The history of LHCb

I. Belyaev1, G. Carboni2, N. Harnew3, C. Matteuzzi4 and F. Teubert5

1 NRC Kurchatov Institute/ITEP, Moscow, 2 INFN and Università di Roma Tor
Vergata, 3 University of Oxford, 4 INFN and Università Milano-Bicocca, 5CERN,
Geneva

In this paper we describe the history of the LHCb experiment over the last
three decades, and its remarkable successes and achievements. LHCb was
conceived primarily as a $b$-physics experiment, dedicated to $C\\!P$
violation studies and measurements of very rare ${b}$ decays, however the
tremendous potential for $c$-physics was also clear. At first data taking, the
versatility of the experiment as a general-purpose detector in the forward
region also became evident, with measurements achievable such as electroweak
physics, jets and new particle searches in open states. These were facilitated
by the excellent capability of the detector to identify muons and to
reconstruct decay vertices close to the primary ${p}{p}$ interaction region.

By the end of the LHC Run 2 in 2018, before the accelerator paused for its
second long shut down, LHCb had measured the CKM quark mixing matrix elements
and $C\\!P$ violation parameters to world-leading precision in the heavy-quark
systems. The experiment had also measured many rare decays of $b$ and $c$
quark mesons and baryons to below their Standard Model expectations, some down
to branching ratios of order 10-9. In addition, world knowledge of ${b}$ and
${c}$ spectroscopy had improved significantly through discoveries of many new
resonances already anticipated in the quark model, and also adding new exotic
four and five quark states.

The paper describes the evolution of the LHCb detector, from conception to its
operation at the present time. The authors’ subjective summary of the
experiment’s important contributions is then presented, demonstrating the wide
domain of successful physics measurements that have been achieved over the
years.

To appear in EPJH

###### Contents

  1. 1 Introduction
    1. 1.1 ${b}$ physics at the end of the XXth century
    2. 1.2 Towards the LHC
  2. 2  The LHCb Detector
    1. 2.1 Overview
    2. 2.2 The Vertex Locator
    3. 2.3 The TT and Downstream Tracking System
    4. 2.4 The RICH system
    5. 2.5 Calorimeters
    6. 2.6 The Muon System
    7. 2.7 The trigger
      1. 2.7.1 Level-0 hardware trigger
      2. 2.7.2 High Level Trigger
  3. 3 LHCb contributions to CKM measurements and $C\\!P$ violation
    1. 3.1 The status of the unitarity triangle before LHCb
    2. 3.2 Heavy quark mixing measurements
    3. 3.3  LHCb measurements of the unitarity triangle parameters
      1. 3.3.1 Measurements of the CKM angle $\beta$
      2. 3.3.2 Measurements of the CKM angle $\alpha$
      3. 3.3.3 Measurements of the CKM angle $\gamma$
      4. 3.3.4 The sides of the triangle
    4. 3.4 Other $C\\!P$ violation measurements
      1. 3.4.1 ${B}_{s}$ weak mixing phase ${\phi_{{s}}}$ in ${{B}^{0}_{s}}\rightarrow J/\psi\phi$
      2. 3.4.2 $C\\!P$ violation in charm
      3. 3.4.3 $C\\!P$ violation in beauty baryons
      4. 3.4.4 $C\\!P$ violation in charmless 3-body ${{B}^{\pm}}$ decays
  4. 4  Rare decays
    1. 4.1 $B^{0}_{\rm{(s)}}\rightarrow\mu^{+}\mu^{-}$
    2. 4.2 ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$
    3. 4.3 Lepton Universality
  5. 5 Spectroscopy
    1. 5.1 Charm hadrons
    2. 5.2 Double-charm baryons
    3. 5.3 Beauty hadrons
    4. 5.4 Conventional charmonia and bottomonia
    5. 5.5 Pentaquarks
    6. 5.6 Charmonium-like exotic states
    7. 5.7 Structures in the ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ mass spectrum
    8. 5.8 Light hadron spectroscopy
  6. 6 Measurements not originally planned in LHCb
    1. 6.1 Production of EW bosons $W$ and $Z$
    2. 6.2 Jets in LHCb
    3. 6.3 Dark Photons
    4. 6.4 Nuclear Collisions
  7. 7  Future prospects
  8. 8 Summary and Conclusions
  9. 9 Acknowledgments

##  1 Introduction

LHCb is an experiment at the CERN LHC, dedicated to the study of heavy
flavours with large statistics. The resulting high precision makes possible
the observation of tiny deviations from the predictions of the Standard Model
(SM) in $C\\!P$ violation and rare phenomena, variations which could hint at
New Physics (NP) processes. LHCb started taking data in 2010. The so-called
Run 1 commenced at an initial centre of mass energy of $\sqrt{s}=7$ TeV which
was then increased to $\sqrt{s}=8$ TeV, collecting an integrated luminosity of
$3.23\,\text{\,fb}^{-1}$ until the end of 2012. After a two-year shutdown, LHC
operation continued from 2015 to 2018 (Run 2), when the experiment took data
at $\sqrt{s}=13$ TeV, recording an integrated luminosity of $\sim
6\,\text{\,fb}^{-1}$. Throughout the running periods, LHCb collected and
analysed an unprecedented number of ${b}$ decays, and also enlarged its scope
to include charm physics, $W$ and $Z$ measurements, jets and nuclear
collisions.

In this paper the motivation for the LHCb experiment is described, recalling
how its design was developed and evolved. The experiment’s major achievements
in terms of physics results are then summarised. Finally we discuss the plans
for the future upgrade, in the period when the Super-KEKB collider will also
operate.

The layout of this paper is as follows. The Introduction (Sect. 1) describes
the status of the ${b}$ physics programme at the time when the LHCb detector
was conceived, and provides an evolution of its design. In Sect. 2 the basic
elements of the detector are described, optimised for the requirements of
heavy-flavour physics measurements, together with a description of the
triggers. The measurements which were originally the major aims of the
experiment, i.e. the CKM matrix measurements and $C\\!P$ violation and very
rare decays of the $b$-quark, are reviewed in Sects. 3 and 4, respectively. In
Sect. 5, a summary of the wide-ranging results in ${b}$\- and ${c}$\-
spectroscopy is presented. A review is given of the many non planned physics
areas in Sect. 6, such as results on jets, electro-weak (EW) physics, and
searches for new particles not associated with heavy flavours. In all these
domains, LHCb proves to be an extremely versatile detector, providing
complementary measurements to those of the LHC General Purpose Detectors
(GPDs). The paper concludes in Sect. 8 with a short description of the upgrade
plans, which will ensure LHCb operation beyond 2030.

The paper presents the authors’ subjective summary of LHCb’s many major
physics results, however the review inevitably omits a substantial number of
important measurements. To this end, additional information can be found in
the paper’s exhaustive bibliography.

###  1.1 ${b}$ physics at the end of the XXth century

In 1970 Glashow, Iliopoulos and Maiani postulated the existence of a fourth
quark, charm, necessary to explain the smallness of ${{K}^{0}}$ oscillations
in the framework of the Standard Model of Weak Interactions [1]. The so-called
Glashow-Iliopoulos-Maiani (GIM) mechanism generalized Cabibbo’s idea of
rotated weak currents to the quark model with two doublets, introducing a
$2\times 2$ mixing matrix written in terms of the Cabibbo angle. An
experimental confirmation of the fourth quark hypothesis came in 1974 with the
discovery of ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ [2, 3], soon identified as a
${{c}{\overline{c}}}$ bound state, and followed by the discovery of open
charm. In 1973 Kobayashi and Maskawa [4] proposed a third heavy-quark doublet
in order to describe $C\\!P$ violation in the framework of the SM, thus
generalizing the Cabibbo matrix to the $3\times 3$ CKM (Cabibbo, Kobayashi,
Maskawa) matrix. The discovery of the $\mathchar 28935\relax$ in 1975,
followed by the charged and neutral ${B}$-mesons in 1983 [5] proved the
validity of their idea and held the prospect of understanding quantitatively
$C\\!P$ violation.

The level of $C\\!P$ violation in ${b}$ quark decays was expected to be orders
of magnitude larger than in the neutral ${K}$ system, but unfortunately the
relevant decay channels had only tiny branching fractions, so the lack of
intense “${b}$-quark sources” slowed the progress of beauty physics: in
particular, in 1986 the PDG only listed five decay modes of ${{B}^{0}}$ and
${{B}^{\pm}}$. However in 1987 a new impetus came from the discovery at ARGUS
of ${B}^{0}-{\overline{{B}}^{0}}$ oscillations [6]. It was clear that the
forthcoming LEP machine, designed for an entirely different purpose, and the
symmetrical CESR collider, could not yield an exhaustive answer to all the
questions related to the CKM hypothesis, despite their valuable contributions
to many facets of ${b}$-physics [7, 8]. When in 1989 P. Oddone proposed an
asymmetric ${e^{+}e^{-}}$ collider [9] operating at the $\mathchar
28935\relax(4S)$ energy with a luminosity above
$10^{33}\;\mathrm{cm^{-2}s^{-1}}$, an intense period of accelerator studies
ensued. This gave birth to the PEP-II and KEKB $B$ Factories, which were
approved in 1994, and started operating in 1998, soon reaching and passing
their design luminosity.

Around this time, proponents pursued the idea of exploiting hadron beams to
attack the problem of detecting $C\\!P$ violation in the ${b}$ sector. The
idea behind this was that the large hadronic ${b}$ production cross-section
plus the high-intensity hadron beams at the already existing and planned
proton accelerators would produce a large number of $b\overline{b}$ pairs,
sufficient to gather evidence for $C\\!P$ violation at least in the so-called
“golden channel”
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{0}_{\mathrm{S}}}$.
To achieve an adequate background rejection, the experimental difficulties
were formidable because of the small ratio of the ${b}$ cross-section to the
total hadronic cross-section at the $\sqrt{s}$ values available in fixed-
target and collider experiments.

In 1985, the fixed-target WA75 hybrid experiment at the CERN SPS [10] observed
in emulsions the first partially reconstructed $b\overline{b}$ pair produced
by an extracted pion beam of 350 GeV/c, confirming that the production cross-
section at such low energies was very small. To circumvent this problem,
simple experiments were proposed [11] for the CERN SPS and for the planned UNK
machine [12] at Serpukhov, with a brute-force approach based on a high-
intensity extracted beam and a minimalist detector designed to reconstruct the
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{0}_{\mathrm{S}}}$
decay and to provide flavour tagging (Fig. 1.1). There was no time-dependent
analysis of the decay since the beam-dump character of the experiments made
the use of a micro vertex detector impossible. There were exploratory fixed-
target experiments, at CERN (WA92) [13] and at Fermilab (E653, E672, E771,
E789) [14], which tried to observe and measure beauty events, albeit without
(or very limited) success.

Figure 1.1: A sketch of a proposed beam-dump style experiment designed to
detect
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{0}_{\mathrm{S}}}$
decays. The ${{K}^{0}_{\mathrm{S}}}$ would traverse the conical slit before
decaying [11].

In 1989 P. Schlein proposed a dedicated Beauty experiment exploiting the large
${b}$ cross-section expected at the CERN SPS proton-antiproton collider
($\sqrt{s}=630$ GeV) [15]. The authors of the proposal (P238) remarked that
the bulk of $b\overline{b}$ production occurred at very small angles with
respect to the beams, therefore making a compact experiment practical. The
heart of the detector was a Silicon Microvertex Detector operating very close
to the beam (1.5 mm) coupled to a fast readout and track-reconstruction
electronics. The Microvertex Detector provided the trigger by requiring that
accepted events had to be inconsistent with a single vertex. P238 was not
approved but the CERN R&D Committee, established to support new detector
developments in view of the LHC, approved in 1991 a test of the Microvertex
Detector [16] in the SPS Collider. This proved very successful [17] and paved
the way towards the future COBEX experimental proposal.

At the time when the ${e^{+}e^{-}}$ colliders were approved, the HERA-B
experiment had been conceived at DESY [18]. Approved in 1994, HERA-B exploited
the 920 GeV HERA proton beam on a fixed target made of metallic wires, placed
inside Roman Pots in the vacuum pipe, and immersed in the beam halo. HERA-B
was approved to take data in 1998, one year before PEP-II and KEKB. The
sophisticated apparatus consisted of a single-arm spectrometer, including a
RICH, a large microvertex silicon detector, a high-resolution tracker, plus an
electromagnetic calorimeter. HERA-B was designed primarily for the detection
of the
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{0}_{\mathrm{S}}}$
decay and its trigger was based on ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$
reconstruction at the first level.

At $\sqrt{s}=40$ GeV, the $b\overline{b}$ cross-section is about $10^{-6}$ of
the total hadronic cross-section, hence HERA-B had to achieve a background
rejection around $10^{-11}$ for the
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{0}_{\mathrm{S}}}$
decay. Data taking conditions were similar to those of the current LHCb
experiment (a 40 MHz interaction rate), as were the requirements of radiation
resistance. HERA-B started data taking in 2000 but it soon emerged that the
detector did not have sufficient rejection power against background and the
track reconstruction was not as efficient as expected. The large number of
detector stations and their total thickness in terms of radiation lengths made
secondary interactions an important issue for event reconstruction. Eventually
HERA-B could not observe ${b}$ events efficiently, but taught several valuable
lessons for any future experiment working in a crowded hadron-collision
environment. These were a need of a robust and efficient tracking and a
flexible trigger systems able to adapt to harsher environments than may have
been expected, as well as the need to design the thinnest and lightest
detector (in terms of radiation and interaction lengths).

###  1.2 Towards the LHC

Over the same period, the planned LHC and SSC machines, with their large
energies, promised spectacular increases of the $b\overline{b}$ cross-section,
thus making the task of background rejection much simpler. This was
particularly true for operation in collider mode, but even in fixed-target
mode ($\sqrt{s}\approx\mathcal{O}(130)$ GeV) the ${b}$ cross-section was
expected to be a respectable $1~{}\mu$b at the LHC and $2~{}\mu$b at the SSC
[19]. The large cross-section and corresponding good background rejection
would facilitate a hadron $B$ Factory, competitive and complementary with
${e^{+}e^{-}}$ colliders, which could focus primarily on the measurement of
$C\\!P$ violation and also allow the study of the spectrum of all ${b}$
particles. Given the intrinsically “democratic” nature of hadronic production,
the new hadron machines would also give access to large samples of
${{B}^{0}_{s}}$ and of $b$ baryons, something not possible at ${e^{+}e^{-}}$
colliders operating at the $\mathchar 28935\relax$ resonances.

Two schools of thought soon emerged: one pursuing a fixed-target (FT) strategy
and the other based on a collider mode. The more favourable ratio of the
$b\overline{b}$ to the total hadronic cross-section, about two orders of
magnitude larger in collider mode, gave this a competitive advantage.

There was, however, a strong reason in favour of the FT concept: a collider
$B$ experiment could not operate at the design luminosity of the machine
($10^{33-34}\,{\rm cm}^{2}{\rm s}^{-1})$ because of the significant number of
overlapping interactions (pile-up) with multiple vertices. This would have
required dedicated low-luminosity running, creating a potential conflict with
the major experiments and considerably reducing the data-taking time. Later it
was ascertained that the individual experiment luminosities could be tuned
over a broad range with an appropriate design of beam optics in the
interaction regions, hence this would become a moot point, but at the time it
was a serious one.

Moreover, for the advocates of the FT approach, the advantage of the larger
cross-section in collider mode was partially offset by the higher event
multiplicity and by the shorter flight path of beauty particles. In addition,
while the $p_{\mathrm{T}}$ of the ${b}$ decay products would to a good
approximation be the same in the two modes, the $p_{\mathrm{T}}$ of the other
collision products would be smaller in FT mode, thus making the trigger
simpler. Active silicon targets were also possible with an extracted beam,
where the track of a charged ${b}$-hadron would be directly measured.

Finally, $b\overline{b}$ production kinematics is forward peaked in the
centre-of-mass (CM) system, but the Lorentz boost of the centre-of-mass in the
FT mode ($\beta\gamma>60$) concentrates the event at smaller angles than in
collider mode. It was therefore possible, in principle, to build a more
compact detector, achieving larger angular acceptance at lower cost. There was
also the possibility of recycling components (in particular dipole magnets)
from existing detectors. The cost was an important consideration, since a
dedicated $B$ experiment at the LHC (or SSC) was generally considered to have
secondary importance with respect to the general-purpose experiments.

The anticipated demise of the SSC led three groups to study and propose
dedicated ${b}$ experiments at the LHC. The three Letters of Intent (LoI) were
presented in 1993. COBEX [20], an acronym for Collider Beauty Experiment, with
P. Schlein as spokesperson, was a collider experiment with a backward-forward
geometry. The other two proposals, Gajet [21] (spokesperson T. Nakada) and LHB
[22] (spokesperson G. Carboni) were fixed target experiments, the first, as
the name suggests, using a gas jet target, the latter exploiting an extracted
beam. Since no traditional beam extraction was foreseen for the LHC, LHB
(Large Hadron Beautyfactory) used a parasitic extraction technique, based on
channeling in a bent silicon crystal placed close to one of the circulating
beams (Fig. 1.2).

Figure 1.2: Proposed LHC beam extraction scheme based on crystal channeling.
The channeled beam, deflected by 0.7 mrad in a bent silicon crystal, was
guided towards the beam transport tunnel by several conventional magnets. The
cost of the extracted beam was a non-negligible fraction of the detector cost.

A dedicated R&D experiment, RD22, approved by the CERN DRDC [23] to test the
feasibility of this idea at the SPS, demonstrated that high-efficiency beam
extraction (larger than 10%) was possible [24]. The three proposed experiments
were presented in their final form in 1994 at the Beauty’94 Conference [25].
It should also be noted that, by then, CDF had already contributed important
${b}$-physics results, offering a glimpse of what would later prove to be the
extraordinary success of $b$-physics at hadronic machines.

Following the submission of the LoIs, the LHC Committee (LHCC) considered the
proposals in June 1994. One of its concerns was the small size of the three
collaborations: the total number of physicists involved was barely one
hundred. In addition the Committee remarked that beam extraction by channeling
could not be guaranteed at that stage, because accelerator experts feared
possible interference with normal LHC operation. Finally the following
recommendation was issued: “The collider model approach has the greater
potential in view of the very high rate of ${b}$ production, the much better
signal/background ratio and the possibility of exploring other physics in the
forward direction at 14 TeV”.

The LHCC encouraged the three collaborations to join together and design a new
experiment, incorporating attributes of each, and operating in collider mode.
The LHCC noted that its ambitious request was justified by the fact that, at
the startup of the LHC, the experiment would not be an exploratory one, since
$C\\!P$ violation would already have been observed “at HERA-B, FNAL or $B$
Factories”. The LHCC also issued guidelines requiring a number of issues to be
addressed and solved. The three experiments combined into the new LHC-B
Collaboration (then named), which published its Letter of Intent in 1995 [26].

The new experiment derived several of its characteristics from the parent
proposals: notably the concept of a silicon vertex detector in retractable
pots, the calorimeter design, and the high $p_{\mathrm{T}}$ first-level
trigger.

In the transition to LHC-B, a number of proponents from the former three
collaborations decided not to continue. This unfortunately included P. Schlein
and his UCLA colleagues who had pushed very strongly for the collider-mode
idea with COBEX, and who had been a driving force up to that point. The LHCC
decision sent a strong signal to the high-energy physics community: that CERN
were prepared to give their strong support to one dedicated $B$ experiment.
This encouraged many physicists from institutions around the world to join the
new collaboration over the subsequent years. T. Nakada, who was an instigator
of the Gajet proposal, was elected LHC-B spokesperson. The 1995 Letter of
Intent of LHC-B established the basis for the new detector design, which was
refined in the following years, until its approval in 1998. By that time, the
name had changed from LHC-B to simply LHCb. Fig. 1.3 shows the detector as it
appeared in the Letter of Intent.

Figure 1.3: The LHC-B detector as it was proposed in 1995. All the basic
components shown would be part of the final detector, albeit with many
refinements and optimisations that will be described in the following Section.

Soon afterwards, a competing ${B}$-physics experiment, BTeV [27], was proposed
to run at the Fermilab Tevatron, incorporating a single magnet, a double arm
spectrometer and a vertex trigger at the first level, in order to recover the
reduced ${b}$ production cross section. Following the decision to shut down
the Tevatron, BTeV was not approved, however several of the experiment’s
innovative ideas were carried through to the future LHCb experiment.

##  2  The LHCb Detector

The basic mechanism for heavy-quark production at the LHC is via gluon-gluon
fusion. The angular distribution of ${{c}{\overline{c}}}$ or $b\overline{b}$
pairs is peaked at small angles with respect to the beam-line, with high
correlation between the constituents of the pair. This allows the detection
with good acceptance of the resulting hadrons in a rather limited solid angle.
QCD calculations give cross-section values of
$\sigma_{{{c}{\overline{c}}}}\simeq 1.5$ mb and $\sigma_{b\overline{b}}\simeq
0.5$ mb, respectively. The large event multiplicity requires a high
granularity of the detector, together with minimal thickness in terms of
radiation and interaction lengths to reduce secondary interactions.

The LHC-B LoI [26] presented a forward spectrometer with 400 mrad acceptance
and a single large dipole magnet. The apparatus would be located inside the
former DELPHI cavern at LEP with little modifications to the existing
infrastructure but, for reasons of cost, the detector sacrificed half of the
solid angle by only being a single arm spectrometer. The LoI design inherited
important features from the three ancestor experiments and from the
contemporary HERA-B, which had rate and radiation issues similar to those
expected at the LHC. In contrast to the latter experiment, LHC-B had in
addition a hadron calorimeter and a second (upstream) RICH with two radiators.
Initially it was thought that an efficient tracking system in a harsh
environment would require a large number of tracking stations, so, paralleling
HERA-B, LHC-B had twelve tracking stations in the large-angle region. This
number was reduced to ten in the Technical Proposal presented in 1998 [28],
which by then had changed its name to LHCb.

The disappointing performance of HERA-B was largely ascribed to the large
amount of material in the detector, which prompted the LHCb collaboration to
perform a thorough review of the apparatus, with the aim to reduce material
without sacrificing performance. A Technical Design Report submitted in 2003
presented the LHCb “Reoptimized” Detector [29]. This is the basis on which the
experiment was eventually built and is described in the following subsections.

###  2.1 Overview

The LHCb detector [30] is a forward spectrometer, shown in Fig. 2.1, and is
installed at Intersection Point 8 of the LHC. A modification to the LHC
optics, shifting the interaction point by about 11 m from the centre, allowed
maximum use of the cavern space. This results in a detector length of
approximately 20 m, and with maximum transverse dimensions about $6\times
5~{}\rm{m}^{3}$. The angular acceptance ranges from approximately 10 mrad to
300 mrad (Fig. 2.2) in the horizontal magnetic-bending plane, and from 10 mrad
to 250 mrad in the vertical plane (Fig. 2.1). With this geometry the detector
is able to reconstruct approximately 20% of all $b\overline{b}$ pairs
produced.

To measure the momenta of charged particles, a dipole magnet producing a
vertical magnetic field is used. It is a warm magnet providing an integrated
field of 4 Tm, with saddle-shaped coils in a window-frame yoke, and with
sloping poles in order to match the required detector acceptance. The design
of the magnet allows for a level of fringe field inside the upstream Ring
Imaging Cherenkov detector (RICH 1, see Sec. 2.4) of less than 2 mT whilst
providing a residual field in the regions between the upstream tracking
stations.

The tracking system consists of a silicon VErtex LOcator detector (VELO) [31],
surrounding the interaction region, and four planar tracking stations, the TT
tracker upstream of the dipole magnet and three tracking stations T1-T3
downstream of the magnet [32, 33]. The T1-T3 stations consist of an Inner
Tracker (IT), located at the centre of the stations and surrounding the beam-
pipe, and an Outer Tracker (OT) for the outer regions. A minimum momentum of
around 1.5$\text{\,Ge\kern-1.00006ptV\\!/}c$ is required for a track to reach
the downstream stations [34].

Figure 2.1: The LHCb detector: side view Figure 2.2: The LHCb detector: top
view

Particle identification (PID) is a fundamental to the goals of the LHCb
experiment by separating pions, kaons and protons produced in heavy-flavour
decays. Accurate reconstruction of electrons and muons is crucial for flavour
tagging. All aspects of PID are accomplished by a set of specialized
detectors.

The PID system is based around two RICH detectors, designed to cover almost
the full momentum range of tracks in LHCb. The upstream detector, RICH 1,
covers the low momentum region from $\simeq 2$ to $60$
$\text{\,Ge\kern-1.00006ptV\\!/}c$ using aerogel (in Run 1 only) and
$\hbox{C}_{4}\hbox{F}_{10}$ radiators, whilst the downstream detector, RICH 2,
covers the high momentum range from 15 $\text{\,Ge\kern-1.00006ptV\\!/}c$ up
to and beyond 100 $\text{\,Ge\kern-1.00006ptV\\!/}c$ using a
$\hbox{C}\hbox{F}_{4}$ radiator.

Two calorimeters, one electromagnetic (ECAL) and the other hadronic (HCAL),
supplemented by a Preshower Detector (SPD/PS) [35], provide identification of
electrons, photons and hadrons and a measurement of their energy. This
measurement is used at the trigger level to select candidates on the basis of
their transverse energy. Muons play a crucial role in many of LHCb’s
measurements because of the cleanliness of the signature. Their identification
is achieved by five muon stations (M1 – M5), interspersed with iron filters.
The muon system also supplies measurements of muon transverse momenta for the
trigger.

###  2.2 The Vertex Locator

The role of the VELO is to measure the impact parameters of all tracks
relative to the primary vertex (PV), to reconstruct the production points and
decay vertices of hadrons containing ${b}$\- and ${c}$-quarks and to allow
precision measurements of their mean lifetimes. The subdetector accepts
particles with pseudorapidities in the range $1.6~{}<~{}\eta~{}<~{}4.9$ and
which have PVs within $|z|~{}<~{}10.6$ cm from the nominal collision point
along the beam direction. The VELO is split into two halves surrounding the
beam-pipe, each containing 21 modules. Each module is then made up of two
silicon half discs of 300 $\mu$m thickness, one with strips in the radial, $r$
coordinate, the other in the polar, $\phi$ coordinate. This cylindrical
geometry allows a fast track- and vertex-reconstruction to be made at the
second stage of trigger. The strip segmentation is such to limit the highest
occupancy of the strips to less than 1.1 %.

The VELO is positioned, with an accuracy better than 4 $\mu$m, at the closest
distance possible from the beam, about 7 mm during data taking, The sensors
operate within a so-called Roman pot configuration, located inside a secondary
vacuum of less than $2\cdot 10^{-7}$ mbar pressure, separated from the primary
LHC vacuum. The sensors are retracted during beam injection and are then
quickly moved in for physics operation when the LHC beams are stable.

The vessel containing the silicon discs and the front-end electronics (RF-box)
has aluminium walls of 300 $\mu$m thickness to minimize multiple scattering.
The average material budget of the detector for tracks in the LHCb acceptance
is 0.22 $X_{0}$. In order to minimize radiation damage and to dissipate the
produced heat, a cooling system keeps the temperature range between -10 to
0∘C.

Fig. 2.3 summarizes the VELO performance in terms of impact parameter and
decay time resolution [31].

Figure 2.3: Left: Projected impact parameter resolution as a function of
$1/p_{T}$. Right: Decay time resolution (points) as a function of momentum for
${{B}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi$ decays.
The superimposed histogram shows the distribution of momentum for the decay.

###  2.3 The TT and Downstream Tracking System

Following the VELO, the tracking system is composed of the TT station, located
between RICH 1 and the magnet, and three stations (T1,T2,T3) downstream of the
magnet. The TT is composed of four stations grouped in pairs, called TTa and
TTb, spaced by 30 cm. Each station consists of silicon microstrip planar
modules covering a rectangular area of 150 cm $\times$ 130 cm (width times
height), covering the LHCb acceptance of 300 mrad in the horizontal plane and
250 mrad in the vertical. The strips of the first and the fourth stations are
vertical and measure the bending $x$ coordinate, whilst the second and third
planes have stereo angles of $\pm 5^{\circ}$, respectively.

Tracking stations T1 - T3 each consist of an inner part (IT) surrounding the
beam pipe, and an outer part (OT) beyond. Each IT station consists of four
overlapping silicon layers, two rotated by a stereo angle of $\pm 5$ deg and
two aligned to the $y$ (vertical) axis. Each layer is made up of four
independent modules placed around the beam pipe, covering about a $120\times
40$ cm2 area, as shown in Fig. 2.4.

Figure 2.4: The LHCb inner detector, (left) a vertically-aligned layer,
(right) a stereo layer.

The spatial resolutions of both the TT and IT are approximately 50 $\mu$m per
hit, with strip pitches of about 200 $\mu$m. The hit occupancies vary between
1.9% for the inner sectors to 0.2% for the outermost modules. To minimize
radiation damage, the sensors operate at 5∘C temperature.

The OT is a drift detector [32] consisting of straw tubes with internal
diameters of 4.9 mm, each filled with an Ar/CO2 gas mixture in a ratio
$70-30$%. The straws provide a 35 ns maximum drift time and 205 $\mu$m spatial
resolution with 17% maximum straw occupancy. Each of the three stations is
made of four modules, shown schematically in Fig. 2.5. A picture of the
assembled OD is shown in Fig. 2.6. In the first and third modules the straw
tubes are aligned to the vertical axis while the third and fourth modules have
stereo angles of $\pm 5$ deg. The total active area is about $5.97\times
4.85\;m^{2}$, covering the full LHCb acceptance.

The overall tracking efficiency for “long” tracks (i.e. those tracks measured
in all the tracking detectors including the VELO) is greater than 96 % for
$5<p<200\;\text{\,Ge\kern-1.00006ptV\\!/}c$. The momentum resolution $dp/p$ is
0.5% at low momentum, increasing to 1.1% at
$240~{}\text{\,Ge\kern-1.00006ptV\\!/}c$. The mass resolution is
$14.3~{}\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ for the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ resonance.

Figure 2.5: Schematic views of the LHCb outer detector.  Figure 2.6: The LHCb
Outer Detector in place on the beam line. Well visible is the beam pipe.

###  2.4 The RICH system

The role of the RICH system [36] is to provide $\pi/K/p$ discrimination for
LHCb, which is essential for most $C\\!P$-violation studies, background
rejection and flavour tagging. The momentum range which contains 90% of kaons,
pions and protons from ${B}$ meson decay is between 2 and 150
$\text{\,Ge\kern-1.00006ptV\\!/}c$ and, to achieve this separation, two
Cherenkov detectors, RICH 1 and RICH 2, are employed.

The RICH 1 detector differentiates particles with low and intermediate
momenta, from 1 to $\sim$60 $\text{\,Ge\kern-1.00006ptV\\!/}c$. It is located
close to the interaction region, upstream of the magnet, and covers the
acceptance from $\pm 25$ mrad to $\pm 300$ mrad (horizontal plane) and to $\pm
250$ mrad (vertical plane). RICH 1 initially contained two different radiator
materials: an aerogel layer 5 cm thick with refractive index $n=1.03$ and a
$\hbox{C}_{4}\hbox{F}_{10}$ gas layer of length 85 cm with refractive index
$n=1.0014$. Aerogel has the power to provide $\pi/K$ discrimination from about
1 up to 10 $\text{\,Ge\kern-1.00006ptV\\!/}c$, however it was removed for Run
2 due to occupancy problems. The $\hbox{C}_{4}\hbox{F}_{10}$ radiator extends
the positive $\pi/K$ identification from about 10
$\text{\,Ge\kern-1.00006ptV\\!/}c$ to 60 GeV/c, however $\pi/K$ discrimination
below 10 $\text{\,Ge\kern-1.00006ptV\\!/}c$ is still possible by operating the
RICH in kaon veto mode.

RICH 2 has a smaller angular acceptance of $\pm 15$ mrad to $\pm 120$ mrad
(horizontal plane) and to $\pm 100$ mrad (vertical plane) and covers the
region where high momentum particles are most abundant. It is located
downstream of the magnet, between T3 and the first muon station M1. RICH 2
uses a $\hbox{CF}_{4}$ gas radiator with refractive index $n=1.00046$.

In both detectors, Cherenkov photons are detected by a combined system of
plane and spherical mirrors to focus photons onto a pair of photo-detector
planes, where Hybrid Photon Detectors are employed to detect the Cherenkov
rings. The photo-detectors are located outside the detector acceptance, in
regions of low magnetic field and relatively low radiation.

The Cherenkov angles for the three different RICH radiators and for different
particles as a function of momentum are shown in Fig. 2.7 (left). A
measurement of RICH performance in LHCb data is shown for the two gaseous
radiators In Fig. 2.7 (right) [36].

Figure 2.7: (Left) Values of Cherenkov angle as a function of momentum for
different particles for the three RICH radiators of refractive index $n$.
(Right) Measured Cherenkov angles in LHCb data [36].

###  2.5 Calorimeters

The calorimeter system identifies hadrons, electrons and photons, and also
measures their energies and positions for the Level-0 trigger. The system is
composed of four sub-detectors: the scintillator Pad Detector (SPD), the
PreShower detector (PS), the Electromagnetic Calorimeter (ECAL), and the
Hadron Calorimeter (HCAL). The SPD and PS are located just upstream of the
ECAL. The ECAL, PS and SPD are segmented into three sections in the $xy$
plane, with active pads growing from the inner to the outer regions. The HCAL
is similarly divided in two sections. The corresponding granularities are
outlined in Fig. 2.8.

Figure 2.8: The (left) LHCb electromagnetic calorimeter and (right) hadronic
calorimeter. The bottom left regions indicate the areas occupied by the beam-
pipe.

The SPD and PS are used at the trigger level and offline, in association with
the ECAL, to indicate the presence of electrons, photons and neutral pions.
The detectors have two plastic scintillator layers separated by a 15 mm thick
lead plate where electrons and photons can radiate; the downstream
scintillator then samples the radiated energy. The light from the
scintillators is sent to photomultipliers by wavelength-shifter (WLS) optical
fibers.

The ECAL employs the Shashlik technology, where independent modules,
constructed from scintillating tiles and lead plates, are alternated (see Fig.
2.9). The ECAL has $66$ layers of such modules consisting of 2 mm of lead
followed by 4 mm of scintillator material. The ECAL also uses WLS optical
fibers to guide the light from the detector to photomultipliers, placed on the
back face of each module. The energy resolution achieved [35] is

$\frac{\sigma\left(E\right)}{E}=\frac{10\%}{\sqrt{E}}\oplus 1\%$ (2.1)

where $E$ is the electron energy expressed in  GeV.

The HCAL is also a sampling calorimeter. It is constituted of iron absorber
with scintillating tiles as the active material. The innovative feature of
this sampling structure is the orientation of the scintillating material: the
tiles run parallel to the beam axis. In the lateral direction, tiles are
spaced with 1 cm iron, while longitudinally the length of the tiles and iron
spacers correspond to the hadron interaction length $\lambda_{I}\simeq 20$ cm
in steel. Light is collected by WLS optical fibres running along the detector
towards the back side where the photomultiplier tubes are located (see Fig.
2.9).

Figure 2.9: A schematic showing the layout and segmentation of the LHCb ECAL
and HCAL calorimeters.

The HCAL is used to measure the hadronic shower transverse energy for the
Level-0 trigger and to improve the high momentum electron/hadron separation.
The energy resolution achieved is

$\frac{\sigma\left(E\right)}{E}=\frac{(69\pm 5)\%}{\sqrt{E}}\oplus(9\pm 2)\%$
(2.2)

where $E$ is the hadron energy in  GeV.

###  2.6 The Muon System

The Muon System consists of five stations, M1-M5, of rectangular shape. The
complete system is made up by 1368 Multi Wire Proportional Chambers
supplemented by 12 Triple GEM Chambers in the inner region of the first
station, to cope with the very high particle rate. The chambers employ a
variety of readouts, optimized for a precise $p_{T}$ measurement for the
trigger. The complete system has an acceptance in the bending plane from 20
mrad to 306 mrad, and in the non-bending plane from 16 mrad to 258 mrad. This
results in a total acceptance of about 20% for muons from semileptonic
inclusive ${b}$ decays.

The M1 station is located in front of the calorimeters and is used in order to
improve the $p_{t}$ measurement for the trigger. The geometry of the five
stations is projective; all the transverse dimensions scale as the distance
from the interaction point. Stations M2-M5 are placed downstream of the
calorimeters and are interleaved with 80 cm thick iron absorbers. The total
absorption thickness, calorimeters included, is about 20 interaction lengths.
In this way the minimum momentum for muons crossing the five stations is about
6 $\text{\,Ge\kern-1.00006ptV\\!/}c$.

Each muon station is designed to achieve an efficiency above 99% in a 20 ns
time window with a noise rate below 1 kHz per physical channel, as described
in [37]. To reach such an efficiency, four chamber layers per station are used
in M2-M5 (two layers in M1). The time resolution is achieved by a fast gas
mixture Ar/CO2/CF4 in the ratio 40:55:5. A ratio 45:15:40 is employed in the
Triple GEM chambers.

###  2.7 The trigger

Even with the relatively large $b\overline{b}$ cross-section at LHC energies,
only approximately 1% of visible $p$ $p$ interactions result in a
$b\overline{b}$ event. Moreover, only about 15% of those events will produce
at least one $b$-hadron with all decay products passing within the acceptance
of the spectrometer. The branching fractions of decays used to study $C\\!P$
violation are typically less than $10^{-3}$. Further reductions are
unavoidable in the offline selection, where stringent cuts must be applied to
enhance signal over background. Therefore the purpose of the LHCb trigger is
to achieve the highest efficiency for the events later selected in the offline
analysis while rejecting drastically most of the uninteresting background
events. To achieve this goal, the trigger uses information from all LHCb sub-
detectors.

The trigger is organised in two different levels: the Level-0 (L0) trigger
based on custom electronic boards, and the High-Level Trigger (HLT),
implemented in a computer farm. Level-0 uses the information from the
calorimeter and muon systems, performing a selection in order to reduce the
event rate from 40 MHz to below 1 MHz, which is the maximum frequency allowed
to read out the entire detector. The HLT is a software application running on
a processor farm that further reduces the rate of events in the kHz range for
storage (see Fig. 2.10).

The HLT has significantly evolved over time from the original design in the
LHCb Technical Proposal (TP) [38] in 1998, to the trigger design in the
Technical Design Report (TDR) [39] in 2003, to the Run 1 (2010-2012) actual
implementation [40] and finally to the additional features introduced during
Run 2 (2015-2018) [41]. In the TP it was assumed that a first HLT trigger
level (L1) would reduce the 1 MHz input rate to a 40 kHz output rate with a
variable latency of less than 256 $\mu$s, using coarse information from the
vertex detector to reconstruct vertices and tracks with no momentum
information (the VELO r-$\phi$ geometry was designed for this purpose). A
second HLT trigger level (L2) was fashioned to extrapolate VELO tracks into
the magnetic field to the tracking stations downstream of the magnet and
reduce the output rate to 5 kHz with an average latency of 10 ms. Finally a
third level (L3) would implement the full event reconstruction and a set of
exclusive selections to bring down the rate to 200 Hz.

By the time of the trigger TDR in 2003, it became clear that the LHC was not
going to start before the end of the decade when much more powerful processing
units would become available. In addition a series of test-beam and detailed
simulation studies convinced the collaboration of the need to have momentum
information at the first stage of the HLT. Therefore a new tracking station
just upstream of the magnet was introduced (the TT station). In addition, a
shield which had been protecting RICH1 from stray magnetic fields was removed
to allow for a rough estimation for the momentum of tracks reconstructed
between the VELO and TT stations. The software trigger then had two levels:
Level-1 able to reduce the output rate to 40 kHz using L0, VELO and TT
information with an average latency of 1 ms, and HLT to reduce the output rate
to 200 Hz with a combination of inclusive and exclusive selections. Between
the time of the trigger TDR and the first physics run (Run 1), the interest in
having a more performing HLT for charm physics (${{c}{\overline{c}}}$ with a
factor 20 larger production cross section than $b\overline{b}$) and a much
more robust system, convinced the collaboration to push for much more
inclusive selections in the final trigger stage and a much larger trigger
output rate (3-5 kHz). This implied a complete redefinition of the offline
data processing model.

Furthermore, it had been assumed that the LHC would operate with a 25 ns bunch
separation, limiting the number of overlapping events to a mean number of
$\mu\simeq 0.4$ per bunch crossing at a luminosity of $2\times 10^{32}$
cm-2s-1. When, from 2011, a separation of 50 ns was adopted for early LHC
operation, the experiment decided to run at $\mu\approx 1.4$ to compensate for
the lower number of bunches. Therefore the HLT had to adapt to running
conditions rather different than first assumed. This was made possible by the
highly flexible design of the HLT.

After the success of the LHCb trigger performance in Run 1, the good
understanding of the trigger reconstruction allowed the introduction of the
”real-time analysis” concept during Run 2. After the first HLT trigger level
(HLT1), events are buffered to disk storage in the online system. This is done
for two purposes, firstly events can be processed further during inter-fill
periods, and secondly the detector can be calibrated and aligned run-by-run
before the HLT2 stage. Once the detector is calibrated and aligned, events are
passed to HLT2, where a full event reconstruction of “offline quality” is
performed. This allows for a wide range of inclusive and exclusive final
states to trigger and obviates the need for further offline processing. In
addition, new techniques to reduce the amount of information saved per event
[42] allowed to increase significantly the output rate to 10-15 kHz, as in
Fig. 2.10, while the output of HLT1 could be increased to ${\mathcal{O}}$(110
kHz). The decrease in requests for offline reconstruction also helped to
mitigate the pressure on the offline computing model.

Figure 2.10: Trigger overview in Run 1 (left) and Run 2 (right)

####  2.7.1 Level-0 hardware trigger

The L0 trigger is divided into three independent components: the
L0-Calorimeter trigger, the L0-Muon trigger and the L0-PileUp trigger. The
latter is used to reject multiple visible interactions in a bunch crossing by
means of the “ad hoc” Pile-Up System detector housed in the VELO. The first
two components are briefly described below.

#####

The L0-Calorimeter part of the trigger obtains informations from the SPD, PS,
ECAL and HCAL subdetectors and computes the transverse energy deposited by
incident particles: $E_{T}=E_{0}\cos\theta$, where $E_{0}$ is the energy of
the particle and $\theta$ is the polar angle given by the cell hit in the
detector. Together with energy information, the total number of hits in the
SPD (SPD multiplicity) is also determined in order to veto large multiplicity
events that would take too large a fraction of the available processing time
in the HLT. From the calorimeter information, three types of candidates are
built and selected according to specific $E_{T}$ criteria: i) Hadron candidate
(L0Hadron); ii) Photon candidate (L0Photon); and iii) Electron candidate
(L0Electron).

#####

The L0-Muon part of the trigger requires a muon candidate to have a hit in all
five muon stations. The L0 muon processor boards select the two highest
$p_{T}$ muon tracks in each quadrant of the muon system with a maximum of
eight candidates. The trigger sets a single threshold either on the largest
muon $p_{T}$ (L0 muon trigger) or on the product of the largest and the 2nd
largest (L0 dimuon trigger). Events with SPD multiplicity $>$ 600 are excluded
in the L0 muon trigger in order to minimize the track multiplicity. This limit
is raised to 900 in the L0 dimuon trigger at the expense of a small increase
in rate.

The total output rate of the L0 trigger is limited to 1 MHz, which is the
maximum rate accepted by the HLT1. Such an output rate consists of about 400
kHz of muon triggers, about 450 kHz of hadron triggers and about 150 kHz of
electron and photon triggers (the individual triggers have an overlap of about
$10\%$).

####  2.7.2 High Level Trigger

Data from L0 are sent to the Event Filter computer Farm (EFF) which runs the
HLT algorithms. The HLT is a software application whose $29500$ instances run
on the EFF. Each instance is made up of independently operating trigger lines;
each line consists of selection parameters for a specific class of events.

The HLT is divided into two stages. The first stage (HLT1) processes the full
L0 rate and uses partial event reconstruction to reduce the rate to about 110
kHz. The second stage (HLT2) reduces the rate to about 12.5 kHz, performimg a
more complete event reconstruction [41].

#####

HLT1 reconstructs the trajectories of charged particles traversing the full
LHCb tracking system which have a $p_{\mathrm{T}}$ larger than 500 MeV. The
hits in the VELO are combined to form straight-line tracks loosely pointing
towards the beam line. Next, at least three hits in the TT are required in a
small region around a straight-line extrapolation from the VELO. The TT is
located in the fringe field of the LHCb dipole magnet, which allows the
momentum to be determined with a relative resolution of about $20\%$, and this
estimate is used to reject low $p_{\mathrm{T}}$ tracks. Tracks are then
extrapolated to the T-stations downstream of the magnet. The search window in
the IT and OT is defined by the maximum possible deflection of charged
particles with $p_{\mathrm{T}}$ larger than 500 MeV. The search is also
restricted to one side of the straight-line extrapolation by the charge
estimate of the track. Subsequently, all tracks are fitted with a Kalman
filter to obtain the optimal parameter estimate using a simplified geometry
description of the LHCb detector. The set of fitted VELO tracks is re-used to
determine the positions of the PVs.

Tight timing constraints in HLT1 mean that most particle-identification
algorithms cannot be executed. The exception is muon identification due to its
clean signature. Hits in the muon stations are searched for in momentum-
dependent regions of interest around the track extrapolations. Tracks with
$p<3$ GeV cannot be identified as muons, as they would not be able to reach
the muon detectors.

HLT1 has two inclusive trigger lines which select events containing a particle
whose decay vertex is displaced from the PV: a line which selects a single
displaced track with high $p_{\mathrm{T}}$, and a line which selects a
displaced two-track vertex with high $p_{\mathrm{T}}$. Both lines start by
selecting good quality tracks that are inconsistent with originating from the
PV. The single-track trigger then selects events based on a hyperbolic
requirement in the 2D plane of the track displacement and $p_{\mathrm{T}}$.
The two-track displaced vertex trigger selects events based on a multivariate
discriminant whose input variables are the vertex-fit quality, the vertex
displacement, the scalar sum of the $p_{\mathrm{T}}$ of the two tracks and the
displacement of the tracks making up the vertex. The two-track line is more
efficient at low $p_{\mathrm{T}}$, whereas the single track line performs
better at high $p_{\mathrm{T}}$, such that in combination they provide high
efficiency over the full $p_{\mathrm{T}}$ range.

The HLT1 muon lines select muonic decays of ${b}$ and ${c}$ hadrons, as well
muons originating from decays of ${W}$ and ${Z}$ bosons. There are four main
lines: one line that selects a single displaced muon with high $p_{T}$, a
second single muon line that selects very high $p_{T}$ muons without
displacement for electroweak physics, a third line that selects a dimuon pair
compatible with originating from a decay of a charmonium or bottonium
resonance or from Drell-Yan production, and a fourth line that selects
displaced dimuons with no requirement on the dimuon mass. During Run 2,
typically about 80 kHz were allocated to the inclusive HLT1 lines, while about
20 kHz to the muon lines. The rest of the HLT1 output is dedicated to special
low multiplicity triggers and calibration trigger lines.

#####

HLT2 can perform the full event reconstruction since the output of HLT1 is
buffered. The full event reconstruction consists of three major steps: the
track reconstruction of charged particles, the reconstruction of neutral
particles and particle identification. The HLT2 track reconstruction exploits
the full information from the tracking sub-detectors, performing additional
steps of the pattern recognition which are not possible in HLT1. Tracks with a
$p_{\mathrm{T}}$ larger than 80 MeV are reconstructed in HLT2, without the
requirement to have hits in the TT station. This is to avoid inefficiencies
due to the TT acceptance, which is crucial for part of the charm and kaon
physics programme. In addition, tracks produced by long-lived resonances that
decay outside the VELO are reconstructed using T-station segments that are
extrapolated backwards through the magnetic field and combined with hits in
the TT. Similarly, the most precise neutral cluster reconstruction algorithms
are executed. Finally, in addition to the muon identification available in
HLT1, HLT2 exploits the full particle identification from the RICH detectors
and calorimeter system.

The HLT2 inclusive ${b}$-hadron trigger lines look for a two-, three-, or
four-track vertex with sizeable $p_{\mathrm{T}}$, significant displacement
from the PV, and a topology compatible with the decay of a ${b}$-hadron, using
a multivariate discriminant. Whenever one or more tracks are identified as
muons, the requirements on the discriminant are relaxed to increase the
efficiency. As in the case of HLT1, several muon lines are used to select
muonic decays of ${b}$ and ${c}$ hadrons and of $W$ and $Z$ bosons. However in
HLT2, the muon reconstruction is identical to the offline procedure, having
access to exactly the same information. During Run 2, typically about 3 kHz of
the trigger rate is from the inclusive ${b}$-hadron trigger while the muon
lines take about 1 kHz. A large fraction of the trigger bandwidth (2-4 kHz) is
allocated to exclusive selection of charm decays, where a reduced amount of
information is saved per event. The rest of the trigger bandwidth is due to
other special triggers and calibration trigger lines.

##  3 LHCb contributions to CKM measurements and $C\\!P$ violation

The violation of the combined operation of charge conjugation and parity,
$C\\!P$, was first observed in 1964 in decays of neutral kaons [43]. The BaBar
[44] and Belle [45] $B$ Factory experiments and the CDF experiment [46]
established $C\\!P$ violation in the decays of neutral ${B}^{0}$ mesons. LHCb
now extends measurements to much greater precision, and also probes the
${B}_{s}$ system, which is vital to explore the full range of $C\\!P$
violation measurements.

In the Standard Model, the Cabibbo-Kobayashi-Maskawa (CKM) unitary matrix [47,
4], $\textrm{V}_{\textrm{CKM}}$, describes the electroweak coupling strength
$V_{ij}$ of the $W$ boson to quarks $i$ and $j$:

$\textrm{V}_{\textrm{CKM}}=\left(\begin{array}[]{ccc}V_{ud}&V_{us}&V_{ub}\\\
V_{cd}&V_{cs}&V_{cb}\\\ V_{td}&V_{ts}&V_{tb}\\\ \end{array}\right).$ (3.1)

$C\\!P$ is violated in the Standard Model if any element of the CKM matrix is
complex. The parametrisation of the CKM matrix due to Wolfenstein [48] is
given by

$\textrm{V}_{\textrm{CKM}}=\left(\begin{array}[]{ccc}1-\frac{1}{2}\lambda^{2}-\frac{1}{8}\lambda^{4}&\lambda&A\lambda^{3}(\rho-i\eta)\\\
-\lambda+\frac{1}{2}A^{2}\lambda^{5}[1-2(\rho+i\eta)]&1-\frac{1}{2}\lambda^{2}-\frac{1}{8}\lambda^{4}(1+4A^{2})&A\lambda^{2}\\\
A\lambda^{3}[1-(1-\frac{1}{2}\lambda^{2})(\rho+i\eta)]&-A\lambda^{2}+\frac{1}{2}A\lambda^{4}[1-2(\rho+i\eta)]&1-\frac{1}{2}A^{2}\lambda^{4}\\\
\end{array}\right)\\\ $ (3.2)

for the four Standard Model parameters $(\lambda,A,\rho,\eta)$. The expansion
parameter, $\lambda$, equal to the sine of the Cabibbo angle, has a value
$|V_{us}|=0.22$ [49], and in Equ. 3.2 the expansion is given for terms up to
order $\lambda^{5}$.

The unitarity of the CKM matrix leads to six orthogonality conditions between
any pair of columns or any pairs of rows of the matrix. The orthogonality
means the six conditions can be represented as six triangles in the complex
plane. The interesting relations for $C\\!P$ violation are those given by:

$V_{ud}V^{*}_{ub}+V_{cd}V^{*}_{cb}+V_{td}V^{*}_{tb}=0~{}~{}~{}:{\mathrm{the}}~{}{\mathrm{unitarity~{}triangle}},$
(3.3)
$V_{us}V^{*}_{ub}+V_{cs}V^{*}_{cb}+V_{ts}V^{*}_{tb}=0~{}~{}~{}:{\mathrm{the}}~{}B_{s}~{}{\mathrm{triangle,~{}and}}$
(3.4)
$V_{cd}V^{*}_{ud}+V_{cb}V^{*}_{ub}+V_{cs}V^{*}_{us}=0~{}~{}~{}:{\mathrm{the~{}charm~{}triangle}}.$
(3.5)

The unitarity triangle has sides with lengths that are the same order in
$\lambda$, namely ${\mathcal{O}}(\lambda^{3})$, which implies large $C\\!P$
asymmetries in ${B}^{0}$ and ${B}^{\pm}$ decays. The ${B}_{s}$ triangle has
two sides of ${\mathcal{O}}(\lambda^{2})$ and the third of
${\mathcal{O}}(\lambda^{4})$. Hence $C\\!P$ violation in ${B}_{s}$ mixing is
significantly smaller than in the ${{B}^{0}}$ system. Moreover, the charm
triangle has two sides of ${\mathcal{O}}(\lambda)$ and the third of
${\mathcal{O}}(\lambda^{5})$, hence $C\\!P$ violation in the charm system is
expected to be extremely small. Note that all three triangles have equal area
[50].

To study $C\\!P$ violation, the ${B}-$physics experiments measure the complex
phases of the CKM elements and measure the lengths of the sides of the
triangles to check for a self-consistent picture. $C\\!P$ violation is
predicted in many (often very rare) ${B}$ hadron decays, hence LHCb utilises
large samples of ${B}$, ${B}_{s}$, ${B}_{c}$ mesons and ${B}-$baryons. New
physics can be discovered and studied when new particles appear in, for
example, virtual loop processes of rare ${B}$ decays, leading to observable
deviations from Standard Model expectations, both in branching ratios and
$C\\!P$ observables. Hence the LHCb strategy is to determine with high
precision the CKM elements and to compare measurements of the same parameters,
especially those where one is sensitive to new physics and the other to
Standard Model processes.

###  3.1 The status of the unitarity triangle before LHCb

The first generation $B$ Factory experiments to study $C\\!P$ violation in the
${B}-$system, BaBar and Belle, made huge in-roads into testing the Standard
Model description of $C\\!P$ violation; the status was summarised extensively
at the Beauty 2009 Conference [51]. CDF and D0 extended these studies at the
Tevatron, and make first explorations in the $B_{s}$ sector. Fig. 3.1 shows
the status of the unitarity triangle measurements compiled by the CKM-Fitter
Group [52] in 2009, when the $B$ Factories had been running for around ten
years. Here graphical results are displayed in the $\rho-\eta$ plane and the
best fit to the apex of the triangle (Equ. 3.3) to the 95% confidence level is
shown. The fit to the CKM parametrisations include measurements of the sides
of the triangle through measurements of the CKM elements and the angles,
information from rare $K$ and ${B}$ meson decays, and
${{B}^{0}_{s}}-\overline{B^{0}_{s}}$ mixing.

Figure 3.1: The knowledge of the unitarity triangle as of autumn 2009 [52].

Before 2009, when the LHC turned on, the $B$ Factory experiments, CDF and, to
a lesser degree D0, measured the parameters of the unitarity triangle with
varying degrees of precision:

  * •

The quantity $\textrm{sin\,}2\beta$ was measured in all channels, including
the “gold plated” channel ${{B}^{0}}\rightarrow{J}/\psi{K}^{0}_{S}$, to a
precision of around $\sim$0.03;

  * •

The sides $|V_{td}/V_{ts}|$ and $|V_{ub}/V_{cb}|$ were known from
${{B}^{0}_{s}}-\overline{B^{0}_{s}}$ mixing and from ${b}\rightarrow\,{u}$
decays, respectively each to $\sim$10%, but limited by theory. The $B_{s}$
mixing phase (${\phi_{{s}}}$) was unmeasured;

  * •

The angle $\alpha$ was measured in the channels ${B}\rightarrow\pi\pi$,
$\rho\pi$ and $\rho\rho$ with a statistical precision of $\sim 5^{\circ}$;

  * •

There was a statistics-limited measurement in ${B}\rightarrow\,{D}{K}$ modes
of the angle $\gamma$ to around $20-25^{\circ}$. A measurement of $\gamma$
from ${B}_{s}$ modes such as ${{B}^{0}_{s}}\rightarrow{{D}^{+}_{s}}{K}^{-}$
had been completely unexplored.

  * •

The parameter ${\varepsilon_{K}}$, measured in kaon decays, provided a very
loose constraint on the triangle vertex;

  * •

The $B_{s}$ sector had been largely unexplored by the first generation
experiments, as had ${b}-$baryons. Running at the $\Upsilon(4S)$, the $B$
Factories produced predominantly ${B}_{u,d}$ meson pairs, however Belle record
a significant sample of $\Upsilon(5S)$ data, allowing some interesting
measurements of ${B}_{s}$ pairs. At the LHC, ${{B}^{0}}$, ${{B}^{\pm}}$,
$B_{s}$, $B_{c}$ and ${b}-$baryons are produced in approximately in the ratios
$\sim 40:40:10:0.1:10\%$;

  * •

The $B$ Factories were statistics limited for very rare processes with
branching ratios $\lesssim 1\times 10^{-6}$, such as ${b}\rightarrow\,{s}$
flavour-changing neutral current (FCNC) transitions, e.g. $b\rightarrow
s\gamma$ and $b\rightarrow sl^{+}l^{-}$. Super-rare transitions such as
${B}_{(s,d)}\rightarrow\mu^{+}\mu^{-}$ were also unobserved.

In contrast, Fig. 3.2 shows the status of the unitarity triangle measurements
today [52].

Figure 3.2: The current knowledge of the unitarity triangle (as of Summer
2019) [52].

###  3.2 Heavy quark mixing measurements

Since flavour is not conserved in the weak interaction, mixing between
${B}^{0}_{q}$ and $\overline{{B}}^{0}_{q}$ mesons (where $q=d$ or $s$) is
possible via the box diagrams shown in Fig. 3.3.

Figure 3.3: The box diagrams representing mixing between ${B}^{0}_{q}$ and
$\overline{{B}}^{0}_{q}$ mesons.

The probability for finding a $\overline{{B}}_{q}$ (or a ${B}_{q}$), given the
initial state was a ${B}_{q}$ (or a $\overline{{B}}_{q}$) at time $\Delta t$
after production, is given by:

$P_{mix}(\Delta t)=\exp{-\dfrac{\Delta t}{\tau_{B^{0}}}}\left(1-\cos(\Delta
m_{q}\Delta t)\right).$ (3.6)

Here $\Delta m_{q}$ is the mass difference $m_{H}-m_{L}$, where $m_{H,L}$ are
the masses of the heavy and light mass eigenstates and $\tau_{B^{0}}$ is the
${B}$ lifetime. At the LHC, the two neutral ${B}$ mesons produced can
oscillate independently at any time after production.

Any $C\\!P$ measurement from a time-dependent analysis of neutral ${B}$ decays
needs the determination of the ${B}$ flavour ($b$ or ${\overline{b}}$) at
production. This requires $b-$quark “tagging”, and several algorithms have
been developed by LHCb involving the combination of so-called opposite side
[53] and same side taggers [54].

  * •

${B}^{0}$ mixing: ${B}^{0}$ oscillations are measured in the channel
${B}^{0}\rightarrow D^{-}\mu^{+}\nu_{\mu}X$ and the charge conjugate modes.
The oscillation measurements rely on tagging, and are shown for 3 fb-1 of data
in Fig. 3.4 (left). The LHCb $\Delta m_{d}$ measurement, which is the world’s
best, is $\Delta m_{d}=(505.0\pm 2.1\pm 1.0)\,\rm{ns}^{-1}$ [55].

  * •

${B}_{s}$ mixing: ${B}^{0}_{s}-{\overline{{B}}^{0}_{s}}$ oscillations were
first observed by CDF [56]. ${B}_{s}$ oscillations are measured at LHCb in the
mode ${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ and its charge
conjugate state. The oscillations are shown for 1 fb-1 of data in Fig. 3.4
(right). The plot shows the proper-time distribution of
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ candidates, in five
different ${{B}^{0}_{s}}$ decay channels, that have been flavour-tagged as not
having oscillated. The LHCb $\Delta m_{s}$ measurement, which is again the
world’s best, is $\Delta m_{s}=(17.768\pm 0.023\pm 0.006)\,{\rm ps}^{-1}$
[57].

Figure 3.4: Proper-time distribution of (left) ${B}^{0}-{\overline{{B}}^{0}}$
oscillations in ${B}^{0}\rightarrow D^{-}\mu^{+}\nu_{\mu}X$ decays, (right)
${B}^{0}_{s}-{\overline{{B}}^{0}_{s}}$ oscillations in
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ decays, and (lower)
${D}^{0}-{\overline{{D}}^{0}}$ oscillations in ${D}^{0}\rightarrow
K^{+}\pi^{-}$ decays. All distributions rely on flavour tagging, and the
curves correspond to the fitted oscillations.

  * •

Charm meson mixing:

${D}^{0}-{\overline{{D}}^{0}}$ mixing was established in a single experiment
in 2013 by LHCb in ${D}^{0}\rightarrow K^{+}\pi^{-}$ decays, although it was
confirmed earlier by combining several $B$ Factory results. The ${D}^{0}$
mixing parameters were measured in LHCb in a decay-time-dependent fit to the
ratio $\frac{N_{{D}^{0}\rightarrow K^{+}\pi^{-}}}{N_{{D}^{0}\rightarrow
K^{-}\pi^{+}}}$. The time-dependent fit is shown in Fig.3.4 (lower) for 1 fb-1
of data. The no-mixing scenario is excluded at $9.1\sigma$ [58].

###  3.3  LHCb measurements of the unitarity triangle parameters

The LHCb experiment performs a high-statistics study of $C\\!P$ violation with
unprecedented precision in many different and complimentary channels,
providing a sensitive test of the Standard Model and physics beyond it.

####  3.3.1 Measurements of the CKM angle $\beta$

The time-dependent decay asymmetry of the channel
${{B}^{0}}\rightarrow{J}/\psi{K}^{0}_{S}$ allows a measurement of the angle
$\beta$. This is known as the “golden” decay mode because the channel is
virtually free of penguin pollution (which enters with the same overall
phase), resulting in very small theoretical uncertainty, of order 1% [59].
$C\\!P$ violation in this channel occurs in the interference between mixing
and decay, where the mixing process introduces a relative $C\\!P$-violating
weak phase of 2$\beta$. Experimentally the $C\\!P$ asymmetry is measured from
the ratio of the numbers of $\overline{{B}}$ and ${B}$ mesons,
$N_{\overline{{B}}\rightarrow f}$ and $N_{{{B}}\rightarrow f}$, decaying into
final state $f$:

$\displaystyle\mathcal{A_{CP}}={{N_{\overline{{B}}\rightarrow
f}-N_{{{B}}\rightarrow f}}\over{N_{\overline{{B}}\rightarrow
f}+N_{{{B}}\rightarrow f}}}=\textrm{sin\,}(\Delta
m_{d}t)~{}\textrm{sin\,}(2\beta)-\textrm{cos\,}(\Delta
m_{d}t)~{}\textrm{cos\,}(2\beta).$ (3.7)

The LHCb measured and fitted asymmetries for the $J/\psi$ $(1S)$ and $(2S)$
states are shown in Fig. 3.5 for 3 fb-1 of data at 7 and 8 TeV [60]. These
measurements are $\textrm{cos\,}(\Delta m_{d}t)=-0.017\pm 0.029$ and
$\textrm{sin\,}(\Delta m_{d}t)=0.760\pm 0.034$, where an observation of a
direct $C\\!P$-violation contribution proportional to $\textrm{cos\,}(\Delta
m_{d}t)$ would be an indication of new physics [59]. The LHCb measurement is
now competitive with BaBar and Belle measurements; the current world average
of $\textrm{sin\,}2\beta=0.695\pm 0.019$ [61] is dominated by LHCb together
with the $B$ Factory measurements in the complementary channels
${{B}^{0}}\rightarrow{J}/\psi{K}^{0}_{S}$ and
${{B}^{0}}\rightarrow{J}/\psi{K}^{0}_{L}$. The measurement by LHCb of
$\textrm{sin\,}2\beta$ in gluonic penguins will further contribute to this
study.

Figure 3.5: The LHCb measured and fitted asymmetry in
${{B}^{0}}\rightarrow{J}/\psi{K}^{0}_{S}$ decays for (left) the $J/\psi(1S)$
and (right) the $\psi(2S)$.

####  3.3.2 Measurements of the CKM angle $\alpha$

The primary method at LHCb for the measurement of $\alpha$ is through an
amplitude analysis via the ${B}\rightarrow\rho\pi$ decay modes [62], however
these channels are difficult at LHCb due to the need to efficiently
reconstruct $\pi^{0}$s. Penguin pollution is present and must be constrained,
with the additional application of isospin symmetry. The precision on $\alpha$
at LHCb is expected to be dominated by systematic uncertainties, and any
measurement is not expected to improve on a combination of the $B$ Factory
measurements, $\alpha=\left(86.4^{+4.5}_{-4,3}\right)^{\circ}$ [52].

####  3.3.3 Measurements of the CKM angle $\gamma$

A precise measurement of the angle $\gamma$ is key to understanding the
closure (or otherwise) of the unitarity triangle. Constraints on the
unitarity-triangle apex largely come from loop decay measurements which are
very sensitive to the presence of new physics. $\gamma$ is the only angle
accessible at tree level and hence forms a SM benchmark to which the loop
measurements can be compared (assuming no significant new physics in tree
decays). The $\gamma$ measurement also relies on theoretical input which is
very well understood [63, 64].

Determination of $\gamma$ from a combined fit to all measured parameters of
the unitarity triangle currently gives a value
$\gamma=\left(65.8^{+1.0}_{-1.7}\right)^{\circ}$ [52]. Conversely the
measurement of $\gamma$ alone from a combination of all direct measurements
from tree decays gives $\gamma=\left(72.1^{+5.4}_{-5.7}\right)^{\circ}$. Hence
reaching degree-level precision from direct $\gamma$ measurements is crucial.

LHCb makes measurements of $\gamma$ by a variety of methods, where
complementary is vital. Examples of the most sensitive LHCb measurements are
outlined below.

  * •

$\gamma$ in the “time integrated” ${{B}^{\pm}}\rightarrow{D}^{0}{K}^{\pm}$
modes

The measurement of $\gamma$ is made in direct CP-violation via
${{B}^{\pm}}\rightarrow{D}^{0}{K}^{\pm}$ by three different methods: the GLW
method (decay into a $C\\!P$ eigenstate) [65, 66], the ADS method (decay into
a flavour-specific mode) [67], and the GGSZ method (Dalitz analysis) [68].
These all access $\gamma$ through interference between the
${{B}^{\pm}}\rightarrow{D}^{0}{K}^{\pm}$ and
${{B}^{\pm}}\rightarrow\overline{{D}}^{0}{K}^{\pm}$ decay paths, where the
${D}^{0}$ and $\overline{{D}}^{0}$ decay to the same final state. When using
these methods, the decay modes are self-tagging. In addition time-dependent
analyses are not necessary. For the ALD and GLW modes, the charge-conjugate
event yields are simply counted to determine the CP asymmetries (i.e.
effectively a “counting experiment”).

Figure 3.6 shows an example of a $C\\!P$ asymmetry in the
${{B}^{\pm}}\rightarrow{D}^{0}{K}^{\pm}$ ADS mode [69]. Here the ${D}^{0}$ is
produced in a Cabibbo favoured mode ($V_{cb}$) but decays via a suppressed
mode ($V_{cd}$) into $K^{+}\pi^{-}$. This interferes with the
$\overline{{D}}^{0}$ charge-conjugate state which is produced in a suppressed
mode ($V_{ub}$) but decays to the same final state $K^{+}\pi^{-}$ via a
favoured mode ($V_{cs}$). The branching fraction for the favoured ${B}$ decay
is only $\sim 10^{-4}$, so these measurements require high statistics. The
asymmetry observed in Fig. 3.6 has a magnitude of around 40% and has a
significance of 7$\sigma$.

Figure 3.6: Example of a $C\\!P$ asymmetry in the ADS mode
${{B}^{\pm}}\rightarrow{D}^{0}{K}^{\pm}$, where the ${D}^{0}$ and
$\overline{{D}}^{0}$ both decay into $K^{+}\pi^{-}$. The event yield is shown
as a function of ${D}{K}$ mass.

A specific example of an LHCb analysis using the GGSZ Dalitz method is in the
decay ${{B}^{\pm}}\rightarrow{D}^{0}{K}^{\pm}$ where ${D}^{0}\rightarrow
K_{S}^{0}\pi^{+}\pi^{-}$ or ${D}^{0}\rightarrow K_{S}^{0}K^{+}K^{-}$ [70].
There is a rich Dalitz plot structure with the presence of large interference
effects. The Dalitz space is divided up into symmetric bins, chosen to
optimise sensitivity. An amplitude analysis can then be used to extract
$\gamma$.

In all ${{B}^{\pm}}\rightarrow{D}^{0}{K}^{\pm}$ modes and decays listed above,
$\gamma$ can also be extracted from the corresponding
${{B}^{\pm}}\rightarrow{D}^{0}{K}^{\star\pm}$ modes, albeit with reduced
$\gamma$ sensitivity. In addition ${{B}^{0}}\rightarrow{D}^{(*)}{K}^{(*)}$
GGSZ modes are also included in the global fit to extract the $\gamma$ average
value.

  * •

$\gamma$ from the “time-dependent”
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{K}^{+}}$ mode

The channel ${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{K}^{+}}$, and its charge
conjugate states, provide a theoretically clean measurement of the angle
($\gamma+{\phi_{{s}}}$) where ${\phi_{{s}}}$ is the (small valued) ${B}_{s}$
mixing phase, with no significant penguin contribution expected [71]. Here
both ${{B}^{0}_{s}}$, and via the mixing diagram $\overline{{B}}^{0}_{s}$ ,
can decay to the same final state ${{D}^{-}_{s}}{{K}^{+}}$, resulting in
interference which is sensitive to $\gamma$. The same is true for decay into
the charge conjugate state ${{D}^{+}_{s}}{{K}^{-}}$. Hence four time-dependent
decay rates are measured: ${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{K}^{+}}$,
${{B}^{0}_{s}}\rightarrow{{D}^{+}_{s}}{{K}^{-}}$,
$\overline{{B}}^{0}_{s}\rightarrow{{D}^{-}_{s}}{{K}^{+}}$ and
$\overline{{B}}^{0}_{s}\rightarrow{{D}^{+}_{s}}{{K}^{-}}$. The method is then
to fit two asymmetries of the form

$\displaystyle\mathcal{A_{CP}}=\dfrac{{N_{\overline{{B}}\rightarrow
f}-N_{{{B}}\rightarrow f}}}{N_{\overline{{B}}\rightarrow
f}+N_{{{B}}\rightarrow f}}\,.$ (3.8)

These measurements yield values for the strong phase difference $\delta_{QCD}$
between the amplitudes ${{B}}\rightarrow f$ and $\overline{{B}}\rightarrow f$,
the amplitude ratio, and ($\gamma+{\phi_{{s}}}$).

The current measurement by LHCb in 1 fb-1 of data yields a value
$\gamma=\left(115^{+28}_{-43}\right)^{\circ}$ [72]. This complements the
measurements in ${{B}^{\pm}}\rightarrow{D}^{0}{K}^{\pm}$, although with less
statistical precision.

  * •

The $\gamma$ combination

The LHCb measurement of $\gamma$ averaged over all the above methods, which
includes all ${B}^{0},B^{\pm}$ and $B_{s}$ modes, is
$\gamma=\left(74.0^{+5.0}_{-5.8}\right)^{\circ}$ [73]. This measurement
dominates the current world average. The confidence limits as a function of
$\gamma$ for the combination is shown in Fig. 3.7 for the various measurement
channels. The agreement between ${B}_{s}$ and ${B}^{\pm}$ initial states is
currently at the 2$\sigma$ level.

Figure 3.7: Confidence limits as a function of $\gamma$ in the LHCb
combination for the various measurement channels.

####  3.3.4 The sides of the triangle

  * •

The side opposite to $\beta$

Currently the closure test of the unitarity triangle is limited mainly by the
side opposite to $\beta$ which has a length proportional to
$|V_{ub}|/|V_{cb}|$ in the Standard Model. This limitation is a consequence of
tension between $B$ Factory inclusive and exclusive $|V_{ub}|$ measurements
which differ by $\sim 3.5\sigma$ [61]. $|V_{ub}|^{2}$ is directly proportional
to the decay rate ${B}^{0}\rightarrow X_{u}\mu^{-}\nu_{\mu}$, where $X_{u}$ is
a meson containing a $u$ quark. Theoretical input from Heavy Quark Effective
Theory and lattice calculations are also necessary to calculate $|V_{ub}|$,
although several of the theoretical uncertainties cancel in the ratio to
calculate the side.

$|V_{ub}|/|V_{cb}|$ is a very difficult measurement at LHCb due to presence of
a neutrino, the identification of which was never in LHCb’s original plans.
Although the $B$ Factory favoured channel
${B}^{0}\rightarrow\pi^{+}\mu^{-}\nu_{\mu}$ cannot currently be identified at
LHCb, the equivalent baryonic channel $\Lambda_{b}\rightarrow
p\mu^{-}\nu_{\mu}$ has been measured. The signal is separated from the lower-
mass backgrounds, shown in Fig. 3.8. Using form factors from lattice
calculations, LHCb measures [74]

$\left|V_{ub}\right|=(3.27\pm 0.15\,{\rm(exp)}\pm 0.17\,{\rm(theory)}\pm
0.06\,(\left|V_{cb}\right|))\times 10^{-3}$

This is to be compared to the world average of $\left|V_{ub}\right|=(3.94\pm
0.36)\times 10^{-3}$ [61].

Figure 3.8: The $(p\mu^{-})$ mass distribution in the measurement of
$|V_{ub}|$, showing the contribution from $\Lambda_{b}\rightarrow
p\mu^{-}\nu_{\mu}$ decays and the various backgrounds.

  * •

The side opposite to $\alpha$

The mass difference $\Delta{m_{s}}$ measured in ${B}^{0}_{s}$ mixing (Fig.
3.3), which is dominated by the top-quark loop, provides a measurement of the
third side of the triangle, ${{V_{td}}\over{\lambda V_{ts}}}$. This is
proportional to the ratio of mixing frequencies $\sqrt{\frac{\Delta
m_{d}}{\Delta m_{s}}}$. Corrections are calculated from the lattice with a
theoretical error of $\sim$5-10% and systematic errors largely cancel in the
ratio.

Following the measurement by LHCb of the mixing parameters presented above,
the ratio $\left|V_{td}/V_{ts}\right|$ is $0.210\pm 0.001\pm 0.008$ [49].
Systematic errors can be reduced in the future by improved lattice QCD
calculations.

###  3.4 Other $C\\!P$ violation measurements

####  3.4.1 ${B}_{s}$ weak mixing phase ${\phi_{{s}}}$ in
${{B}^{0}_{s}}\rightarrow J/\psi\phi$

It can be seen in Fig. 3.3 that $V_{ts}$ appears twice in the
${{B}^{0}_{s}}-\overline{{B}}^{0}_{s}$ mixing process, introducing a relative
“weak mixing phase”, of ${\phi_{{s}}}$ to fourth order in $\lambda$. The
${B}_{s}$ mixing phase can be measured in the channel
${{B}^{0}_{s}}\rightarrow J/\psi\phi$, which is governed by a single tree-
level diagram with a negligible penguin contribution. Hence this mode is the
strange-quark analogue of the golden mode
${{B}^{0}}\rightarrow{J}/\psi{K}^{0}_{S}$ in the ${B}^{0}$ system. In the
${B}_{s}$ system $C\\!P$ asymmetry arises from the interference of the
${{B}^{0}_{s}}\rightarrow J/\psi\phi$ with the mixed process
${{B}^{0}_{s}}\rightarrow\overline{{B}}^{0}_{s}\rightarrow J/\psi\phi$. In the
Standard Model, ${\phi_{{s}}}$ is expected to be very small, $\sim 0.036\pm
0.002$ rad [52], hence this channel is a very sensitive probe for new physics.

LHCb reconstructs ${{B}^{0}_{s}}\rightarrow J/\psi\phi$ events in the decay
modes $J/\psi\rightarrow\mu^{+}\mu^{-}$, and $\phi\rightarrow K^{+}K^{-}$
[75]. This ${{B}^{0}_{s}}$ final state is an admixture of $C\\!P$-even and odd
contributions, therefore an angular analysis of decay products is required.
Good tagging performance of ${{B}^{0}_{s}}$ and $\overline{{B}}^{0}_{s}$ is
important, with a total tagging power in this analysis of $4.73\pm 0.34$.

The fitted value of ${\phi_{{s}}}$ is correlated with ${\Delta\Gamma_{{s}}}$,
the width difference of the ${{B}^{0}_{s}}$ mass eigenstates. The decay
${{B}^{0}_{s}}\rightarrow J/\psi\pi^{+}\pi^{-}$ is also added to improve the
sensitivity [76]. Contours in the
$\left({\phi_{{s}}},{\Delta\Gamma_{{s}}}\right)$ plane are plotted in Fig.
3.9. The LHCb measurements are ${\Delta\Gamma_{{s}}}=0.0816\pm
0.0048~{}\mathrm{ps}^{-1}$ with the $C\\!P$-violating phase
${\phi_{{s}}}=-0.041\pm 0.025~{}\mathrm{rad}$. Figure 3.9 also shows the
world-averaged measurement of
${\phi_{{s}}}~{}\mathrm{versus}~{}{\Delta\Gamma_{{s}}}$ showing the LHCb
result in combination with those from other experiments and the Standard Model
expectation [61].

Figure 3.9: The measurement of the ${{B}^{0}_{s}}$ weak mixing phase in the
$\left({\phi_{{s}}},{\Delta\Gamma_{{s}}}\right)$ plane, showing the LHCb
result in combination with those from other experiments, the Standard Model
expectation, and the $1\sigma$ contour of the combination [61].

####  3.4.2 $C\\!P$ violation in charm

The Standard Model prediction of $C\\!P$ violation in the charm system is
expected to be very small
${\mathcal{O}}\left(10^{-4}\right)\rightarrow{\mathcal{O}}\left(10^{-3}\right)$,
where $C\\!P$ violation can arise in Cabibbo-suppressed (CS) decays in the
interference between tree and penguin amplitudes. In particular LHCb has
measured asymmetries in the direct $C\\!P$-violating channels
${D}^{0}(\overline{{D}}^{0})\rightarrow\pi^{+}\pi^{-}$ and
${D}^{0}(\overline{{D}}^{0})\rightarrow K^{+}K^{-}$ [77].

In the LHCb analysis, ${D}^{0}$ and $\overline{{D}}^{0}$ decays are identified
via two self-tagging decay paths. “Prompt” decays (${D}$ decays originating
from the primary vertex) are characterised by the presence of a “soft” low-
momentum pion from a ${D}^{*}$ i.e. ${D}^{*+}\rightarrow{D}^{0}\pi^{+}_{\rm
soft}$ and the charge-conjugate mode. “Semileptonic” decays are secondary
${D}$’s which originate from prompt ${B}$ decays, i.e.
${B}^{+}\rightarrow\overline{{D}}^{0}\mu^{+}X$ and its charge-conjugate state.

The raw asymmetry $(\mathcal{A})$ for ${D}^{0}\rightarrow h^{+}h^{-}$ decays
$(h=K$ or $\pi)$ is defined as

$\mathcal{A}(D\rightarrow f)=\frac{N(D\rightarrow
f)-N(\overline{D}\rightarrow\overline{f})}{N(D\rightarrow
f)+N(\overline{D}\rightarrow f)}$ (3.9)

which includes both physics and detector terms:
$\mathcal{A}=\mathcal{A}_{{C\\!P}}+\mathcal{A}_{D}+\mathcal{A}_{P}$. Detection
asymmetry arises from small charge differences associated with the
$\pi^{\pm}_{{\rm soft}}$ or $\mu^{\pm}$. Production asymmetry arises from
different production rates of ${D}^{*}$ and ${B}$ in $pp$ collisions. To
eliminate these two contributions and cancel associated systematics, the
$\Delta\mathcal{A}_{{C\\!P}}$ parameter is measured in LHCb:

$\Delta\mathcal{A}_{{C\\!P}}=\mathcal{A}\left(K^{+}K^{-}\right)-\mathcal{A}\left(\pi^{+}\pi^{-}\right)=\mathcal{A}_{{C\\!P}}\left(K^{+}K^{-}\right)-\mathcal{A}_{{C\\!P}}\left(\pi^{+}\pi^{-}\right).$
(3.10)

The raw symmetries are obtained from mass fits, and then by simply counting
the numbers of ${D}$’s decaying to $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$,
respectively.

A measurement performed with Run 1 and Run 2 LHCb data combined gives
$\Delta\mathcal{A}_{{C\\!P}}=(-15.4\pm 2.9)\times 10^{-4}$. This is a
5.3$\sigma$ measurement of $C\\!P$ violation in the charm system and opens a
new window for the study of $C\\!P$ violation in the future.

####  3.4.3 $C\\!P$ violation in beauty baryons

$C\\!P$ violation has been observed in ${B}$, $K$, and ${D}$ decays, but not
yet in baryon decays. A search for $C\\!P$ violation in the multi-body mode
$\Lambda_{b}^{0}\rightarrow p^{+}\pi^{-}\pi^{+}\pi^{-}$ decays was performed
on LHCb Run 1 data [78]. This decay proceeds via tree and loop diagrams with
similar contributions and through numerous intermediate resonances, enhancing
the possibility for $C\\!P$ violation, although in areas where re-scattering
effects can play a role. A 3.3$\sigma$ deviation from $C\\!P$ symmetry was
observed, however introducing 6.6 fb-1 of Run 2 data has not confirmed this
result [79]. Hence this measurement awaits further statistics, and will be
improved when cleaner 2-body ${B}$-baryon decays can be added to the study.

####  3.4.4 $C\\!P$ violation in charmless 3-body ${{B}^{\pm}}$ decays

Yields of ${{{B}^{+}}}\rightarrow\pi^{+}K^{+}K^{-}$ and
${{{B}^{-}}}\rightarrow\pi^{-}K^{-}K^{+}$ decays show striking asymmetries in
the region of phase space dominated by re-scattering effects [80]. Similarly
huge $C\\!P$-violating effects between
${{{B}^{+}}}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ and
${{{B}^{-}}}\rightarrow\pi^{-}\pi^{-}\pi^{+}$ decays are observed in a region
of phase space including the $\rho^{0}(770)$ and $f_{2}(1270)$ resonances [81,
82]. Figure 3.10 shows an example of the spectacular asymmetries observed in
${B}\rightarrow\pi KK$ decays, which exceed 50% at low values of $K^{+}K^{-}$
mass.

Figure 3.10: Yields of (top) ${{{B}^{+}}}\rightarrow\pi^{+}K^{+}K^{-}$ and
(bottom) ${{{B}^{-}}}\rightarrow\pi^{-}K^{-}K^{+}$ decays as a function of
$K^{+}K^{-}$ mass squared.

##  4  Rare decays

Within the SM the interplay of weak and Higgs interactions implies that
Flavour Changing Neutral Currents (FCNCs) can occur only at higher orders in
the electroweak interactions and are strongly suppressed by the GIM mechanism.
This strong suppression makes FCNC processes natural candidates to search for
physics beyond the SM. If the new degrees of freedom do not have the same
flavour structure of the quarks/leptons-Higgs interactions present in the SM,
then they could contribute to FCNCs at a comparable (or even larger) level to
the SM amplitudes.

In $B$-meson decays, experimenters have measured ${b}\rightarrow{s}$ and
${b}\rightarrow{d}$ quark transitions, while ${c}\rightarrow{u}$ and
${s}\rightarrow{d}$ transitions have been measured in $D$-meson and $K$-meson
decays, respectively. At first order, these transitions can occur through two
kinds of Feynman diagram shown in Fig. 4.1. The first corresponds to the so-
called “box” diagram and describes the mixing between neutral mesons,
discussed in Section 3, the example of Fig. 4.1 shows ${{B}^{0}_{s}}$ mixing.
The second kind of diagram, the so-called ”penguin” diagram, is responsible
for a large variety of FCNC rare decays. The example shown in Fig. 4.1 is that
of a ${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$ transition. In particular, if the
radiated bosons are of electroweak type ($Z$, $W$ or $\gamma$-like), the
uncertainties in the calculation of the SM predictions due to non-perturbative
QCD effects are drastically reduced as compared with the case where a gluon is
radiated. These ”electroweak penguins” are the subject of the discussions in
this section.

Figure 4.1: Examples of loop processes within the SM that allow FCNC
${b}\rightarrow{s}$ quark transition. On the left is an example of a box
diagram and on the right an example of a penguin diagram.

Before the first physics run of the LHC accelerator in 2010, the main
contributors to the study of rare $B$ and $D$-meson decays were the $B$
Factory experiments (BaBar and Belle) and the Tevatron experiments (CDF and
D0). However the production rate of $b$ $\overline{b}$ pairs in the $e^{+}$
$e^{-}$ $B$ Factories was typically five orders of magnitude smaller than at
the LHC. In addition, the lower $p$ $p$ collision energy of the Tevatron (with
a correspondingly reduced $b$ $\overline{b}$ cross section which is
proportional to the collision energy), and the detector’s reduced trigger
acceptance for rare $B$ and $D$-meson decays, implied that LHCb would already
be the most sensitive experiment even after 1 $\rm{fb}^{-1}$ accumulated in
2011. For example, prior to the LHC, the rarest $B$-meson decay ever measured
was ${\mathcal{B}}({{B}^{+}}\rightarrow{{K}^{+}}{\mu^{+}}{\mu^{-}})\sim
5\times 10^{-7}$ with a $20\%$ precision; Belle and BaBar had analyzed
${\mathcal{O}}(200)$ ${{B}^{0}}\rightarrow{{K}^{*0}}\ell^{+}\ell^{-}$ events
and CDF had reached a sensitivity of $\mathcal{B}$
(${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$) $<3\times 10^{-8}$ at the
$95\%$ C.L. As will be clear from the next sections, LHCb has already reached
after Run 1 of the LHC an order of magnitude larger statistical power.

###  4.1 $B^{0}_{\rm{(s)}}\rightarrow\mu^{+}\mu^{-}$

The pure leptonic decays of $K$, $D$ and $B$ mesons are a particular
interesting case of electroweak penguins, where the final quark leg in Fig.
4.1 (right) needs to be swapped to the initial state. The helicity
configuration of the purely leptonic final state suppresses the vector and
axial-vector contributions by a factor in terms of masses proportional to
$\left[\tfrac{m_{\mu}}{m_{{K},{D},{B}}}\right]^{2}$. Therefore, these decays
are particularly sensitive to new (pseudo-) scalar interactions. In the case
of ${B}^{0}$ and ${B}^{0}_{s}$-meson decays, the contribution of the
absorptive part can be safely neglected. As a consequence, the rate is well
predicted theoretically [83] :
${\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}})=(3.66\pm
0.05)\times 10^{-9}$ and
${\mathcal{B}}({{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}})=(1.03\pm 0.05)\times
10^{-10}$. In the ${B}^{0}_{s}$ case, this prediction corresponds to a
flavour-averaged time-integrated measurement, taking into account the
correction due to the non-negligible width difference.

The experimental signature is sufficiently clean to reach an expected signal
over background ratio $S/B\sim 3$ for the ${B}^{0}_{s}$ decay, assuming the SM
branching fraction. The main background in the region around the ${B}^{0}_{s}$
invariant mass is due to combinations of uncorrelated muons and can be
estimated from the mass sidebands. The most important handle to reduce this
combinatorial background is the invariant mass resolution of the experiment,
$\sigma\sim 23$ MeV, which is also crucial to differentiate between ${B}^{0}$
and ${B}^{0}_{s}$ decays ($\Delta m\sim 87$ MeV). Moreover, the large fraction
of ${B}\\!\rightarrow h^{+}h^{\prime-}$ decays is an important source of
background due to hadrons being misidentified as muons in the region around
the ${B}^{0}$ mass (this background is very small in the ${B}^{0}_{s}$ mass
region).

Given their experimental detector resolution and trigger acceptance during Run
1 and Run 2, the CMS experiment with $61~{}\rm{fb}^{-1}$ of data has similar
sensitivity to the LHCb experiment with $4.4~{}\rm{fb}^{-1}$ of data collected
in the same period. Both experiments have provided a clear observation of the
decay ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ (CMS and LHCb observe
about 90 and 30 ${B}^{0}_{s}$ signal candidates with a background of about 40
and 10 events respectively within the invariant mass interval expected to
contain $95\%$ of the signal). Neither experiment has yet reached the
sensitivity to observe the decay ${{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$.
The invariant mass distribution obtained by the LHCb experiment is shown in
Fig. 4.2 and the LHCb [84] and CMS [85] results are
${\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}})=(3.0\pm
0.6_{-0.2}^{+0.3})\times 10^{-9}$ and
${\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}})=(2.9_{-0.6}^{+0.7}\pm
0.2)\times 10^{-9}$ respectively, while for the ${B}^{0}$ decay only limits
can be quoted. The ATLAS collaboration has also published after a
measurement[86] using $51~{}\rm{fb}^{-1}$ of data,
${\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}})=(2.8_{-0.7}^{+0.8})\times
10^{-9}$ in agreement with the previous results.

Figure 4.2: Invariant mass distribution of dimuons selected from Ref. [84].
Superimposed on the data points in blue (solid line) is the combined fit, and
its components as quoted in the insert.

The ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ very rare decay has been
searched for ever since the discovery of $B$ mesons, around 40 years ago.
Thanks to the ingenuity and persistence of the experimenters, it has been
eventually measured at the LHC and found to be in agreement with the SM within
current uncertainties, as shown in Fig. 4.3. Over the next decade it will be
extremely interesting to see how the measurement of $\mathcal{B}$
(${{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$) evolves, for which only upper
limits are currently available.

Figure 4.3: Upper limits (${{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$) and
eventual measurements of the ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
branching ratio by several experiments over the last 40 years.

###  4.2 ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$

The language of effective field theory is used to parameterise NP
contributions in terms of a sum of local four-fermion operators ($Q_{i}$)
which depend only on SM fermions modulated by Wilson coefficients ($C_{i}$)
which in turn depend on the heavy degrees of freedom i.e. NP particles. The
decay ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ is the so-called
“golden mode” to test new vector (axial-vector) couplings, i.e. the $C_{9}$
and $C_{10}$ Wilson coefficients contributing to the ${b}\rightarrow{s}$
transition. The ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ channel
also complements the ${b}\rightarrow{s}{\gamma}$ decay which is mostly
sensitive to NP dipole operators (i.e. $C_{7}$). and also the
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ decay which is mostly
sensitive to NP (pseudo-) scalar operators (i.e. $C_{S}$ and $C_{P}$). The
charge of the pion in the decay ${{K}^{*}}\rightarrow K\pi$ defines the
flavour of the $B$ meson and an angular analysis can be performed
unambiguously to test the helicity structure of the electroweak penguin.

The above system is completely defined by four variables: $q^{2}$, the square
of the invariant mass of the dimuon system, $\theta_{l}$, the angle between
the positive lepton and the direction opposite to the $B$-meson in the dimuon
rest frame, $\theta_{K}$, the equivalent angle of the $K^{+}$ in the $K^{*}$
rest frame and $\phi$ the angle between the two planes defined by $(K,\pi)$
and $(\mu^{+},\mu^{-})$ in the $B$-meson rest frame. The four-fold
differential distribution contains a total of eleven angular terms that can be
written in terms of seven $q^{2}$-dependent complex decay amplitudes. These
amplitudes can be expressed in terms of five complex Wilson coefficients
($C_{S},C_{P},C_{7},C_{9}$ and $C_{10}$), their five helicity counterparts and
six form-factors, which play a role of nuisance parameters in the fit.

The LHCb experiment, with $3~{}\rm{fb}^{-1}$ of data collected in Run 1, has
triggered and selected about 2400
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ candidates in the range
$0.1<q^{2}<19$ GeV2 with signal over background $(S/B)>5$. This is about one
order of magnitude larger than the samples available at previous experiments
(BaBar, Belle and CDF) and similar to the samples collected by ATLAS and CMS
with ten times the luminosity, however with significantly worse $S/B$.

The statistics and the quality of the data accumulated by the LHCb experiment
allows for a full angular analysis of
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ decays to be performed
for the first time. The results [87] of this “tour de force” analysis mostly
agree with SM predictions, however with some hints of disagreement for some
specific distributions. In Fig.4.4, two examples of the $C\\!P$-averaged
angular coefficients (i.e. the average of the coefficients measured with
${B}^{0}$ and $\overline{\rm B}{}^{0}$ decays) are shown as a function of
$q^{2}$. For these two examples, $A_{\rm{FB}}$ (modulating the
$\sin^{2}\theta_{K}\times\cos\theta_{l}$ angular term) and $S_{5}$ (modulating
the $\sin(2\theta_{K})\times\sin\theta_{l}\times\cos\phi$ angular term) seem
to agree less well with SM predictions. However these are early days, and more
data will be required (the Run 2 data analysis will be released soon). Also a
careful reassessment of the SM uncertainties are needed before drawing
definitive conclusions.

Several authors have already attempted to see if the overall pattern of the
angular measurements is consistent with a given value of the relevant Wilson
coefficients. As previously discussed, the inclusive ${b}\rightarrow{s}\gamma$
measurements strongly constrain non-SM values for $C_{7}$. The scalar $C_{S}$
and pseudo-scalar $C_{P}$ coefficients are constrained, for example, by the
measurement of the branching fraction of the very rare decay
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$. Therefore, the small
disagreements observed in the angular analysis of the decay
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ and other decays, seem
to be consistent with a non-SM value of the $C_{9}$ Wilson coefficient, as can
be seen in Fig. 4.5 taken from Ref. [88].

Figure 4.4: Two examples of the $C\\!P$-averaged coefficients in the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ angular terms as a
function of $q^{2}$. The shaded boxes show the SM predictions taken from Ref.
[89]. Figure 4.5: Constrains on the contribution of NP to the real parts of
$C_{9}$ and $C_{10}$ at the $1\sigma$ and $2\sigma$ level taken from Ref.
[88], corresponding to different measurements as indicated in the insert. The
dotted lines correspond to the status before the latest updates for the 2019
physics winter conferences.

###  4.3 Lepton Universality

In the SM, the electroweak couplings of leptons are flavour independent, or
lepton “universal”. However, this may not necessarily be the case for new
particles beyond the SM. In particular, if the hints described in the previous
section are an indication of new particles modifying the penguin diagram in
Fig. 4.1, it is interesting to measure the ratios of branching fractions in
decays of different lepton families. For example, the ratios

$R_{X}=\int\frac{d\Gamma(B^{0}\rightarrow
X\mu\mu)}{dq^{2}}dq^{2}\Bigg{/}\int\frac{d\Gamma(B^{0}\rightarrow
Xee)}{dq^{2}}dq^{2}$ (4.1)

between $B$ decays to final states with muons and electrons, where $X$ is a
hadron containing an $s$-quark or $d$-quark, are predicted to be very close to
unity in the SM [90, 91, 92]. The uncertainties from QED corrections are found
to be at the percent level [93].

LHCb has measured the above ratio in several channels. Using Run 1 and part of
Run 2 data ($4.4~{}\rm{fb}^{-1}$), LHCb measures [94]
$R_{K}=0.846^{+0.060}_{-0.054}\rm{(stat)}^{+0.014}_{-0.016}\rm{(syst)}$ in the
range $1.1<q^{2}<6$ GeV2, about $2.5\sigma$ below the SM prediction, and using
only Run 1 data ($3~{}\rm{fb}^{-1}$) LHCb measures [95]
$R_{{K}^{*}}=0.69^{+0.11}_{-0.07}\rm{(stat)}\pm 0.05\rm{(syst)}$ in the same
$q^{2}$ range, with a similar level of disagreement with the SM prediction.
The latest $R_{K}$ results from LHCb in bins of $q^{2}$ can be seen in Fig.
4.6, compared with previous results from the BaBar and Belle collaborations.
The consistency between different experiments, and different channels,
although with very different precision, has motivated many theoretical studies
that relate these hints of lepton non-universality with the discrepancies
described in the previous section. Figure 4.5 shows the status of the
compatibility of both sets of measurements when assuming that only the
${b}{s}\mu\mu$ Wilson coefficients are modified (and ${b}{s}ee$ coefficients
are as predicted by the SM). Whilst the initial results showed remarkable
consistency between different sets of measurements as shown by the dotted
lines in Fig. 4.5, the 2019 latest updates show a less clear picture. As of
today, it is difficult to draw reliable conclusions, and more data is eagerly
awaited.

Figure 4.6:  Measurement of $R_{K}$ as a function of the $q^{2}$ bins by
different experiments.

##  5 Spectroscopy

A deep understanding of quantum chromodynamics, the theory of strong
interactions, is vital for precision tests of the Standard Model and in
searches for new physics beyond. QCD is intensively tested in deep-inelastic-
scattering processes and heavy vector boson production, however in the low-
energy regime, there is a lack of precise QCD predictions. QCD, being a non-
pertubative theory, does not calculate hadron properties, namely masses and
decay widths from first principles. Alternative theoretical approaches are
developed, such as heavy quark effective theory, heavy quark expansion or
lattice calculations. These approaches require verification with experiment in
various regimes, e.g. testing the agreement with data for hadrons with
different quark content and quantum numbers. Spectroscopic measurements of
hadron masses and widths or lifetimes provide a wide variety of tests for QCD
models.

Huge production cross sections of charm and beauty in high-energy ${p}{p}$
collisions in the forward region at LHCb [96, 97, 98, 99, 100, 101, 102, 103,
104, 105, 106, 107, 108, 109, 110, 111, 112], together with a good
reconstruction efficiency, versatile trigger scheme and an excellent momentum
and mass resolution, opens up exciting opportunities for spectroscopy
measurements. The employment of LHCb’s powerful hadron identification system
[113, 36, 34] enables a substantial reduction in the combinatorial background
specific to high-energy hadron-hadron collisions. The unique hadron
identification is also especially important for spectroscopy measurements
involving charged kaons and/or protons in the final state. The excellent
momentum and vertex resolutions provided by the LHCb tracking system allows
unprecedented precision on mass and width measurements: indeed the most
precise measurements of mass for all open beauty particles and lifetimes of
all open heavy flavour particles, currently result from the LHCb experiment
[114]. Additional good control over the momentum scale and the detector
alignment [115, 116] also allows the natural widths of hadronic resonances to
be probed with world-leading sub- MeV precision [117, 118, 119, 120, 121, 122,
123].

Almost 30 new hadrons have been discovered [124] using the Run 1&2 data-set,
among them several open charm mesons and baryons, double-charm baryon, a
charmonium state, several beauty mesons, fifteen beauty baryons, and
pentaquark-like $P_{{c}}(4312)^{+}$, $P_{{c}}(4400)^{+}$ and
$P_{{c}}(4457)^{+}$ states. The quantum numbers of many hadrons, especially
for charm mesons and charmonia-like exotic candidates, have been determined
using amplitude-analysis techniques. In the case of exotic particles such as
pentaquark and tetraquark candidates, the determination of quantum numbers is
vital for the understanding of their nature. Large statistics and low levels
of background in decays of beauty hadrons with
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ or
${\psi{(2S)}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ final states allow the study of
the spectroscopy of light unflavoured or strange hadrons [125, 126, 127, 128,
129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143,
144]. Selected spectroscopy highlights are discussed in this chapter.

###  5.1 Charm hadrons

Two complementary methods for the study of spectroscopy of charm hadrons have
been exploited in LHCb:

  * •

the study of promptly produced charm hadrons,

  * •

the study of charm particles produced in weak decays of beauty hadrons, e.g.
from exclusive ${B}\\!\rightarrow{D}^{(\ast)}\pi\pi$ decays.

The first technique allows the most efficient exploitation of the huge prompt
${c}{\overline{c}}$ production cross section in high energy hadron collisions,
but this is usually affected by a large background from light hadrons produced
at the ${p}{p}$ collision vertex. The second technique exploits a full
amplitude analysis of the exclusive decays of beauty hadrons and therefore
involves much lower statistics, however the method often allows the
determination of quantum numbers of the charm hadrons. The study of $D\pi$
final states enables a search for natural spin-parity resonances,
($P=(-1)^{J}$, labelled as ${D}^{*}$) whilst the study of ${D}^{*}\pi$ final
states provides the possibility of studying both natural and unnatural spin-
parity states, except for the $J^{P}=0^{+}$ case, which is forbidden because
of angular momentum and parity conservation. In inclusive $D^{(*)}\pi$
production, the production of any $J^{P}$ state is permitted. An amplitude
analysis of $B$ decays allows a full spin-parity analysis of the charmed
mesons present in the decay.

Both the above approaches are complementary, and have resulted in discoveries
of several new charm hadrons, amongst them the excited charm mesons. Many
previous meson and baryon states, discovered earlier by other experiments,
have been confirmed with high statistical significance, and their masses and
widths have been measured with high precision. For many, the quantum numbers
were either measured or constrained.

##### Excited charm mesons in the ${{D}^{*+}}{{\pi}^{-}}$,
${{D}^{+}}{{\pi}^{-}}$ and ${{D}^{0}}{{\pi}^{+}}$ spectra

have been studied in LHCb using a 1$\text{\,fb}^{-1}$ data-set collected at
$\sqrt{s}=7\text{\,Te\kern-1.00006ptV}$ [145]. Four resonances labelled
${D}_{0}(2550)$, ${D}^{\ast}_{J}(2600)$, ${D}(2740)$ and
${D}_{3}^{\ast}(2750)$ have been observed. The ${D}_{0}(2550)$, and
${D}(2740)$ decay angular distributions are consistent with an unnatural spin-
parity, whilst the ${D}^{\ast}_{J}(2600)$, and ${D}_{3}^{\ast}(2750)$ states
are assigned natural parities. For the ${D}_{0}(2550)$, meson, angular
distributions are consistent with a $J^{P}=0^{-}$ assignment, however for the
other states, no definite assignment exists.

The excited charm mesons were also studied in $B$-decay amplitude analyses of
${{B}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{{B}^{-}}}\\!\rightarrow{{D}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ decays using the
Run 1 data-set [146, 147] and
${{{B}^{-}}}\\!\rightarrow{{D}^{*+}}{{\pi}^{-}}{{\pi}^{+}}$ [148] using a
data-set corresponding to 4.7$\text{\,fb}^{-1}$, collected at $\sqrt{s}=7$, 8
and 13 TeV. The ${D}^{\ast}_{3}(2760)^{0}$ and ${D}^{\ast}_{2}(3000)^{0}$
states were observed for the first time [147], and the most precise
determination of masses, widths and quantum numbers has been performed for the
${D}^{\ast}_{0}(2400)^{-}$, ${D}^{\ast}_{2}(2660)^{-}$,
${D}^{\ast}_{3}(2760)^{-}$ [146], ${D}^{\ast}_{2}(2660)^{0}$,
${D}^{\ast}_{1}(2680)^{0}$, ${D}^{\ast}_{3}(2760)^{0}$,
${D}^{\ast}_{2}(3000)^{0}$ [147] and for the ${D}_{1}(2420)^{0}$,
${D}_{1}(2430)^{0}$, ${D}_{0}(2550)^{0}$, ${D}^{\ast}_{1}(2600)^{0}$,
${D}_{2}(2740)^{0}$, ${D}^{\ast}_{3}(2750)^{0}$ [148] states.

##### Excited ${{D}^{+}_{s}}$ mesons

were studied both in inclusive production in the
${{D}^{+}}{{K}^{0}_{\mathrm{S}}}$ and ${{D}^{0}}{{K}^{+}}$ spectra using a
data-set of 1$\text{\,fb}^{-1}$ collected at
$\sqrt{s}=7\text{\,Te\kern-1.00006ptV}$ [149], and through the amplitude
analysis of ${{B}^{0}_{s}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{-}}{{\pi}^{+}}$ decays using
Run 1 data [150]. It has been shown that the previously reported
${D}_{{s}J}^{\ast}(2860)^{+}$ state [151, 152, 149] is an admixture of spin-1
and spin-3 resonances. The masses and width of new states, dubbed
${D}_{{s}1}^{\ast}(2860)^{+}$ and ${D}_{{s}3}^{\ast}(2860)^{+}$, are precisely
measured, as well as the masses and widths of the
${D}_{{s}2}^{\ast}(2573)^{+}$ and ${D}_{{s}1}^{\ast}(2700)^{+}$ [149, 150].

##### Excited ${\mathchar 28931\relax}^{+}_{c}$ baryons

were studied in their decays to the ${{D}^{0}}{p}$ final state via the
amplitude analysis of ${\mathchar 28931\relax}^{0}_{b}$ baryon decays using an
integrated luminosity sample of 3$\text{\,fb}^{-1}$ collected at $\sqrt{s}=7$
and 8 TeV [153]. The analysis uses a sample of $11\,212\pm 126$ signal
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{D}^{0}}{p}{{\pi}^{-}}$ decays,
where the ${D}^{0}$ mesons are reconstructed in the${{K}^{-}}{{\pi}^{+}}$
final state. The amplitude fit is performed in the four-phase space regions in
the Dalitz plot. For the near-threshold $m_{{{D}^{0}}{p}}$ region, an
enhancement in the ${{D}^{0}}{p}$ amplitude is studied. The enhancement is
consistent with being a resonant state, dubbed the $\mathchar
28931\relax_{{c}}(2860)^{+}$, with quantum numbers $J^{P}=\tfrac{3}{2}^{+}$,
and with the parity measured relative to that of the $\mathchar
28931\relax_{{c}}(2880)^{+}$ state. The other quantum numbers are excluded
with a significance greater than 6 standard deviations. The phase motion of
the $\tfrac{3}{2}^{+}$ component with respect to the non-resonant amplitudes
is obtained in a model-independent way and is consistent with resonant
behavior. The mass of the $\mathchar 28931\relax_{{c}}(2860)^{+}$ state is
consistent with predictions for an orbital D-wave ${\mathchar
28931\relax}^{+}_{c}$-excitation with quantum numbers $\tfrac{3}{2}^{+}$,
based on the nonrelativistic heavy quark-light diquark model and from QCD sum
rules in the HQET framework. Also the fit allowed the most precise
determination of the masses and widths of the known resonances $\mathchar
28931\relax_{{c}}(2880)^{+}$ and $\mathchar 28931\relax_{{c}}(2940)^{+}$, as
well as constraining their quantum numbers.

##### Three excited $\mathchar 28932\relax_{{c}}^{\ast 0}$ baryons

have been observed in the ${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}$ mass
spectrum using the Run 2 data-set [154]. The mass difference $\delta m\equiv
m({{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}})-m({{\mathchar
28931\relax}^{+}_{c}})-m({{K}^{-}})$ spectrum for selected ${{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}$ combinations is shown in Fig. 5.1. Three
narrow structures, denoted $\mathchar 28932\relax_{{c}}(2923)^{0}$, $\mathchar
28932\relax_{{c}}(2939)^{0}$ and $\mathchar 28932\relax_{{c}}(2965)^{0}$ are
clearly visible with a significance exceeding $20\sigma$ for each signal. The
data and fit show the least compatibility in the region $\delta m\approx
110\text{\,Me\kern-1.00006ptV}$, that could be evidence for a fourth
$\mathchar 28932\relax_{{c}}^{\ast 0}$ state. Figure 5.1(right) shows the
$\delta m$ distribution for the signal samples, where a structure in this
region is added into the fit. A large improvement in the fit quality is
achieved.

$\begin{array}[]{c}\mbox{LHCb}\\\
\text{Run~{}2}\end{array}$$\begin{array}[]{c}\mbox{LHCb}\\\
\text{Run~{}2}\end{array}$ Figure 5.1:  Mass difference $\Delta
m=m_{{{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}}-m_{{{\mathchar
28931\relax}^{+}_{c}}}-m_{{{K}^{-}}}$ distributions [154]. Fits accounting for
(left) three and (right) four excited $\mathchar 28932\relax_{{c}}^{\ast 0}$
states are superimposed.

##### Five narrow excited $\mathchar 28938\relax_{{c}}^{\ast}$ baryons

have been observed in the ${{\mathchar 28932\relax}^{+}_{c}}{{K}^{-}}$ mass
spectrum using the Run 1 data-set [155]. A large sample of ${\mathchar
28932\relax}^{+}_{c}$ candidates were reconstructed in their Cabibbo-
suppressed mode ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$. In total around
$1.05\times 10^{6}$ ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ candidates with
with a purity of 83% were selected, shown in Fig. 5.2(left). The mass
distribution for ${{\mathchar 28932\relax}^{+}_{c}}{{K}^{-}}$ combinations is
shown in Fig. 5.2(right), and five narrow peaks are clearly visible. The
natural widths of the peaks are found to be between 0.8 and
$8.7\text{\,Me\kern-1.00006ptV}$, and two of them, named $\mathchar
28938\relax_{{c}}(3050)^{0}$ and $\mathchar 28938\relax_{{c}}(3119)^{0}$, are
found to be extremely narrow, with 95%CL limits of 1.2 and 2.8 MeV,
respectively. It is found that the fit improves if an additional broad
Breit$-$Wigner function is included in the 3188 MeV mass region. This broad
structure may represent a single resonance, be the superposition of several
resonances, be a feed-down from higher states, or some combination of the
above. The interpretation of the narrow states is still an open question. The
naive quark model expects five states in the region, but some have to be
relatively broad. The molecular model predicts two states with
$J^{P}=\frac{1}{2}^{+}$ and two with $J^{P}=\frac{1}{2}^{+}$, and three of the
observed states are in remarkable agreement, both in mass and width, with this
hypothesis [156].

${{\mathchar 28932\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$
$\begin{array}[]{c}\mbox{LHCb}\\\
\text{Run~{}1}\end{array}$$\begin{array}[]{c}\mbox{LHCb}\\\
\text{Run~{}1}\end{array}$ Figure 5.2:  (left) Distribution of the
reconstructed invariant mass $m_{{p}{{K}^{-}}{{\pi}^{+}}}$ for all ${\mathchar
28932\relax}^{+}_{c}$ candidates. The solid (red) curve shows the result of
the fit, and the dashed (blue) line indicates the fitted background. (right)
Distribution of the reconstructed invariant mass of ${{\mathchar
28932\relax}^{+}_{c}}{{K}^{-}}$ combinations; the solid (red) curve shows the
result of the fit, and the dashed (blue) line indicates the fitted background
[155]. The shaded (red) histogram shows the corresponding mass spectrum from
the ${\mathchar 28932\relax}^{+}_{c}$ side-bands and the shaded (light grey)
distributions indicate the feed-down from partially reconstructed ${\mathchar
28938\relax}_{c}(X)^{0}$ resonances.

###  5.2 Double-charm baryons

Three weakly decaying states with charm number $C=2$ are expected in the quark
model: one isospin doublet $\mathchar 28932\relax_{{c}{c}}$ and one isospin
singlet $\mathchar 28938\relax_{{c}{c}{s}}$, each with spin-parity
$J^{P}=\tfrac{1}{2}^{+}$. The properties of these baryons have been calculated
with a variety of theoretical models. In most cases, the masses of the
$\mathchar 28932\relax_{{c}{c}}$ states are predicted to lie in the range 3500
to 3700$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ [157]. The masses of the
${\mathchar 28932\relax}^{++}_{{c}{c}}$ and ${\mathchar
28932\relax}^{+}_{{c}{c}}$ states are expected to differ by only a few
$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ due to approximate isospin symmetry.
Most predictions for the lifetime of the ${\mathchar
28932\relax}^{+}_{{c}{c}}$ baryon are in the range 50 to 250 fs, and the
lifetime of the ${\mathchar 28932\relax}^{++}_{{c}{c}}$ baryon is expected to
be three to four times longer at 200 to 700 fs, While both are expected to be
produced at hadron colliders the longer lifetime of the ${\mathchar
28932\relax}^{++}_{{c}{c}}$ baryon should make it significantly easier to
experimentally observe than the ${\mathchar 28932\relax}^{+}_{{c}{c}}$ baryon.

Experimentally, there is a longstanding puzzle in the $\mathchar
28932\relax_{{c}{c}}$ system. Observations of the ${\mathchar
28932\relax}^{+}_{{c}{c}}$ baryon in the ${{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}$ final state at a mass of $3519\pm
2$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ with signal yields of 15.9 events
over $6.1\pm 0.5$ events background ($6.3\sigma$ significance), and 5.62
events over $1.38\pm 0.13$ events background in the final state
${p}{{D}^{+}}{{K}^{-}}$ ($4.8\sigma$ significance), were reported by the SELEX
collaboration [158, 159]. The SELEX results included a number of unexpected
features, notably a short lifetime and a large production rate relative to
that of the singly charmed ${\mathchar 28931\relax}^{+}_{c}$ baryon. The
lifetime was reported to be shorter than $33\text{\,fs}$ at the 90% confidence
level, and SELEX concluded that 20% of all ${{\mathchar 28931\relax}^{+}_{c}}$
baryons observed by the experiment originated from ${{\mathchar
28932\relax}^{+}_{{c}{c}}}$ decays, implying a relative $\mathchar
28932\relax_{{c}{c}}$ production rate several orders of magnitude larger than
theoretical expectations. Searches from the FOCUS [160], BaBar [161], and
Belle [162] experiments did not find evidence for a state with the properties
reported by the SELEX collaboration, and neither did a search at LHCb with
data corresponding to an integrated luminosity of $0.65$$\text{\,fb}^{-1}$
[163]. However, because the production environments at all the above
experiments differ from that of SELEX, which studied collisions of a hyperon
beam on fixed nuclear targets, these null results do not exclude the original
observations.

LHCb has searched for the ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ decaying
into ${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ using
a sample of ${p}{p}$ collision data at 13 TeV, corresponding to an integrated
luminosity of 1.7$\text{\,fb}^{-1}$. A highly significant structure is
observed in the mass spectrum, where the ${\mathchar 28931\relax}^{+}_{c}$
baryon is reconstructed into ${p}{{K}^{-}}{{\pi}^{+}}$, shown in Fig.
5.3(left). The structure is consistent with originating from a weakly decaying
particle, identified as the doubly charmed baryon ${\mathchar
28932\relax}^{++}_{{c}{c}}$. The observation of the state is confirmed using
an additional sample of data collected at 8 TeV. Soon after, the observation
was further confirmed by observing the same state in the decay ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$, shown in Fig. 5.3(right). The mass of the
${\mathchar 28932\relax}^{++}_{{c}{c}}$ state was measured to be in very good
agreement with the value measured in the ${{\mathchar
28932\relax}^{++}_{{c}{c}}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay channel [164]. The
lifetime and mass of the ${{\mathchar 28932\relax}^{++}_{{c}{c}}}$ baryon were
also precisely measured [165, 166], where the lifetime favours smaller values
in the range of the theoretical predictions.

$\begin{array}[]{l}\mbox{LHCb}~{}1.7\text{\,fb}^{-1}\\\
\sqrt{s}=13\text{\,Te\kern-0.80005ptV}\end{array}$
$\begin{array}[]{l}\mbox{LHCb}~{}1.7\text{\,fb}^{-1}\\\
\sqrt{s}=13\text{\,Te\kern-0.80005ptV}\end{array}$ Figure 5.3: (Left) Mass
distribution of ${{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ candidates with fit
projections overlaid [167]. (Right) Mass distribution of ${{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{+}}$ candidates [164].

###  5.3 Beauty hadrons

##### Excited ${B}^{+}$ and ${B}^{0}$ mesons

have been investigated in the mass distributions of ${{{B}^{+}}}{{\pi}^{-}}$
and ${{B}^{0}}{{\pi}^{+}}$ combinations using a $3\text{\,fb}^{-1}$ data
sample at 7 and $8\text{\,Te\kern-1.00006ptV}$. The ${B}^{+}$ and ${B}^{0}$
candidates were reconstructed through the ${{B}^{+}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\pi^{+}$,
${{B}^{+}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\pi^{+}\pi^{+}\pi^{-}$,
${{B}^{+}}\rightarrow J/\psi K^{+}$, ${{B}^{0}}\rightarrow{{D}^{-}}\pi^{+}$,
${{B}^{0}}\rightarrow{{D}^{-}}\pi^{+}\pi^{+}\pi^{-}$ and ${{B}^{0}}\rightarrow
J/\psi{{K}^{*0}}$ decay chains. Samples of about 1.2 million ${B}^{0}$ and 2.5
million ${B}^{+}$ candidates have been obtained with purity depending on decay
mode, but always better than 80%. The ${{{B}^{+}}}{{\pi}^{-}}$ and
${{B}^{0}}{{\pi}^{+}}$ mass spectra with requirements that
$p_{\mathrm{T}}>2\text{\,Ge\kern-1.00006ptV\\!/}c$ are shown in Fig. 5.4,
where ten peaking structures are reconstructed. Out of these, six narrow low-
mass structures correspond to the decays of the four ${B}_{1}(5721)^{0,+}$ and
${B}_{2}^{*}(5747)^{0,+}$ states observed by the CDF and D0 collaborations
[168, 169, 170]: ${B}_{1}(5721)\rightarrow{B}^{\ast}\pi$ and
${B}_{2}(5747)\rightarrow{B}^{(\ast)}\pi$. The high statistics of LHCb has
allowed the most precise measurements of the masses and widths of the
${B}_{1}(5721)^{0,+}$ and ${B}_{2}^{*}(5747)^{0,+}$ states to be made.

In addition to the six low-mass structures, four wider high-mass structures
are observed, particularly prominent at high pion transverse momentum. These
structures are consistent with the presence of four new excited $B$ mesons,
labeled ${B}_{J}(5840)^{0,+}$ and ${B}_{J}(5960)^{0,+}$, whose masses and
widths are obtained under different hypotheses of their quantum numbers [171].

LHCbRun 1LHCbRun 1 Figure 5.4:  Spectra of $\Delta m\equiv
m_{{B}\pi}-m_{{B}}-m_{\pi}$ for (left) ${{{B}^{+}}}{{\pi}^{-}}$ and (right)
${{B}^{0}}{{\pi}^{+}}$ candidates with
$p_{\mathrm{T}}(\pi)>2\text{\,Ge\kern-0.90005ptV\\!/}c$ [171].

##### Orbitally excited ${B}^{0}_{s}$ mesons

have been studied using only 1$\text{\,fb}^{-1}$ of data, collected at
$\sqrt{s}=7\text{\,Te\kern-1.00006ptV}$. The ${{B}^{+}}{{K}^{-}}$ mass spectra
were investigated with the ${B}^{+}$ mesons being reconstructed in four decay
modes. Previously, two narrow peaks had been observed in the
${{B}^{+}}{{K}^{-}}$ mass distribution by the CDF and D0 collaborations [172,
173], named the ${B}_{{s}1}^{*}(5830)^{0}$ and ${B}_{{s}2}^{*}(5840)^{0}$.
They are putatively identified as members of $j_{q}=\tfrac{3}{2}$ HQET doublet
[174]. The two states are also visible in LHCb data, here as three narrow
peaks shown in Fig. 5.5(left), corresponding to the decays
${B}_{{s}1}(5830)^{0}\rightarrow{B}^{\ast+}{{K}^{-}}$,
${B}_{{s}2}^{\ast}(5840)^{0}\rightarrow{B}^{\ast+}{{K}^{-}}$, and
${B}_{{s}2}^{\ast}(5840)^{0}\rightarrow{{B}^{+}}{{K}^{-}}$, where a soft
photon from ${B}^{\ast+}\rightarrow{{B}^{+}}{\gamma}$ is undetected. This is
the first observation of the
${B}_{{s}2}^{\ast}(5840)^{0}\rightarrow{B}^{\ast+}{{K}^{-}}$ decay mode and a
$J^{P}=2^{+}$ assignment is favoured for this state. Large statistics, low
background and LHCb’s excellent mass resolution has allowed the first
determination of the ${B}_{{s}2}^{\ast}(5840)^{0}$ width as well as the most
precise mass measurements of both states. Due to the small energy release, the
position and the shape of the peaks depends on the mass of the ${B}^{\ast+}$
states, allowing the most precise determination of the $m_{{B}^{\ast+}}$ mass,
as well as the mass difference $m_{{B}^{\ast+}}-m_{{{B}^{+}}}$.

$\begin{array}[]{l}\mbox{LHCb}~{}1\text{\,fb}^{-1}\\\
\sqrt{s}=7\text{\,Te\kern-0.70004ptV}\end{array}$ LHCbRun
1&2$\begin{array}[]{l}{B}_{{c}}(2S)^{(\ast)+}\\\
\rightarrow{B}_{{c}}^{(\ast)+}{{\pi}^{+}}{{\pi}^{-}}\end{array}$ Figure 5.5:
(left) The mass difference distribution
$m_{{{B}^{+}}{{K}^{-}}}-m_{{{B}^{+}}}-m_{{{K}^{-}}}$ in the
${{B}^{+}}{{K}^{-}}$ mass spectra. The three peaks are identified as (left-to-
right) ${B}_{{s}1}(5830)^{0}\rightarrow{B}^{\ast+}{{K}^{-}}$,
${B}_{{s}2}^{\ast}(5840)^{0}\rightarrow{B}^{\ast+}{{K}^{-}}$, and
${B}_{{s}2}^{\ast}(5840)^{0}\rightarrow{{B}^{+}}{{K}^{-}}$. The total fit
function is shown as a solid blue line, while the shaded red region is the
spectrum of like-charge ${{B}^{+}}{{K}^{+}}$ combinations. The inset shows an
expanded view of the
${B}_{{s}1}(5830)^{0}/{B}_{{s}2}^{\ast}(5840)^{0}\rightarrow{B}^{\ast+}{{K}^{-}}$
region. (right) Distribution of $\Delta M\equiv
m_{{{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}}-m_{{{B}_{c}^{+}}}$ in the
${{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ mass spectra with fit results overlaid.
The same-sign distribution has been normalised to the data in the
${B}_{{c}}(2S)$ side band region.

##### Excited ${B}_{c}^{+}$ mesons

have been searched for via their decays into the
${{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ final state. A wide peak, interpreted as
the ${B}_{{c}}(2S)^{+}$, was observed by the ATLAS collaboration using a
sample of about 300 reconstructed
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{\pi}^{+}}$
candidates [175], with a large relative production rate with respect to the
base ${B}_{c}^{+}$ state. LHCb has searched for this state using
2$\text{\,fb}^{-1}$ of data collected at
$\sqrt{s}=8\text{\,Te\kern-1.00006ptV}$, observing a sample of $3325\pm 73$
reconstructed
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{\pi}^{+}}$
decays. No signal is observed and an upper limit on the relative production
rate has been obtained [176]. This upper limit is smaller than the relative
production rate reported by ATLAS.

In 2018, the CMS collaboration, using a huge data set corresponding to
143$\text{\,fb}^{-1}$ collected at $\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$
and containing of $7629\pm 225$ signal
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{\pi}^{+}}$
decays, reported observation of a doublet of two narrow states, interpreted as
spin-triplet ${B}_{{c}}(2S)^{\ast+}$ and spin-singlet ${B}_{{c}}(2S)^{+}$
states [177]. These observations were confirmed by LHCb using a
$8.5\text{\,fb}^{-1}$ data-set collected at $\sqrt{s}=7,8$ and 13 TeV [178],
with $3785\pm 73$ signal
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{\pi}^{+}}$
decays. Two narrow peaks with a width compatible with the detector resolution
are seen in the mass-difference
$m_{{{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}}-m_{{{B}_{c}^{+}}}$ spectrum, shown in
Fig. 5.5(right). The local (global) significances of the two peaks are
estimated to be $6.8\sigma$ ($6.3\sigma$) and $3.2\sigma$ ($2.2\sigma$) for
the low-mass and high-mass states, respectively. The low-mass signal is
interpreted as the spin-triplet state ${B}_{{c}}(2S)^{\ast+}$, decaying into
${B}_{{c}}^{\ast+}{{\pi}^{+}}{{\pi}^{-}}$ with the subsequent decay of the
${B}_{{c}}^{\ast+}$ into ${{B}_{c}^{+}}{\gamma}$. The high-mass peak is
attributed to the decay of the spin-singlet ${B}_{{c}}(2S)^{+}$ state into the
${{B}_{c}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ final state.

##### Excited ${{\mathchar 28931\relax}^{0}_{b}}$ baryons

were discovered in the ${{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}}$ mass spectrum using only
1$\text{\,fb}^{-1}$ of LHCb data, accumulated at
$\sqrt{s}=7\text{\,Te\kern-1.00006ptV}$ in 2011 [117]. The ${\mathchar
28931\relax}^{0}_{b}$ candidates were reconstructed via ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$ followed by ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$. In total
$(70.54\pm 0.33)\times 10^{3}$ signal ${{\mathchar 28931\relax}^{0}_{b}}$
decays were selected and then combined with ${{\pi}^{+}}{{\pi}^{-}}$ pairs.
The ${{\mathchar 28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}}$ spectrum in the
region $5.90\leq m_{{{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{+}}}\leq
5.95\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ is shown in Fig. 5.6(left), where
two narrow peaks, consistent with detector resolution, are visible. The
significances of the observations are 5.2 and 10.2 standard deviations for the
low-mass and high-mass peaks, respectively. The observed states are
interpreted as the doublet of orbitally-excited $\mathchar
28931\relax_{{b}}(1P)^{0}$ states with quantum numbers
$J^{P}=\tfrac{1}{2}^{+}$ and $\tfrac{3}{2}^{+}$. In 2020 the analysis was
updated using the full Run 1 and 2 data-sets, shown in Fig. 5.6(right). The
masses of these states are measured with unprecedented precision [121].

$\begin{array}[]{l}\mbox{LHCb}~{}1\text{\,fb}^{-1}\\\
\sqrt{s}=7\text{\,Te\kern-0.80005ptV}\end{array}$ LHCbRun 1&2 $m({{\mathchar
28931\relax}^{0}_{b}}\pi\pi)$$\left[\\!\text{\,Ge\kern-0.90005ptV\\!/}c^{2}\right]$
Figure 5.6:  (Left) Mass spectrum of ${{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}}$ combinations [117]. The points
with error bars are the data, the solid line is the result of a fit, and the
dashed line is the background contribution. (Right) Mass spectrum of (top)
${{\mathchar 28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}}$, (middle)
${{\mathchar 28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{+}}$ and (bottom)
${{\mathchar 28931\relax}^{0}_{b}}{{\pi}^{-}}{{\pi}^{-}}$ combinations in the
full Run 1 and 2 data-sets [121].

Using the full Run 1 and 2 data-sets, the mass spectrum of ${{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{+}}$ combinations, where ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{p}{{K}^{-}}$,
was explored for higher masses of ${{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}}$ combinations. A significant broad
structure is found at $m~{}\approx 6.150\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$
with a width around 10 MeV, shown in Fig. 5.7(left) [179]. The mass and width
agree well when measured in the two decay modes ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{p}{{K}^{-}}$,
where the significance exceeds 26 and 9 standards deviations, respectively.
Since the mass of the new structure is above the $\mathchar
28934\relax_{{b}}^{(\ast)\pm}\pi^{\mp}$ kinematic thresholds, the ${{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}}$ mass spectrum is investigated in
the ${{\mathchar 28931\relax}^{0}_{b}}\pi^{\pm}$ mass regions populated by the
$\mathchar 28934\relax_{{b}}^{(\ast)\pm}$ resonances.

The data are split into three non overlapping regions: candidates with a
${{\mathchar 28931\relax}^{0}_{b}}\pi^{\pm}$ mass within the natural width of
the known $\mathchar 28934\relax_{{b}}^{\pm}$ mass, candidates with a
${{\mathchar 28931\relax}^{0}_{b}}\pi^{\pm}$ mass within the natural width of
the known $\mathchar 28934\relax_{{b}}^{\ast\pm}$ mass, and the remaining
nonresonant (NR) region. The ${{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}}$ mass spectra in these three
regions are shown in Fig. 5.7(right). The spectra in the $\mathchar
28934\relax_{{b}}$ and $\mathchar 28934\relax_{{b}}^{\ast}$ regions look
different and suggest the presence of two narrow peaks with very similar
widths. The two-signal hypothesis is favoured over the single-signal
hypothesis with a statistical significance exceeding seven standard
deviations. The masses of the two states measured are consistent with
predictions for the doublet of $\mathchar 28931\relax_{{b}}(1D)^{0}$ states
with quantum numbers $J^{P}=\tfrac{3}{2}^{+}$ and $\tfrac{5}{2}^{+}$.

LHCbRun 1&2 LHCbRun 1&2 $m({{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}})$$\left[\\!\text{\,Ge\kern-0.90005ptV\\!/}c^{2}\right]$
$m({{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}})$$\left[\\!\text{\,Ge\kern-0.90005ptV\\!/}c^{2}\right]$
Figure 5.7:  (left) The mass distribution of selected ${{\mathchar
28931\relax}^{0}_{b}}{{\pi}^{+}}{{\pi}^{-}}$ candidates for (top) the
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$ and (bottom) the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{p}{{K}^{-}}$