 28931\relax}$ | $>15\sigma$ | $4338.2\pm 0.7\pm 0.4$ | $7.0\pm 1.2\pm 1.3$

##  3 Rare beauty hadron decays

By rare decays we mainly refer to flavour-changing-neutral-current (FCNC)
processes, which are highly suppressed in the SM by the Glashow-Iliopoulos-
Maiani (GIM) mechanism [647]. Rare decays of hadrons could receive significant
contributions from new particles or new interactions beyond the SM. Precision
measurements of their properties play a special role in search of physics
beyond the SM.

The LHCb collaboration has given priority to the study of FCNC $b\rightarrow
s$ transitions, focusing on theoretically clean observables such as decay
rates of purely leptonic $B$-meson decays, angular coefficients in
$b\rightarrow s\ell^{+}\ell^{-}$ decays, and ratio of decay rates between
$b\rightarrow s\ell^{+}\ell^{-}$ processes with different lepton flavours.
Analyses of $pp$ collision data collected in the Run 1 and Run 2 periods have
already led to some very important and interesting findings, including but not
limited to the first observation of the purely leptonic decay
${{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}}$, anomalous angular distribution in
the decay ${{B}^{0}}\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$, and intriguing
results from extensive tests of lepton flavour universality in $B\rightarrow
K^{(*)}\ell^{+}\ell^{-}$ decays.

###  3.1 Effective field theory for $b\rightarrow s$ transitions

The effective Hamiltonian describing the quark-level $b\rightarrow s$
transitions is given by [648, 649, 650, 651, 652, 653, 654]

$H_{\rm eff}(b\rightarrow
s)=-{\frac{4G_{F}}{\sqrt{2}}}{V_{{t}{b}}^{\phantom{\ast}}}{V_{{t}{s}}^{\ast}}\sum_{i=1}^{10}C_{i}\mathcal{O}_{{i},}$
(3)

with $\mathcal{O}_{i}$ denoting the local operators in the SM and $C_{i}$
indicating the corresponding Wilson coefficients. Of particular interest is
the electromagnetic dipole operator corresponding to penguin diagrams mediated
by photons,

$\mathcal{O}_{7}={\dfrac{e}{16\pi^{2}}}m_{b}({\overline{s}}\sigma_{\mu\nu}P_{R}b)F^{\mu\nu},$
(4)

and semileptonic operators corresponding to loop diagrams mediated by $Z^{0}$
or ${W^{\pm}}$ bosons,

$\mathcal{O}_{9}={\dfrac{e^{2}}{16\pi^{2}}}({\overline{s}}\gamma_{\mu}P_{L}b)(\overline{\ell}\gamma^{\mu}\ell),\,\,\mathcal{O}_{10}={\dfrac{e^{2}}{16\pi^{2}}}({\overline{s}}\gamma_{\mu}P_{L}b)(\overline{\ell}\gamma^{\mu}\gamma_{5}\ell),$
(5)

where $P_{L}=(1-\gamma_{5})/2$ and $P_{R}=(1+\gamma_{5})/2$. Contribution from
physics beyond the SM can either alter the values of the Wilson coefficients
and/or give rise to new operators that are absent or highly suppressed in the
SM, such as the scalar and pseudo-scalar operators

$\mathcal{O}_{S}={\dfrac{e^{2}}{16\pi^{2}}}({\overline{s}}P_{L}b)(\overline{ell}\ell),\,\,\mathcal{O}_{P}={\dfrac{e^{2}}{16\pi^{2}}}({\overline{s}}P_{L}b)(\overline{\ell}\gamma_{5}\ell)\;,$
(6)

and the chirality-flipped operators $\mathcal{O}_{7,8,9,10,S,P}^{\prime}$,
which are obtained by changing $P_{L(R)}$ to $P_{R(L)}$ in
$\mathcal{O}_{7,8,9,10,S,P}$.

The Wilson coefficients $C^{(\prime)}_{7}$ can be probed in radiative
$b$-hadron decays such as ${{B}^{0}_{s}}\rightarrow\phi\gamma$ and
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}\gamma$,
$C^{(\prime)}_{9,10}$ probed in semileptonic decays such as
$B^{0}\rightarrow{{K}^{+}}\ell^{+}\ell^{-}$ and
$B^{+}\rightarrow{{K}^{*0}}\ell^{+}\ell^{-}$, and $C^{(\prime)}_{S,P}$ probed
in purely leptonic decays such as $B^{0}_{s}\rightarrow\ell^{+}\ell^{-}$.
Recent LHCb results on $b\rightarrow s$ transitions are summarised in the
remainder of this section.

###  3.2 Purely leptonic $B$ meson decays

The decays ${{B}_{({s})}^{0}}\rightarrow\ell^{+}\ell^{-}(\ell=\mu,\,e)$ are
among the most interesting probes of new physics. They are theoretically clean
and are expected to be extremely rare in the SM due to helicity suppression in
addition to the FCNC loop suppression. Their branching fractions in the SM are
precisely predicted to be [655]

$\displaystyle{\mathcal{B}}({{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}})$
$\displaystyle=(3.66\pm 0.14)\times 10^{-9},$ (7)
$\displaystyle{\mathcal{B}}({{B}^{0}}\rightarrow{\mu^{+}\mu^{-}})$
$\displaystyle=(1.03\pm 0.05)\times 10^{-10},$
$\displaystyle{\mathcal{B}}({{B}^{0}_{s}}\rightarrow{e^{+}e^{-}})$
$\displaystyle=(8.60\pm 0.36)\times 10^{-14},$
$\displaystyle{\mathcal{B}}({{B}^{0}}\rightarrow{e^{+}e^{-}})$
$\displaystyle=(2.41\pm 0.13)\times 10^{-15}.$

Note the ${{B}^{0}}\rightarrow\ell^{+}\ell^{-}$ decays proceed via
$b\rightarrow d$ transitions, thus are further suppressed with respect to the
${{B}^{0}_{s}}\rightarrow\ell^{+}\ell^{-}$ decays by a factor of
$|{V_{{t}{d}}^{\phantom{\ast}}}/{V_{{t}{s}}^{\phantom{\ast}}}|^{2}\sim\lambda^{2}$.
The decay rates of ${{B}_{({s})}^{0}}\rightarrow\ell^{+}\ell^{-}$ processes
are highly sensitive to (pseudo-)scalar interactions beyond the SM.

A joint analysis of data from the LHCb and CMS experiments collected in Run 1
led to the observation of the ${{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}}$ decay
with a significance exceeding six standard deviations, and determined the
branching fraction to be
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}})=(2.8^{+0.7}_{-0.6})\times
10^{-9}$ [656]. This result was later updated by LHCb [657] and CMS [658] by
adding the 2016 data. The ATLAS collaboration reported an evidence for the
decay ${{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}}$ with a significance of
$4.6\sigma$ using the data collected between 2011 and 2016 [659]. However, no
significant signal for the decay ${{B}^{0}}\rightarrow{\mu^{+}\mu^{-}}$ has
been found by any experiment yet. A combination of the results from ATLAS, CMS
and LHCb gives the branching fraction
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}})=(2.69^{+0.37}_{-0.35})\times
10^{-9}$, and sets an upper limit of
${\mathcal{B}}({{B}^{0}}\rightarrow{\mu^{+}\mu^{-}})<1.9\times 10^{-10}$ at
95% confidence level [660]. Figure 13 shows the constraints on
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}})$ and
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}})$ from the three
experiments and the combined results, which are compatible with the SM
predictions [655] within 2.1 $\sigma$. The difference is mainly driven by the
ATLAS results, which have an optimal solution outside the physical region and
are slightly in tension with the SM predictions.

Very recently, LHCb reported updated results on the
${{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}}$ and
${{B}^{0}}\rightarrow{\mu^{+}\mu^{-}}$ decays using all data collected in Run
1 and Run 2. The results are
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}})=(3.09^{+0.46+0.15}_{-0.43-0.11})\times
10^{-9}$, ${\mathcal{B}}({{B}^{0}}\rightarrow{\mu^{+}\mu^{-}})<2.6\times
10^{-10}$ at 95% confidence level [661, 662], which are in good agreement with
the SM expectations, as shown in Fig. 14.

In addition to the decay rate, other interesting quantities, such as effective
lifetime and CP asymmetry, can also be measured to search for possible non-SM
contribution to the decay ${{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}}$ [663].
Pioneering studies of its effective lifetime using all data collected in Run 1
and Run 2 have been performed at LHCb, leading to the result of $\tau^{\rm
eff}({{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}})=2.07\pm 0.29\pm 0.03$ ps [661,
662]. As a long-term goal, the effective lifetime and time-dependent CP
violation of the decay ${{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}}$ will be
fully exploited in LHCb upgrade II, which aims to accumulate a $pp$ collision
data sample of 300$\text{\,fb}^{-1}$ [664].

Figure 13:  (Left) Likelihood contours in the
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}})$ \-
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}})$ plane corresponding
to $-2\Delta\ln L=2.3,\,6.2$ and $11.8$, for ATLAS, CMS and LHCb experiments
and the combination. (Right) Likelihood contours for the combination
corresponding to $-2\Delta\ln L=2.3,\,6.2,\,11.8,\,19.3$ and $30.2$. The data
sets used were collected from 2011 to 2016. Figures are extracted from Refs.
[660].

Figure 14:  (Left) Invariant mass distribution of the selected
${{B}_{({s})}^{0}}\rightarrow{\mu^{+}\mu^{-}}$ candidates with output of the
used multivariate classifier above 0.5, superimposed with the fit result.
(Right) Confidence intervals in the plane of the branching fractions of
${{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}}$ and
${{B}^{0}}\rightarrow{\mu^{+}\mu^{-}}$. The data sets used for the blue solid
(brown dotted) contours were collected from 2011 to 2018 (2016). Figures are
extracted from Refs. [661, 662].

The decays ${{B}_{({s})}^{0}}\rightarrow{e^{+}e^{-}}$ are even rarer than
${{B}_{({s})}^{0}}\rightarrow{\mu^{+}\mu^{-}}$ and can provide powerful tests
of lepton flavour universality. To date, the searches performed by the CDF and
LHCb experiments have found no evidence for either
${{B}^{0}_{s}}\rightarrow{e^{+}e^{-}}$ or ${{B}^{0}}\rightarrow{e^{+}e^{-}}$.
The most stringent upper limits on their branching fractions are
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow{e^{+}e^{-}})<11.2\times 10^{-9}$ and
${\mathcal{B}}({{B}^{0}}\rightarrow{e^{+}e^{-}})<3.0\times 10^{-9}$ at 95%
confidence level set by LHCb [665].

In addition to decays to dileptons, the LHCb experiment has also searched for
decays of neutral $B$ mesons to four leptons. A recent search using the full
Run 1 and Run 2 data sample found no hint of such decays, and upper limits at
90% confidence level for the nonresonant decays are determined to be
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}}{\mu^{+}\mu^{-}})<8.6\times
10^{-10}$ and
${\mathcal{B}}({{B}^{0}}\rightarrow{\mu^{+}\mu^{-}}{\mu^{+}\mu^{-}})<1.8\times
10^{-10}$ [666]. More stringent limits are set for decays involving the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ resonance or a promptly decaying
intermediate scalar particle with a mass of
$1\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$.

###  3.3 Semileptonic $b\rightarrow s\ell^{+}\ell^{-}$ decays

Semileptonic ${b}\rightarrow{s}\ell^{+}\ell^{-}$ decays provide valuable
insight into possible non-SM contributions that affect the Wilson coefficients
$C_{9}$ and $C_{10}$ of the electromagnetic operators. The presence of hadrons
in the final state makes the search for new physics in semileptonic decays
more complicated than that in purely leptonic decays. The challenges in
hadronic form-factor calculations lead to significant uncertainties in the SM
predictions of their decay rates. Fortunately, a number of relatively clean
observables that are less affected by the form factors than the total decay
rates have been identified, including some special observables in angular
distributions and observables for lepton universality test. A comprehensive
study of these observables in a series of $b\rightarrow s\ell^{+}\ell^{-}$
processes has been pursued by the LHCb collaboration and the results are
summarized below.

####  3.3.1 Differential decay rates with respect to $q^{2}$

The differential branching fraction $d{\mathcal{B}}/dq^{2}$ can be measured in
intervals of $q^{2}$, the invariant mass squared of the lepton pair, and
compared with SM predictions. Calculations of form factors are needed for
making the SM predictions. Such calculations are challenging, and require
different treatments depending on the $q^{2}$ regions. Light-cone sum rule
calculations [667, 668, 669] and lattice QCD calculations [670, 671] are often
used to determine the form factors in low- and high-$q^{2}$ regions,
respectively.

After measuring the branching fraction of the decay
${{B}^{0}_{s}}\rightarrow\phi{\mu^{+}\mu^{-}}$ to be about $3\sigma$ below the
SM expectation value [672], the LHCb experiment further studied its
differential branching fraction as a function of $q^{2}$. The top left plot in
Fig. 15 shows the latest results of
$d{\mathcal{B}}({{B}^{0}_{s}}\rightarrow\phi{\mu^{+}\mu^{-}})/dq^{2}$ obtained
using Run 1 data [673], where the ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ and
${\psi{(2S)}}$ regions are excluded. A puzzle appears in the range
$1<q^{2}<6\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$. The branching fraction
integrated over this range is measured to be $(2.88\pm 0.22)\times 10^{-8}$.
Currently, the most precise SM prediction for this range is $(5.37\pm
0.66)\times 10^{-8}$, obtained from a combination of light-cone sum rule and
lattice QCD calculations. A discrepancy of $3.6\,\sigma$ is observed. Similar
patterns are also seen in the LHCb measurements of differential branching
fractions in the decays ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
[674], ${{B}^{0}}\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ [675],
${{{B}^{+}}}\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}}$,
${{B}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{\mu^{+}\mu^{-}}$ and
${{{B}^{+}}}\rightarrow{{K}^{*+}}{\mu^{+}\mu^{-}}$ [676], as shown in Fig. 15.

Figure 15:  Top left: measured
$d{\mathcal{B}}({{B}^{0}_{s}}\rightarrow\phi{\mu^{+}\mu^{-}})/dq^{2}$ [673]
overlaid with the SM predictions based on light-cone sum rule calculations
[677, 678, 667] at low $q^{2}$ and lattice QCD calculations [670, 671] at high
$q^{2}$; top right: measured $d{\mathcal{B}}({{\mathchar
28931\relax}^{0}_{b}}\rightarrow{\mathchar
28931\relax}{\mu^{+}\mu^{-}})/dq^{2}$ [674] compared with the SM predictions
using form factors from lattice QCD calculations [679]; middle left: measured
$d{\mathcal{B}}({{B}^{0}}\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}})/dq^{2}$ [675]
compared with the SM predictions using form factors from line-cone sum rule
and lattice QCD calculations [667, 670]; middle right, bottom left and bottom
right: measured
$d{\mathcal{B}}({{{B}^{+}}}\rightarrow{{K}^{*+}}{\mu^{+}\mu^{-}})/dq^{2}$,
$d{\mathcal{B}}({{B}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{\mu^{+}\mu^{-}})/dq^{2}$
and $d{\mathcal{B}}({{{B}^{+}}}\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}})/dq^{2}$
[676] compared with the SM predictions using form factors from light-cone sum
rule and lattice QCD calculations [680, 681].

####  3.3.2 Angular distributions

Angular distributions in $b\rightarrow s\ell^{+}\ell^{-}$ decays contain rich
information about interference between the SM and non-SM contributions that
may not be accessible via decay rates integrated over angular variables. A set
of $q^{2}$-dependent angular coefficients can be extracted from the angular
distributions and used as probes for new physics, which are complementary to
branching fractions and $d{\mathcal{B}}/dq^{2}$. Based on these coefficients,
we can define some theoretically clean observables with reduced dependency on
the form factors.

Of particular interest is the angular distribution of the ${B}^{0}$
$\rightarrow$ ${K}^{*0}$ ($\rightarrow$ ${K}^{+}$ ${\pi}^{-}$)
$\mu^{+}\mu^{-}$ decay, which has been extensively studied by BaBar [682],
Belle [683], ATLAS [684], CMS [685] and LHCb [686, 687, 688, 689]. Following
the definitions in Ref. [651], the CP-averaged angular distribution of the
decay ${{B}^{0}}\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ with
${{K}^{*0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ is given by

$\displaystyle\frac{1}{d(\Gamma+\bar{\Gamma})/dq^{2}}\frac{d^{4}(\Gamma+\bar{\Gamma})}{dq^{2}d\vec{\Omega}}=\frac{9}{32\pi}$
$\displaystyle\left[\tfrac{3}{4}(1-F_{\rm L})\sin^{2}\theta_{K}+F_{\rm
L}\cos^{2}\theta_{K}\right.$ (8) $\displaystyle+\tfrac{1}{4}(1-F_{\rm
L})\sin^{2}\theta_{K}\cos 2\theta_{\ell}-F_{\rm L}\cos^{2}\theta_{K}\cos
2\theta_{\ell}$ $\displaystyle+S_{4}\sin 2\theta_{K}\sin
2\theta_{\ell}\cos\phi+S_{5}\sin 2\theta_{K}\sin\theta_{\ell}\cos\phi$
$\displaystyle+\tfrac{4}{3}A_{\rm
FB}\sin^{2}\theta_{K}\cos\theta_{\ell}+S_{7}\sin
2\theta_{K}\sin\theta_{\ell}\sin\phi$ $\displaystyle+\left.S_{8}\sin
2\theta_{K}\sin
2\theta_{\ell}\sin\phi+S_{9}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\sin
2\phi\right],$

where $\vec{\Omega}=(\cos\theta_{K},\cos\theta_{\ell},\phi)$, $\theta_{K}$ is
the angle between the directions of the ${{K}^{+}}$ (${{K}^{-}}$) and
${{B}^{0}}$ (${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$) in the
rest frame of the ${{K}^{*0}}$ (${{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$) system, $\theta_{\ell}$ is the
angle between the direction of the ${\mu^{+}}$ (${\mu^{-}}$) and the opposite
direction of the ${{B}^{0}}$ (${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$) in the rest frame of the
${\mu^{+}\mu^{-}}$ system, $\phi$ is the angle between the plane defined by
the muon pair and the plane defined by the kaon and pion in the ${{B}^{0}}$
(${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$) rest frame. Eight
observables can be extracted, including the fraction of the longitudinal
polarisation of the ${{K}^{*0}}$ meson ($F_{\rm L}$), the forward-backward
asymmetry of the ${\mu^{+}\mu^{-}}$ system ($A_{\rm FB}$), and six other
angular coefficients ($S_{i}$, $i=3,4,5,7,8,9$). Using the $S_{i}$
coefficients, new observables less sensitive to form factor uncertainties are
defined, such as $P^{\prime}_{i}=S_{i}/\sqrt{F_{\rm L}(1-F_{\rm L})}$
$(i=3,4,5)$ [690].

The latest LHCb results on angular analysis of the
${{B}^{0}}\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ decay are obtained using data
collected in 2011, 2012 and 2016 [689]. The majority of the angular
observables are consistent with the SM predictions [677] based on form factors
obtained from a combination of light-cone sum rule calculations [667] for
low-$q^{2}$ regions and lattice QCD calculations [670, 671] for high-$q^{2}$
regions. A clear exception is seen with the robust observable $P^{\prime}_{5}$
defined using $S_{5}$, as shown in Fig. 16. The measured values of
$P^{\prime}_{5}$ in the intervals
$4.0<q^{2}<6.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ and
$6.0<q^{2}<8.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ are found to be higher
than the SM predictions [690, 668] by $2.5\,\sigma$ and $2.9\,\sigma$,
respectively. These results confirm the discrepancy in $P^{\prime}_{5}$
observed in an earlier LHCb analysis with Run 1 data [688]. According to
model-independent fits using the FLAVIO software package [691], the overall
tension with the SM is increased from $3.0\,\sigma$ to $3.3\,\sigma$. The fits
reveal that the current measurements of the angular observables in
${{B}^{0}}\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ can be accommodated by
shifting the real part of the Wilson coefficient $C_{9}$ from its SM value by
$0.99^{+0.25}_{-0.21}$ [689].

Figure 16:  Results of the observables $F_{\rm L}$, $A_{\rm FB}$, $S_{5}$ and
$P^{\prime}_{5}$ in bins of $q^{2}$, compared with the SM predictions. Figures
are extracted from Ref. [689].

Recently, LHCb reported results of angular analysis of the $B^{+}\rightarrow
K^{*+}(\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}){\mu^{+}\mu^{-}}$ decay
using Run 1 and Run 2 data[692]. A trend of deviations from the SM predictions
in $P^{\prime}_{5}$, similar to that in the isospin partner decay
$B^{0}\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$, is shown in the left of Fig. 17.
Meanwhile, a large discrepancy in the measurement of $P_{2}=\frac{2}{3}A_{\rm
FB}/(1-F_{\rm L})$ has also been observed in the
$6.0<q^{2}<8.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ region, where the
measurement deviates from its SM prediction [677, 690] by $3.0\,\sigma$ (Fig.
17 right ).

Figure 17:  Results for the observables $P^{\prime}_{5}$ and $P_{2}$ in
$B^{+}\rightarrow K^{*+}{\mu^{+}\mu^{-}}$, compared with SM predictions.
Figures are extracted from Ref. [692].

An untagged time-integrated angular analysis of the decay
${{B}^{0}_{s}}\rightarrow\phi{\mu^{+}\mu^{-}}$ has been performed by the LHCb
collaboration using data collected in 2011 to 2012 and 2016 to 2018 [693]. In
this channel, the same particles ($\mu^{+}$ and ${{K}^{+}}$) are used to
define the angular variables for both ${{B}^{0}_{s}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$ decays, since the final
state is not self-tagging as in the
${{B}^{0}}\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ case. With this convention,
the coefficients of the terms in the CP-averaged time-integrated angular
distributions corresponding to the interference between CP-even ($0$ or $||$)
and CP-odd ($\perp$ or $S$) amplitudes are CP asymmetries, $A^{{C\\!P}}_{\rm
FB}$ and $A_{5,8,9}$, rather than the CP-average observables $A_{\rm FB}$ and
$S_{5,8,9}$ in Eq.8. These asymmetries can arise from either direct CP
violation or nonzero effective mixing phase, with the latter contribution
suppressed by the small value of $\Delta\Gamma_{s}/\Gamma_{s}$. They are
predicted to be close to zero in the SM but has some sensitivity to new
physics contributions [694]. The measurements of $A^{CP}_{\rm FB}$ and
$A_{5,8,9}$ in intervals of $q^{2}$ are shown in Fig. 18, which are consistent
with CP invariance. Much more information on CP violation can be obtained from
time-dependent angular analysis of tagged
${{B}^{0}_{s}}\rightarrow\phi{\mu^{+}\mu^{-}}$ decays [695], which may become
feasible with the huge amount of data that will be collected with the upgraded
LHCb detector in the coming data-taking periods.

Figure 18:  CP asymmetries $A^{CP}_{\rm FB}$ and $A_{5,8,9}$ in intervals of
$q^{2}$ in the decay ${{B}^{0}_{s}}\rightarrow\phi{\mu^{+}\mu^{-}}$ measured
using Run 1 and Run 2 data (black) in Run 1 data only (grey). Figures are
extracted from Ref. [693].

Angular coefficients in the $B^{+}\rightarrow K^{+}{\mu^{+}\mu^{-}}$ decay
have been measured by LHCb [696] and CMS [697] using Run 1 data. Angular
observables in ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow
p{{K}^{-}}{\mu^{+}\mu^{-}}$ decays have been determined by LHCb [698] from a
moment analysis using data collected between 2011 and 2016. These results are
consistent with the SM predictions but limited by statistical uncertainties.

####  3.3.3 Lepton flavour universality tests

In the SM, the couplings of the three generations of leptons to the
electroweak gauge bosons $Z^{0}$ and ${W^{\pm}}$ are assumed to be identical.
This is known as lepton flavour universality (LFU). Under this assumption,
processes involving the three flavours of charged leptons, $e$, $\mu$ and
$\tau$, have equal rates up to corrections caused by different lepton masses,
which can be trivially taken into account. Contributions of new particles or
new interactions may lead to violation of LFU, particularly in FCNC processes
such as $b\rightarrow s\ell^{+}\ell^{-}$ decays. Stringent tests of LFU can be
performed by measuring the ratio of the branching fractions between
$B\rightarrow X{\mu^{+}\mu^{-}}$ and $B\rightarrow X{e^{+}e^{-}}$ decays [699,
700, 701] outside the charmonium regions in the dilepton mass spectrum, with
$X$ indicating the hadron(s) in the decays. The ratio is denoted by

$R_{X}\equiv\frac{{\mathcal{B}}(B\rightarrow
X{\mu^{+}\mu^{-}})}{{\mathcal{B}}(B\rightarrow X{e^{+}e^{-}})}\;.$ (9)

In the SM, $R_{X}$ is expected to be very close to unity with negligible
theoretical uncertainty, due to the small and precisely known difference
between the muon and electron masses. On the experimental side, reconstruction
of electrons is challenging due to the Bremsstrahlung radiation. The decays
$B\rightarrow
X{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{\mu^{+}\mu^{-}})$ and
$B\rightarrow X{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{e^{+}e^{-}})$
are used as control channels for cancellation of systematic uncertainties
associated with electron reconstruction. Practically, the ratio $R_{X}$ is
measured using a double-ratio technique following the equation

$R_{X}=\frac{{\mathcal{B}}(B\rightarrow
X{\mu^{+}\mu^{-}})/{\mathcal{B}}(B\rightarrow
X{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{\mu^{+}\mu^{-}}))}{{\mathcal{B}}(B\rightarrow
X{e^{+}e^{-}})/{\mathcal{B}}(B\rightarrow
X{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow e^{+}e^{-}))}\;,$ (10)

where ${\mathcal{B}}(B\rightarrow
X{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{\mu^{+}\mu^{-}}))/{\mathcal{B}}(B\rightarrow
X{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{e^{+}e^{-}}))$ is known to be
very close to unity [326].

The LHCb collaboration previously measured
$R_{K}=0.846^{+0.060}_{-0.054}\text{\,(stat)}{}^{+0.016}_{-0.014}\text{\,(syst)}$
in ${{{B}^{+}}}\rightarrow{{K}^{+}}\ell^{+}\ell^{-}\;(\ell=e,\,\mu)$ decays in
the dilepton mass-squared range
$1.1<q^{2}<6.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ using Run 1 and part
of Run 2 data [702]. The result was below the SM expectation [691] by
$2.5\,\sigma$. Recently, LHCb updated the $R_{K}$ measurement using the full
Run 1 and Run 2 sample [703]. The mass distributions of the
${{{B}^{+}}}\rightarrow{{K}^{+}}\ell^{+}\ell^{-}$ candidates are shown in Fig.
19. The $R_{K}$ value is measured to be

$R_{K}(1.1<q^{2}<6.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4})=0.846^{+0.042}_{-0.039}\text{\,(stat)}{}^{+0.013}_{-0.012}\text{\,(syst)}\;,$

which are lower than the SM prediction, $1.00\pm 0.01$ [704, 705, 700, 706,
691], by $3.1\,\sigma$. A comparison of the LHCb $R_{K}$ result with the
values measured by BaBar [707] and Belle [708] is shown in Fig. 20.

Tests of LFU have also been performed in other $b\rightarrow
s\ell^{+}\ell^{-}\;(\ell=e,\mu)$ decays. Based on the Run 1 data sample, the
LHCb collaboration has determined the ratios of branching fractions of
$B^{0}\rightarrow{{K}^{*0}}\ell^{+}\ell^{-}\;(\ell=e,\mu)$ decays in two
regions of dilepton mass-squared below the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ resonance to be [709]

$\displaystyle
R_{{{K}^{*0}}}(0.045<q^{2}<1.1\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4})$
$\displaystyle=0.66^{+0.11}_{-0.07}\text{\,(stat)}\pm 0.03\text{\,(syst)}\;,$
$\displaystyle
R_{{{K}^{*0}}}(1.1<q^{2}<6.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4})$
$\displaystyle=0.69^{+0.11}_{-0.07}\text{\,(stat)}\pm 0.05\text{\,(syst)}\;.$

These results are in tension with the SM predictions [700, 704, 710, 711, 712,
706, 691, 667, 713] at the level of $2.1-2.3\,\sigma$ and $2.4-2.5\,\sigma$,
respectively. LHCb has also measured

$R_{pK}(0.1<q^{2}<6.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4})=0.86^{+0.14}_{-0.11}\text{\,(stat)}\pm
0.05\text{\,(syst)}$

in ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow
p{{K}^{-}}\ell^{+}\ell^{-}\;(\ell=e,\mu)$ decays using Run 1 and part of Run 2
data [714].

Very recently, LHCb reported the observation of the decays
${{B}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{e^{+}e^{-}}$ and
${{{B}^{+}}}\rightarrow{{K}^{*+}}{e^{+}e^{-}}$ and the measurements of LFU
observables $R_{{{K}^{0}_{\mathrm{S}}}}$ and $R_{{{K}^{*+}}}$ using the full
Run 1 and Run 2 data samples [715]. The obtained results of
$R_{{{K}^{0}_{\mathrm{S}}}}$ and $R_{{{K}^{*+}}}$ are

$\displaystyle
R_{{{K}^{0}_{\mathrm{S}}}}(1.1<q^{2}<6.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4})$
$\displaystyle=0.66^{+0.20}_{-0.14}\text{\,(stat)}{}^{+0.02}_{-0.04}\text{\,(syst)}\;,$
$\displaystyle
R_{{{K}^{*+}}}(0.045<q^{2}<6.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4})$
$\displaystyle=0.70^{+0.11}_{-0.07}\text{\,(stat)}{}^{+0.03}_{-0.04}\text{\,(syst)}\;,$

which are lower than but consistent with the SM predictions at $1.5\,\sigma$
and $1.4\,\sigma$, respectively.

The anomalous results that the LHCb collaboration has obtained in the study of
LFU and angular distributions in $b\rightarrow s\ell^{+}\ell^{-}$ decays are
highly interesting but not well understood yet. These results have prompted
extensive theoretical studies of potential new physics effects in
$b\rightarrow s\ell^{+}\ell^{-}$ transitions [716, 717, 718, 719, 720, 721,
722, 723, 724, 725]. Particularly, new physics scenarios that mainly affect
$b\rightarrow s{\mu^{+}\mu^{-}}$ transitions are preferred, according to
global analysis in the framework of SM effective field theory [718, 719, 720,
721, 724, 725].

It should be noted with caution that the current measurements of these rare
decays are largely limited by the statistical precision, and the non-trivial
background contamination and reconstruction efficiency in $b\rightarrow
s{e^{+}e^{-}}$ decays may also need to be further scrutinized. An improved
understanding of FCNC $b$-hadron decays can be achieved using the huge amount
of data that will be recorded following Upgrade I and Upgrade II. This will
not only significantly increase the accuracy of the benchmark measurements in
$b\rightarrow s$ transitions, but also provide great opportunities to explore
new observables and new rare decay modes that are currently inaccessible, such
as time-dependent observables in
${{B}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{\mu^{+}\mu^{-}}$ [726, 727] and
${{B}^{0}_{s}}\rightarrow\phi{\mu^{+}\mu^{-}}$ [695] decays, and lepton
universality ratios and angular observables in heavily suppressed
$b\rightarrow d\ell^{+}\ell^{-}$ transitions [728, 729, 730, 731].

Figure 19:  Invariant mass distributions of the selected (left)
${{{B}^{+}}}\rightarrow{{K}^{+}}{e^{+}e^{-}}$ and (right)
${{{B}^{+}}}\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}}$ candidates, superimposed by
the fit results. Figures are extracted from Ref. [703].  Figure 20:
Comparison of the LHCb $R_{K}$ result with the measurements by B-factories
[707, 708]. The vertical line indicates the SM expected value. Figures are
extracted from Ref. [703].

###  3.4 Radiative $b\rightarrow s\gamma$ decays

The effective Hamiltonian for $b\rightarrow s\gamma$ transitions can be
approximately written as

$H_{\rm eff}(b\rightarrow
s\gamma)=-{\dfrac{4G_{F}}{\sqrt{2}}}V_{tb}V^{*}_{ts}(C_{7}\mathcal{O}_{7}+C^{\prime}_{7}\mathcal{O}^{\prime}_{7}),$
(11)

where only the leading operator $\mathcal{O}_{7}$ and its chirality-flipped
counterpart $\mathcal{O}^{\prime}_{7}$ are included. In the SM, the
coefficient $C^{\prime}_{7}$ is given by
$C^{\prime}_{7}=\frac{m_{s}}{m_{b}}C_{7}$ due to the chiral $V-A$ structure of
the weak interaction, where $m_{s}$ ($m_{b}$) indicates the mass of the $s$
($b$) quark. Consequently, the photons emitted in radiative $b$-hadron
(${\overline{b}}$-hadron) decays are predominantly left-handed (right-handed).
Amplitudes with right-handed photons, $A_{R}$, are suppressed by the ratio
$m_{s}/m_{b}$ compared with those with left-handed photons, $A_{L}$, but could
be enhanced in new physics scenarios with right-handed charged current, such
as supersymmetric grand unified theories and left-right symmetric models [732,
733, 734, 735, 736, 737, 738].

Rich information on photon polarisation can be obtained from time-dependent
analysis of $B_{q}^{0}(\bar{B}^{0}_{q})\rightarrow f_{\rm CP}\gamma$ decays
[732, 739, 740], where $f_{\rm CP}$ is a CP eigenstate with eigenvalue $\eta$.
The time-dependent decay rates summing over left-handed and right-handed
photons are expressed as

$P(t)=P_{0}e^{{-\Gamma_{q}t}}\left[\cosh(\Delta\Gamma_{q}t/2)-A^{\Delta}\sinh(\Delta\Gamma_{q}t/2)+\xi
C\cos(\Delta m_{q}t)-\xi S\sin(\Delta m_{q}t)\right]\;,$ (12)

where $\xi$ takes the value of $+1$ ($-1$) for an initial $B^{0}_{q}$
($\bar{B}^{0}_{q}$) meson. The coefficient $C$ quantifies CP violation in the
decay. This type of CP violation has been constrained to be small in radiative
$B$ meson decays by BaBar, Belle and LHCb [741, 742, 743]. Assume no CP
violation in the decay for simplicity, it is convenient to write

$\displaystyle\mathrm{Favoured:}\,\,\,\phantom{0}A(\bar{B}^{0}_{q}\rightarrow
f_{\rm
CP}\gamma_{L})=a_{L}e^{i\delta_{L}}e^{i\phi_{L}}\,\;\Rightarrow\;A(B^{0}_{q}\rightarrow
f_{\rm CP}\gamma_{R})=\eta\,a_{L}e^{i\delta_{L}}e^{-i\phi_{L}}\,,$ (13)
$\displaystyle\mathrm{Suppressed:\,}A(\bar{B}^{0}_{q}\rightarrow f_{\rm
CP}\gamma_{R})=a_{R}e^{i\delta_{R}}e^{i\phi_{R}}\;\Rightarrow\;A(B^{0}_{q}\rightarrow
f_{\rm CP}\gamma_{L})=\eta\,a_{R}e^{i\delta_{R}}e^{-i\phi_{R}}\,,$

where $a_{L(R)}$, $\delta_{L(R)}$ and $\phi_{L(R)}$ are the size, strong phase
and weak phase of $A(\bar{B}^{0}_{q}\rightarrow f_{\rm CP}\gamma_{L(R)})$,
respectively. The terms $\sin(\Delta m_{q})$ and $\cosh(\Delta\Gamma_{q}t/2)$
in Eq. 12 arise from interference of the amplitudes of direct decay,
$A(\bar{B^{0}_{q}}\rightarrow f_{\rm CP}\gamma_{L})$ or
$A(B^{0}_{q}\rightarrow f_{\rm CP}\gamma_{R})$, and the decay via
$B^{0}_{q}$-$\bar{B}^{0}_{q}$ mixing, $p/qA(B^{0}_{q}\rightarrow f_{\rm
CP}\gamma_{L})$ or $q/pA(\bar{B}^{0}_{q}\rightarrow f_{\rm CP}\gamma_{R})$.
The mixing-induced observables $S$ and $A^{\Delta}$ are given by [740]

$\displaystyle S$
$\displaystyle\approx\frac{2\eta\,r}{1+r^{2}}\cos(\delta_{L}-\delta_{R})\sin(\phi_{q}-\phi_{L}-\phi_{R})\,,$
(14) $\displaystyle A^{\Delta}$
$\displaystyle\approx\frac{2\eta\,r}{1+r^{2}}\cos(\delta_{L}-\delta_{R})\cos(\phi_{q}-\phi_{L}-\phi_{R})\;,$

where $r\equiv|a_{R}/a_{L}|\approx|C^{\prime}_{7}/C_{7}|$, $\phi_{q}$ is the
$B^{0}_{q}$-$\bar{B}^{0}_{q}$ mixing phase. The values of $S$ and $A^{\Delta}$
are expected to be small in the SM due to the suppression by the ratio
$r\approx m_{s}/m_{b}$. Since $S$ and $A^{\Delta}$ are approximately linearly
dependent on $r$, they are sensitive to even a small increase of right-handed
photons.

Currently, the observables $S$ and $A^{\Delta}$ are only weakly constrained.
The ${{B}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{0}}\gamma$ decay is a
golden channel to study photon polarisation at $B$ factories. The mixing-
induced CP asymmetry in this channel has been measured to be
$S_{{{K}^{0}_{\mathrm{S}}}{{\pi}^{0}}\gamma}=-0.10\pm 0.31\text{\,(stat)}\pm
0.07\text{\,(syst)}$ [744] and
$S_{{{K}^{0}_{\mathrm{S}}}{{\pi}^{0}}\gamma}=-0.78\pm 0.59\text{\,(stat)}\pm
0.09\text{\,(syst)}$ [745] by the Belle and BaBar collaborations,
respectively, both consistent with the SM expectation value of roughly
$\frac{m_{s}}{m_{b}}\sin 2\beta$. The coefficient of the
$\sinh{\Delta\Gamma_{d}t/2}$ term, $A^{\Delta}$, is inaccessible in
${{B}^{0}}$ decays due to the tiny value of the ${{B}^{0}}$ width difference,
$\Delta\Gamma_{d}$.

Reconstruction of
${{B}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{0}}\gamma$ decays is
challenging at the LHCb experiment. Alternatively, LHCb can measure mixing-
induced CP violation in
$B^{0}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}\gamma$ decays
through a time-dependent amplitude analysis [664]. A more promising channel to
probe right-handed NP is the decay ${{B}^{0}_{s}}\rightarrow\phi\gamma$. Both
$S$ and $A^{\Delta}$ can be measured in this channel and they are predicted to
be close to zero in the SM [740]:

$S_{\phi\gamma}({\rm SM})=0.000\pm 0.002\;,\;A^{\Delta}_{\phi\gamma}({\rm
SM})=0.047\pm 0.039\;.$

Using data collected in Run 1, the LHCb collaboration studied the tagged time-
dependent decay rates of ${{B}^{0}_{s}}\rightarrow\phi\gamma$, which are shown
in Fig. 21. The mixing-induced observables are measured to be [746]

$S_{\phi\gamma}=0.43\pm 0.30\text{\,(stat)}\pm
0.11\text{\,(syst)}\;,\;A^{\Delta}_{\phi\gamma}=-0.67^{+0.37}_{-0.41}\text{\,(stat)}\pm
0.17\text{\,(syst)}\;,$

which are in agreement with the SM expectations.

Figure 21:  Decay-time distributions of the tagged (left)
${{B}^{0}_{s}}\rightarrow\phi\gamma$ and (right) ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow\phi\gamma$
candidates, superimposed by the fit projections. Plots are taken from Ref.
[746].

The ${{B}^{0}}\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ decay in the low-$q^{2}$
region offers a powerful probe of right-handed new physics. In the vicinity of
the photon pole, the decay amplitudes are dominated by contributions from the
electromagnetic Wilson coefficients $C^{(\prime)}_{7}$. An angular analysis
can be performed in a similar way as that in the
${{B}^{0}}\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ case. In the angular
distribution, there are two terms arising from interference of the left-handed
and right-handed decay amplitudes that are proportional to $C_{7}$ and
$C^{\prime}_{7}$, respectively. For small values of $r$
($\equiv|C_{7}^{\prime}/C_{7})|$), the coefficients of these two terms,
denoted $A^{(2)}_{\rm T}$ and $A^{\rm Im}_{\rm T}$, are approximately
expressed as [747, 738]

$A^{(2)}_{\rm T}\approx r\,\cos(\phi_{L}-\phi_{R})\;,\;A^{\rm Im}_{\rm
T}\approx r\,\sin(\phi_{L}-\phi_{R})\;,$ (15)

where $\phi_{L}$ and $\phi_{R}$ represent the phases of $C_{7}$ and
$C^{\prime}_{7}$, respectively. Like the mixing-induced observables in
${{B}^{0}_{s}}\rightarrow\phi\gamma$, $A^{(2)}_{\rm T}$ and $A^{\rm Im}_{\rm
T}$ depend approximately linearly on $r$, thus can provide high sensitivity to
right-handed currents in the small $r$ region. Using data collected in Run 1
and Run 2, the LHCb collaboration has measured $A^{(2)}_{\rm T}$ and $A^{\rm
Im}_{\rm T}$ to be [748]

$A^{(2)}_{\rm T}=0.11\pm 0.10\text{\,(stat)}\pm 0.02\text{\,(syst)}\;,\;A^{\rm
Im}_{\rm T}=0.02\pm 0.10\text{\,(stat)}\pm 0.01\text{\,(syst)}\;.$

These results are compatible with the following SM predictions calculated
using the FLAVIO software package [691]:

$A^{(2)}_{\rm T}({\rm SM})=0.033\pm 0.020\;,\;A^{\rm Im}_{\rm T}({\rm
SM})=-0.00012\pm 0.00034\;,$

and provide the most stringent constraint on the $b\rightarrow s\gamma$ photon
polarisation.

Photon polarisation in $b\rightarrow s\gamma$ transitions can also be probed
by exploiting the angular correlations in radiative decays of $b$ baryons or
charged $b$ mesons. Since current detection technology cannot distinguish
left-handed and right-handed photons, the final states with both left-handed
and right-handed photons are summed together. The left-handed amplitude
$A_{L}$ and right-handed amplitude $A_{R}$ add incoherently in the form of
$|A_{L}|^{2}+|A_{R}|^{2}$, without any interference. In certain cases, the
angular distributions allow for determining a parity violation parameter,
$A_{\rm parity}$, which is proportional to the photon polarisation [749, 750],

$A_{\rm
parity}\propto\lambda_{\gamma}\equiv\frac{|A_{L}|^{2}-|A_{R}|^{2}}{|A_{L}|^{2}+|A_{R}|^{2}}\approx\frac{1-r^{2}}{1+r^{2}}\;.$
(16)

This approach is powerful in probing large right-handed currents but has
limited sensitivity to any small right-handed component.

The LHCb collaboration observed a significantly non-zero up-down asymmetry of
the photons in $B^{-}\rightarrow K^{-}{{\pi}^{+}}{{\pi}^{-}}\gamma$ decays in
the range $m_{K\pi\pi}=[1.1,\,1.3]\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ [751]
with respect to the plane defined by the three final-state hadrons in their
rest frame. This observation demonstrates that the photons are indeed
polarised. However, it is nontrivial to translate the measured asymmetry into
a constraint on the polarisation parameter $\lambda_{\gamma}$, due to
currently limited knowledge of the structure and decay dynamics of the
intermediate resonances involved in this process. A recent theoretical study
pointed out that this $m_{K\pi\pi}$ range is dominated by the $K_{1}(1270)$
resonance and proposed to exploit the charm decay $D\rightarrow K_{1}e\nu$ to
quantify the hadronic effects in $K_{1}\rightarrow
K^{-}{{\pi}^{+}}{{\pi}^{-}}$ [752], which can be studied at a future Super
$\tau$-charm factory [753].

The baryonic decay ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{\mathchar
28931\relax}\gamma$, observed by the LHCb experiment using data collected in
2016 [754], provides a more convenient way to measure the photon polarisation
in $b\rightarrow s\gamma$ transitions [755, 756, 757]. The angular
distribution of this process is given by the differential rate

$\frac{d\Gamma}{d\cos\theta_{p}}\propto 1-\alpha_{{\mathchar
28931\relax}}\lambda_{\gamma}\cos\theta_{p}\;,$ (17)

where $\theta_{p}$ is the helicity angle of the proton in the ${\mathchar
28931\relax}$ rest frame with respect to the opposite direction of the photon,
$\alpha_{{\mathchar 28931\relax}}$ is the decay parameter of the weak process
${\mathchar 28931\relax}\rightarrow p{{\pi}^{-}}$. The photon polarisation
parameter has recently been measured to be
$\lambda_{\gamma}=0.82^{+0.17}_{-0.26}(\rm stat){}^{+0.04}_{-0.13}(\rm syst)$
[758] by the LHCb experiment using all data from Run 2 and the average of the
decay parameter values of ${\mathchar 28931\relax}$ and ${\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}$ measured by BESIII,
$\alpha_{{\mathchar 28931\relax}}=0.754\pm 0.004$ [759]. This result is in
agreement with the SM predictions from Refs. [760, 761, 762]. The LHCb
experiment also searched for the decay $\Xi_{b}^{-}\rightarrow\Xi^{-}\gamma$
using Run 2 data and found no signal [763].

###  3.5 Other rare decays of beauty hadrons

Besides the $b\rightarrow s\ell^{+}\ell^{-}$ decays discussed above, the LHCb
experiment has also performed studies of other rare decay processes of beauty
hadrons. These include: lepton-flavour violating decays
${{B}^{0}}\rightarrow{{K}^{*0}}\tau^{\pm}\mu^{\mp}$ [764],
${{B}^{0}}\rightarrow{{K}^{*0}}\mu^{\pm}e^{\mp}$ and
${{B}^{0}_{s}}\rightarrow\phi\mu^{\pm}e^{\mp}$ [765], ${{{B}^{+}}}\rightarrow
K^{+}{\mu^{-}}\tau^{+}$ [766], ${{{B}^{+}}}\rightarrow K^{+}\mu^{\pm}e^{\mp}$
[767], $B^{0}_{(s)}\rightarrow\tau^{\pm}\mu^{\mp}$ [768],
$B^{0}_{(a)}\rightarrow e^{\pm}\mu^{\mp}$ [769]; lepton- and baryon-number
violating decays $B^{0}_{(s)}\rightarrow p{\mu^{-}}$ [770]; $b\rightarrow
d\ell^{+}\ell^{-}$ decays ${{B}^{0}_{s}}\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$
[771], ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow
p{{\pi}^{-}}{\mu^{+}\mu^{-}}$ [772],
${{{B}^{+}}}\rightarrow{{\pi}^{+}}{\mu^{+}\mu^{-}}$ [773]; annihilation-type
decays ${{B}^{0}}\rightarrow\phi{\mu^{+}\mu^{-}}$ [774],
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi$ [775]. Due to the
limited space, the results of these studies are not included in this review.

##  4 CP violation in beauty and CKM parameters

CP violation is a necessary condition to explain the matter-dominated
universe. While the SM with the CKM mechanism can account for the current
experimental results on CP violation, it fails to explain the cosmological
matter–antimatter imbalance. Searching for new sources of CP violation is one
of the primary goals of flavour physics. This can be done by overconstraining
the CKM matrix using measurements of the matrix elements in many different
processes.

The decays of $b$-hadrons provide a number of key measurements to access the
five CKM matrix elements related to the $b$ or $t$ quark. Taking advantages of
the intense source of b-hadrons at the LHC and a detector designed to probe CP
violation in heavy-flavour decays, the LHCb experiment has been the leading
experiment in the field of $B$ physics in the past ten years, and achieved
some of the most precise measurements of CP violation and mixing of $B$
mesons. Particularly, the precision of the CKM angle $\gamma$ is now
approaching that of the indirect determination; the CP violation parameter
$\phi_{s}$ and mixing parameter ${\Delta m_{{s}}}$ of the ${B}^{0}_{s}$
system, which are key observables for NP searches, have been pinned down with
unprecedented precision.

This section describes the key measurements of the CKM elements in the beauty
sector by the LHCb experiment using the data taken in Run 1 and Run 2. The
parameters $\gamma$, $\phi_{s}$ and ${\Delta m_{{s}}}$, which have received
the most significant improvements, are discussed in detail. Other observables
related to CKM global fit will also be discussed briefly while many other
interesting topics, such as CP violation in $b$-baryon decays, are not
mentioned.

###  4.1 CKM angle $\gamma$

The angle $\gamma$, defined as $\arg[-(V_{ud}V_{ub}^{*})/(V_{cd}V_{cb}^{*})]$,
is one of the key observables related to the CKM matrix. As can be seen from
Fig. 29, one of the main limitations of global constraints comes from the
angle $\gamma$. To be noted, the fits have already included recent $\gamma$
measurements from LHCb which improves the sensitivity on $\gamma$ from
$14^{\circ}$, established at the era of $B$-factories, to around $5^{\circ}$.
Future improvements on the sensitivities on $\gamma$ can be foreseen with the
upgrade of LHCb and running of the Belle II experiment. In the following
sections, we briefly overview the main developments on $\gamma$ measurements
in the past several years from the LHCb experiment.

The direct determination of the angle $\gamma$ is obtained through
interference between $b\rightarrow c$ and $b\rightarrow u$ tree-level
processes, where new physics hardly enters [776]. The hadronic parameters of
the system are all determined from experimental data and related theoretical
uncertainty is negligible [777]. The direct $\gamma$ measurements thus serve
as key inputs for SM predictions, which can be compared with other NP
sensitive measurements to search for physics beyond the SM.

Several methods have been proposed to measure the angle $\gamma$, based on the
types of $D$ decays. In this paper, when not specified, $D$ means a mixture
state of $D^{0}$ and $\overline{D}^{0}$. The GLW method [778, 779] refers to
those decays with $D$ into a CP eigenstate or multi-body $D$ decays which can
be effectively considered as a CP eigenstate using a CP-even fraction $F_{+}$.
The ADS method [780] refers to two-body $D$ decays or multi-body $D$ decays
where the detailed structures over phase space are considered by introducing a
global coherent factor $R_{D}$ and an effective strong phase $\delta_{D}$. In
this case, the interference happens between $b\rightarrow c$ transition, with
$D$ decaying into doubly-Cabibbo-suppressed final states, and $b\rightarrow u$
transition, with $D$ decaying into Cabibbo favoured final state. The BPGGSZ
method [781, 782, 783] refers to $D$ decaying into multi-body final state
where the phase space is binned to make full use of the statistic power of the
decay. In addition, the angle $\gamma$ can also be extracted from time-
dependent CP violation measurements of ${{B}^{0}_{s}}$ decays. The
measurements from the LHCb experiments with these methods are discussed in
detail in the following section.

####  4.1.1 GLW and ADS measurements

The GLW and ADS channels are usually considered together for their
similarities in their final states and experimental treatments. The GLW and
ADS measurements have been performed in many decay channels to obtain the best
sensitivity on $\gamma$. The full list of the measured channels can be found
in Ref. [784].

The decay rate of the GLW method is

$\Gamma(B^{-}\rightarrow D[\rightarrow f_{\textrm{GLW}}]h^{-})\propto
1+r_{B}^{2}+2\kappa_{B}r_{B}(2F_{+}-1)\cos(\delta_{B}-\gamma),$ (18)

where $F_{+}=1(0)$ means a pure CP-even (CP-odd) state while decays with
$F_{+}=0.5$ does not have any sensitivity on $\gamma$. The value of $F_{+}$
can be determined from quantum-coherent data from BESIII and CLEO-c
experiments in a model-independent way. For example, the decays
$B^{-}\rightarrow DK^{-}$, $D\rightarrow\pi^{+}\pi^{-}\pi^{0}$ and
$D\rightarrow K^{+}K^{-}\pi^{0}$ have been used to measure the angle $\gamma$
in LHCb using the measured $F_{+}^{\pi^{+}\pi^{-}\pi^{0}}$ and
$F_{+}^{K^{+}K^{-}\pi^{0}}$ of $0.973\pm 0.017$ and $0.73\pm 0.06$,
respectively [785]. One can see that almost full sensitivity can be achieved
in the $\pi^{+}\pi^{-}\pi^{0}$ channel without considering structures over the
phase space while further binning the phase space of $K^{+}K^{-}\pi^{0}$ will
help get more sensitivity due to small $F_{+}^{K^{+}K^{-}\pi^{0}}$. In fact,
the amplitude analyses of $D^{0}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ and
$D^{0}\rightarrow K^{+}K^{-}\pi^{0}$ channels have both been performed [786,
787, 788, 789]. The relative strong phase between
$D^{0}\rightarrow\rho^{+}\pi^{-}$ and $D^{0}\rightarrow\rho^{-}\pi^{+}$ is
$(-2.0\pm 0.8)^{\circ}$ while that between $D^{0}\rightarrow
K^{*}(892)^{+}K^{-}$ and $D^{0}\rightarrow K^{*}(892)^{-}K^{+}$ is $(-37.0\pm
2.9)^{\circ}$, this is consistent with the fact that
$F_{+}^{\pi^{+}\pi^{-}\pi^{0}}$ is close to 1 and is much larger than
$F_{+}^{K^{+}K^{-}\pi^{0}}$. The parameters $r_{B}$ and $\delta_{B}$ are the
amplitude ratio and phase difference between $b\rightarrow u$ and
$b\rightarrow c$ processes, $\kappa_{B}$ is the coherent factor to take into
account the sensitivity lost due to contamination of other contributions when
the bachelor $h$ is a broad resonant structure, e.g. $K^{*}(892)$. The
coherent factor $\kappa_{B}$ is obtained based on amplitude models used to
describe multi-body $B$ decays. In Eq. 18 only $B^{-}$ is written; however, it
also applies for $B^{0}$ decays. The sensitivity of $\gamma$ is directly
linked to the size of $r_{B}$ and $\kappa_{B}$, larger $r_{B}\times\kappa_{B}$
gives better sensitivity on $\gamma$. The CP measurements from the GLW
channels are only sensitive to $\cos(\delta_{B}-\gamma)$ which has four-fold
ambiguity on determination of angle $\gamma$.

In the GLW modes, as $D$ decays into CP eigenstates, in case of no CP
violation in $D$ decays, the decay amplitudes and their phases are the same
for $D^{0}$ and $\overline{D}^{0}$. However, LHCb has discovered direct CP
violation in $D^{0}\rightarrow K^{+}K^{-}$ and
$D^{0}\rightarrow\pi^{+}\pi^{-}$ by looking at the difference of CP violation
between the two decay channels [790]. This indicates small difference of
amplitudes and phases between the two channels, which affects $\gamma$
determination. Studies [791] show that the effect on $\gamma$ determination is
smaller than 0.5∘. The CP violation measured between $D^{0}\rightarrow
K^{+}K^{-}$ and $D^{0}\rightarrow\pi^{+}\pi^{-}$ has been considered in the
LHCb $\gamma$ combinations shown later.

Based on Eq. 18, the angle $\gamma$ can be accessed by measuring the relative
decay rate difference for $B^{+}$ and $B^{-}$ mesons. An example of the decay
rate difference can be seen directly from the raw yields of the invariant mass
distributions of the decays $B^{\pm}\rightarrow D(\rightarrow