LHCb-PROC-2023-003, Nikhef-2023-005

# Flavour Physics at LHCb — 50 years of the KM paradigm

Patrick Koppenburg111on behalf of the LHCb collaboration
<EMAIL_ADDRESS>Nikhef, Amsterdam, The Netherlands

###### Abstract

The LHCb experiment is in operation since 2009 and has provided measurements
of the KM matrix with unprecedented precision. 50 years after the original
paper we are in the position of pinning down the parameters of the theory, or
possibly to show its limitations. In these proceedings the status of LHCb is
shown in a historical perspective, along with some anecdotes.

xxxx, xxx

Figure 1: The briefest history of $C\\!P$ violation Kobayashi:1973fv ;
Christenson:1964fg ; Wu:1957 ; Aubert:2001nu ; Abe:2001xe ; LHCb-
PAPER-2013-018 ; LHCb-PAPER-2019-006

##  1 Introduction

It took Kobayashi and Maskawa Kobayashi:1973fv  a bit short of a decade since
the observation of $C\\!P$ violation in neutral kaons Christenson:1964fg  to
provide a theoretical description, the KM model. In hindsight, this time-lapse
seems relatively short as one needed to wait until the twenty-first century to
get confirmation of the model in the $b$ and $c$ sectors (Fig. 1). Since then
however, $C\\!P$ violation has been observed in interference of mixing and
decay (also known as “indirect”) in the ${B}^{0}$ Aubert:2001nu ; Abe:2001xe
and ${B}^{0}_{s}$ LHCb-PAPER-2013-018  systems, and $C\\!P$ violation in
decays (also known as “direct”) of ${K}^{0}$ KTeV:1999kad ; NA48:2002tmj ,
${B}^{0}$ BaBar:2004gyj ; Chao:2004mn , ${B}^{0}_{s}$ LHCb-PAPER-2020-029 ,
${{B}^{+}}$ LHCb-PAPER-2012-001  and ${D}^{0}$ LHCb-PAPER-2019-006  mesons.

The original observations of $C\\!P$ violation in the ${B}^{0}$ system that
led to the Nobel Prize awarded to Kobayashi and Maskawa in 2008 were the work
of the BaBar and Belle experiments. Since then all of the following “firsts”
were performed by LHCb. Barring the still missing observation of $C\\!P$
violation in baryons (and leptons, but this is another story), we are now
entering the precision regime in which the KM paradigm no longer needs to be
established, but is tested to its ultimate precision.

Figure 2: The LHCb detector. Figure adapted from Ref. LHCb-DP-2022-002 .

##  2 LHCb

The LHCb experiment was designed at a time before the $B$ factories Belle and
BaBar came into operations, and when the amount of $C\\!P$ violation in the SM
was poorly constrained Schmidtler:1991tv . The benchmark observables were the
$\sin 2{\maybebm{\phi_{1}}}$ $C\\!P$ asymmetry amplitude222See Sec. 3. As
these proceedings are published in PTEP, the Belle convention is used in the
text: $\phi_{1}=\beta$, $\phi_{2}=\alpha$, $\phi_{3}=\gamma$. Readers who are
allergic to this convention can download the source, change the argument of
$\backslash$def$\backslash$CKMconventionvalue{j} to anything but j and
recompile.  obtained from the decay
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{0}_{\mathrm{S}}}$
— because it may be that the $B$ factories wouldn’t be able to measure $C\\!P$
violation Anonymous  — and the frequency of ${B}^{0}_{s}$ oscillations — as it
was hoped that LHCb would see them before Tevatron. Incidentally both
measurements were performed before LHCb came into operation Aubert:2001nu ;
Abe:2001xe ; Abulencia:2006ze , but these benchmarks set constraints on the
design of the experiment from which the physics programme largely profits
today. In particular the excellent vertex resolution is a legacy of the
requirement that ${B}^{0}_{s}$ oscillations should be resolvable for
frequencies up to 60$\text{\,ps}^{-1}$.

The LHCb detector LHCb-DP-2008-001 ; LHCb-DP-2014-002  depicted in Fig. 2 is a
single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$,
designed for the study of particles containing $b$ or $c$ quarks. It consists
of a silicon-strip vertex detector surrounding the $pp$ interaction region
that allows $c$ and $b$ hadrons to be identified from their characteristically
long flight distance; a tracking system that provides a measurement of the
momentum of charged particles; and two ring-imaging Cherenkov detectors that
are able to discriminate between different species of charged hadrons.

Lessons from past experiments have taught LHCb to have a versatile trigger
(Sec. 6), which allowed the collaboration to quickly adapt to varying
experimental conditions. These variations are kept minimal by offsetting the
LHC beams in order to keep the luminosity constant throughout a fill, and
throughout a data-taking period. The nominal value was around $2$ to $4\times
10^{32}\rm\>\text{cm}^{-2}\text{s}^{-1}$.

Table 1: Operational statistics of LHCb running conditions in Runs 1 and 2.
The Run 1, Run 2 and last column contain sums or weighted averages depending
on which is more appropriate (or none if neither makes sense).
$f_{\text{LHC}}\simeq 27\text{\,km}/c\simeq 11\text{\,kHz}$ is the LHC
frequency.

Quantity | unit | TDR | $2011$ | $2012$ | Run 1 | $2015$ | $2016$ | $2017$ | $2018$ | Run 2 | Tot/Avg
---|---|---|---|---|---|---|---|---|---|---|---
Peak Luminosity | ${\cal L}_{\text{peak}}$ | $\,\upmu\text{b}^{-1}/\rm s$ | $280$ | $461$ | $492$ |  | $302$ | $422$ | $453$ | $493$ |  |
Average Luminosity | ${\cal L}_{\text{avg}}$ | $\,\upmu\text{b}^{-1}/\rm s$ | $200$ | $250$ | $330$ | $298$ | $140$ | $240$ | $280$ | $310$ | $268$ | $278$
Seconds of running | $t$ | $10^{6}\>\rm s$ | $10.0$ | $4.3$ | $6.2$ | $10.5$ | $1.6$ | $6.9$ | $4.6$ | $6.9$ | $20.0$ | $30.5$
Integrated luminosity | $\maybebm{\int\\!\\!{\cal L}{\rm d}t}$ | $\text{fb}^{-1}$ | $2.0$ | $1.1$ | $2.1$ | $3.2$ | $0.5$ | $1.9$ | $1.5$ | $2.5$ | $6.4$ | $9.6$
Bunches | $N_{\text{b}}$ |  | $2600$ | $1320$ | $1320$ |  | $1710$ | $2036$ | $2332$ | $2332$ |  |
Energy | $E$ |  TeV | $14$ | $7$ | $8$ |  | $13$ | $13$ | $13$ | $13$ |  |
Inelastic cross-section | $\sigma_{\text{inel}}$ |  mb | $80$ | $64$ | $67$ |  | $77$ | $77$ | $77$ | $77$ |  |
Charged multiplicity | $\frac{{\rm d}N_{\text{ch}}}{{\rm d}\eta}$ |  | $6$ | $6$ | $6$ | $6$ | $6$ | $6$ | $6$ | $6$ | $6$ | $6$
$b$$\overline{b}$ cross-section (acc.) | $\sigma_{{b}{}{\overline{b}}}$ | $\,\upmu\text{b}$ | $150$ | $72$ | $83$ |  | $144$ | $144$ | $144$ | $144$ |  |
$p$$p$ interactions/BX | $\mu=\frac{{\cal L}\sigma_{\text{inel}}}{f_{\text{LHC}}N_{\text{b}}}$ |  | $0.55$ | $1.08$ | $1.49$ | $1.32$ | $0.56$ | $0.81$ | $0.82$ | $0.91$ | $0.83$ | $0.99$
Non-empty rate | $f_{\text{LHC}}N_{\text{b}}(1-e^{-\mu})$ |  MHz | $12.3$ | $9.8$ | $11.5$ | $10.8$ | $8.3$ | $12.7$ | $14.7$ | $15.7$ | $14.0$ | $12.9$
Avg. MB rate | ${\sigma_{\text{inel}}{\cal L}_{\text{avg}}}$ |  MHz | $16.0$ | $16.0$ | $22.1$ | $19.7$ | $10.8$ | $18.5$ | $21.6$ | $23.9$ | $20.7$ | $20.3$
MB events | ${\sigma_{\text{inel}}\int{\cal L}_{\text{avg}}{\rm d}t}$ | $10^{12}$ | $160$ | $70$ | $141$ | $211$ | $38$ | $146$ | $116$ | $192$ | $493$ | $704$
Peak particle flow | ${\frac{{\rm d}N_{\text{ch}}}{{\rm d}\eta}\sigma_{\text{inel}}{\cal L}_{\text{peak}}}$ | $10^{6}$ | $134$ | $177$ | $198$ |  | $140$ | $195$ | $209$ | $228$ |  |
Irradiation | ${\frac{{\rm d}N_{\text{ch}}}{{\rm d}\eta}\sigma_{\text{inel}}\maybebm{\int\\!\\!{\cal L}{\rm d}t}}$ | $10^{15}$ | $1.0$ | $0.4$ | $0.8$ | $1.3$ | $0.2$ | $0.9$ | $0.7$ | $1.2$ | $3.0$ | $4.2$
$b$$\overline{b}$ rate | ${\sigma_{\text{{$b$}{}{$\overline{b}$}}}}{\cal L}_{\text{avg}}$ |  kHz | $30$ | $18$ | $27$ | $24$ | $20$ | $35$ | $40$ | $45$ | $39$ | $34$
$b$$\overline{b}$ yield | ${\sigma_{\text{{$b$}{}{$\overline{b}$}}}\maybebm{\int\\!\\!{\cal L}{\rm d}t}}$ | $10^{9}$ | $300$ | $79$ | $174$ | $254$ | $72$ | $274$ | $216$ | $360$ | $922$ | $1175$
Output rate | $\lambda_{\text{HLT}}$ |  kHz | $2.0$ | $2.6$ | $4.5$ | $3.7$ | $10.4$ | $6.1$ | $7.5$ | $5.8$ | $6.6$ | $5.7$
Stored events (bkk) | $\lambda_{\text{HLT}}t$ | $10^{9}$ | $20$ | $11$ | $28$ | $39$ | $17$ | $42$ | $35$ | $40$ | $133$ | $172$
Event size | $S_{\text{ev}}$ | $\rm kB$ | $2$ | $53$ | $59$ | $56$ | $48$ | $55$ | $58$ | $58$ | $56$ | $56$
HLT B/W | $S_{\text{ev}}\lambda_{\text{HLT}}$ | $\rm MB/s$ | $5$ | $136$ | $263$ | $212$ | $501$ | $333$ | $438$ | $333$ | $371$ | $319$
Total storage | $S_{\text{ev}}\lambda_{\text{HLT}}t$ | $\rm EB$ | $0.1$ | $0.6$ | $1.6$ | $2.2$ | $0.8$ | $2.3$ | $2.0$ | $2.3$ | $7.4$ | $9.6$

In the startup year of 2010, LHCb was able to identify the first $B$ meson at
the LHC333I had the honour of presenting the LHC’s first $B$ meson at the LHCC
open session in May 2010 LHCC2010 , and then killed it by declaring the 1 hour
run period during which it was recorded as of too low physics quality. It is
thus lost for posterity. with low luminosity and then cope with a rapid
increase of the collision rate. Eventually LHCb capped the luminosity and
collected data corresponding to 1, 2 and 6$\text{\,fb}^{-1}$ at 7, 8 and 13
TeV collision energies, respectively. Together they form the
9$\text{\,fb}^{-1}$ “legacy” sample. The huge cross-section at the LHC
provided more than $10^{12}$ $b$$\overline{b}$ pairs in the LHCb acceptance,
in rough proportion of 4:4:2:1 ${{B}^{+}}$:${B}^{0}$:${\mathchar
28931\relax}^{0}_{b}$:${B}^{0}_{s}$ hadrons LHCb-PAPER-2020-046 . In total
LHCb collected more than $10^{11}$ events between 2010 and 2018, a number that
can be compared to the 770 million $B$ ${\kern
1.79993pt\overline{\kern-1.79993ptB}}$ pairs collected by Belle. More numbers
can be found in Table 1.

Figure 3: Status of the KM ${B}^{0}$ unitarity triangle as of March 2021
CKMfitter2005 .

##  3 Unitarity triangle

Unitarity relations of the KM matrix are exploited to draw triangles in the
complex plane, the most renowned being the “${B}^{0}$” unitarity triangle
(“UT”, Fig. 3) formed from the first and last columns of the KM matrix. The
two non-unit sides intersect at an apex, which defines three angles: ϕ_1, ϕ_2,
and ϕ_3. All sides and angles are observables while only two are
independent.444There is also a constraint from kaon physics, noted
$\epsilon_{K}$, which is beyond the scope of these proceedings. This opens the
way to over-constraining the triangle and thus putting the KM paradigm to
test.

LHCb has measured the top-right side — proportional to
$|{V_{{t}{d}}^{\phantom{\ast}}}{V_{{t}{b}}^{\ast}}|$ — by precision
measurements of the ${B}^{0}$ and ${B}^{0}_{s}$ mixing frequencies. Such a
mixing plot is shown in Fig. 4. LHCb obtained the most precise values of
$\Delta m_{s}=17.766\pm 0.006\text{\,ps}^{-1}$ LHCb-PAPER-2021-005  and
$\Delta m_{d}=505.0\pm 2.3\text{\,ps}^{-1}$ LHCb-PAPER-2015-031 . Their
conversion into constraints on the KM UT is however limited by hadronic
uncertainties.

Figure 4: (left) Oscillation pattern of
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\pm}_{s}}{{\pi}^{\mp}}$ decays. Figure from
Ref. LHCb-PAPER-2021-005 . (right) Constraints on
$V_{{u}{b}}^{\phantom{\ast}}$ and $V_{{c}{b}}^{\phantom{\ast}}$ from Refs.
LHCb-PAPER-2020-038 ; LHCb-PAPER-2015-031 .

The other side is obtained from semileptonic $b$ decays to $c$ and $u$ quarks
to determine $V_{{c}{b}}^{\phantom{\ast}}$ and $V_{{u}{b}}^{\phantom{\ast}}$,
respectively. Such processes are members of a category known colloquially as
“things LHCb cannot do (but still does)”. LHCb was never designed for decays
with missing neutrinos. However the precise vertexing allows the determination
of the primary production (PV) and secondary decay vertices (SV) with
resolution of a few 10 microns.

Figure 5: Corrected mass inputs.

The corrected mass $m_{\text{corr}}=\sqrt{m^{2}+p_{\perp}^{2}}+p_{\perp}$ is
determined from the momentum missing along the direction of flight and peaks
at the $b$ hadron mass if the lost particle is of small mass, see Fig. 5. LHCb
measured the ratio
$|{V_{{u}{b}}^{\phantom{\ast}}}|/|{V_{{c}{b}}^{\phantom{\ast}}}|$ in decays of
${\mathchar 28931\relax}^{0}_{b}$ baryons LHCb-PAPER-2015-031  and
${B}^{0}_{s}$ mesons LHCb-PAPER-2020-038 , which are inaccessible to $B$
factories at the $\mathchar 28935\relax{(4S)}$ resonance. The resulting
constraints are shown as diagonal bands in Fig. 4. It is particularly striking
that the constraints from ${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{\nu}$
in the low and high range of dilepton masses squared, ${q^{2}}$, are
inconsistent, owing to inconsistent form factors from light-cone sum rules
Khodjamirian:2017fxg  and Lattice QCD Bazavov:2019aom , respectively. The
figure also shows the discrepancy between the inclusive
${b}\\!\rightarrow{q}\ell{\nu}$ and exclusive ${B}\\!\rightarrow X\ell{\nu}$
determinations of $|{V_{{u}{b}}^{\phantom{\ast}}}|$ and
$|{V_{{c}{b}}^{\phantom{\ast}}}|$. This puzzle is still unresolved in spite of
multiple measurements and computations spanning several decades. Let us hope
we won’t have to wait for an $e^{+}e^{-}$ collider to run at the $W$ $W$
threshold to resolve the issue.

The situation in the measurements of the angles is somewhat clearer. The angle
ϕ_1 is obtained from final states that are $C\\!P$ eigenstates reached via
interference of ${b}\\!\rightarrow{c}{\overline{c}}{s}$ and their charged-
conjugated processes after ${B}^{0}$ to ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ mixing Bigi:1983cj . The angle is
known with sub-degree precision: LHCb now holds the most precise value of
$\sin 2{\maybebm{\phi_{1}}}=0.717\pm 0.013\pm 0.008$, from a combination of
the decay modes
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{0}_{\mathrm{S}}}$
and $\psi{(2S)}$ ${K}^{0}_{\mathrm{S}}$ LHCb-PAPER-2023-013 .

The angle ϕ_3 is often referred to as a standard candle of the Standard Model
— to the extent that one is certain that no new physics affects tree-dominated
decays, which is a bold assumption Gligorov:2023mji . The $C\\!P$ asymmetry is
generated from interference of ${b}\\!\rightarrow{c}{\overline{u}}{q}$ and
${b}\\!\rightarrow{u}{\overline{c}}{q}$ where the $D$ mesons formed by $c$
$\overline{u}$ ($\overline{c}$ $u$) decay to a common final state. If it is
assumed that such tree-level processes are free from new physics
contributions, then the determination of the angle ϕ_3 will be dominated by
the experimental resolution in the foreseeable future and beyond Brod:2013sga
.

Table 2: Used integrated luminosities and references of measurements with
sensitivity to ϕ_3 split by final states. Entries in red are not yet included
in the combination LHCb-CONF-2022-003 .

| Time-integrated measurements
---|---
Decays | ${{{B}^{+}}}\\!\rightarrow{D}h^{+}$ | ${{{B}^{+}}}\\!\rightarrow{{D}^{*0}}h^{+}$ | ${{{B}^{+}}}\\!\rightarrow{D}{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$ | ${{B}^{0}}\\!\rightarrow{D}{{K}^{+}}{{\pi}^{-}}$ | ${{{B}^{+}}}\\!\rightarrow{D}{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$
${D}\\!\rightarrow h^{+}h^{-}$ |  $9\text{\,fb}^{-1}$ LHCb-PAPER-2020-036  |  $9\text{\,fb}^{-1}$ LHCb-PAPER-2020-036  | 5$\text{\,fb}^{-1}$ LHCb-PAPER-2017-030  | 5$\text{\,fb}^{-1}$ LHCb-PAPER-2019-021  | 3$\text{\,fb}^{-1}$ LHCb-PAPER-2015-020
${D}\\!\rightarrow h^{+}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ | 3$\text{\,fb}^{-1}$ LHCb-PAPER-2016-003  |  $9\text{\,fb}^{-1}$ LHCb-PAPER-2019-021  | 5$\text{\,fb}^{-1}$ LHCb-PAPER-2017-030  | 5$\text{\,fb}^{-1}$ LHCb-PAPER-2019-021  |
${D}\\!\rightarrow{{K}^{\pm}}{{\pi}^{\mp}}{{\pi}^{+}}{{\pi}^{-}}$ |  $9\text{\,fb}^{-1}$ LHCb-PAPER-2022-017  |  |  |  |
${D}\\!\rightarrow h^{+}h^{-}{{\pi}^{+}}{{\pi}^{-}}$ | $9\text{\,fb}^{-1}$ LHCb-PAPER-2022-037  |  |  |  |
${D}\\!\rightarrow h^{+}h^{-}{{\pi}^{0}}$ |  $9\text{\,fb}^{-1}$ LHCb-PAPER-2021-036  |  |  |  |
${D}\\!\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ |  $9\text{\,fb}^{-1}$ LHCb-PAPER-2020-019  |  $9\text{\,fb}^{-1}$ LHCb-PAPER-2023-012  |  |  $9\text{\,fb}^{-1}$ LHCb-PAPER-2023-009  |
${D}\\!\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{+}}{{\pi}^{-}}$ |  $9\text{\,fb}^{-1}$ LHCb-PAPER-2019-044  |  |  |  |

| Time-dependent measurements
---|---
Decays | ${{B}^{0}}\\!\rightarrow{{D}^{\mp}}{{\pi}^{\pm}}$ | ${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ | ${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}$
${{D}^{\pm}}\\!\rightarrow{{K}^{\pm}}h^{+}h^{-}$ | 3$\text{\,fb}^{-1}$ LHCb-PAPER-2018-009  | N/A | N/A
${{D}^{\pm}_{s}}\\!\rightarrow h^{\pm}h^{\mp}{{\pi}^{\pm}}$ | N/A |  $9\text{\,fb}^{-1}$ LHCb-PAPER-2017-047 ; LHCb-CONF-2023-004  |  $9\text{\,fb}^{-1}$ LHCb-PAPER-2020-030

However, unlike with
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{0}_{\mathrm{S}}}$
for $\sin 2{\maybebm{\phi_{1}}}$ there is no single process that yields a good
resolution on its own. Multiple $B$ meson (and potentially $b$-baryon) decays
can be used, as well as many $D$ decays. The ultimate precision will be
reached once all combinations are analysed with the full data set. The present
status is shown in Table 2, where multiple possible analyses are still
missing. It is however unlikely that this table will ever be fully filled, as
some combination of $B$ and $D$ decay modes have marginal sensitivities. In
addition to the interference pattern above, ϕ_3 can also be measured from
interference of mixing (both ${B}^{0}$ and ${B}^{0}_{s}$) and decay. These
processes are however potentially affected by new physics entering the $B$
mixing loop. A comparison of values of ϕ_3 obtained by from the two tables in
Tab. 2 thus constitutes an additional test of the Standard Model.

A precise value of
${\maybebm{\phi_{3}}}=(63.8{\>}^{+{\>}3.5}_{-{\>}3.7})^{\circ}$ is obtained
from a combination the analyses reported in Ref. LHCb-CONF-2022-003 , which
does not yet include some of the latest LHCb results LHCb-PAPER-2022-037 ;
LHCb-PAPER-2023-012 ; LHCb-PAPER-2023-009 . Presently, the time-dependent
measurements yield
${\maybebm{\phi_{3}}}=(79{\>}^{+{\>}21}_{-{\>}23})^{\circ}$, compatible with
the value from time-integrated analyses.

The values reported in this combination also make use of the charm mixing
LHCb-PAPER-2022-020  and $C\\!P$ violation LHCb-PAPER-2019-006 ; LHCb-
PAPER-2022-024  measurements (that slightly affect the ϕ_3 determination for
${B}\\!\rightarrow{D}$ modes). The first observation of $C\\!P$ violation in
charm LHCb-PAPER-2019-006  — with a precision in units of $10^{-4}$ — is a
major achievement but its direct understanding in terms of KM matrix elements
is not yet within theoretical reach.

The third angle, ϕ_2, is obtained from interference in ${b}\\!\rightarrow{u}$
transitions. LHCb’s main contribution is the most precise time-dependent
measurement of $C\\!P$ asymmetries in
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ LHCb-PAPER-2020-029 . A full
determination of ϕ_2 needs an isospin analysis of several decay modes
involving ${\pi}^{0}$ mesons Gronau:1990ka , which are hard to reconstruct at
LHCb.555See however Ref. LHCb-CONF-2015-001 .

The combination of these two sides and three angles yields the over-
constrained triangle depicted in Fig. 3 CKMfitter2005 . This is however only
one of several possible triangles, dominated by $B$ decay modes.

Figure 6: Preliminary $\phi_{{s}}$ combination by HFlav HFLAV21 .

Another triangle, dubbed the ${B}^{0}_{s}$ triangle, features the angle
$\phi_{{s}}$. In analogy to ϕ_1, it is obtained from ${B}^{0}_{s}$ oscillation
followed by a ${B}^{0}_{s}$ decay to a $c$ $\overline{c}$ $s$ $\overline{s}$
$C\\!P$ eigenstate, e.g. ${J\mskip-3.0mu/\mskip-2.0mu\psi}$$\phi$. It used to
be said that the SM expectation is close to vanishing, making any measurement
of a non-zero value a sign of new physics. We are no longer in this regime:
LHCb recently released their legacy measurement with the full
9$\text{\,fb}^{-1}$ dataset and obtain ${\phi_{{s}}}=-0.039\pm 0.022\pm
0.006\text{\,rad}$ LHCb-PAPER-2023-016  from
${{B}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}{{K}^{-}}$
with the ${K}^{+}$ ${K}^{-}$ mass in the vicinity of the $\phi$ meson. This
value is combined with LHCb measurements using previous datasets or other
decays LHCb-PAPER-2019-003 ; LHCb-PAPER-2017-008 ; LHCb-PAPER-2016-027 ; LHCb-
PAPER-2014-051  as well as results from Tevatron CDF:2012nqr ; D0:2011ymu  and
other LHC experiments ATLAS:2020lbz ; CMS:2020efq . The result is
${\phi_{{s}}}=-0.050\pm 0.016\text{\,rad}$, which is now significantly
deviating from zero. The expectation from KM fits is ${\phi_{{s}}}=-0.037\pm
0.001$ CKMfitter2005 ; UTfit-UT , with which the world average is well
compatible, as shown in Fig. 6.

There is some disagreement in the determination of the total decay width of
the ${B}^{0}_{s}$ meson $\Gamma_{s}$ and of the difference of the decays
widths of the heavy and light ${B}^{0}_{s}$ mass eigenstates,
$\Delta\Gamma_{s}$, which are by-products of these analyses. The update of
ATLAS with the full Run 2 sample is eagerly awaited, hopefully resolving the
issue.

In 2003 there was some excitement due to inconsistent values of $\sin
2{\maybebm{\phi_{1}}}$ obtained from ${b}\\!\rightarrow{c}{\overline{c}}{s}$
processes and from ${b}\\!\rightarrow{s}{\overline{s}}{s}$, as
${{B}^{0}}\\!\rightarrow\phi{{K}^{0}_{\mathrm{S}}}$ Belle:2003vik . The latter
are also sensitive to the same KM phases, but proceed via penguin diagrams and
are thus more likely affected by heavy degrees of freedom in the loop. The
issue turned out to be a fluke but the penguin decay modes remained on the
radar.

A similar test can be done by measuring the time-dependent $C\\!P$-violating
phase in from ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$, that is strictly zero in
the SM. LHCb obtain $\phi_{s}^{{s}{\overline{s}}{s}}=-0.18\pm 0.09$ in a
combination of all data samples LHCb-PAPER-2023-001 , which is compatible with
the SM expectation.

Figure 7: Schematics of information used in flavour tagging. Figure from Ref.
LHCb-FIGURE-2020-002 .

Resolution of the above-mentioned interference patterns require flavour
tagging, i.e. the identification of the original flavour of the ${B}^{0}$ or
${B}^{0}_{s}$ meson. LHCb uses the flavour of the accompanying $b$ from the
$b$ $\overline{b}$ pair, using muon, electron and kaon particles, as well as
the vertex charge (Fig. 7). In addition, accompanying pions, kaons and protons
from fragmentation are used to determine the $B$ meson flavour directly LHCb-
PAPER-2016-039 .

Improved understanding of proton-proton collisions at LHC energies has allowed
continuous increase of the effective tagging power from less than 2% in 2011
LHCb-PAPER-2011-021  to more than 6% for selected modes nowadays LHCb-
PAPER-2023-001 ; LHCb-PAPER-2023-013 .

A (somewhat outdated) comparison of several decays modes is shown in Fig. 8.
Note however that the tagging performance anti-correlates with the hardware
trigger efficiency: Modes with muons profit from low trigger thresholds while
those with hadrons or electrons have lower efficiencies.

Figure 8: Tagging performance for selected analyses. The lines indicate
constant values of the effective tagging power
$\epsilon_{\text{tag}}(1-2\omega_{\text{tag}})^{2}$. Figure from Ref. LHCb-
FIGURE-2020-002 .

These thresholds however bias the transverse momentum of the accompanying $B$
and the tagging particles from fragmentation, which improves the tagging
performance. As an example, the best tagging performance at the LHC is
obtained by CMS CMS:2020efq  thanks for their triggering on the tagging muon:
Trigger efficiency is traded off for better tagging.

##  4 Anomalies coming and going

After 30 years of unsuccessful searches at multiple experiments, the
observation of ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ was one of the
benchmark goals at the LHC. This loop-induced, GIM and helicity-suppressed
process is very rare in the SM — its branching fraction is 3 in a billion
Beneke:2019slt ; DeBruyn:2012wj  — which makes it sensitive to any new physics
contribution at this or higher level. In supersymmetric extensions its rate is
enhanced proportionally to the sixth power of $\tan\beta$ — the ratio of the
two vacuum expectations of the two neutral Higgs bosons. Its non-observation
increasingly shattered hopes for large-$\tan\beta$ supersymmetry as the limit
on the branching fraction decreased (Fig. 9). The first evidence was then
finally reported by LHCb in 2013 LHCb-PAPER-2012-043 , the first observation
was achieved via from a joint fit to LHCb and CMS data in 2014 LHCb-
PAPER-2014-049 666The fit to a joint dataset of two experiments was a major
enterprise. It required first to align all definitions and treatments of
backgrounds in the two experiments. Once this was done the complex
simultaneous fit to multiple datasets returned the same result as one would
have obtained from combining the LHCb and CMS likelihoods LHCb-PAPER-2014-049
. This teaches a lesson on the relative importance of agreeing between
experiments versus developing complex fitting frameworks. and LHCb reported
the first single-experiment observation in 2017 LHCb-PAPER-2017-001 , soon
followed by CMS CMS:2019bbr . ATLAS are just a bit short of an observation
ATLAS:2018cur . The LHCb and CMS results were in the meantime updated using
the full Run 1–2 data sets LHCb-PAPER-2021-007 ; LHCb-PAPER-2021-008 ;
CMS:2022mgd . The average of the
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ measurements Allanach:2022iod
is consistent with the Standard Model expectation, which sets strong
constraints on new physics affecting ${b}{\overline{s}}\ell\overline{\ell}$
operators.

Figure 9: (left) Summary of all limits and measurements of
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ and
${{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ branching fractions. Figure
courtesy F. Dettori adapted from Ref. LHCb-PAPER-2014-049 .

The further suppressed ${{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ mode still
escapes clear detection, but one can nevertheless determine a ratio of
branching fractions which is precisely predicted in the SM as it is
essentially a measurement of KM matrix elements:

${\cal
R}_{\mu^{+}\mu^{-}}\equiv\frac{{\mathcal{B}}({{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}})}{{\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}})}\stackrel{{\scriptstyle\text{SM}}}{{=}}\frac{\Gamma_{s}^{\text{H}}}{\Gamma_{d}}\left(\frac{f_{{B}^{0}}}{f_{{B}^{0}_{s}}}\right)^{2}\frac{|{V_{{t}{d}}^{\phantom{\ast}}}|^{2}}{|{V_{{t}{s}}^{\phantom{\ast}}}|^{2}}\frac{\sqrt{m^{2}_{{B}^{0}}-4m^{2}_{\mu}}}{\sqrt{m^{2}_{{B}^{0}_{s}}-4m^{2}_{\mu}}},$
(1)

and is measured as LHCb-PAPER-2021-007 ; LHCb-PAPER-2021-008

${{\cal
R}_{\mu^{+}\mu^{-}}^{\text{exp}}=\left(3.9{\>}^{+{\>}3.0}_{-{\>}2.4}{\>}^{+{\>}0.6}_{-{\>}0.4}\right)\times
10^{-2}.}$

Another decay mode sensitive to new physics in these operators is
${B}\\!\rightarrow{{K}^{*}}\ell^{+}\ell^{-}$. While
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ is essentially probing axial
currents, ${B}\\!\rightarrow{{K}^{*}}{\mu^{+}\mu^{-}}$ also probes vector
currents, and their interference, as shown in Fig. 10.

Figure 10: Feynman diagrams of the dominant Standard Model contributions to
${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$: (left) electroweak loop, (centre) box,
(right) $c$$\overline{c}$ loop diagram.

The exclusive decay ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\ell^{+}}{\ell^{-}}$,
with ${{K}^{*0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$, provides a rich set of
observables with different sensitivities to new physics, and for which
theoretical predictions are available. This process is complicated by a
dependence on $q^{2}$, the dilepton mass squared. At low $q^{2}$,
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\ell^{+}\ell^{-}$ behaves like
${{B}^{0}}\\!\rightarrow{{K}^{*}}{\gamma}$, with a slightly off-shell photon
decaying to two leptons. At higher $q^{2}$ values, there is an interference of
the amplitudes controlled by the ${\cal O}_{9}$ and ${\cal O}_{10}$ operators
Buchalla:1995vs , related to the $Z$ loop and $W$ box diagrams, respectively.
This “low-$q^{2}$” region between 1 and 6
$\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ is the most interesting and
theoretically cleanest. The observation of high mass resonances above the
$\psi{(2S)}$ meson by the LHCb collaboration LHCb-PAPER-2013-039  is an
indication that a lot of care is needed when interpreting the high-$q^{2}$
region.

Branching fraction predictions are affected by hadronic uncertainties (see
also below), but selected ratios of observables benefit from cancellations of
uncertainties, thus providing a cleaner test of the Standard Model
Eilam:1986fs ; Ali:1999mm ; Kruger:2005ep ; Altmannshofer:2008dz ;
Egede:2008uy ; Bobeth:2008ij ; Descotes-Genon:2013vna . The observable
$P_{5}^{\prime}$ LHCb-PAPER-2020-002  for instance is in tension with the
theoretical prediction, as seen in Fig. 11, but the jury is still out on
determining what the cause is. In order to address this question, LHCb
recently performed an amplitude analysis in the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ decay in which the short-
distance Wilson coefficients and long-distance nuisance parameters are
determined from the data LHCb-PAPER-2023-033 . The overall level of
discrepancy with the Standard Model is at the level of $2\sigma$.

Figure 11: Experimental results on the (left)
${B}\\!\rightarrow{K}\ell^{+}\ell^{-}$ differential decay rate Lees:2012tva ;
Wei:2009zv ; Aaltonen:2011qs ; LHCb-PAPER-2012-011 ; LHCb-PAPER-2022-045 ;
LHCb-PAPER-2022-046  compared to prediction of Ref. Parrott:2022zte  and
(right) $P_{5}^{\prime}$ Lees:2015ymt ; Wehle:2016yoi ; Khachatryan:2015isa ;
Sirunyan:2017dhj ; Aaboud:2018krd ; LHCb-PAPER-2020-002 ; LHCb-PAPER-2020-041
asymmetry compared to predictions from Refs. Straub:2015ica ;
Altmannshofer:2014rta  (sea green) and Refs. Descotes-Genon:2014uoa ;
Khodjamirian:2010vf  (green).

Until recently there was excitement about the evidence of lepton-universality
breaking in the decays ${B}\\!\rightarrow{K}{e^{+}e^{-}}$ and
${B}\\!\rightarrow{K}{\mu^{+}\mu^{-}}$ LHCb-PAPER-2021-004 ; LHCb-
PAPER-2021-038 , and similarly but less significantly in
${B}\\!\rightarrow{{K}^{*}}{e^{+}e^{-}}$ and
${B}\\!\rightarrow{{K}^{*}}{\mu^{+}\mu^{-}}$ LHCb-PAPER-2017-013 ; LHCb-
PAPER-2021-038 . With these decays one defines the ratio Hiller:2003js

$R_{X}=\frac{\int\limits_{q_{\text{min}}^{2}}^{q_{\text{max}}^{2}}{\rm
d}{q^{2}}\frac{d\Gamma\left({B}\\!\rightarrow X{\mu^{+}\mu^{-}}\right)}{{\rm
d}{q^{2}}}}{\int\limits_{q_{\text{min}}^{2}}^{q_{\text{max}}^{2}}{\rm
d}{q^{2}}\frac{d\Gamma\left({B}\\!\rightarrow X{e^{+}e^{-}}\right)}{{\rm
d}{q^{2}}}}$ (2)

in a well chosen range of $q^{2}$ (usually 1 to
6$\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$) in order to avoid charmonium
resonances and photon poles. These ratios should be identical to unity at a
level of precision well below experimental resolution Isidori:2020acz .

Initially Hiller and Krüger Hiller:2003js  had introduced these ratios as they
are linearly correlated with the branching fraction of
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ (under some reasonable
assumptions, notably minimal flavour violation). In 2003 the hope was thus
that Belle and BaBar could see sign of new physics in these ratios before the
LHC would observe ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$. At a high-
luminosity workshop at SLAC in 2003, Hiller convinced me to check whether LHCb
could measure this ratio. I showed that LHCb could so Koppenburg:1027442
thereby demonstrating that LHCb had physics potential with
electrons.777Inspection of old reports to the LHCC Dijkstra:691698 ; CERN-
LHCC-98-004 ; :2009ny ; LHCb-TDR-009  shows that the use of electrons was
essentially for flavour tagging, and adding some data to $\phi_{{s}}$ and
$\sin 2{\maybebm{\phi_{1}}}$ measurements.  A first proof-of-concept analysis
was the topic of a master thesis Skidmore  that exceeded my expectations and
led to an LHCb measurement that was “compatible with the SM prediction within
2.6 standard deviations” LHCb-PAPER-2014-024 .888The result was submitted to
PRL after the editor had expressed their interest in this result following a
CERN news update.  The result was below unity,999Since Hiller and Krüger
expected $R_{X}\geq 1$ they defined it as muon/electrons. This is an annoyance
for experiments as the uncertainty on $R_{X}$ is dominated by the electron
modes, which leads to asymmetric uncertainties when expressed in the ratio. In
hindsight one should have defined $R_{X}^{-1}$, which is what is reported in
Ref. LHCb-PAPER-2019-040 . meaning we saw a deficit of muons; a case that had
not been foreseen by Hiller in the original publication.

Three years later the analysis of
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\ell^{+}\ell^{-}$ yielded a similar
deviation from unity. However this result already contained a hint that
something was not quite right: the ratio $R_{{K}^{*}}$ in the low $q^{2}$ bin
should have been unity, as this region is dominated by the photon pole and
lepton-universality is known to be respected in electromagnetic decays. In
spite of lengthy investigations nothing could be found that explained the
discrepancy and the result was published as is LHCb-PAPER-2017-013 . The value
of $R_{K}$ in ${{B}^{+}}\\!\rightarrow{{K}^{+}}\ell^{+}\ell^{-}$ was updated
twice with Run 2 data LHCb-PAPER-2019-009 ; LHCb-PAPER-2021-004 , the latter
of which exhibited a $3\sigma$ evidence. This was exciting news for model
builders. Indeed, no hadronic effects can mimic values of $R_{K}$ different
from unity, unlike other deviations, which can be generated by stretching some
not-so-certain QCD predictions. It was known that the discrepancy with the SM
was driven by ${b}\\!\rightarrow{s}{\mu^{+}\mu^{-}}$ channels, for which all
rates are measured below the SM expectation LHCb-PAPER-2012-011 ; LHCb-
PAPER-2021-014 , while the electrons were thought to be SM-like.

The anomaly disappeared after a reappraisal of the LHCb result LHCb-
PAPER-2022-045 ; LHCb-PAPER-2022-046 . More stringent particle identification
requirements reduced the amount of backgrounds with two hadrons mis-identified
as electrons, and these and similar backgrounds were determined from data and
incorporated in the mass fit. The measurements of two bins in $q^{2}$ (below
and above 1$\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$) in
${{B}^{+}}\\!\rightarrow{{K}^{+}}\ell^{+}\ell^{-}$ and
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\ell^{+}\ell^{-}$ are now all compatible
with unity.

Figure 12: Fits to C_9 and C_10 Wilson coefficients from
${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$ modes. Figure from Ref. Alguero:2023jeh
.

However, the demise of $R_{K}$ replaced one anomaly with another. After the
correction, the electron modes also have measured decay rates below the SM
expectation. Some lattice QCD groups report discrepancies in excess of
$4\sigma$ Parrott:2022zte  for the muon modes, while the electron and muon
modes are experimentally compatible. Other groups however determine form
factors from the data and get much lower tensions Gubernari:2022hxn .

Input from all ${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$,
${B}\\!\rightarrow{\mu^{+}\mu^{-}}$ and ${b}\\!\rightarrow{s}{\gamma}$ modes
is used to perform fits to Wilson coefficients C_9 and C_10 corresponding to
the vector and axial operators, respectively. Multiple groups DeBlas:2019ehy ;
Isidori:2021vtc ; Hurth:2021nsi ; Altmannshofer:2021qrr ; Geng:2021nhg ;
Angelescu:2021lln ; Ciuchini:2022wbq ; Greljo:2022jac ; Gubernari:2022hxn ;
Alguero:2023jeh  consistently hint at a modified vector
$b{\overline{s}}\ell\bar{\ell}$ operator — with varying significance — as
shown in in Fig. 12 Alguero:2023jeh .

The elephant in the room are non-local $c$$\overline{c}$ contributions (Fig.
10, right). Unlike the local form factors that are in principle calculable,
the charm loops are nonlocal — i.e. the $b$ $\overline{s}$$c$$\overline{c}$
and $c$$\overline{c}$$\ell{}\bar{\ell}$ operators appear at different points
in spacetime — which requires involved integrals. In the
${B}\\!\rightarrow{K}{\mu^{+}\mu^{-}}$ channel, light-cone-sum-rule (LCSR)
methods predict small nonlocal effects Khodjamirian:2012rm , and attempts to
constrain charm loops from data Gubernari:2022hxn  reach the conclusion that
they are too small to explain the observed discrepancies. Are the
discrepancies due to form factors? The latest evidence by Belle II for the
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{\nu}{\overline{\nu}}$ decay B2Knunu , at a
rate above the SM predicted branching fraction, tends to indicate that the
${B}\\!\rightarrow{K}$ form factor is not low compared to the SM expectation.
In ${B}\\!\rightarrow{{K}^{*}}{\mu^{+}\mu^{-}}$ decays the form factors and
nonlocal effects are less well controlled Khodjamirian:2010vf .

Figure 13: Experimental constraints on $R({D})$ and $R({{D}^{*}})$. Figure
from Ref. HFLAV21 .

Another test of lepton universality in tree decays is done with semileptonic
decays to $\tau$ leptons. The ratio $R({D})$ of the
${B}\\!\rightarrow{D}\tau{\nu}$ to ${B}\\!\rightarrow{D}{\mu}{\nu}$ decay
rates — and mutatis mutandis for $R({{D}^{*}})$ — are not unity because of
phase-space factors, but well predicted in the SM. There has been a long-
standing discrepancy at the level of $3\sigma$, mostly driven by a measurement
from BaBar BaBar:2012obs ; BaBar:2013mob , that pulls the experimental average
away from the SM, see Fig. 13. In the spirit of “things LHCb cannot do (but
still does)”, LHCb is contributing to this programme: in spite of the missing
neutrino and the multiplicity of backgrounds affecting these decays, LHCb is
able to reconstruct the missing-mass, $q^{2}$ and muon energy in the $B$
frame, owing to the missing-momentum correction coming from the $B$ pointing
requirement. The LHCb results are compatible with the SM in $R({{D}^{*}})$,
but show a slight tension in $R({D})$ LHCb-PAPER-2022-039 ; LHCb-
PAPER-2022-052 . The full legacy dataset is not yet exploited, so more updates
will be coming, while waiting for the first results from Belle II.

Figure 14:  Cherry-picked selection of measurements in flavour physics. The SM
prediction is placed at zero. The experimental value is then offset its
deviation from the SM in units of standard deviations. The quadratic sum of
the uncertainties is unity by construction. Figure from ref. PKanomalies .
Values from Refs. LHCb-PAPER-2014-006 ; LHCb-PAPER-2022-045 ; LHCb-
PAPER-2022-046 ; Parrott:2022zte ; LHCb-PAPER-2021-014 ; Horgan:2015vla ;
Allanach:2022iod ; LHCb-PAPER-2021-007 ; LHCb-PAPER-2021-008 ; ATLAS:2018cur ;
CMS:2022mgd ; HFLAV21 ; Muong-2:2021ojo ; Aoyama:2020ynm ; LHCb-PAPER-2017-035
; Harrison:2020nrv ; LHCb-PAPER-2021-044 ; Bernlochner:2018bfn ; PDG2022 ;
UTfit:2006vpt ; LHCb-PAPER-2021-005 ; DiLuzio:2019jyq ; ALEPH:2005ab . Figure
from Ref. Alguero:2023jeh .

Figure 14 shows a cherry-picked selection of measurements which are of
particular interest. For each measurements, the SM prediction is placed at
zero. The experimental value is then offset its deviation from the SM in units
of standard deviations. The quadratic sum of the two uncertainties is
therefore unity by construction. This presentation shows which observables
have an uncertainty dominated by experiment ($R({D})$,
${\mathcal{B}}({{{B}^{+}}}\\!\rightarrow{\tau^{+}}{\nu}$ stick out) or by
theory (${\mathcal{B}}({{{B}^{+}}}\\!\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}}$),
which tells where efforts are needed to reduce
uncertainties.101010Additionally there are anomalies like the muon $g-2$ value
where multiple theory determinations do not agree (we report Ref.
Aoyama:2020ynm ) or where there is disagreement between experiments (as the
$W$ mass, which is not reported here).

##  5 Spectroscopy

The LHC has become a hadron discovery machine ConversationHadrons  with 72
states observed so far, 64 of which were discovered by LHCb. A timeline is
shown in Figure 15 LHCb-FIGURE-2021-001 .

Figure 15: Masses and date of arxiv submission for all states observed at the
LHC. Figure from Ref. LHCb-FIGURE-2021-001 .

Many of these new states are exotic; but let us rewind history to 2003. I
remember well the internal discussions about a bump in the
${J\mskip-3.0mu/\mskip-2.0mu\psi}$ ${\pi}^{+}$ ${\pi}^{-}$ mass that was seen
in Belle data of the
${{{B}^{+}}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
decay, an considered it likely to be yet another charmonium state. Instead,
this accidental observation Choi:2003ue  of what turned out to be the first
tetraquark, called the $X(3872)$ and then the $\chi_{c1}(3872)$ meson, started
the whole new field of exotic spectroscopy. Multiple similar states — some of
which charged — have then been discovered by the $B$ factories; see Ref.
Lebed:2016hpi  for a history. We now know that $\chi_{c1}(3872)$ is an
isospin-singlet that proceeds mostly in an isospin-violating decay to
${D}^{0}$ ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ ${\pi}^{0}$ or
${D}^{*0}$ ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ Belle:2006olv
, or to ${J\mskip-3.0mu/\mskip-2.0mu\psi}$ ${\pi}^{+}$ ${\pi}^{-}$ via a
${\rho}^{0}$ Belle:2011vlx , but with a sizeable $\omega$ component LHCb-
PAPER-2021-045 .111111This $\omega$ component has been underestimated for a
long time because of a bug in the EvtGen model which made the generated
${\pi}^{+}$ ${\pi}^{-}$ mass from pure
${\chi_{c1}(3872)}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{\rho}^{0}}$
decays look like what was seen in data; see e.g. mass plots in Refs.
CMS:2013fpt ; ATLAS:2016kwu ; LHCb-PAPER-2015-015 . LHCb showed that
$\chi_{c1}(3872)$ meson has quantum numbers $J^{PC}=1^{++}$ LHCb-
PAPER-2013-001 ; LHCb-PAPER-2015-015  and is consistent with ${D}^{*0}$
${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ bound state with 24 keV
binding energy and a width of $1.39\pm 0.26\text{\,Me\kern-1.00006ptV}$ LHCb-
PAPER-2020-008  (see also Ref. Belle:2023zxm ). This makes it look like a
molecule, while its production mode is that of charmonium LHCb-PAPER-2021-026
. If both possibilities exist, reality must be a superposition of the two
Weinberg:1965zz ; LHCb-PAPER-2020-008 .

In spite of some early papers (notably Ref. LHCb-PAPER-2011-033 ), most
members of the LHCb experiment only realised the potential of exotic
spectroscopy when we stumbled over the ${J\mskip-3.0mu/\mskip-2.0mu\psi}$ $p$
pentaquarks with minimal quark content $c$ $\overline{c}$ $u$ $u$ $d$ LHCb-
PAPER-2015-029 . These states were first seen in ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{p}{{K}^{-}}$
decays,121212simultaneously by multiple people including a CERN summer student
supervised by the author. and established in an amplitude analysis. The
3$\text{\,fb}^{-1}$ data set was best fit with a wide $P_{\psi}^{N}(4380)^{+}$
and a narrow $P_{\psi}^{N}(4450)^{+}$ state (using the naming scheme of Ref.
Gershon:2022xnn ). A subsequent simplified analysis LHCb-PAPER-2022-031  of
the 9$\text{\,fb}^{-1}$ legacy data showed that the latter state is split in
two states, $P_{\psi}^{N}(4440)^{+}$ and $P_{\psi}^{N}(4457)^{+}$, and that
another is needed at $4312\text{\,Me\kern-1.00006ptV}$. The full amplitude
analysis is underway.

Table 3: Selected tetraquark states listed quark content, beyond $c$$\overline{c}$$q$$\overline{q}$ LHCb-PAPER-2020-011 ; CMS:2013jru ; LHCb-PAPER-2016-019 ; CMS:2023owd ; ATLAS:2023bft ; LHCb-PAPER-2022-040 ; LHCb-PAPER-2020-044 ; LHCb-PAPER-2022-026 ; LHCb-PAPER-2022-027 ; LHCb-PAPER-2020-025 ; LHCb-PAPER-2021-031 ; LHCb-PAPER-2021-032 ; Choi:2003ue . |  $c$ $\overline{c}$ |  $c$ $\overline{d}$ |  $s$ $\overline{s}$ |  $\overline{s}$ $u$ |  $\overline{s}$ $d$ |  $s$ $\overline{u}$
---|---|---|---|---|---|---
| $T_{\psi\psi}(6900)$ |  | $X(4140)$ | $T^{\theta}_{\psi s1}(4000)^{-}$ | $T^{\theta}_{\psi s1}(4000)^{0}$ |
$c$ $\overline{c}$ | $T_{\psi\psi}(6600)$ |  | +5 more | $T_{\psi s1}(4220)^{-}$ |  |
$\pagecolor{gray}{c}{\overline{u}}$ |  | $T_{cc}(3875)^{+}$ |  |  | $T^{a}_{c{\overline{s}}0}(2900)^{0}$ |
|  |  |  |  | $T^{a}_{c{\overline{s}}0}(2900)^{++}$ | $T_{c{\overline{s}}0}(2900)^{0}$
$c$ $\overline{d}$ |  |  |  |  |  | $T_{c{\overline{s}}1}(2900)^{0}$
$\pagecolor{gray}{q}{\overline{q}}$ | ${\chi_{c1}(3872)}\dots$ |  |  |  |  |

With the full legacy data sample, LHCb also discovered doubly charmed states
$T_{cc}^{+}$ LHCb-PAPER-2021-031 ; LHCb-PAPER-2021-032 , which differ from
previously discovered tetraquarks in that they have two charm quarks and two
light anti-quarks, the $T^{a}_{cs0,1}(2900)$ states with a single charm and a
strange quark LHCb-PAPER-2022-026 ; LHCb-PAPER-2022-027 , and the heavy
$T_{\psi\psi}$ with two charm and two anti-charm quarks LHCb-PAPER-2020-011 ;
ATLAS:2023bft ; CMS:2023owd . Table 3 attempts to classify these states by
quark content.

Mapping out existing (and eventually non-existing) quark contents helps
understanding the internal structure of exotic hadrons. The dispute between
proponents of the molecular picture and those of the compact tetra- and
pentaquarks has calmed down lately. Most likely there are representatives of
both kinds. Let’s look at states with two heavy quarks $Q$, and $Q$ or
$\overline{Q}$. If $Q\overline{Q}$ is in a colour-singlet configuration, it
will immediately hadronise into quarkonium. If the quarks are in different
colourless hadrons, they may form a molecule. On the other hand $QQ$ can never
be in a singlet configuration. A $QQ{\overline{q}}{\overline{q}}$ state may
thus be compact. A $Q\overline{Q}{q}{\overline{q}}$ may not Karliner . It is
thus likely that all kinds of structures exist in nature: hadronic molecules,
compact multi-quark objects, superposition of those, as well as rescattering
effects. Sorting them out will be an enterprise for the next decades
Brambilla:2022ura .

Figure 16: Evolution of LHCb trigger schemes in Run 1, 2 and 3. Figures from
Ref. LHCb-FIGURE-2020-016 .

##  6 LHCb status and prospects

The LHCb experiment has just undergone a major upgrade LHCb-DP-2022-002 . The
detector layout is hardly changed — actually the image in Fig. 2 is that of
the new detector — but many components have been changed. The goal of the
upgrade was to allow for an increased instantaneous luminosity, in the
$10^{33}\>\text{cm}^{-2}\text{s}^{-1}$ range. In order to achieve that, the
hardware trigger needed to be removed: Meeting a 1 MHz bottleneck would
require $p_{\mathrm{T}}$ thresholds of several  GeV, which starts to remove as
much $B$ signal as backgrounds. This in turn required to change all readout
electronics of the detectors not included in the hardware trigger, namely the
vertex detector, the tracking system, and the RICH. The silicon strip vertex
detector was replaced by a pixel detector and the trackers consisting of
silicon strips near the beam and straw tubes elsewhere were replaced by
scintillating fibres.

All this detector is read out at 40 MHz, of which 30 MHz are non-empty events.
These data are sent to a first trigger farm of graphical processing units. The
selected events are then buffered on disk while a calibration and alignment
procedure is run. Only when all calibration constants are available are the
data processed by the second trigger level. The final selected data thus have
the full offline-level quality and no further reconstruction is required. Most
of the events are saved partially, only keeping the objects of interest for
the analysis. The evolution of the trigger scheme is shown in Fig. 16.

The data stored by the trigger are massaged by a “sprucing” process, which may
for instance add neighbouring tracks to a selected $B$ candidate to form a
potential excited state to be used in spectroscopy studies. These neighbouring
tracks have to be duly requested by the relevant trigger selection as they may
otherwise be lost. The spruced candidates are stored in data streams that are
analysed by centrally managed user analysis productions, as shown in Fig. 17.

Figure 17: LHCb offline dataflow. Figure from Ref. LHCb-FIGURE-2020-016

The detector is now fully installed and commissioning is ongoing. There was a
recent set-back as the vacuum inside the vertex detector exceeded the
specifications due to a faulty sensor. The pressure difference with the vacuum
of the LHC deformed the thin RF foil that separates the two volumes. Luckily
no sensors were affected, but the vertex detector cannot fully close to its
nominal position until the foil is replaced during the 2023–24 winter
shutdown. In the meantime the experiment can run with a partially open vertex
detector and a consequently degraded resolution. Here degraded means a
resolution similar to that of the decommissioned Run 1–2 vertex detector. The
main issue is rather the lack of simulation perfectly matching this sub-
optimal situation and thus determining the geometrical acceptance of the
partially open detector. It should all be recovered in 2024.

Table 4: Expected sensitivities for selected observables. Table adapted from Refs. LHCb-PII-Physics ; LHCbCollaboration:2806113 . | Legacy | $2026$ | U2
---|---|---|---
Observable | (9$\text{\,fb}^{-1}$) | $(23\text{\,fb}^{-1})$ | $(300\text{\,fb}^{-1})$
$\sin 2{\maybebm{\phi_{1}}}\text{, with ${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{0}_{\mathrm{S}}}$}$ | $0.015$ | LHCb-PAPER-2023-013  | $0.011$ | $0.003$
${\phi_{{s}}}\text{, with ${{B}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}{{K}^{-}}$}\ [\\!\text{\,mrad}]$ | $23$ | LHCb-PAPER-2014-059  | $14$ | $4$
$\phi_{s}^{s{\bar{s}s}}\text{, with ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$}\ [\\!\text{\,mrad}]$ | $80$ | LHCb-PAPER-2023-001  | $39$ | $11$
${\maybebm{\phi_{3}}}$ | $4^{\circ}$ | LHCb-CONF-2022-003  | $1.5^{\circ}$ | $0.35^{\circ}$
$|{V_{{u}{b}}^{\phantom{\ast}}}|/|{V_{{c}{b}}^{\phantom{\ast}}}|$ | $6\%$ | LHCb-PAPER-2015-013  | $3\%$ | $1\%$
${\cal R}_{\mu^{+}\mu^{-}}$ | $90\%$ | LHCb-PAPER-2021-007  | $34\%$ | $10\%$
$R_{K}\ (1<q^{2}<6\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4})$ | $0.1$ | LHCb-PAPER-2022-045  | $0.025$ | $0.007$
$R_{K^{\ast}}\ (1<q^{2}<6\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4})$ | $0.1$ | LHCb-PAPER-2022-045  | $0.031$ | $0.008$
$R(D^{\ast})$ | $0.022$ | LHCb-PAPER-2022-052  | $0.0072$ | $0.002$
$R(J/\psi)$ | $0.24$ | LHCb-PAPER-2017-035  | $0.071$ | $0.02$
$\Delta A_{{C\\!P}}(KK-\pi\pi)\ [10^{-5}]$ | $85$ | LHCb-PAPER-2015-055  | $17$ | $3.0$

##  7 Future prospects

After Runs 3 and 4, LHCb plans to upgrade again the detector in order to keep
up with the requirements of the High Luminosity LHC. A luminosity in the
vicinity of $10^{34}\>\text{cm}^{-2}\text{s}^{-1}$ will however generate
several tens of $pp$ collisions per bunch crossings. Fishing out $b$ and $c$
hadrons from such busy events will required 4D tracking including timing LHCb-
TDR-023 . With such a detector one could achieve unprecedented sensitivities,
some of which are listed in Table 4.

Figure 18: (left) The unitarity triangle in 1995 CKMfitter2005 . (right)
Possible status after Run 5 of the LHC LHCb-PII-Physics .

##  8 Conclusion

Such fantastic accuracies will require enormous efforts on understanding the
detector, the backgrounds, the simulation. The prize will be an extremely
precise knowledge of the flavour sector, including of course the KM matrix at
the origin of this meeting. Figures 18 (left) and 3 show the progress between
the early dates of KM metrology and now, and Fig. 18 (right) shows how much
there is still to do.

Precision metrology should not be dismissed. It is about the best possible
understanding of Nature and the best possible exploitation of the machines we
have built. It is a prerequisite to discovery of new physics. Who knows if at
this level of precision the triangle will still close? Either way, it is a
journey worth taking.

## Acknowledgements

I would like to thank Hashimoto-san for the kind invitation to the workshop.
It was a pleasure to visit KEK after 20 years of absence. Many thanks for
Carla Göbel for commenting on the manuscript.

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