Heavy flavour physics and CP violation at LHCb: A ten-year review

doi:10.1007/s11467-022-1247-1

Front. Phys.

2023, Vol. 18

Issue (4) : 44601 https://doi.org/10.1007/s11467-022-1247-1

REVIEW ARTICLE

Heavy flavour physics and CP violation at LHCb: A ten-year review

Shanzhen Chen¹, Yiming Li¹(

), Wenbin Qian², Zhihong Shen³, Yuehong Xie⁴, Zhenwei Yang³, Liming Zhang⁵, Yanxi Zhang³

¹. Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
². University of Chinese Academy of Sciences, Beijing 100049, China
³. State Key Laboratory of Nuclear Physics and Technology & School of Physics, Peking University, Beijing 100871, China
⁴. Key Laboratory of Quark and Lepton Physics of Ministry of Education & Institute of Particle Physics, Central China Normal University, Wuhan 430079, China
⁵. Department of Engineering Physics & Center for High Energy Physics, Tsinghua University, Beijing 100084, China

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Abstract

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

Keywords LHCb flavour physics CP vioation

Corresponding Author(s): Yiming Li

Issue Date: 27 March 2023

Cite this article:

Shanzhen Chen,Yiming Li,Wenbin Qian, et al. Heavy flavour physics and CP violation at LHCb: A ten-year review[J]. Front. Phys. , 2023, 18(4): 44601.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-022-1247-1
https://academic.hep.com.cn/fop/EN/Y2023/V18/I4/44601

Fig.1 Layout of the LHCb detector [5].

System	$s (N N)$ [ $TeV$ ]
System	$5 (2.76)$	$7$	$8 (8.16)$	$13$

$p p$	$J / ψ$ [84], $Υ$ [88], $D$ [100]	$η c$ [91], $J / ψ$ [7], $χ c 1 (3872)$ [78], $χ c / J / ψ$ [77], $χ c 2 / χ c 1$ [76, 86], $ψ (2 S)$ [81, 111], $Υ$ [79, 97], $χ b 2 / χ b 1$ [92, 93], $D, Λ c +$ [9], $B$ [80, 85, 105, 117], $Λ b 0$ [89, 96], $Ξ b$ [109], $B c +$ [83], $J / ψ J / ψ$ [75], $Υ D$ [98], $J / ψ D, D D, D D ¯ 0$ [82]	$η c$ [91], $J / ψ$ [95], $χ b 2 / χ b 1$ [92, 93], $χ c 1 (3872) / ψ (2 S)$ [116], $χ c 1 (3872)$ [119], $Υ$ [95, 97], $B s 0 / B 0$ [117], $Λ b 0$ [96], $Ξ b$ [109], $Υ D$ [98], $B c +$ [94]	$η c$ [112], $J / ψ$ [10, 102], $χ c 1 (3872)$ [119], $ψ (2 S)$ [111], $Υ$ [106], $D$ [11], $Ξ c c + +$ [114], $B +$ [105], $Ξ b$ [109], $B c +$ [113], $B s 0 / B 0$ [117], $J / ψ J / ψ$ [101]
$p$ Pb	$J / ψ$ [87], $ψ (2 S)$ [99], $Υ$ [90], $D 0$ [104], $Λ c +$ [107]		$J / ψ$ [103], $Υ$ [108], $D 0$ [74], $B +, B 0, Λ b 0$ [110], $χ c 2 / χ c 1$ [118], $D D, D D ¯ 0$ [115]

Tab.1 LHCb measurements of production cross-sections (or ratios) for various heavy hadrons in

p p

and

p

Pb collisions at different centre-of-mass energies.

Fig.2 (Left) Differential cross-section of

J / ψ

p p

collisions at

s = 13 TeV

, reproduced from Ref. [10]. (Right)

J / ψ

polarisation in

p p

collisions at

s = 7 TeV

, reproduced from Ref. [123]. The measurements are compared with NRQCD [124-126] and colour singlet calculations [125].

Fig.3 (Left) Distribution of relative azimuthal angle,

Δ ϕ

, between

J / ψ

and

D

mesons in pair production, showing flat behaviour. (Right) the

p T

distribution of

J / ψ

mesons in

J / ψ D

pair production, compared with that in inclusive production (shown in black open circle) [7]. Reproduced from Ref. [82].

Fig.4 (Left) The parameter

σ e f f

measured in

p

Pb collisions at

s NN = 8.16 TeV

using (green)

J / ψ D 0

and (brown)

D 0 D 0

production for (positive rapidity)

p

beam and (negative rapidity) Pb-beam direction, reproduced from Ref. [115]. The prediction shown in shaded area is from Ref. [152]. (Right) The ratio of cross-sections between

Υ (3 S)

and

Υ (1 S)

over that in

p p

data at

s NN = 8 TeV

, reproduced from Ref. [108], is compared with the comovers model [153]. The rapidities are defined in the rest frame of two colliding nucleons with respect to proton-beam direction.

Fig.5 New hadrons that are discovered by the LHC experiments, reproduced from [170].

Fig.6 Experimentally identified bottom baryons, grouped according to spin and flavour symmetry of the light quarks as well as the excitation between the light quark system and

b

quark. The assignments of quantum numbers are based on measured properties and relevant predictions. Note that for recently discovered states their

J P

need confirmation and the assignments of excited

Ω b

are not certain, though they likely have

J P = 1 −

. In the figure, 1

P

excitation of flavour symmetric bottom baryons are gathered into a single multiplet (shown in purple). Only

λ

mode radial and orbital angular momentum excitation are considered. Missing states are marked as question marks in the figure. The notation

s q q ′

indicates the total spin of the light quark system.

Fig.7 (Left) Invariant mass distribution of

Ξ c + K −

, overlaid with the fit including five narrow

Ω c

states, reproduced from Ref. [200]. (Right) Invariant mass distribution of

Λ c + K −

, overlaid with the fit including three narrow

Ξ c

states (red, brown and purple dashed lines) and partially reconstructed components. Reproduced from Ref. [215].

Fig.8 (Left) Invariant mass distribution of

Δ M ≡ m B c + π + π − − m B c +

, overlaid with the fit including two narrow structures, corresponding to

B c ∗ (2 S) +

and

B c (2 S) +

repectively, reproduced from Ref. [257]. (Right) Distribution of the reconstructed invariant mass

m c a n d (Ξ c c + +) ≡ m (Λ c + K − π +) − m c a n d (Λ c +) + m P D G (Λ c +)

, overlaid with the fit projections. Reproduced from Ref. [258].

Contribution	Significance [ $× σ$ ]	$M 0$ [ $MeV$ ]	$Γ 0$ [ $MeV$ ]	FF [%]

$X (2 −)$
$X (4150)$	4.8 (8.7)	$4146 ± 18 ± 33$	$135 ± 28 − 30 + 59$	$2.0 ± 0.5 − 1.0 + 0.8$
$X (1 −)$
$X (4630)$	5.5 (5.7)	$4626 ± 16 − 110 + 018$	$174 ± 27 − 073 + 134$	$2.6 ± 0.5 − 1.5 + 2.9$
All $X (0 +)$				$20 ± 5 − 07 + 14$
$X (4500)$	20 (20)	$4474 ± 3 ± 3$	$77 ± 6 − 08 + 10$	$5.6 ± 0.7 − 0.6 + 2.4$
$X (4700)$	17 (18)	$4694 ± 4 − 03 + 16$	$87 ± 8 − 06 + 16$	$8.9 ± 1.2 − 1.4 + 4.9$
All $X (1 +)$				$26 ± 3 − 10 + 08$
$X (4140)$	13 (16)	$4118 ± 11 − 36 + 19$	$162 ± 21 − 49 + 24$	$17 ± 3 − 6 + 19$
$X (4274)$	18 (18)	$4294 ± 4 − 6 + 3$	$53 ± 5 ± 5$	$2.8 ± 0.5 − 0.4 + 0.8$
$X (4685)$	15 (15)	$4684 ± 7 − 16 + 13$	$126 ± 15 − 41 + 37$	$7.2 ± 1.0 − 2.0 + 4.0$
All $T ψ s (1 +)$				$25 ± 5 − 12 + 11$
$T ψ s (4000)$	15 (16)	$4003 ± 6 − 14 + 04$	$131 ± 15 ± 26$	$9.4 ± 2.1 ± 3.4$
$T ψ s (4220)$	5.9 (8.4)	$4216 ± 24 − 30 + 43$	$233 ± 52 − 73 + 97$	$10 ± 4 − 07 + 10$

Tab.2 Spin-parity, significance, masses, widths, and fit fractions of exotic candidates observed in the

B + → J / ψ ϕ K +

decay by LHCb [429].

Fig.9 Distributions of (left)

ϕ K +

, (middle)

J / ψ ϕ

and (right)

J / ψ K +

invariant masses for (black data points) the

B + → J / ψ ϕ K +

candidates compared with (red solid lines) the fit results using (top row) the final model and (bottom row) the model of LHCb Run 1 analysis [401]. reproduced from Ref. [429].

Fig.10 (Top left) Invariant-mass distribution of

D − K +

B + → D + D − K +

decays in data and the fit projections, reproduced from Ref. [435]. The yellow dot-dashed and red dotted line correspond to

T c s 0 (2900) 0

and

T c s 1 (2900) 0

, respectively, and more details can be found in Ref. [435]. (Top right) The distribution of the invariant mass of the

D s + D s −

system in the

B + → D s + D s − K +

decay. A near-threshold structure, denoted as

X (3960)

, is marked in red [453]. The distribution of the invariant mass of (bottom left) the

D s + π −

pair in the

B 0 → D ¯ 0 D s + π −

and (bottom right)

D s + π +

system in the

B + → D − D s + π +

decay [438].

Fig.11 (Left) Distribution of

D 0 D 0 π +

invariant mass overlaid with fit projections. The contribution of the non-

D 0

background has been statistically subtracted. Vertical dashed lines show the

D ∗ + D 0

and

D ∗ 0 D +

mass thresholds respectively for a comparison, reproduced from Ref. [457]. (Right) Invariant mass distribution of the combination of two

J / ψ

mesons overlaid with fit projections, reproduced from Ref. [487]. The fit model does not contain interference between any components.

State	Decays	Significance [ $σ$ ]	Mass [ $MeV/ c 2$ ]	Width [ $MeV$ ]

$P ψ N (4312) +$	$J / ψ p$	$7.3 σ$	$4311.9 ± 0.7 − 0.6 + 6.8$	$0 9.8 ± 2.7 − 0 4.5 + 0 3.7$
$P ψ N (4440) +$	$J / ψ p$	$5.4 σ$	$4440.3 ± 1.3 − 4.7 + 4.1$	$20.6 ± 4.9 − 10.1 + 0 8.7$
$P ψ N (4457) +$	$J / ψ p$	$5.4 σ$	$4457.3 ± 0.6 − 1.7 + 4.1$	$0.53 ± 2.0 − 0 1.9 + 0 5.7$
$P ψ N (4337) +$	$J / ψ p$	$3.1 σ$	$04337 0 − 04 + 07 0 ± 2$	$029 0 − 012 + 026 ± 14$
$P ψ s Λ (4459) 0$	$J / ψ Λ$	$3.1 σ$	$4458.8 ± 2.9 − 1.1 + 4.7$	$17.3 ± 6.5 − 0 5.7 + 0 8.0$
$P ψ s Λ (4338) 0$	$J / ψ Λ$	$> 15 σ$	$4338.2 ± 0.7 ± 0.4$	$7.0 ± 1.2 ± 1.3$

Tab.3 Detection decay channels, experimental significance, masses and widths of pentaquark states reported by LHCb.

Fig.12 (Left) Invariant mass distribution of

J / ψ p

in the

Λ b 0 → J / ψ p K −

decay fitted with three BW functions plus polynomial background, reproduced from Ref. [487]. The vertical lines mark the

Σ c + D ¯ 0

and the

Σ c + D ¯ ∗ 0

mass thresholds respectively. (Right) Invariant mass distribution of the

J / ψ Λ

system in the

Ξ b → J / ψ Λ K −

decay overlaid with fit projections, reproduced from Ref. [564].

Fig.13 (Left) Likelihood contours in the

B (B 0 → μ + μ −)

−

B (B s 0 → μ + μ −)

plane corresponding to

− 2 Δ ln ? L = 2.3, 6.2

and

11.8

, for ATLAS, CMS and LHCb experiments and the combination. (Right) Likelihood contours for the combination corresponding to

− 2 Δ ln ? L = 2.3, 6.2, 11.8, 19.3

and

30.2

. The data sets used were collected from 2011 to 2016. Reproduced from Refs. [660].

Fig.14 (Left) Invariant mass distribution of the selected

B (s) 0 → μ + μ −

candidates with output of the used multivariate classifier above 0.5, superimposed with the fit result. (Right) Confidence intervals in the plane of the branching fractions of

B s 0 → μ + μ −

and

B 0 → μ + μ −

. The data sets used for the blue solid (brown dotted) contours were collected from 2011 to 2018 (2016). Reproduced from Refs. [661, 662].

Fig.15 Top left: Measured

d B (B s 0 → ϕ μ + μ −) / d q 2

[673] overlaid with the SM predictions based on light-cone sum rule calculations [667, 677, 678] at low

q 2

and lattice QCD calculations [670, 671] at high

q 2

; Top right: Measured

d B (Λ b 0 → Λ μ + μ −) / d q 2

[674] compared with the SM predictions using form factors from lattice QCD calculations [679]; Middle left: Measured

d B (B 0 → K ∗ 0 μ + μ −) / d q 2

[675] compared with the SM predictions using form factors from line-cone sum rule and lattice QCD calculations [667, 670]; Middle right, bottom left and bottom right: Measured

d B (B + → K ∗ + μ + μ −) / d q 2

d B (B 0 → K S 0 μ + μ −) / d q 2

and

d B (B + → K + μ + μ −) / d q 2

[676] compared with the SM predictions using form factors from light-cone sum rule and lattice QCD calculations [680, 681].

Fig.16 Results of the observables

F L

A F B

S 5

and

P 5 ′

in bins of

q 2

, compared with the SM predictions. Reproduced from Ref. [689].

Fig.17 Results for the observables

P 5 ′

and

P 2

B + → K ∗ + μ + μ −

, compared with SM predictions. Reproduced Ref. [692].

Fig.18 CP asymmetries

A F B C P

and

A 5, 8, 9

in intervals of

q 2

in the decay

B s 0 → ϕ μ + μ −

measured using Run 1 and Run 2 data (black) in Run 1 data only (grey). Reproduced from Ref. [693].

Fig.19 Invariant mass distributions of the selected (left)

B + → K + e + e −

and (right)

B + → K + μ + μ −

candidates, superimposed by the fit results. Reproduced from Ref. [703].

Fig.20 Comparison of the LHCb

R K

result with the measurements by B-factories [707, 708]. The vertical line indicates the SM expected value. Reproduced from Ref. [703].

Fig.21 Decay-time distributions of the tagged (left)

B s 0 → ϕ γ

and (right)

B ¯ s 0 → ϕ γ

candidates, superimposed by the fit projections. Reproduced from Ref. [746].

Fig.22 Global fit results of the CKM matrix from different measurements, provided by the CKMfitter group [846]. Different colours and labels on the plot indicate constraints from measurements of different observables.

Fig.23 The invariant mass distributions of

(K + K −) D K +

(left) and

(K + K −) D K −

(right) from Ref. [792].

Fig.24 The invariant mass distributions of

(K + π −) D K +

(left) and

(π + K −) D K −

(right) from Ref. [792].

Fig.25 (Left) Binning scheme of

D → K S 0 π + π −

decays used for

γ

measurements; (right) Observed raw CP asymmetries and the predicted values using obtained CP parameters in different bins [802].

Fig.26

γ

combinations from different

B

hadrons [784].

Fig.27 Combinations of

γ

from all LHCb measurements, 2D contours of

r B

vs.

γ

(left) and

δ B

vs.

γ

(right) have been shown at 68.3% confidence level [784].

Fig.28 Combination of

ϕ s c c ¯ s

measurements by HFLAV [812], where 2D contours of (left)

ϕ s c c ¯ s

Δ Γ s

and (right)

Γ s

Δ Γ s

are displayed at 68% confidence level.

Fig.29 Global fits of

| V u b |

and

| V c b |

from the CKMfitter group [846] where measurements using different methods have been displayed.

Fig.30 Oscillation of

D 0

mesons via (top)

W ±

exchanges or (bottom) rescattering effect.

Data sample	no CPV		CPV allowed
Data sample	$x ′ 2$ ( $× 10 − 3$ )	$y ′$ ( $× 10 − 3$ )	$x ′ 2$ ( $× 10 − 3$ )	$y ′$ ( $× 10 − 3$ )

1.0 fb⁻¹, $D ∗$ tag [863]	$− 0.09 ± 0.13$	$7.2 ± 2.4$	−	−
3.0 fb⁻¹, $D ∗$ tag [865]	$0.055 ± 0.049$	$2.8 ± 1.0$	$D 0$ : $0 0.049 ± 0.070$	$5.1 ± 1.4$
3.0 fb⁻¹, $D ∗$ tag [865]	$0.055 ± 0.049$	$2.8 ± 1.0$	$D ¯ 0$ : $0 0.060 ± 0.068$	$4.5 ± 1.4$
3.0 fb⁻¹, $B$ tag [866]	$0.028 ± 0.310$	$4.6 ± 3.7$	$D 0$ : $− 0.019 ± 0.447$	$5.81 ± 5.26$
3.0 fb⁻¹, $B$ tag [866]	$0.028 ± 0.310$	$4.6 ± 3.7$	$D ¯ 0$ : $0 0.079 ± 0.433$	$3.32 ± 5.23$
5.0 fb⁻¹, $D ∗$ tag [867]	$0.039 ± 0.027$	$5.28 ± 0.52$	$D 0$ : $0 0.061 ± 0.037$	$5.01 ± 0.74$
5.0 fb⁻¹, $D ∗$ tag [867]	$0.039 ± 0.027$	$5.28 ± 0.52$	$D ¯ 0$ : $0 0.016 ± 0.039$	$5.54 ± 0.74$

Tab.4 Summary of the charm mixing parameters measured by LHCb using

D 0 → K + π −

decays. Results under the assumptions of CP invariance and CP violation are given.

Fig.31 Ratio of the

D 0 → K + π

D 0 → K − π +

decay rates,

R

, as a function of the

D 0

decay time normalised by the average

D 0

lifetime [863]. The solid (dashed) line indicates the projection of the mixing (no-mixing) fits.

Data sample	CP-averaged parameters
Data sample	$x$ ( $× 10 − 3$ )	$y$ ( $× 10 − 3$ )

1.0 fb⁻¹, $D ∗$ tag [868]	$− 8.6 ± 5.3 ± 1.7$	$0.3 ± 4.6 ± 1.3$
3.0 fb⁻¹, $B$ tag [869]	$2.7 ± 1.6 ± 0.4$	$7.4 ± 3.6 ± 1.1$
5.4 fb⁻¹, $D ∗$ tag [870]	$3.97 ± 0.46 ± 0.29$	$4.59 ± 1.20 ± 0.85$
	CP-violating parameters
Data sample	$Δ x$ ( $× 10 − 3$ )	$Δ y$ ( $× 10 − 3$ )
3.0 fb⁻¹, $B$ tag [869]	$− 0.53 ± 0.70 ± 0.22$	$0.6 ± 1.6 ± 0.3$
5.4 fb⁻¹, $D ∗$ tag [870]	$− 0.27 ± 0.18 ± 0.01$	$0.20 ± 0.36 ± 0.13$

Tab.5 Summary of the charm mixing parameters measured by LHCb using

D 0 → K S 0 π + π −

decays.

Fig.32 Profile likelihood contours (from 1 to 5

σ

) of the charm mixing parameters

x

and

y

. The solid (brown) contours are determined from the simultaneous combination [784], while the dashed (blue) indicate the current world average from Ref. [812].

Data sample	$Δ A C P$ ( $× 10 − 3$ )

0.62 fb⁻¹, $D ∗$ tag [879]	$− 8.2 ± 4.1 ± 0.6$
1.0 fb⁻¹, $B$ tag [880]	$4.9 ± 3.0 ± 1.4$
3.0 fb⁻¹, $B$ tag [871]	$1.4 ± 1.6 ± 0.8$
3.0 fb⁻¹, $D ∗$ tag [872]	$− 1.0 ± 0.8 ± 0.3$
5.9 fb⁻¹, $B$ or $D ∗$ tag [790]	$− 1.54 ± 0.29$

Tab.6 Summary of the

Δ A C P

results from the LHCb measurements in the charm sector.

Fig.33 (Left) Distributions of local per-bin asymmetry significance

S C P

and (right) corresponding one-dimensional distributions obtained by the binned

χ 2

method for

Ξ c + → p K − π +

decays. Reproduced from Ref. [896].

Decay channel	Data sample	Method

$D + → K − K + π +$ [892]	$35 pb − 1$	binned $χ 2$
$D 0 → K − K + π − π +$ [893]	1.0 fb⁻¹, $D ∗$ tag	binned $χ 2$
$D 0 → π − π + π − π +$ [893]	1.0 fb⁻¹, $D ∗$ tag	binned $χ 2$
$D + → π − π + π +$ [894]	1.0 fb⁻¹	binned $χ 2$
$D 0 → K − K + π − π +$ [895]	3.0 fb⁻¹, $B$ tag	binned $χ 2$
$D 0 → π − π + π 0$ [897]	2.0 fb⁻¹, $D ∗$ tag	energy test
$D 0 → π − π + π − π +$ [898]	3.0 fb⁻¹, $D ∗$ tag	energy test
$Λ c + → p h − h +$ [901]	3.0 fb⁻¹	$Δ A C P$
$D 0 → K − K + π − π +$ [891]	3.0 fb⁻¹, $B$ tag	amplitude analysis
$Ξ c + → p K − π +$ [896]	3.0 fb⁻¹	binned $χ 2$

Tab.7 Summary of LHCb direct CP violation searches in phase space of charm decays.

Fig.34 (Left) The global test statistic compared with the distribution of the statistic calculated from a large number of random permutations obtained from the energy test applied in the CP violation searches in

D 0 → π − π + π 0

decays. (Right) Dalitz plot distribution of significance of local test statistics. Reproduced from Ref. [897].

Data sample	Final state(s)	$y C P$ [%]	$A Γ$ ( $× 10 − 3$ )

29 pb⁻¹, $D ∗$ tag [902]	$K + K −$	$0.55 ± 0.63 ± 0.41$	$− 5.9 ± 5.9 ± 2.1$
1.0 fb⁻¹, $D ∗$ tag [903]	$π + π −$	−	$0.33 ± 1.06 ± 0.14$
1.0 fb⁻¹, $D ∗$ tag [903]	$K + K −$	−	$− 0.35 ± 0.62 ± 0.12$
3.0 fb⁻¹, $B$ tag [874]	$π + π −$	−	$− 0.92 ± 2.6 − 0.33 + 0.25$
3.0 fb⁻¹, $B$ tag [874]	$K + K −$	−	$− 1.34 ± 0.77 − 0.34 + 0.26$
3.0 fb⁻¹, $B$ tag [874]	$π + π −$ & $K + K −$	−	$− 1.25 ± 0.73$
3.0 fb⁻¹, $B$ tag [873]	$π + π −$ & $K + K −$	$0.57 ± 0.13 ± 0.09$	−
3.0 fb⁻¹, $D ∗$ tag [875]	$π + π −$	−	$0.46 ± 0, 58 ± 0.12$
3.0 fb⁻¹, $D ∗$ tag [875]	$K + K −$	−	$− 0.30 ± 0.32 ± 0.10$
3.0 fb⁻¹, $D ∗$ tag [875]	$π + π −$ & $K + K −$	−	$− 0.13 ± 2.0 ± 0.7$
5.4 fb⁻¹, $B$ tag [876]	$π + π −$	−	$0.22 ± 0.70 ± 0.08$
5.4 fb⁻¹, $B$ tag [876]	$K + K −$	−	$− 0.43 ± 0.36 ± 0.05$
6 fb⁻¹, $D ∗$ tag [877]	$π + π −$	−	$0.4 ± 0.28 ± 0.04$
6 fb⁻¹, $D ∗$ tag [877]	$K + K −$	−	$0.23 ± 0.15 ± 0.03$
8.4 fb⁻¹, $D ∗$ or $B$ tag [877]	$π + π −$	−	$0.36 ± 0.24 ± 0.04$
8.4 fb⁻¹, $D ∗$ or $B$ tag [877]	$K + K −$	−	$0.03 ± 0.13 ± 0.03$
8.4 fb⁻¹, $D ∗$ or $B$ tag [877]	$π + π −$ & $K + K −$	−	$0.10 ± 0.11 ± 0.03$
6 fb⁻¹, $D ∗$ tag [904]	$π + π −$	$0.657 ± 0.053 ± 0.016$ ¹⁾	−
6 fb⁻¹, $D ∗$ tag [904]	$K + K −$	$0.708 ± 0.030 ± 0.014$ ²⁾	−
6 fb⁻¹, $D ∗$ tag [904]	$π + π −$ & $K + K −$	$0.696 ± 0.026 ± 0.013$ ³⁾	−

Tab.8 Summary of LHCb

y C P

and

A Γ

measurements.

Fig.35 The world averages of

y C P

and

A Γ

. Reproduced from Ref. [812].

Fig.36 Instantaneous and integrated luminosities of LHCb as functions of year.

Observable	LHCb	LHCb	LHCb	Belle II	ATLAS
Observable	current	(23 fb^-1)	(300 fb^-1)	( $50 ab − 1$ )	& CMS

CKM tests
$γ$ (all modes)	$4 ∘$ [784, 931]	$1.5 ∘$	$0.35 ∘$	$1.5 ∘$	$−$
$γ$ ( $B s 0 → D s + K −$ )	$(− 22 + 17) ∘$	$4 ∘$	$1 ∘$	$−$	$−$
$sin ? 2 β$	0.04 [932]	0.011	0.003	0.005	$−$
$ϕ s$ ( $B s 0 → J / ψ ϕ$ )	49 mrad [933]	14 mrad	4 mrad	−	22 mrad [934]
$ϕ s$ ( $B s 0 → J / ψ ϕ$ )	49 mrad [933]	14 mrad	4 mrad	−	5−6 mrad [935]
$ϕ s$ ( $B s 0 → D s + D s −$ )	170 mrad [825]	35 mrad	9 mrad	$−$	$−$
$ϕ s s s ¯ s$ ( $B s 0 → ϕ ϕ$ )	154 mrad [936]	39 mrad	11 mrad	$−$	feasible [937]
$a s l s$	$33 × 10 − 4$ [938]	$10 × 10 − 4$	$3 × 10 − 4$	$−$	$−$
$\| V u b \| / \| V c b \|$	6% [847]	3%	1%	1%	$−$
Charm
$Δ A C P$	$2.9 × 10 − 4$ [790]	$1.7 × 10 − 4$	$3.0 × 10 − 5$	$5.4 × 10 − 4$	$−$
$A Γ$	$1.3 × 10 − 4$ [877]	$4.2 × 10 − 5$	$1.0 × 10 − 5$	$3.5 × 10 − 4$	$−$
$B (s) 0 → μ + μ −$
$B (B 0 → μ + μ −) B (B s 0 → μ + μ −)$	71% [661, 662]	34%	10%	$−$	21% [939, 940]
$τ B s 0 → μ + μ −$	14% [661, 662]	8%	2%	$−$	$−$
EW penguins
$R K$ ( $B + → K + ℓ + ℓ −$ )	0.044 [703]	0.025	0.007	0.036	$−$
$R K ∗$ ( $B 0 → K ∗ 0 ℓ + ℓ −$ )	0.10 [709]	0.031	0.008	0.032	$−$
LFU tests
$R D ∗$ ( $B 0 → D ∗ − ℓ + ν$ )	0.026 [941, 942]	0.007	0.002	0.005	$−$
$R J / ψ$ ( $B c + → J / ψ ℓ + ν$ )	0.24 [943]	0.07	0.02	$−$	$−$

Tab.9 Sensitivity of selected key flavour observables for LHCb, ATLAS and CMS, reproduced from Ref. [664] with updates when available, and Belle II sensitivities from Ref. [930].

Fig.37 Sensitivity to probe key CP violating variables, rare decay and lepton flavour universality tests expected from LHCb upgrades. Anticipated results from Belle II, ATLAS or CMS are listed when available. Reproduced from Ref. [664].

Fig.38 LHCb constraints to the unitarity triangle with anticipated improvement from (left) Upgrade I and (right) Upgrade II. Reproduced from Ref. [664].

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