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

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.

Graphical abstract

Keywords

LHCb / flavour physics / CP vioation

1 Introduction

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

The LHCb detector [5] is optimised for the study of the decays of heavy-flavour hadrons, i.e., hadrons containing heavy quarks (b or c quarks, often collectively referred to as Q). In proton−proton (pp) collisions at LHC energies, the b\overline{b} pairs are produced dominantly through the gluon fusion process gg\to b\overline{b}. Due to the large Lorentz boost along the proton beam in the laboratory frame, the b and \overline{b} quarks generated in a pair are highly correlated in their momentum directions, either both in the forward region or both in the backward region in the majority of cases. In order to take advantage of this characteristic of the b\overline{b} pair at the LHC, the LHCb detector is designed to have a forward geometry as shown in Fig.1 to cover the forward region of pp collisions. The flavour of the b-hadron under study can usually be tagged by the other b-hadron that is also inside the LHCb acceptance. This enhances the potential of the LHCb experiment in the study of CP violation and mixing with B^0 and B_s^0 decays.

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

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

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

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

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

2 Heavy-flavour production and spectroscopy

Rich information on QCD dynamics can be deciphered from measurements of heavy-flavour production [13-49] and studies of heavy-flavour spectroscopy [50-71]. The mass of a heavy quark, which is much larger than the nonperturbative QCD scale {\Lambda }_{\mathrm{Q}\mathrm{C}\mathrm{D}}, provides an energy scale that allows for perturbative calculation of heavy-quark production. The production of heavy quark pairs, Q\overline{Q}, is predominantly in the initial stage of the collision, thus can be used to probe properties of the colliding system and the possibly created QCD medium [72, 73]. The presence of heavy quark(s) also provides practical benefits for theoretical and experimental studies of spectroscopy. Heavy quarks are approximately nonrelativistic in hadrons, which makes it possible to simplify theoretical calculations. The large mass and the weak decay of heavy-flavour hadrons offer essential features, such as decay products with high transverse momentum, p_T, and vertices displaced from the PVs, which can be exploited to reject the huge QCD background at hadron colliders.

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

2.1 Production

A summary of LHCb production measurements for open heavy-flavour hadrons, heavy quarkonia and pairs of heavy-flavour hadrons at LHCb [7, 9–11, 74–119] are listed in Tab.1. Inclusive hadroproduction of open heavy-flavour hadrons (H_Q) factorises into three components in perturbative QCD (pQCD) calculations: the parton distribution function (PDF) in the two initial projectiles f_{i,j}, the parton level cross-section {\sigma }_{ij\to Q+X} of a heavy-quark Q production, and the heavy-quark fragmentation function D_{Q\to H_Q} [120]. Differential cross-section for H_Q production in A−B collisions is expressed as

(1)\mathrm{d}{\sigma }_{AB\to H_Q}={\displaystyle \sum _{i,j}(f_i^A\otimes f_j^B)\otimes \mathrm{d}{\sigma }_{ij\to Q+X}\otimes D_{Q\to H_Q},}

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

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

2.1.1 Associated production

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

(2){\sigma }^{ab}=\kappa {\displaystyle \frac{{\sigma }^a{\sigma }^b}{{\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}},}

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

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

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

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

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

2.1.2 Production in pPb collisions

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

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

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

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

2.2 Spectroscopy

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

2.2.1 Conventional hadrons

LHCb have filled many gaps in conventional heavy meson and baryon spectra. This section will focus on more recent highlights in baryon spectroscopy and discovery of doubly heavy hadrons. Though not discussed in detail, it should not be ignored that multiple excited beauty [171, 172] and charm [173] mesons are observed in high energy pp collisions by LHCb. Moreover, the beauty hadron decay is an ideal place to study excited charm hadron as b\to c is the dominant transition of the b quark. Among the many new charm mesons discovered in B decay at LHCb [174-176], an interesting example is a new excited D_s^{+} meson, D_s(2590{)}^{+}, consistent with the radial excited state D_s(2^1{S}_0{)}^{+}. Its mass and width will help to understand the excitation spectrum of D_s^{+} mesons which are found to be not fully consistent with the quark model predictions [177-182].

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

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

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

Bottom baryons. A summary of all bottom baryons with clear experiment evidences are shown in Fig.6, organised in multiplets of q q^{\prime } flavour symmetry and \lambda-mode excitation when possible. The spin-parity quantum numbers for most of these states are not measured, so we rely on theoretical calculations as a guidance to make the classification. Actually, there is not always a consensus on the J^P of each state, in particularly for {\Omega }_b baryons [224-239].

In total, five states have been reported in the {\Lambda }_b^0{\pi }^{+}{\pi }^{-} mass spectrum: {\Lambda }_b( 5912{)}^0 and {\Lambda }_b(5920{)}^0 with widths below 1\text{MeV} [240], {\Lambda }_b( 6146{)}^0 and {\Lambda }_b(6152{)}^0 with widths of about 2\text{MeV} [241], and {\Lambda }_b(6072{)}^0 with a width around 70\text{MeV} [242]. Their masses match two 1P, two 1D and 2S {\Lambda }_b^0 \lambda-mode excitation respectively, though other assignments are also discussed [243-247]. It is useful to note that intermediate {\Sigma }_b^{(\ast )\pm }(\to {\Lambda }_b^0{\pi }^{\pm }) states are found to be present in the {{\Lambda }_b}^{\ast (\ast )0}\to {\Lambda }_b^0{\pi }^{+}{\pi }^{-} decays.

The ground {\Sigma }_b^{\pm } and {\Sigma }_b^{\ast \pm } states were first detected in the {\Lambda }_b^0{\pi }^{\pm } mass spectrum by CDF [248]. In the same final state, two new ones, {\Sigma }_b(6097 {)}^{+} and {\Sigma }_b( 6097{)}^{-}, are observed by LHCb [249], whose widths are about 30\text{MeV}. These two new states belong to the P-wave family, and many more of them are still missing, like for charm baryons.

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

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

Doubly heavy hadrons. LHCb has observed a few new hadrons with two heavy quarks. In the final states of D^0{\overline{D}}^0 and D^{+}{D}^{-} mesons produced promptly, a new particle X(3842), with a width of about 3 \text{MeV}, is discovered [259]. It is consistent with the spin-3 conventional charmonium {\psi }_3 ({}^3{D}_3) with J^{PC}=3^{--} [260].

As shown on the left of Fig.8, two narrow structures are detected in the B_c^{+}{\pi }^{+}{\pi }^{-} invariant mass spectrum of LHCb data [257]. The left one corresponds to the B_c^{\ast }(2^3{S}_1{)}^{+}\to B_c^{\ast }(1^3{S}_1{)}^{+}{\pi }^{+}{\pi }^{-} decay with the photon in decay of B_c^{\ast }(1^3{S}_1{)}^{+}\to B_c^{+}\gamma not detected. The peak on the right is consistent with the B_c( 2^1{S}_0{)}^{+}\to B_c^{+}{\pi }^{+}{\pi }^{-} decay. Almost in parallel, these two states are independently observed by CMS [261]. As the B_c^{+} meson is composed of two heavy quarks, its excitation mass spectroscopy can be calculated using models similar to those applied to heavy quarkonia [262-267] despite their hadroproduction mechanisms are very different [268, 269].

LHCb opens a new era in studies of doubly heavy baryons by observing the {\Xi }_{cc}^{++} baryon in the {\Xi }_{cc}^{++}\to {\Lambda }_c^{+}{K}^{-}{\pi }^{+}{\pi }^{+} mass spectrum [258], as shown on the right of Fig.8. This discovery decay mode is predicted to have a relatively large branching fraction [270, 271]. The {\Xi }_{cc}^{++} state is later confirmed using the {\Xi }_{cc}^{++}\to {\Xi }_c^{+}{\pi }^{+} decay [272]. Its lifetime is measured to be about 0.25 \text{ps} [273] and its mass is precisely determined to be (3621.55 \pm 0.38) \text{MeV/}{c}^2 [273]. Its SU(3) partners, {\Xi }_{cc}^{+} and {\Omega }_{cc}^{+}, are also searched for by LHCb, but with no sign of observation yet [274-276]. Theoretically, an important question is to understand the mass spectrum of the doubly charm baryons and related systems, which depends on the binding energy between the two heavy quarks [277-289]. The successful discovery of {\Xi }_{cc}^{++} has triggered wide theoretical work to understand the properties of baryons with more than a heavy quark [290-312]. In addtion, the {\Xi }_{cc}^{++} baryon mass is used to study the stability of tetraquark states with QQ contents [313, 314] with the assumption that QQ form a heavy diquark [315-319].

2.2.2 Exotic hadrons

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

Tetraquark states. LHCb provides essential information for the understanding of previously known tetraquark states. For example LHCb determined the quantum number of the X(3872) state, first reported by Belle [324], to be J^{PC}=1^{++} through a full amplitude analysis [325]. LHCb also precisely measured the mass of the X(3872) state (referred to as {\chi }_{c1}(3872 ) in Ref. [326]) to be m_{{\chi }_{c1}(3872)}-m_{\psi (2S)}=185.49\pm 0.06\pm 0.03\text{MeV/}{c}^2 [327, 328]. Its Breit−Wigner (BW) width is determined by LHCb to be {\mathrm{\Gamma }}_{{\chi }_{c1}(3872)}^{\mathrm{B}\mathrm{W}}={0.96}_{-0.18}^{+0.19}\pm 0.21\text{MeV} [327, 328]. Evidence of its decay to \psi (2S)\gamma is found and the branching fraction relative to J/\psi \gamma is measured to 2.46 \pm 0.64\pm 0.29 [329], disfavouring a pure D{\overline{D}}^{\ast } molecule intepretation. The {\rho }^0 and \omega contributions are disentangled in its decay to the J/\psi {\pi }^{+} {\pi }^{-} and a sizeable contribution from \omega is confirmed [330]. The {\chi }_{c1} (3872) state is the mostly studied exotic candidate, and its exotic behaviours include a extremely narrow width, isospin breaking decays and a mass close to the D^{\ast }D threshold. Despite all the available information we are still not sure whether it is a compact tetraquark state, a D^{\ast 0}{\overline{D}}^0 hadron molecule, a mixture of D^{\ast 0}{\overline{D}}^0 molecule with a {\chi }_{c1}(2P) charmonium component or just caused by kinematic rescattering effect [331-373].

The exotic candidate T_{\psi 1}^b(4430{)}^{+} was first observed by Belle in the \psi (2S){\pi }^{-} mass spectrum in B^0\to \psi (2S)K^{+}{\pi }^{-} decays [374].) An amplitude analysis of the B^0\to \psi (2S)K^{+}{\pi }^{-} decay is performed at LHCb, confirming the existence of the T_{\psi 1}^b(4430{)}^{+} state and determining it to be consistent with a Breit−Wigner resonance with J^P=1^{+} [376]. The quark contents of T_{\psi 1}^b(4430{)}^{+}, c\overline{c}\overline{u} d, are the same as the T_{\psi 1}^b(3900{)}^{+} state observed by the BESIII experiment [377]. Being charged, they are definitely not consistent with conventional charmonia and many phenomenological calculations are performed to explain their internal structure and properties [378-395].

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

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

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

{\displaystyle \frac{\mathrm{\Gamma }\left(X\to D^{+}{D}^{-}\right)}{\mathrm{\Gamma }\left(X\to D_s^{+}{D}_s^{-}\right)}}=0.29\pm 0.09\pm 0.10\pm 0.08,

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

Prompt production of di-charm hadrons has been suggested to search for exotic states containing multiple charm quarks [455]. In the invariant mass spectrum of D^0{D}^0{\pi }^{+}, shown in Fig.11, a new structure is observed close to the D^{\ast +}{D}^0 mass threshold [456, 457]. The structure is measured to be consistent with the ground state of a T_{cc}^{+} isoscalar tetraquark, with J^P=1^{+} and quark contents cc\overline{u}\overline{d}. Its Breit-Wigner mass is measured to be -273\pm 61\pm 5_{-14}^{+11}\text{keV/} c^2 below m_{D^{\ast +}}+m_{D^0}, and its Breit-Wigner width is {\mathrm{\Gamma }}_{\mathrm{B}\mathrm{W}}=410\pm 165\pm {43}_{-38}^{+18}\text{keV}. The same state also appears in the D^0{D}^0 and D^0{D}^{+} mass spectra, with a {\pi }^{+}, {\pi }^0 or \gamma in the T_{cc}^{+}\to D^0{D}^0{\pi }^{+}, T_{cc}^{+}\to D^0{D}^{+}{\pi }^0 or T_{cc}^{+}\to D^0{D}^{+}\gamma decays undetected, respectively. Dedicated studies of the T_{cc}^{+} resonance lineshape are performed using a unitarised Breit−Wigner distribution, considering T_{cc}^{+} decays in D^0{D}^0{\pi }^{+},D^0{D}^{+}{\pi }^0 and D^0{D}^0\gamma final states. The pole mass of the resonance in this advanced model is measured to be -360\pm 4_{-0}^{+4} \text{keV/}{c}^2 below m_{D^{\ast }}+m_{D^0}, and the pole width is {\mathrm{\Gamma }}_{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}}=-48\pm 2_{-14}^{+0} \text{keV}. This extremely narrow width has attracted many theoretical interests [458-463]. The T_{cc}^{+} state is the first observed tetraquark candidates with two heavy quarks of the same flavour. Many theoretical models have been applied to explain the existence and structure of such a tetraquark state with a large fraction of them favouring a D^{\ast +}{D}^0 hadron molecule interpretation [464-486].

In addition to two open charm final states, di-J/\psi mass spectrum of prompt production is also studied using full LHCb data [487]. Two peaking structures are observed in the mass range , where fully charmed tetraquark states are predicted [488-512]. The first structure (referred to as the threshold peak) covers the range between 6.2 and 6.6 \text{GeV/}{c}^2 close to the di-J/\psi mass threshold, and the other one sits at 6.9 \text{GeV/}{c}^2, as shown on the left of Fig.11. The di-J/\psi mass spectrum is modelled with a combination of BW functions for the two peaking structures and empirical smooth functions for SPS, DPS production of nonpeaking background. When no interference between BW and SPS is applied, the threshold structure can be described by two BW functions and the one at 6.9\text{GeV/} c^2 is well described by a BW. The narrow structure, denoted as T_{\psi \psi }(6900), is measured to have a mass of m_{T_{\psi \psi }(6900)}=6905 \pm 11\pm 7 \text{MeV/}{c}^2 and a width of {\mathrm{\Gamma }}_{T_{\psi \psi }(6900)}= 80\pm 19 \pm 33 \text{MeV}. Interpretations of the threshold peak using feeddown decays from excited quarkonium pairs are also possible. For various fit models without any interference, the dip around 6.8 \text{GeV/}{c}^2 cannot be well described. Advance fit studies are performed introducing interference between SPS and resonant structures. In one such fit, two BW functions are considered: a broad BW interfering with SPS used to describe the threshold structure, and a stand-alone narrow one used to model the 6.9 \text{GeV/}{c}^2 peak. This new model could fit well the overall spectrum, and the broad structure is now measured to have a mass around 6.7 \text{GeV/}{c}^2 and a width of about 0.3 \text{GeV}, while the T_{\psi \psi }(6900) structure has a mass consistent with the no-interference fit model, but its width becomes about twice larger. As the fit results for the broad structure is not stable in different models, its nature is not fully resolved and more data are needed to provide better information. The T_{\psi \psi }(6900) and higher resonances have been recently confirmed by CMS [513] and ATLAS [514]. The T_{\psi \psi }(6900) state is consistent with a genuine fully charmed tetraquark, however, in some models possible origins due to rescatterings of multiple charmonia or dibaryon molecules etc. are also discussed [515-551]. Other fully heavy tetraquarks, such as b\overline{b}c\overline{c} and b\overline{b} b\overline{b}, can be searched for in {\rm Y}J/\psi and {\rm Y}{\rm Y} or similar final states, however current limited data are estimated to have small sensitivities for these states, demanding the increased luminosity in LHCb upgrades.

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

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

Observations of pentaquark states with heavy-quark contents have triggered many studies from theorists, with the purpose of understanding their nature. In general, these states are considered to either be compact tetraquarks, hadronic molecules or simple bumps due to kinematic rescatterings [565-638]. Measurements of their J^P and production properties, and finding their flavour partners will shed light on the problem. However, it is likely that debates on the nature of exotic hadrons will continue before we have a complete and coherent theory to explain all of them. In the past years, phenomenology models on the hadron spectroscopy evolved quickly and some patterns have been revealed, for example, dynamics close to two-hadron mass thresholds [639]. Besides, lattice QCD simulations have improved in computational performances and application scopes substantially over the years and will play more and more important roles in our understanding of low-energy QCD [640-646]. Hopefully, one day we can predict low energy QCD phenomena as precise as other parts of the SM.

3 Rare beauty hadron decays

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

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

3.1 Effective field theory for b→s transitions

The effective Hamiltonian describing the quark-level b\to s transitions is given by [648-654]

(3)H_{\mathrm{e}\mathrm{f}\mathrm{f}}(b\to s)=-{\displaystyle \frac{4G_F}{\sqrt{2}}}{V}_{tb}^{\phantom{\ast }}{V}_{ts}^{\ast }{\displaystyle \sum _{i=1}^{10}{C}_i{\mathcal{O}}_i,}

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

(4){\mathcal{O}}_7={\displaystyle \frac{e}{16{\pi }^2}}{m}_b( \overline{s}{\sigma }_{\mu \nu }{P}_{R}b)F^{\mu \nu },

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

(5)\begin{array}{rl}& {\mathcal{O}}_9={\displaystyle \frac{e^2}{16{\pi }^2}}(\overline{s}{\gamma }_{\mu }{P}_{L}b)(\overline{\ell }{\gamma }^{\mu }\ell ),\\ & {\mathcal{O}}_{10}={\displaystyle \frac{e^2}{16{\pi }^2}}(\overline{s}{\gamma }_{\mu }{P}_{L}b)(\overline{\ell }{\gamma }^{\mu }{\gamma }_5\ell ),\end{array}

where P_L=(1-{\gamma }_5)/2 and P_R=(1+{\gamma }_5)/2. Contribution from physics beyond the SM can either alter the values of the Wilson coefficients and/or give rise to new operators that are absent or highly suppressed in the SM, such as the scalar and pseudo-scalar operators

(6){\mathcal{O}}_S={\displaystyle \frac{e^2}{16{\pi }^2}}(\overline{s} P_{L}b)(\overline{\ell }\ell ),{\mathcal{O}}_P={\displaystyle \frac{e^2}{16{\pi }^2}}(\overline{s} P_{L}b)(\overline{\ell }{\gamma }_5\ell ),

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

The Wilson coefficients C_7^{(\mathrm{\prime })} can be probed in radiative b-hadron decays such as B_s^0\to \varphi \gamma and {\Lambda }_b^0\to \Lambda \gamma, C_{9,10}^{(\mathrm{\prime })} probed in semileptonic decays such as B^0\to K^{+}{\ell }^{+}{\ell }^{-} and B^{+}\to K^{\ast 0}{\ell }^{+}{\ell }^{-}, and C_{S,P}^{(\mathrm{\prime })} probed in purely leptonic decays such as B_s^0\to {\ell }^{+}{\ell }^{-}. Recent LHCb results on b\to s transitions are summarised in the remainder of this section.

3.2 Purely leptonic B meson decays

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

(7)\begin{array}{rl}\mathcal{B}(B_s^0\to {\mu }^{+}{\mu }^{-})& =(3.66\pm 0.14)\times {10}^{-9},\\ \mathcal{B}(B^0\to {\mu }^{+}{\mu }^{-})& =(1.03\pm 0.05)\times {10}^{-10},\\ \mathcal{B}(B_s^0\to e^{+}{e}^{-})& =(8.60\pm 0.36)\times {10}^{-14},\\ \mathcal{B}(B^0\to e^{+}{e}^{-})& =(2.41\pm 0.13)\times {10}^{-15}.\end{array}

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

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

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

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

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

In addition to decays to dileptons, the LHCb experiment has also searched for decays of neutral B mesons to four leptons. A recent search using the full Run 1 and Run 2 data sample found no hint of such decays, and upper limits at 90% confidence level for the nonresonant decays are determined to be and [666]. More stringent limits are set for decays involving the J/\psi resonance or a promptly decaying intermediate scalar particle with a mass of 1 \text{GeV/}{c}^2.

3.3 Semileptonic b→sℓ⁺ℓ⁻ decays

Semileptonic b\to s{\ell }^{+}{\ell }^{-} decays provide valuable insight into possible non-SM contributions that affect the Wilson coefficients C_9 and C_{10} of the electromagnetic operators. The presence of hadrons in the final state makes the search for new physics in semileptonic decays more complicated than that in purely leptonic decays. The challenges in hadronic form-factor calculations lead to significant uncertainties in the SM predictions of their decay rates. Fortunately, a number of relatively clean observables that are less affected by the form factors than the total decay rates have been identified, including some special observables in angular distributions and observables for lepton universality test. A comprehensive study of these observables in a series of b\to s{\ell }^{+}{\ell }^{-} processes has been pursued by the LHCb collaboration and the results are summarized below.

3.3.1 Differential decay rates with respect to q²

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

After measuring the branching fraction of the decay B_s^0\to \varphi {\mu }^{+}{\mu }^{-} to be about 3\sigma below the SM expectation value [672], the LHCb experiment further studied its differential branching fraction as a function of q^2. The top left plot in Fig.15 shows the latest results of \mathrm{d}\mathcal{B}(B_s^0\to \varphi {\mu }^{+}{\mu }^{-})/\mathrm{d}{q}^2 obtained using Run 1 data [673], where the J/\psi and \psi (2S) regions are excluded. A puzzle appears in the range . The branching fraction integrated over this range is measured to be (2.88\pm 0.22)\times {10}^{-8}. Currently, the most precise SM prediction for this range is (5.37\pm 0.66)\times {10}^{-8}, obtained from a combination of light-cone sum rule and lattice QCD calculations. A discrepancy of 3.6\sigma is observed. Similar patterns are also seen in the LHCb measurements of differential branching fractions in the decays {\Lambda }_b^0\to \Lambda {\mu }^{+}{\mu }^{-} [674], B^0\to K^{\ast 0}{\mu }^{+}{\mu }^{-} [675], B^{+}\to K^{+}{\mu }^{+}{\mu }^{-}, B^0\to K_{\mathrm{S}}^0{\mu }^{+}{\mu }^{-} and B^{+}\to K^{\ast +}{\mu }^{+}{\mu }^{-} [676], as shown in Fig.15.

3.3.2 Angular distributions

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

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

(8)\begin{array}{rl}& \frac{1}{\mathrm{d}(\mathrm{\Gamma }+\overline{\mathrm{\Gamma }})/\mathrm{d}{q}^2}\frac{{\mathrm{d}}^4(\mathrm{\Gamma }+\overline{\mathrm{\Gamma }})}{\mathrm{d}{q}^2\mathrm{d}\overrightarrow{\mathrm{\Omega }}}\\ & =\frac{9}{32\pi }[{\displaystyle \frac{3}{4}}(1-F_{\mathrm{L}}){\sin }^2{\theta }_K+F_{\mathrm{L}}{\cos }^2{\theta }_K\\ & +{\displaystyle \frac{1}{4}}(1-F_{\mathrm{L}}){\sin }^2{\theta }_K\cos 2{\theta }_{\ell }-F_{\mathrm{L}}{\cos }^2{\theta }_K\cos 2{\theta }_{\ell }\\ & +S_3{\sin }^2{\theta }_K{\sin }^2{\theta }_{\ell }\cos 2\varphi +S_4\sin 2{\theta }_K\sin 2{\theta }_{\ell }\cos \varphi \\ & +S_5\sin 2{\theta }_K\sin {\theta }_{\ell }\cos \varphi +{\displaystyle \frac{4}{3}}{A}_{\mathrm{F}\mathrm{B}}{\sin }^2{\theta }_K\cos {\theta }_{\ell }\\ & +S_7\sin 2{\theta }_K\sin {\theta }_{\ell }\sin \varphi +S_8\sin 2{\theta }_K\sin 2{\theta }_{\ell }\sin \varphi \\ & +S_9{\sin }^2{\theta }_K{\sin }^2{\theta }_{\ell }\sin 2\varphi ],\end{array}

where \mathbf{\Omega } =(\cos {\theta }_K,\cos {\theta }_{\ell },\varphi ), {\theta }_K is the angle between the directions of the K^{+} (K^{-}) and B^0 ({\overline{B}}^0) in the rest frame of the K^{\ast 0} ({\overline{K}}^0) system, {\theta }_{\ell } is the angle between the direction of the {\mu }^{+} ({\mu }^{-}) and the opposite direction of the B^0 ({\overline{B}}^0) in the rest frame of the {\mu }^{+}{\mu }^{-} system, \varphi is the angle between the plane defined by the muon pair and the plane defined by the kaon and pion in the B^0 ({\overline{B}}^0) rest frame. Eight observables can be extracted, including the fraction of the longitudinal polarisation of the K^{\ast 0} meson (F_{\mathrm{L}}), the forward-backward asymmetry of the {\mu }^{+}{\mu }^{-} system (A_{\mathrm{F}\mathrm{B}}), and six other angular coefficients (S_i, i=3,4,5,7,8,9). Using the S_i coefficients, new observables less sensitive to form factor uncertainties are defined, such as P_i^{\mathrm{\prime }}=S_i/\sqrt{F_{\mathrm{L}}(1-F_{\mathrm{L}})} (i=3,4,5) [690].

The latest LHCb results on angular analysis of the B^0\to K^{\ast 0}{\mu }^{+}{\mu }^{-} decay are obtained using data collected in 2011, 2012 and 2016 [689]. The majority of the angular observables are consistent with the SM predictions [677] based on form factors obtained from a combination of light-cone sum rule calculations [667] for low-q^2 regions and lattice QCD calculations [670, 671] for high-q^2 regions. A clear exception is seen with the robust observable P_5^{\mathrm{\prime }} defined using S_5, as shown in Fig.16. The measured values of P_5^{\mathrm{\prime }} in the intervals and are found to be higher than the SM predictions [668, 690] by 2.5 \sigma and 2.9 \sigma, respectively. These results confirm the discrepancy in P_5^{\mathrm{\prime }} observed in an earlier LHCb analysis with Run 1 data [688]. According to model-independent fits using the FLAVIO software package [691], the overall tension with the SM is increased from 3.0 \sigma to 3.3 \sigma. The fits reveal that the current measurements of the angular observables in B^0\to K^{\ast 0}{\mu }^{+}{\mu }^{-} can be accommodated by shifting the real part of the Wilson coefficient C_9 from its SM value by {0.99}_{-0.21}^{+0.25} [689].

Recently, LHCb reported results of angular analysis of the B^{+}\to K^{\ast +}(\to K_{\mathrm{S}}^0{\pi }^{+}){\mu }^{+}{\mu }^{-} decay using Run 1 and Run 2 data [692]. A trend of deviations from the SM predictions in P_5^{\mathrm{\prime }}, similar to that in the isospin partner decay B^0\to K^{\ast 0}{\mu }^{+}{\mu }^{-}, is shown in the left of Fig.17. Meanwhile, a large discrepancy in the measurement of P_2=\frac{2}{3}{A}_{\mathrm{F}\mathrm{B}}/(1-F_{\mathrm{L}}) has also been observed in the region, where the measurement deviates from its SM prediction [677, 690] by 3.0\sigma (Fig.17 right).

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

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

3.3.3 Lepton flavour universality tests

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

(9)R_X\equiv {\displaystyle \frac{\mathcal{B}(B\to X{\mu }^{+}{\mu }^{-})}{\mathcal{B}(B\to X{e}^{+}{e}^{-})}}.

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

(10)R_X={\displaystyle \frac{\mathcal{B}(B\to X{\mu }^{+}{\mu }^{-})/\mathcal{B}(B\to XJ/\psi (\to {\mu }^{+}{\mu }^{-}))}{\mathcal{B}(B\to X{e}^{+}{e}^{-})/\mathcal{B}(B\to XJ/\psi (\to e^{+}{e}^{-}))}},

where \mathcal{B}(B\to XJ /\psi (\to {\mu }^{+}{\mu }^{-}))/\mathcal{B}(B\to XJ/\psi ( \to e^{+}{e}^{-})) is known to be very close to unity [326].

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

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

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

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

in {\Lambda }_b^0\to p{K}^{-}{\ell }^{+}{\ell }^{-}(\ell =e, \mu ) decays using Run 1 and part of Run 2 data [714].

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

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

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

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

3.4 Radiative b→sγ decays

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

(11)H_{\mathrm{e}\mathrm{f}\mathrm{f}}(b\to s\gamma )=- {\displaystyle \frac{4G_F}{\sqrt{2}}}{V}_{tb} V_{ts}^{\ast }(C_7 {\mathcal{O}}_7+C_7^{\mathrm{\prime }}{\mathcal{O}}_7^{\mathrm{\prime }}) ,

where only the leading operator {\mathcal{O}}_7 and its chirality-flipped counterpart {\mathcal{O}}_7^{\mathrm{\prime }} are included. In the SM, the coefficient C_7^{\mathrm{\prime }} is given by C_7^{\mathrm{\prime }}=\frac{m_s}{m_b}{C}_7 due to the chiral V-A structure of the weak interaction, where m_s (m_b) indicates the mass of the s (b) quark. Consequently, the photons emitted in radiative b-hadron (\overline{b}-hadron) decays are predominantly left-handed (right-handed). Amplitudes with right-handed photons, A_R, are suppressed by the ratio m_s/m_b compared with those with left-handed photons, A_L, but could be enhanced in new physics scenarios with right-handed charged current, such as supersymmetric grand unified theories and left-right symmetric models [732-738].

Rich information on photon polarisation can be obtained from time-dependent analysis of B_q^0({\overline{B}}_q^0)\to f_{\mathrm{C}\mathrm{P}}\gamma decays [732, 739, 740], where f_{\mathrm{C}\mathrm{P}} is a CP eigenstate with eigenvalue \eta. The time-dependent decay rates summing over left-handed and right-handed photons are expressed as

(12)\begin{array}{rl}P(t)=& P_0{\mathrm{e}}^{-{\mathrm{\Gamma }}_{q}t}[\cosh (\mathrm{\Delta }{\mathrm{\Gamma }}_{q}t/2)-A^{\mathrm{\Delta }}\sinh (\mathrm{\Delta }{\mathrm{\Gamma }}_{q}t/2)\\ & +\xi C\cos (\mathrm{\Delta }{m}_{q}t)-\xi S\sin (\mathrm{\Delta }{m}_{q}t)],\end{array}

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

(13)\begin{array}{rl}& \mathrm{F}\mathrm{a}\mathrm{v}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}:\phantom{0}A({\overline{B}}_q^0\to f_{\mathrm{C}\mathrm{P}}{\gamma }_L)=a_L{\mathrm{e}}^{\mathrm{i}{\delta }_L}{\mathrm{e}}^{\mathrm{i}{\varphi }_L}\\ & \Rightarrow A(B_q^0\to f_{\mathrm{C}\mathrm{P}}{\gamma }_R)=\eta a_L{\mathrm{e}}^{\mathrm{i}{\delta }_L}{\mathrm{e}}^{-\mathrm{i}{\varphi }_L},\\ & \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{d}:A({\overline{B}}_q^0\to f_{\mathrm{C}\mathrm{P}}{\gamma }_R)=a_R{\mathrm{e}}^{\mathrm{i}{\delta }_R}{\mathrm{e}}^{\mathrm{i}{\varphi }_R}\\ & \Rightarrow A(B_q^0\to f_{\mathrm{C}\mathrm{P}}{\gamma }_L)=\eta a_R{\mathrm{e}}^{\mathrm{i}{\delta }_R}{\mathrm{e}}^{-\mathrm{i}{\varphi }_R},\end{array}

where a_{L(R)}, {\delta }_{L(R)} and {\varphi }_{L(R)} are the size, strong phase and weak phase of A( {\overline{B}}_q^0\to f_{\mathrm{C}\mathrm{P}}{\gamma }_{L(R)} ), respectively. The terms \sin (\mathrm{\Delta }{m}_q) and \cosh (\mathrm{\Delta }{\mathrm{\Gamma }}_{q}t/2) in Eq. (12) arise from interference of the amplitudes of direct decay, A(\overline{B_q^0} \to f_{\mathrm{C}\mathrm{P}}{\gamma }_L ) or A (B_q^0\to f_{\mathrm{C}\mathrm{P}}{\gamma }_R), and the decay via B_q^0-{\overline{B}}_q^0 mixing, p /q A(B_q^0\to f_{\mathrm{C}\mathrm{P}}{\gamma }_L) or q/pA({\overline{B}}_q^0\to f_{\mathrm{C}\mathrm{P}}{\gamma }_R). The mixing-induced observables S and A^{\mathrm{\Delta }} are given by [740]

(14)\begin{array}{rl}& S\approx \frac{2\eta r}{1+r^2}\cos ({\delta }_L-{\delta }_R)\sin ({\varphi }_q-{\varphi }_L-{\varphi }_R),\\ & A^{\mathrm{\Delta }}\approx \frac{2\eta r}{1+r^2}\cos ({\delta }_L-{\delta }_R)\cos ({\varphi }_q-{\varphi }_L-{\varphi }_R),\end{array}

where r\equiv |a_R/a_L|\approx | C_7^{\mathrm{\prime }}/ C_7|, {\varphi }_q is the B_q^0−{\overline{B}}_q^0 mixing phase. The values of S and A^{\mathrm{\Delta }} are expected to be small in the SM due to the suppression by the ratio r\approx m_s/m_b. Since S and A^{\mathrm{\Delta }} are approximately linearly dependent on r, they are sensitive to even a small increase of right-handed photons.

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

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

S_{\varphi \gamma }( \mathrm{S}\mathrm{M})=0.000\pm 0.002,A_{\varphi \gamma }^{\mathrm{\Delta }}(\mathrm{S}\mathrm{M}) =0.047\pm 0.039 .

Using data collected in Run 1, the LHCb collaboration studied the tagged time-dependent decay rates of B_s^0\to \varphi \gamma, which are shown in Fig.21. The mixing-induced observables are measured to be [746]

\begin{array}{rl}& S_{\varphi \gamma }=0.43\pm 0.30\text{(stat)}\pm 0.11\text{(syst)},\\ & A_{\varphi \gamma }^{\mathrm{\Delta }}=-{0.67}_{-0.41}^{+0.37}\text{(stat)}\pm 0.17\text{(syst)},\end{array}

which are in agreement with the SM expectations.

The B^0\to K^{\ast 0}{e}^{+}{e}^{-} decay in the low-q^2 region offers a powerful probe of right-handed new physics. In the vicinity of the photon pole, the decay amplitudes are dominated by contributions from the electromagnetic Wilson coefficients C_7^{(\mathrm{\prime })}. An angular analysis can be performed in a similar way as that in the B^0\to K^{\ast 0}{\mu }^{+}{\mu }^{-} case. In the angular distribution, there are two terms arising from interference of the left-handed and right-handed decay amplitudes that are proportional to C_7 and C_7^{\mathrm{\prime }}, respectively. For small values of r (\equiv |C_7^{\mathrm{\prime }}/ C_7)|), the coefficients of these two terms, denoted A_{\mathrm{T}}^{(2)} and A_{\mathrm{T}}^{\mathrm{I}\mathrm{m}}, are approximately expressed as [738, 747]

(15)A_{\mathrm{T}}^{(2)}\approx r \cos ({\varphi }_L-{\varphi }_R) ,A_{\mathrm{T}}^{\mathrm{I}\mathrm{m}}\approx r\sin ({\varphi }_L-{\varphi }_R) ,

where {\varphi }_L and {\varphi }_R represent the phases of C_7 and C_7^{\mathrm{\prime }}, respectively. Like the mixing-induced observables in B_s^0\to \varphi \gamma, A_{\mathrm{T}}^{(2)} and A_{\mathrm{T}}^{\mathrm{I}\mathrm{m}} depend approximately linearly on r, thus can provide high sensitivity to right-handed currents in the small r region. Using data collected in Run 1 and Run 2, the LHCb collaboration has measured A_{\mathrm{T}}^{(2)} and A_{\mathrm{T}}^{\mathrm{I}\mathrm{m}} to be [748]

\begin{array}{rl}& A_{\mathrm{T}}^{(2)}=0.11\pm 0.10\text{(stat)}\pm 0.02\text{(syst)},\\ & A_{\mathrm{T}}^{\mathrm{I}\mathrm{m}}=0.02\pm 0.10\text{(stat)}\pm 0.01\text{(syst)}.\end{array}

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

\begin{array}{rl}& A_{\mathrm{T}}^{(2)}(\mathrm{S}\mathrm{M})=0.033\pm 0.020,\\ & A_{\mathrm{T}}^{\mathrm{I}\mathrm{m}}(\mathrm{S}\mathrm{M})=-0.00012\pm 0.00034,\end{array}

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

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

(16)A_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}}\propto {\lambda }_{\gamma } \equiv {\displaystyle \frac{|A_L{|}^2-|A_R{|}^2}{|A_L{|}^2+|A_R{|}^2}}\approx {\displaystyle \frac{1-r^2}{1+r^2}}.

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

The LHCb collaboration observed a significantly non-zero up-down asymmetry of the photons in B^{-} \to K^{-} {\pi }^{+}{\pi }^{-} \gamma decays in the range m_{K\pi \pi }=[1.1,1.3 ] \text{GeV/}{c}^2 [751] with respect to the plane defined by the three final-state hadrons in their rest frame. This observation demonstrates that the photons are indeed polarised. However, it is nontrivial to translate the measured asymmetry into a constraint on the polarisation parameter {\lambda }_{\gamma }, due to currently limited knowledge of the structure and decay dynamics of the intermediate resonances involved in this process. A recent theoretical study pointed out that this m_{K\pi \pi } range is dominated by the K_1(1270) resonance and proposed to exploit the charm decay D \to K_{1}e \nu to quantify the hadronic effects in K_1\to K^{-}{\pi }^{+}{\pi }^{-} [752], which can be studied at a future Super \tau-charm factory [753].

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

(17){\displaystyle \frac{\mathrm{d}\mathrm{\Gamma }}{\mathrm{d}\cos {\theta }_p}}\propto 1-{\alpha }_{\Lambda }{\lambda }_{\gamma }\cos {\theta }_p,

where {\theta }_p is the helicity angle of the proton in the \Lambda rest frame with respect to the opposite direction of the photon, {\alpha }_{\Lambda } is the decay parameter of the weak process \Lambda \to p{\pi }^{-}. The photon polarisation parameter has recently been measured to be {\lambda }_{\gamma }={0.82}_{-0.26}^{+0.17}(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}){}_{-0.13}^{+0.04}(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}) [758] by the LHCb experiment using all data from Run 2 and the average of the decay parameter values of \Lambda and \overline{\Lambda } measured by BESIII, {\alpha }_{\Lambda }=0.754 \pm 0.004 [759]. This result is in agreement with the SM predictions from Refs. [760-762]. The LHCb experiment also searched for the decay {\Xi }_b^{-}\to {\Xi }^{-}\gamma using Run 2 data and found no signal [763].

3.5 Other rare decays of beauty hadrons

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

4 CP violation in beauty and CKM parameters

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

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

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

4.1 CKM angle γ

The angle \gamma, defined as \arg [-(V_{ud} V_{ub}^{\ast })/(V_{cd} V_{cb}^{\ast })], is one of the key observables related to the CKM matrix. As can be seen from Fig.22, one of the main limitations of global constraints comes from the angle \gamma. To be noted, the fits have already included recent \gamma measurements from LHCb which improves the sensitivity on \gamma from {14}^{\circ }, established at the era of B-factories, to around 5^{\circ }. Future improvements on the sensitivities on \gamma can be foreseen with the upgrade of LHCb and running of the Belle II experiment. In the following sections, we briefly overview the main developments on \gamma measurements in the past several years from the LHCb experiment.

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

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

4.1.1 GLW and ADS measurements

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

The decay rate of the GLW method is

(18)\begin{array}{r}\mathrm{\Gamma }(B^{-}\to D[\to f_{\mathrm{G}\mathrm{L}\mathrm{W}}]h^{-})\propto 1+r_B^2\\ +2{\kappa }_B{r}_B(2F_{+}-1)\cos ({\delta }_B-\gamma ),\end{array}

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

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

Based on Eq. (18), the angle \gamma can be accessed by measuring the relative decay rate difference for B^{+} and B^{-} mesons. An example of the decay rate difference can be seen directly from the raw yields of the invariant mass distributions of the decays B^{\pm }\to D(\to K^{+}{K}^{-})K^{\pm } as shown in Fig.23. The figures are from the latest LHCb measurements using 9 fb⁻¹ data [792]. By further considering the production asymmetry of B^{+} and B^{-} and detection efficiency difference between K^{+} and K^{-}, the size of the CP violation of the decay can be determined.

The decay rate of the ADS channel has a similar form of

(19)\begin{array}{r}\mathrm{\Gamma }(B^{-}\to D[\to f_{\mathrm{A}\mathrm{D}\mathrm{S}}]h^{-})\propto r_D^2+r_B^2\\ +2{\kappa }_B{r}_B{r}_D{R}_D\cos ({\delta }_D+{\delta }_B-\gamma ),\end{array}

where r_D and {\delta }_D are the average amplitude ratio and phase difference between doubly Cabibbo-suppressed and Cabibbo-favoured D decays. The value of r_D is at similar magnitude as r_B and thus leads to larger CP violation. However, the statistics of the ADS channel is also suppressed. This can be seen from the measurements done by the LHCb experiment using B^{\pm }\to (K^{\mp }{\pi }^{\pm })K^{\pm } decays as show in Fig.24 [792]. The measurements also have four-fold ambiguity, however, as {\delta }_D is not zero, combining with GLW mode gives two-fold ambiguity on determination of angle \gamma.

The coherent factor R_D equals to one for two-body final states and is less than one for multi-body final states to take into account dilutions due to different resonant contributions from Cabibbo-favoured and doubly-Cabibbo-suppressed D decays. The values of r_D, R_D and {\delta }_D can also be determined using the quantum-coherent data from BESIII and CLEO-c experiments. For example, the coherent factor of D^0({\overline{D}}^0)\to K^{+}{\pi }^{+}{\pi }^{-}{\pi }^{-}, R_{K3\pi } is measured by the two experiments to be {0.43}_{-0.13}^{+0.17} [793], which is significantly lower than one. Further binning the phase space according to the variation of strong phase determined from amplitude analysis can improve the sensitivity on \gamma [794], and a first measurement using this approach is recently published [795].

The GLW and ADS measurements have been performed by the LHCb experiment with D decaying mainly into charged final states. However, final states with neutral particles are also studied, either by reconstructing the neutral particles {\pi }^0 or \gamma as in recent study of B^{\pm }\to D{h}^{\pm },D\to h^{\pm }{h}^{{}^{\prime }\mp }{\pi }^0 [796], or in a partially reconstructed method, where {\pi }^0 and \gamma from D^{\ast 0} are not reconstructed [792]. In the second case, the reconstruction efficiency is much higher, however, the sensitivity of \gamma is limited by the background contributions, especially those from B^0\to D^{\ast -}{K}^{+} decays which has similar line-shapes as signal.

4.1.2 Measurements using the GGSZ method

As has discussed in the above section, for multi-body D decays, if CP even fraction F_{+} for self-conjugated decay or coherent factor R for semi-flavour tagged decay is significantly smaller than one, further sensitivity can be achieved by considering the variation of r_D and {\delta }_D over the phase space. One can model the D decays with an amplitude model which provides the information of strong phase {\delta }_D over the phase space. However, this may suffer from large systematic uncertainties due to modelling of amplitude distributions.

An alternative method [782, 783] is to bin the phase space and the effective r_D^i and {\delta }_D^i (or c_i and s_i) in bin i are defined as

(20)c_i+i s_i\equiv R_D^i{\mathrm{e}}^{\mathrm{i}{\delta }_D^i}\equiv {\displaystyle \frac{{\int }_{\mathrm{b}\mathrm{i}\mathrm{n}i}{A}_f(p){\overline{A}}_f(p{)}^{\ast }\mathrm{d}p}{\sqrt{{\int }_{\mathrm{b}\mathrm{i}\mathrm{n}i}|A_f(p){|}^2\mathrm{d}p}\sqrt{{\int }_{\mathrm{b}\mathrm{i}\mathrm{n}i}|{\overline{A}}_f(p){|}^2\mathrm{d}p}}}.

The GGSZ mode refers to decays into self-conjugated final states. BPGGSZ is used when referring to the analysis method using a binned strategy. In the BPGGSZ analysis, c_i and s_i are used instead of effective R_D and {\delta }_D for the benefit of better statistical performance. The values of c_i and s_i can be determined using quantum coherent data collected by BESIII and CLEO-c experiments, where a mixture of D^0 and {\overline{D}}^0 can be achieved from \psi (3770)\to D^0{\overline{D}}^0. As \psi (3770) is a parity-odd state, the D^0 and {\overline{D}}^0 are in quantum-correlated state of (|D^0{\overline{D}}^0 ⟩- | {\overline{D}}^0{D}^0⟩)/\sqrt{2} [797-800]. Using the measured c_i and s_i values, the angle \gamma and strong parameters r_B and {\delta }_B can be extracted. The method has been applied to B^{-}\to D{K}^{-}, D \to K_S^0{\pi }^{+} {\pi }^{-} and D \to K_S^0{K}^{+} K^{-} decays by LHCb [801, 802], where 16 bins and 4 bins are used according to the statistics of the decays, respectively. In order to optimise the sensitivity, binning schemes are chosen according to the strong phase variation over the Dalitz plot considering possible background contamination and efficiency effects. Symmetry between K_S^0{\pi }^{+}( K_S^0{K}^{+}) and K_S^0{\pi }^{-}( K_S^0{K}^{-} ) is also used to increase the sensitivity on c_i and s_i. The binning scheme optimised for the K_S^0{\pi }^{+} {\pi }^{-} decays is shown in Fig.25 [798, 799].

Each bin of the BPGGSZ method can offer constraints to the angle \gamma and to the strong parameters r_B and {\delta }_B. The number of measurements is much more than the number of unknown parameters, thus global production asymmetry and detection asymmetry can be treated as fit variables and related systematic uncertainties are reduced. While for the ADS and GLW methods, production asymmetry and detection asymmetry have to be considered using control channels. The measured yield difference in each bin is shown in Fig.25 for the K_S^0{\pi }^{+} {\pi }^{-} and K_S^0{K}^{+}{K}^{-} decays. Clear CP violation can be found. By combining the statistical power of these bins, the angle \gamma is measured to be ({68.7}_{-5.1}^{+5.2}{)}^{\circ } [802].

4.1.3 Multi-body B decays

Similar to multi-body D decays, multi-body B decays can also be used where r_B and {\delta }_B now is a function of B decay Dalitz plot. However, in this case, one can only use a model to describe different resonant contributions, where some can only be obtained through b\to c process and some can be obtained through both b\to c and b \to u processes and the interference between them gives sensitivity to the angle \gamma. The measurements have been performed by the LHCb collaboration in B^0\to D{K}^{+} {\pi }^{-} decays with D \to K^{+}{K}^{-}( {\pi }^{+}{\pi }^{-}) [803] using 3 fb⁻¹ Run 1 data. However, due to limited statistics, the sensitivity on \gamma is still low. Further measurements with all the data collected by the LHCb experiment will be very interesting.

4.1.4 Time-dependent B_s⁰ decays

The angle \gamma can be measured through time-dependent B_s^0 and B^0 decays where the weak phases extracted are ( \gamma -2{\beta }_s) and (\gamma +2\beta ), respectively. In hadron collider experiments like LHCb, the golden channels are B_s^0\to D_s^{\pm }{K}^{\mp }, B_s^0\to D_s^{\pm }{K}^{\mp }{\pi }^{+}{\pi }^{-} and B_s^0\to D\varphi decays. The time-dependent analyses have been performed for the first two channels [804, 805], while only branching fraction has been measured for the B_s^0\to {\overline{D}}^0\varphi decay [806, 807].

As B_s^0 mixing is involved, a time-dependent analysis is needed to extract CP parameters. The time-dependent decay rate of the B_s^0 decay into a final state f is given by

(21)\begin{array}{rl}& \frac{\mathrm{d}{\mathrm{\Gamma }}_{B_s^0\to f}(t)}{\mathrm{d}t}\propto {\mathrm{e}}^{-{\mathrm{\Gamma }}_{s}t}[\cosh \left(\frac{\mathrm{\Delta }{\mathrm{\Gamma }}_{s}t}{2}\right)+A_f^{\mathrm{\Delta }\mathrm{\Gamma }}\sinh \left(\frac{\mathrm{\Delta }{\mathrm{\Gamma }}_{s}t}{2}\right)\\ & +C_f\cos (\mathrm{\Delta }{m}_{s}t)-S_f\sin (\mathrm{\Delta }{m}_{s}t)],\end{array}

where \mathrm{\Delta }{\mathrm{\Gamma }}_s={\mathrm{\Gamma }}_{B_L}-{\mathrm{\Gamma }}_{B_H} and \mathrm{\Delta }{m}_s=m_{B_H}-m_{B_L} are the decay-width and mass differences between the light (B_L) and heavy (B_H) B_s^0 mass eigenstates and {\mathrm{\Gamma }}_s is the average B_s^0 decay width. For the decays to the CP-conjugated final states, the CP violation parameters C_f, S_f and A_f^{\mathrm{\Delta }\mathrm{\Gamma }} are replaced with C_{\overline{f}}, S_{\overline{f}} and A_{\overline{f}}^{\mathrm{\Delta }\mathrm{\Gamma }}. These CP violation parameters are related with \gamma through

(22)\begin{array}{rl}& C_f=-C_{\overline{f}}=\frac{1-r_B^2}{1+r_B^2},\\ & S_f=\frac{2r_B\sin ({\delta }_B-(\gamma -2{\beta }_s))}{1+r_B^2},\\ & S_{\overline{f}}=\frac{-2r_B\sin ({\delta }_B+(\gamma -2{\beta }_s))}{1+r_B^2},\\ & A_f^{\mathrm{\Delta }\mathrm{\Gamma }}=\frac{-2r_B\cos ({\delta }_B-(\gamma -2{\beta }_s))}{1+r_B^2},\\ & A_{\overline{f}}^{\mathrm{\Delta }\mathrm{\Gamma }}=\frac{-2r_B\cos ({\delta }_B+(\gamma -2{\beta }_s))}{1+r_B^2}\end{array}

following the definitions in Ref. [808]. In the formulae above, we have assumed no CP violation in either the mixing and in the decay amplitude.

The time-dependent CP violation has been measured using 3.0 \mathrm{f}{\mathrm{b}}^{-1} and 9.0 \mathrm{f} {\mathrm{b}}^{-1} data for B_s^0\to D_s^{\pm }{K}^{\mp } [805] and B_s^0\to D_s^{\pm }{K}^{\mp }{\pi }^{+}{\pi }^{-} [804], respectively. The measured values of the angle \gamma, using the world-average value of -2 {\beta }_s, are \gamma =({128}_{-22}^{+17}{)}^{\circ } and \gamma =(44\pm 12{)}^{\circ }, respectively (modulo {180}^{\circ }). The B_s^0\to D_s^{\pm }{K}^{\mp }{\pi }^{+}{\pi }^{-} is complicated due to multiple bachelor particles, and an amplitude analysis is needed. However, as discussed before, multi-body B decays effectively introduce a dilution factor \kappa, as Eq. (22) have five constraints while together with \kappa, there are four unknown variables. The measurement is performed in the same paper in a model-independent way, which leads to \gamma =({44}_{-13}^{+20}{)}^{\circ }. The sensitivity is worse than model-dependent results as expected.

Sensitivity studies with B_s^0\to D\varphi using a time-integrated method has also been performed [809], the expected statistical sensitivity of \gamma is about (8-19 {)}^{\circ } using the 9 fb⁻¹ pp collision data collected by the LHCb experiment. It is pointed out that additional sensitivity on \gamma can be achieved using the longitudinal polarised part of B_s^0\to D^{\ast }\varphi decays with a partially reconstructed technique [807]. Besides, further sensitivities on \gamma can also be obtained by using other quasi-two-body decays in B_s^0\to D{K}^{+}{K}^{-} decays [810].

4.1.5 Combination on γ

A combination of the parameter \gamma was recently performed using all available results of \gamma measurements from the LHCb experiment [784]. The determination of \gamma relies on the inputs from charm decays; on the other hand, precise \gamma measurements and strong parameters of B decays can offer valuable constraints on the charm parameters, which in turn helps constraining the mixing parameters of D^0 mesons. Therefore, the measurements that are sensitive to the charm mixing parameters are also used in the combination. The CKM angle \gamma and charm mixing parameters are simultaneously determined with significant improvements. Here we focus on the \gamma combination results, leaving those on the charm mixing parameters to be discussed later in Section 5.1.

Fig.26 shows the \gamma contributions from different B decays. The sensitivity on \gamma mainly comes from B^{+} decays. For B^0 and B_s^0 decays, where multi-body decays are usually involved, only a few channels have been studied due to the complications in analysis procedures. The small sample sizes and small number of available measurements limit their sensitivity on \gamma. The central values of \gamma determination from B^0 and B_s^0 are around 20 {}^{\circ } higher, which motivates further measurements with the B^0 and B_s^0 mesons to check overall consistency between different B mesons. Special efforts to the B_s^0 meson are well worth, since the B_s^0−{\overline{B}}_s^0 mixing is involved and new physics contributions can easily enter inside.

The 1\sigma contour of the constraint on \gamma and strong parameters of B^{+} decays are shown in Fig.27. Ambiguities on determination of \gamma from two-body D decays can be seen as has been discussed in Section 4.1.1, however, they constrain the parameter space into a very narrow region and together with the unique determination of \gamma from the BPGGSZ channel, \gamma can be precisely obtained. The combined \gamma is found to be \gamma =({65.4}_{-4.2}^{+3.8}{)}^{\circ } and is the most precise determination from a single experiment.

4.2 CKM angle β

The angle \beta, defined as arg(-V_{cd}{V}_{cb}^{\ast } /V_{td}{V}_{tb}^{\ast }), is approximately the phase of V_{td}^{\ast } in the Wolfenstein parameterisation [811]. It enters the decay time distributions of B^0 and {\overline{B}}^0 meson decays due to B^0−{\overline{B}}^0 oscillation. The effective value of \sin (2\beta ), which could have been altered by NP contributions in B^0−{\overline{B}}^0 mixing, can be extracted from the time-dependent CP asymmetries of B^0 decays via b\to c\overline{c}s transitions following the relation

(23)\begin{array}{rl}{A}_{\mathrm{C}\mathrm{P}}(t)& =\frac{\mathrm{\Gamma }({\overline{B}}^0(t)\to f_{\mathrm{C}\mathrm{P}})-\mathrm{\Gamma }(B^0(t)\to f_{\mathrm{C}\mathrm{P}})}{\mathrm{\Gamma }({\overline{B}}^0(t)\to f_{\mathrm{C}\mathrm{P}})+\mathrm{\Gamma }(B^0(t)\to f_{\mathrm{C}\mathrm{P}})}\\ & \approx {\eta }_f\sin (2\beta )\sin (\mathrm{\Delta }mt),\end{array}

where f_{\mathrm{C}\mathrm{P}} is a CP eigenstate with eigenvalue {\eta }_f, and the approximation assumes no CP violation in the mixing or decay. Any significant deviation of the measured \sin (2\beta ) value from the indirect determination of \sin (2\beta ) through a global CKM fit excluding \sin (2\beta ) measurements is a clear sign of NP.

The c\overline{c} pair could appear either in a charmonium meson or in two charmed mesons in the final state. The precision of the \beta is mainly driven by B⁰ decays to charmonium final-states due to their large decay rates and distinct characteristics for identification. The world-average of \sin (2\beta ) of all the charmonium measurements is sin(2\beta )=0.699\pm 0.017 [812]. The LHCb experiment has performed measurements of \sin (2\beta ) in the decays B^0\to J/ \psi (\to {\mu }^{+}{\mu }^{-})K_S^0 [813], B^0\to J/ \psi (\to e^{+}{e}^{-})K_S^0 and B^0\to \psi ( 2S)(\to {\mu }^{+}{\mu }^{-})K_S^0 [814]. The combined value is \sin (2\beta ) =0.760\pm 0.034, the precision of which is already comparable to that of the BaBar result \sin (2\beta ) =0.09\pm 0.03 \pm 0.01 [815] and the Belle result \sin (2\beta )=0.67\pm 0.02\pm 0.01 [816]. An improvement by a factor of two is expected from measurements including LHCb Run 2 data.

The presence of small penguin contributions in b\to c\overline{c}s processes may shift the measured values of \sin (2\beta ) by up to few percent [817]. The decays via tree-level b\to c\overline{u} d transitions, though having smaller signal yields due to the small branching fractions of D decays, are free of the penguin effects and thus theoretically clean. A recent joint analysis of the decay {\overline{B}}^0 \to D{h}^0 with D\to K_{\mathrm{S}}^0{h}^{+} h^{-} by the BaBar and Belle experiments measured \sin (2\beta ) =0.80\pm 0.14 \pm 0.04\pm 0.03 and \cos (2\beta ) =0.91\pm 0.22 \pm 0.09\pm 0.07, which ruled out the other solution of \beta at 7.3\sigma [818, 819]. Analysis of this decay is challenging at LHCb due to the presence of K_{\mathrm{S}}^0 and {\pi }^0 mesons in the final state. On the other hand, the decay {\overline{B}}^0\to D{\pi }^{+} {\pi }^{-} followed by D \to K^{+}{K}^{-}( {\pi }^{+}{\pi }^{-}) only involves charged particle and thus is ideal for LHCb to pursue.

4.3 CKM angle β_s

The angle {\beta }_s, defined as −arg(- V_{cb}{V}_{cs}^{\ast }/V_{tb}{V}_{ts}^{\ast }), is approximately the phase of V_{ts} in the Wolfenstein parameterisation. The effective value of - 2{\beta }_s can be measured in the time-dependent CP asymmetries of B_s^0 decays to CP eigenstates via b\to c\overline{c}s transitions, and is denoted {\varphi }_s^{c\overline{c}s}. In contract to the angle \beta, {\beta }_s is very small. The SM prediction for {\varphi }_s is {\varphi }_s^{\mathrm{S}\mathrm{M}}=-2{\beta }_s=-0.03696\pm 0.0004 [820], which is subject to small corrections due to the neglected penguin contributions in b\to c\overline{c}s decays. Presence of new particles in B_s^0−{\overline{B}}_s^0 mixing diagrams may have a sizeable effect on {\varphi }_s, making it a sensitive probe of physics beyond the SM.

The LHCb experiment has performed measurements of {\varphi }_s^{c\overline{c}s} in the decays B_s^0\to J/ \psi (\to {\mu }^{+}{\mu }^{-})\varphi [821], B_s^0\to J/ \psi (\to {\mu }^{+}{\mu }^{-}){\pi }^{+}{\pi }^{-} [822], B_s^0\to J/ \psi (\to {\mu }^{+}{\mu }^{-})K^{+}{K}^{-} with m(K^{+}{K}^{-})>1.05 \text{GeV/}{c}^2 [823], B_s^0\to \psi (2S)(\to {\mu }^{+}{\mu }^{-})\varphi [824], B_s^0\to D_s^{+}{D}_s^{-} [825] and more recently in B_s^0\to J/ \psi (\to e^{+}{e}^{-})\varphi [826]. A combination of the measurements in B_s^0\to J/ \psi (\to {\mu }^{+}{\mu }^{-})\varphi and B_s^0\to J/\psi (\to {\mu }^{+}{\mu }^{-} ){\pi }^{+}{\pi }^{-} obtained using data taken during 2011−2016 gives {\varphi }_s^{c\overline{c}s}=- 0.042\pm 0.025 rad [821].

The average of all LHCb measurements of {\varphi }_s^{c\overline{c}s} is compared with the ATLAS [827] and CMS [828] results in Fig.28, where 2-dimensional contours in the plane of {\varphi }_s^{c\overline{c}s} versus the B_s^0 decay width difference (\mathrm{\Delta }{\mathrm{\Gamma }}_s) are displayed at 68% confidence level [812]. Note ATLAS and CMS have only performed measurements in B_s^0\to J/ \psi (\to {\mu }^{+}{\mu }^{-})\varphi decays, due to constraints from their trigger systems,. The current world-average value is {\varphi }_s^{c\overline{c}s}=-0.050\pm 0.019 rad [812]. In Fig.28, one can see a good agreement in the {\varphi }_s^{c\overline{c}s} measurements from different measurements. However, some tension is observed for \mathrm{\Delta }{\mathrm{\Gamma }}_s and {\mathrm{\Gamma }}_s. Factors of 2.5 and 1.77 have been applied to scale up the uncertainties of {\mathrm{\Gamma }}_s and \mathrm{\Delta }{\mathrm{\Gamma }}_s in the combination. Further investigations by the relevant experiments are needed to solve this problem. With the uncertainty of {\varphi }_s^{c\overline{c}s} well below its SM value, the study of CP violation in B_s^0 decays enter an era of precision test, and control of the penguin pollution in {\varphi }_s^{c\overline{c}s} using data-driven methods is essential for identification of NP signals in B_s^0−{\overline{B}}_s^0 mixing [829-836]. In this regard, the LHCb experiment has measured CP violation in the penguin-enhanced b\to c\overline{c}d decays B^0\to J/ \psi {\rho }^0 [837] and B_s^0\to J/ \psi {\overline{K}}^0 [838] decays. The measurements are used to estimate the penguin shift of {\varphi }_s^{c\overline{c}s} measured in B_s^0\to J/ \psi \varphi, assuming SU(3) flavour symmetry. The shift is found to be compatible with zero [838], with an uncertain well below the statistical uncertainties of the current {\varphi }_s^{c\overline{c}s} measurements.

Similar CP-violating phases can also be measured in decays of B_s^0 mesons via b \to s\overline{s}s and b\to d\overline{d}s transitions, denoted {\varphi }_s^{s\overline{s}s} and {\varphi }_s^{d\overline{d}s}, respectively. Since these decays are dominated by penguin diagrams with internal top quarks, the phases {\varphi }_s^{s\overline{s}s} and {\varphi }_s^{d\overline{d}s} receive contributions from the decay amplitudes that cancel out the contribution of -2\beta from the B_s^0−{\overline{B}}_s^0 mixing, resulting vanishing values in the SM predictions. Measurements of these quantities can probe NP in these FCNC decays. LHCb has measured {\varphi }_s^{s\overline{s}s} and {\varphi }_s^{d\overline{d}s} in the B_s^0\to \varphi \varphi and B_s^0\to K^{\ast }(892) K^{\ast }(892) decays using data taken during 2011−2016, and the results are {\varphi }_s^{s\overline{s}s}=- 0.073\pm 0.115\pm 0.027 rad [839] and {\varphi }_s^{d\overline{d}s}=-0.10\pm 0.13\pm 0.14 rad [840], respectively, where the first uncertainties are statistical and the second systematic.

4.4 CKM elements V_ub and V_cb

The amplitudes of the CKM matrix elements V_{ub} and V_{cb} are measured through semi-leptonic transitions of b\to u\ell \nu and b\to c\ell \nu. They have been extensively studied previously in B-factories using the so-called exclusive and inclusive methods, which infer to whether a specific decay channel is used or not. The two approaches suffer from different theoretical and experimental uncertainties and offer important cross-checks between each other. In the inclusive measurements, the Heavy Quark Expansion (HQE) is used as {\Lambda }_{\mathrm{Q}\mathrm{C}\mathrm{D}}/m_b is small and the Operator Product Expansion (OPE) calculates non-perturbative contributions involved. For the exclusive measurements, parameterisation of the form factor of the corresponding decay is needed, where inputs are obtained from light-cone sum rules (LCSR) [669, 841–845] or from lattice QCD (LQCD). However, tensions have been found between the inclusive and exclusive results as can be seen in Fig.29 [846]. Efforts from both experimental and theoretical sides are needed to understand the discrepancy.

Unlike B-factories, where full kinematics can be obtained, LHCb can not obtain kinematic information of neutrinos from energy-momentum conservation as it covers only forward region. In addition, background contributions from other b and c hadrons, and also huge combinatorial backgrounds randomly combined from tracks other than signal makes the analyses of semi-leptonic decays very complicated. Despite of these difficulties, the LHCb experiment has successfully measured the ratio of |V_{ub}| and |V_{cb}| using {\Lambda }_b^0\to p{\mu }^{-}{\nu }_{\mu } (q^2>15{\text{GeV}}^2/c^4) and {\Lambda }_b^0\to {\Lambda }_c^{+}{\mu }^{-}{\nu }_{\mu } (q^2>7{\text{GeV}}^2/c^4) [847]. The usage of the control channel {\Lambda }_b^0\to {\Lambda }_c^{+}{\mu }^{-}{\nu }_{\mu } not only cancels out common systematic uncertainties between the two channels, but also offers a global scale needed to determine |V_{ub} | from branching fractions. Using the updated branching fraction measurement of {\Lambda }_c^{+}\to p{K}^{-}{\pi }^{+}, benefiting from the {\Lambda }_c^{+}{\overline{\Lambda }}_c^{-} data collected by the BESIII experiment [848], the ratio is determined to be |V_{ub}|/|V_{cb}|=0.079\pm 0.009.

Using the same approach, the ratio of |V_{ub} | and |V_{cb}| is also determined using B_s^0\to K^{-}{\mu }^{+} {\nu }_{\mu } and B_s^0\to D_s^{-}{\mu }^{+}{\nu }_{\mu } [849]. In the measurement, two q^2 regions are used, q^2>7 {\text{GeV}}^2/c^4 and where the form factors are obtained from LQCD and LCSR, respectively. However, the ratios of |V_{ub}| and |V_{cb}| obtained from the two methods differ significantly, and |V_{ub}|/|V_{ub}|/ |V_{cb}{|}_{q^2>7{\text{GeV}}^2/c^4}=0.0946 \pm {0.0030}_{-0.0025}^{+0.0024}\pm 0.00013\pm 0.0068, where the first uncertainty of each result is statistical, the second systematic, the third due to D_s^{-} branching fraction, and the last one from the form factor. The discrepancy between the two results clearly indicates that more efforts from theoretical side are needed to resolve the tension on |V_{ub} | and |V_{cb}| measurements.

In addition to the ratio between the two CKM matrix elements, the LHCb experiment is also exploring its potential in determine the |V_{cb}| alone using B_s^0\to D_s^{(\ast )-} {\mu }^{+}{\nu }_{\mu } [850]. The branching fraction of B_s^0\to D_s^{(\ast )-}{\mu }^{+} {\nu }_{\mu } is needed to set the global scale for the determination of |V_{cb}|. This is obtained using the control channel B^0\to D^{(\ast )-} \mu {\nu }_{\mu }, where the ratios of

(24)R={\displaystyle \frac{B_s^0\to D_s^{-}{\mu }^{+}{\nu }_{\mu }}{B^0\to D^{-}\mu {\nu }_{\mu }}},

(25)R^{\ast }= {\displaystyle \frac{B_s^0\to D_s^{\ast -}{\mu }^{+}{\nu }_{\mu }}{B^0\to D^{\ast -}\mu {\nu }_{\mu }}},

are determined. The data from LHCb offer q^2 dependence needed to extract |V_{cb} | together with non-perturbative inputs. Using form factor parameterisations from Caprini, Lellouch and Neubert [851] or from Boyd, Grinstein and Lebed [852, 853], the measured values of |V_{cb} | are (41.4\pm 0.6\pm 0.9 \pm 1.2) \times {10}^{-3} and (42.3\pm 0.8\pm 0.9 \pm 1.2) \times {10}^{-3}, respectively, where the first uncertainty is statistical, the second systematic, and the last one due to external inputs such as theoretical inputs on form factors, branching fractions of D_s^{-} or D^{-} decays, B_s^0 lifetime.

4.5 ∆m_d and ∆m_s

The parameters \mathrm{\Delta }{m}_d and \mathrm{\Delta }{m}_s denote the mass differences between the heavy and light mass eigenstate of the B^0 and B_s^0 systems, and define the oscillation frequencies of B^0 mixing and B_s^0 mixing, respectively. Currently, The most precise determination of \mathrm{\Delta }{m}_d comes from the LHCb measurements in semileptonic decays with a D^{-} or D^{\ast -} meson using 3 \mathrm{f}{\mathrm{b}}^{-1} of data. Combining the results obtained in the two decay modes yields \mathrm{\Delta }{m}_d=(0.5050 \pm 0.0021\pm 0.0010) ps⁻¹ [854], where the first uncertainty is statistical and the second systematic. The world-average is \mathrm{\Delta }{m}_d=0.5065 \pm 0.0019 ps⁻¹ [812]. The determination of \mathrm{\Delta }{m}_s is also led by the LHCb experiment. Combining the recent measurements in the decays B_s^0\to D_s^{-} {\pi }^{+} [855] and B_s^0\to D_s^{-} {\pi }^{+}{\pi }^{-}{\pi }^{+} [804] and earlier measurements yields \mathrm{\Delta }{m}_s=17.7656 \pm 0.0057 ps⁻¹.

However, the constraints on the CKM matrix elements provided by the \mathrm{\Delta }{m}_d and \mathrm{\Delta }{m}_s measurements rely on the decay constants and Bag parameters of the the B^0 and B_s^0 mesons, which are obtained from Lattice QCD calculations [856]. The precision of these hadronic parameters is much worse than the experimental precision, thus limits the constraining power of \mathrm{\Delta }{m}_d and \mathrm{\Delta }{m}_s on the CKM global fit. Further improvements in Lattice QCD calculations are eagerly awaited.

4.6 Global fit

The four parameters of the CKM matrix, namely A, \lambda, \rho and \eta in the Wolfenstein parameterization [811], are measured in different processes and some of the key observables have been discussed in the above sections. A global fit is needed to get the best sensitivity and to probe NP effects. The {\chi }^2 value of the fit provides a measure of the overall consistency between the different measurements, while the pull value for each measurement quantifies the difference between the measured value and value predicted by the fit results. Clues for new physics can be identified from large {\chi }^2 or pull values.

The constraining of the CKM matrix is usually illustrated in complex planes using triangles defined using unitarity relations of the CKM matrix. The most commonly quoted CKM triangle corresponds to the relation V_{ud}^{\phantom{\ast }}{V}_{ub}^{\ast }+V_{cd}^{\phantom{\ast }}{V}_{cb}^{\ast }+V_{td}^{\phantom{\ast }}{V}_{tb}^{\ast }=0, where V_{ij} is the CKM matrix element between the quarks of the flavours i and j. The results of the state-of-the-art global fit are shown in Fig.22 provided by the CKMfitter group [820]. Inside, \overline{\rho } and \overline{\eta } [857] are used to ensure the relationship \overline{\rho }+\mathrm{i}\overline{\eta }=- (V_{ud} V_{ub}^{\ast })/(V_{cd} V_{cb}^{\ast }).

Within current precision, different measurements cross on a single point and give an overall consistent picture. However, the argument of matter-antimatter asymmetry suggests that CP violation from sources beyond the CKM matrix may break the consistency. One of the main efforts of the LHCb experiment is to search for such a discrepancy by further improving measurement precision in the beauty and charm sectors.

In the following decade, both the LHCb and Belle II experiments will accumulate much more data to further constrain the CKM matrix. With data collected till 2025, either experiment will be able to reduce the uncertainty of \gamma to around 1.5 {}^{\circ }, and further improve it to 0.3{}^{\circ } after 2030s. Other CKM angles and matrix elements will also be significantly improved. Details on the future outlook can be found in Section 6.2. Together with improvements of other measurements and lattice calculations, NP may be observed from inconsistency between different measurements.

5 Charm mixing and CP violation

Charm physics covers the studies of hadrons containing charm quarks. CP violation in the charm sector is expected to be incredibly small in the SM, of the order \mathcal{O}({10}^{-3}) or less [326]. However, the presence of new physics may enhance the amount of CP violation, which can be probed using the enormously large sample of charmed hadrons at LHCb. Particularly, the study of mixing and CP violation of neutral D mesons can provide unique probes of NP in FCNC transitions in the up-type quark sector, complementary to the study of mixing and CP violation in neutral B and K mesons, which are sensitive to NP in FCNC transitions of down-type quarks. For this reason, this section mainly focuses on results in mixing and CP violation of D^0 mesons from the LHCb experiment.

5.1 Neutral D meson mixing

Similar to neutral K^0 and B_{(s)}^0 mesons, the neutral charmed meson, D^0, can oscillate to its antiparticle partner, {\overline{D}}^0, via the short-distance W^{\pm } exchange or long-distance rescattering diagrams, as shown in Fig.30. This phenomenon of oscillation or mixing can be characterised by the normalised (dimensionless) mixing parameters x and y, defined as

(26)x\equiv {\displaystyle \frac{\mathrm{\Delta }M}{\mathrm{\Gamma }}},y \equiv {\displaystyle \frac{\mathrm{\Delta }\mathrm{\Gamma }}{2\mathrm{\Gamma }}},

where \mathrm{\Delta }M (\mathrm{\Delta }\mathrm{\Gamma }) is the mass (decay width) difference of the heavy and light mass eigenstates, and \mathrm{\Gamma } is the average decay width. Unlike in the case of the K^0 or B_{(s)}^0 system, both x and y in the D^0 system are significantly smaller than unity, thus very large data samples are required to observe D^0 mixing and determine the tiny values of x and y.

Evidence of D^0−{\overline{D}}^0 mixing was first reported by BaBar [858] and Belle [859] in 2007, and later also seen by CDF [860] in 2008. Subsequent measurements by BaBar [861, 862] with different D^0 decay channels provided more evidences of the mixing. The combination of these measurements confirmed the existence of charm mixing with a significance more than 5\sigma. The first observation of D^0−{\overline{D}}^0 mixing in a single measurement was achieved by LHCb [863] in 2012 by using the data taken in 2011 to study the time-dependent ratio of D^0\to K^{+}{\pi }^{-} (doubly Cabibbo-suppressed, DCS) to D^0\to K^{-}{\pi }^{+} (Cabibbo favoured, CF) decay rates. The D^0 candidates are selected from the D^{\ast +}\to D^0{\pi }^{+} decays, where the charge of the pion directly from each D^{\ast +} decay is used to determine the D^0 flavour at its production time.

The D^{\ast +}\to D^0(\to K^{-}{\pi }^{+}){\pi }^{+} process, referred to as the right-sign (RS) process, is dominated by a CF decay, contaminated with a small contribution from the D^0- {\overline{D}}^0 mixing followed by the DCS decay; the D^{\ast +}\to D^0(\to K^{+}{\pi }^{-} ){\pi }^{+} process, referred to as wrong-sign (WS) process, includes contributions from both the DCS decay and the D^0- {\overline{D}}^0 mixing followed by the CF decay. Under the assumption of small mixing and negligible CP violation, the time-dependent ratio of the WS to the RS decay rates, R, is given by [864]

R(t)\approx R_D+\sqrt{R_D}{y}^{\mathrm{\prime }}{\displaystyle \frac{t}{\tau }}+{\displaystyle \frac{{x^{\mathrm{\prime }}}^2+{y^{\mathrm{\prime }}}^2}{4}} {\left({\displaystyle \frac{t}{\tau }}\right)}^2,

where t/\tau is the decay time normalised to the average D^0 lifetime, R_D is the ratio between the DCS and CF decay rates, and x^{\mathrm{\prime }} and y^{\mathrm{\prime }} are the mixing parameters “rotated” by the strong phase difference \delta between the DCS and CF amplitudes: x^{\prime }=x\cos \delta +y\sin \delta and y^{\prime }=y\cos \delta -x\sin \delta. The time evolution of the ratio R is shown in Fig.31. Further studies with larger data samples have also been performed by LHCb [863, 865–867] and the results are summarized in Tab.4.

The measurements in the D^0\to K^{-}{\pi }^{+} decay are sensitive to the normalised decay-width difference y and the sum x^2+y^2 (under the assumption of negligible CP violation), but not to the sign of the normalised mass difference x. One approach to solve this problem is to study the Dalitz distributions of three-body decays. The “golden channel” at LHCb for such studies is the decay D^0\to K_{\mathrm{S}}^0{\pi }^{+}{\pi }^{-}, where the decay to the K_{\mathrm{S}}^0{\pi }^{+}{\pi }^{-} final state proceed mainly via the following three processes with different intermediate resonances: i) the K_{\mathrm{S}}^0{\rho }^0 process with {\rho }^0 \to {\pi }^{+}{\pi }^{-}, which is common for both D^0 and {\overline{D}}^0 mesons; ii) the K^{\ast -}{\pi }^{+} process with K^{\ast -}\to K_{\mathrm{S}}^0{\pi }^{-}, which is a CF decay; and iii) the K^{\ast +}{\pi }^{-} process with K^{\ast +}\to K_{\mathrm{S}}^0{\pi }^{+}, which is either a DCS decay or D^0-{\overline{D}}^0 oscillation followed by a CF decay.

In the Dalitz phase space, the DCS and CF decay amplitudes of the D^0\to K_{\mathrm{S}}^0{\pi }^{+}{\pi }^{-} decay populate the same space and interfere. Therefore, the parameters x and y can be determined by measuring the strong phase difference between the contributing amplitudes in an amplitude analysis, or by importing the average strong-phase difference in regions of phase space obtained by e^{+}{e}^{-} experiments operating at the energy of the \psi (3770) resonance. The latter approach is employed in several LHCb measurements [868-870], and the most recent measurement [870] led to the first observation of a nonzero mass difference between the two mass eigenstates in the D^0−{\overline{D}}^0 system. The results of these measurements using D^0\to K_{\mathrm{S}}^0{\pi }^{+}{\pi }^{-} decays are summarised in Tab.5.

As mentioned in Section 4.1.5, different from the past LHCb \gamma combinations, the recent combination exploited the LHCb measurements that are sensitive to the CKM angle \gamma and to the charm mixing parameters, and the \gamma angle and charm mixing parameters are simultaneously determined [784]. The motivation for the simultaneous combination is as follows:

● The \gamma angle and the strong phase difference between the interfering B decays are now so precisely constrained by the large B-meson samples that the strong phase difference, {\delta }_D^{K\pi }, between the decays D^0\to K^{-}{\pi }^{+} and {\overline{D}}^0\to K^{-} {\pi }^{+} can achieve a precision of about a factor of two better than the previous world average [812]. This improvement can then be used to improve the precision of the charm mixing parameters x and y.

● Due to non-negligible effects originating from charm-meson mixing, a simultaneous combination is needed to obtain an unbiased determination of the \gamma angle and the charm mixing parameters x and y.

In the charm sector, the inputs used in the combination are obtained from the time-dependent measurements of D^0\to h^{+}{h}^{-}, D^0\to K^{+}{\pi }^{-}, D^0\to K^{\pm }{\pi }^{\mp }{\pi }^{+}{\pi }^{-}, and D^0\to K_{\mathrm{S}}^0{\pi }^{+}{\pi }^{-} decays performed by LHCb [790, 866–878]. Fig.32 shows the two-dimensional profile likelihood contours in the x−y plane. The values of x and y, determined in the simultaneous combination, are found to be

x=(0.400{}_{-0.053}^{+0.052}) \mathrm{\%}, y=(0.630 {}_{-0.030}^{+0.033})\mathrm{\%}.

These results provide the most precise determinations of the parameters x and y. Particularly, the precision of y is improved by a factor of two with respect to the current world average [812].

5.2 CP violation

5.2.1 Time-integrated CP violation

The time-integrated CP asymmetry, {\mathcal{A}}_{CP}, in the decay D\to f is dominated by the direct CP asymmetry. Its measurement follows the formula

(27){\mathcal{A}}_{CP}={\mathcal{A}}_{raw}-{\mathcal{A}}_{prod}- {\mathcal{A}}_{det},

where {\mathcal{A}}_{prod} denotes the meson production asymmetry between the c-hadron and its antiparticle, {\mathcal{A}}_{det} represents the detection asymmetry, and {\mathcal{A}}_{raw} is the raw asymmetry between the yields of D\to f and \overline{D}\to \overline{f} decays. Often the difference of CP asymmetries between two different decay processes are measured, which is defined as

(28)\mathrm{\Delta }{\mathcal{A}}_{CP}\equiv {\mathcal{A}}_{CP}(D\to f_1)-{\mathcal{A}}_{CP} (D\to f_2),

where f_1 and f_2 are two different final states with similar topologies. The effects of production asymmetry and CP asymmetries in mixing as well as part of the detection asymmetries on \mathrm{\Delta }{\mathcal{A}}_{CP} are largely cancelled.

Two-body decays of D mesons are particularly interesting due to their super large sample sizes, which are crucial for probing the tiny CP violating effects. The first observation of CP violation in the charm sector was reported by LHCb in 2019 using the Run 2 data [790]. The difference of {\mathcal{A}}_{CP} between the D^0\to K^{+}{K}^{-} and D^0\to {\pi }^{+}{\pi }^{-} decays, \mathrm{\Delta }{\mathcal{A}}_{CP}\equiv {\mathcal{A}}_{CP}(K^{+}{K}^{-})-{\mathcal{A}}_{CP}({\pi }^{+}{\pi }^{-}), was measured with a deviation from zero corresponding to a significance of 5.3\sigma. Tab.6 summarises the \mathrm{\Delta } {\mathcal{A}}_{CP} results of a series of LHCb measurements [790, 871, 872, 879, 880].

Several {\mathcal{A}}_{CP} measurements using two-body decays have been performed by LHCb as well [881-890], and the first evidence of direct CP violation in a specific charm hadron decay was reported in Ref. [890].

While multi-body charm decays often have much smaller sample sizes compared to two-body charm decays, they can provide excellent opportunities for CP violation measurements. The presence of intermediate resonances can lead to large variation of the strong phase difference between the interfering amplitudes, which can lead to sizeable local CP asymmetries. Besides the \mathrm{\Delta }{\mathcal{A}}_{CP} method, several techniques to search for CP violation in multi-body charm decays are exploited by LHCb, including amplitude analysis [891], the binned {\chi }^2 technique [892-896], and an unbinned technique called the energy test [897, 898]. For the latter two methods, model-dependent analyses are eventually required to pin down the source in case significant CP violation were observed.

The binned {\chi }^2 technique computes the distribution of local asymmetries and compare it with a normal distribution to judge if CP violation were observed. An example of binned {\chi }^2 distribution in a Dalitz plot is shown in Fig.33. This method relies on the optimal choice of the binning scheme. Wide bins across resonances can lead to the cancellation of real CP asymmetries within a bin.

The LHCb collaboration has developed a novel unbinned method, energy test [899, 900], to perform model-independent search for CP violation in many-body decays. With this method, a test statistic, T, is defined. For a given data sample, a p-value for the hypothesis of CP invariance is assigned by comparing the observed value of T to the distribution of T obtained from many random permutations of the data. This method has been applied to search for CP violation in decays of charm mesons and beauty baryons. As an example, Fig.34 shows the global test statistic compared with the distribution of the statistic from many random permutations, and the Dalitz plot distribution of significance of local test statistics in D^0\to {\pi }^{-}{\pi }^{+}{\pi }^0 decays. Despite the many efforts made by LHCb and the significant improvements in the measurement precision, no evidence of CP violation in multi-body charm decays has ever been found to date. Tab.7 summarises the searches for direct CP violation in phase space of charm decays by LHCb.

5.2.2 y_CP and A_Γ measurements

The amplitudes of the direct decay of D^0 to a CP eigenstate and the decay after mixing can interfere and lead to indirect CP asymmetry. Its contribution to the time-integrated CP asymmetry is denoted {\mathcal{A}}_{CP}^{ind}.

Due to the D^0−{\overline{D}}^0 mixing, the effective decay width of D^0 decays to a CP-even final state (e.g., f=K^{+}{K}^{-} or {\pi }^{+}{\pi }^{-}), {\mathrm{\Gamma }}_{CP+}, differs from the average decay width \mathrm{\Gamma }. We can define the parameter y_{CP}\equiv {\mathrm{\Gamma }}_{CP+}/\mathrm{\Gamma }- 1 to represent the amount of CP-violation in mixing. The quantity y_{CP} is related to x and y, |q/p|, and \varphi \equiv arg(q\overline{A}/pA),

y_{CP}\approx {\displaystyle \frac{1}{2}}\left(\left|{\displaystyle \frac{q}{p}}\right|+\left|{\displaystyle \frac{p}{q}}\right|\right)y\cos \varphi -{\displaystyle \frac{1}{2}}\left(\left|{\displaystyle \frac{q}{p}}\right|-\left|{\displaystyle \frac{p}{q}}\right|\right)x\sin \varphi .

Only if CP is conserved, y_{CP} is equal to y. The decay rate asymmetry is defined as

A_{\mathrm{\Gamma }}\approx {\displaystyle \frac{1}{2}}\left(\left|{\displaystyle \frac{q}{p}}\right|-\left|{\displaystyle \frac{p}{q}}\right|\right)y\cos \varphi -{\displaystyle \frac{1}{2}}\left(\left|{\displaystyle \frac{q}{p}}\right|+\left|{\displaystyle \frac{p}{q}}\right|\right)x\sin \varphi .

The asymmetry A_{\mathrm{\Gamma }} is related to the indirect CP asymmetry {\mathcal{A}}_{CP}^{ind} through {\mathcal{A}}_{CP}^{ind}=-A_{\mathrm{\Gamma }}.

The quantities y_{CP} and A_{\mathrm{\Gamma }} can be determined by measuring the ratio of the effective lifetimes of D^0 and \overline{D^0} decays to the same CP eigenstate:

(29)y_{CP}= {\displaystyle \frac{2\tau (D^0\to f_{\text{⧸}C\text{⧸}P})}{\tau ({\overline{D}}^0\to f_{CP})+\tau (D^0\to f_{CP})}}-1,

(30)A_{\mathrm{\Gamma }}={\displaystyle \frac{\tau ({\overline{D}}^0\to f_{CP})-\tau (D^0\to f_{CP})}{\tau ({\overline{D}}^0\to f_{CP})+\tau (D^0\to f_{CP})}},

where f_{\text{⧸}C\text{⧸}P} denotes a non-CP-eigenstate, such as K^{-}{\pi }^{+}.

In recent years, LHCb has preformed several measurements of y_{CP} and A_{\mathrm{\Gamma }}, which are summarized in Tab.8 [873–876, 902–904]. A recent study shows that using the average decay width of D^0\to K^{-}{\pi }^{+} and {\overline{D}}^0\to K^{+}{\pi }^{-} decays as a proxy to the average decay width of the neutral charm meson mass eigenstates D_1 and D_2 does not give direct access to y_{CP} but rather corresponds to y_{CP}- y_{CP}^{K\pi } [905], where y_{CP}^{K\pi } is approximately equal to -0.4\times {10}^{-3} [904]. In Ref. [877], an LHCb legacy result of A_{\mathrm{\Gamma }} combined with both D^{\ast } and B flavour tag using 2011−2012 and 2015−2018 data sample is obtained. None of these measurements shows any indication of CP violation in D^0−{\overline{D}}^0 mixing or in the interference between mixing and decay. Fig.35 compares the y_{CP} and A_{\mathrm{\Gamma }} measurements performed by different experiments, and the averages provided by the Heavy Flavour Averaging Group [812]. The world averages are dominated by the measurements by the LHCb experiment.

6 Prospects and summary

6.1 Upgrade plan of LHCb

The physics output discussed in this review shows that LHCb has successfully deepened our understanding of flavour physics with experimental data taken up to the year of 2018. Most of the the key flavour observables are measured to an unprecedented precision, yet it is generally true that the uncertainties are still dominated by statistical fluctuation. To further increase the availability of high-quality collision data, the LHCb detector is currently under a major upgrade [906], known as Upgrade Ia or simply Upgrade I. The installation has almost completed by the end of the second Long Shutdown (LS2) of the LHC, and the upgraded detector is starting to take data in 2022 with an instantaneous luminosity of 2 \times {10}^{33}{\mathrm{c}\mathrm{m}}^{-2}{\mathrm{s}}^{-1}, five times the value achieved so far. Fig.36 shows the plan for LHCb operation after upgrade. LHCb aims to accumulate an integrated luminosity of approximately 23 fb⁻¹ by the end of Run 3 around 2025, and a total of 50 fb⁻¹ by the end of Run 4. Note that during the Long Shutdown 3 (LS3) between Run 3 and 4, intensive work will be done on the machine configuration to prepare the High Luminosity Large Hadron Collider (HL-LHC) [907]. During this period consolidation work (Upgrade Ib) will be carried out at LHCb with only minor change on the detector configuration or performance. In order to fully exploit the HL-LHC potential in flavour physics, the collaboration plans another major upgrade, Upgrade II [908], to enable the detector to operate at luminosity as high as 1.5\times {10}^{34}{\mathrm{c}\mathrm{m}}^{-2}\cdot {\mathrm{s}}^{-1}. This will allow for an integrated luminosity of ~300 fb⁻¹ to be achieved in the lifetime of the (HL-)LHC.

To put the LHCb prospects in context, other players in the field of flavour physics study have to be mentioned. The Belle II experiment [909], the B-factory at the superKEKB [910], has started to take collision data of e^{+}{e}^{-}\to {\rm Y} (4S)\to B\overline{B} since 2018, aiming to collect an integrated luminosity of 50{\text{ab}}^{-1} by 2025. Belle II and LHCb are expected to be competitive in pushing the measurement precision, though with very different systematic uncertainties, of a number of key flavour observables such as the CKM unitarity angles, the Wilson coefficients and the tests on lepton flavour universality. Given the difference in production mechanism and corresponding detector features, their complementarity should be more appreciated. With the beam energy constraints Belle II will be generally good at treating final states with neutral particles (\gamma, {\pi }^0) or missing energy (neutrinos); it has unique advantage in studying the \tau leptons through e^{+}{e}^{-}\to {\tau }^{+}{\tau }^{-} process. With the huge cross section of heavy hadrons, LHCb will have larger yield in most of the final states especially those with charged tracks. The beauty hadrons produced at high-energy pp collisions are highly boosted, hence their decay vertices could be well separated from the primary vertices. With an excellent vertex detector LHCb fully exploits this phenomenon to suppress background for most B signatures to an extremely low level. In addition to B mesons, the studies at LHCb extend to all type of heavy hadrons like B_s^0, B_c^{+} and beauty baryons. Future electron positron colliders proposed primarily for Higgs study, such as CEPC [911] and FCC-ee [912], also plan to operate at Z pole. With yields of B-mesons comparable or higher than Belle II depending on the machine design, future Z factories produce all spectrum of beauty hadrons with large boost and efficient trigger, hence will also contribute to heavy flavour physics with unique advantages. At the HL-LHC era ATLAS and CMS will keep pushing the high-energy frontier by studying the Higgs and searching for signature of new physics beyond the Standard Model, meanwhile the yield of beauty hadrons will be so considerable that measurement of a few key flavour observables can be made precisely, especially those with a pair of muons in their final state. They are designed to perform well at high p_{\mathrm{T}} and central rapidity, perfectly complementing the kinematic range of LHCb. The BESIII experiment [913] will continue to operate for another 10 years [914] and accumulate a sample of charm mesons corresponding to ~20 fb⁻¹, which will not only allow more mixing and CP violation study in the charm sector but also help to reduce the uncertainties related to the charm strong phase in measurement of the CKM \gamma angle [915]. Even larger D\overline{D} samples at the order of \sim {\mathrm{a}\mathrm{b}}^{-1} are expected at proposed future tau-charm factories SCTF [916] and STCF [917].

6.1.1 Upgrade I

The ongoing Upgrade I aims to increase the instantaneous luminosity of LHCb from the current value of 4\times {10}^{32}{\mathrm{c}\mathrm{m}}^{-2}\cdot {\mathrm{s}}^{-1} to 2\times {10}^{33}{\mathrm{c}\mathrm{m}}^{-2}\cdot {\mathrm{s}}^{-1}. Naïvely it would mean a five-fold increase in all signal yield, which will be roughly true for final states with muons. Actually the gain for all-hadronic final state will be more than that due to a major change in the trigger system [918]. The hardware trigger L0, which reduces the data rate from 40 MHz to 1 MHz, will be completely removed after Upgrade I, allowing a more flexible full software trigger. Generally the trigger efficiencies for all-hadronic final states are expected to be doubled, as taken in simulation study of upgrade performances, however this number could vary depending on the individual channel.

The increased pile-up causes much higher combinatorial background and more challenging track reconstruction, therefore the tracking systems [919, 920] have been completely redesigned with higher granularity and better radiation tolerance so as to provide uncompromised tracking performance at higher pile-up. Components of the particle identification systems [921] will be reused as much as possible, yet the readout electronics will be replaced in accordance with the 40 MHz readout rate. As a result of higher luminosity, improved trigger rate and larger number of output channels, the data volume to be treated either in real time or offline will be substantially higher, hence new software infrastructure and computing models have been developed correspondingly [922-924] to ensure physics data to be processed and stored in a timely manner. A new subsystem has been installed to enhance the detector’s capability in fixed target and heavy-ion studies without disturbing the main physics program [925]. The simulation study shows that detector performance after upgrade will be at least as good as before, with improvement at some areas.

6.1.2 Upgrade II

By the end of Run 4 LHCb will have accumulated 50 fb⁻¹ pp collision data, with many subdetectors reaching end of lifetime. Operation at the same condition beyond that point would be less attractive. To fully exploit the HL-LHC potential in flavour physics study, the collaboration proposed Upgrade II towards an integrated luminosity of 300 fb⁻¹ [908]. The physics cases with the luminosity an order of magnitude higher than before HL-LHC time have been studied extensively by the collaboration and summarised in a document in 2018 [664]. A few benchmarks will be discussed below. Note that the HL-LHC baseline design assumes LHCb running condition to be the same as in Run 3, the HL-LHC experts recently released a report on the upgrade feasibility from the machine side [926] showing that possible solution of operating at a luminosity of 1.5\times {10}^{34}{\mathrm{c}\mathrm{m}}^{-2}{\mathrm{s}}^{-1} will allow the target of 300 fb⁻¹ to be met.

The seven-fold increase of luminosity will again impose more technical challenges for the experiment. The expected number of interactions per crossing is around 40, twenty times of current situation (or a hundred times of the LHCb initial design). Fast timing resolution will be required in most subsystems to fight against the combinatorial backgrounds caused by the pile-up. Finer granularity in all tracking detectors is compulsory under much higher multiplicity. Radiation hardness will be more of concern especially for areas close to the beampipe. A daunting amount of 200 Tb of data will be produced every second, and has to be reduced by four orders of magnitude before stored permanently. New subsystems are being proposed in order to extend geometrical acceptance for low-momentum tracks, and to improve K/\pi separation at lower momentum. A lot of development activities have been launched driven by these requirements, while exploiting new technologies in detector and computing. A framework Technical Design Report summarising these activities was released recently [927].

6.2 Physics prospects

Before the HL-LHC or by the end of Run 3, LHCb will have taken 23 fb⁻¹ data, drastically reducing the statistical uncertainties for most of channels compared with current measurements. The expected projections are studied in detail [906] and updated with inputs from experiences in Run I [928, 929]. The physics opportunities in Upgrade II with 300 fb⁻¹ have also been studied [664], which concludes that the energy scale probed by flavour observables will be doubled with respect to pre-HL-LHC era. The sensitivity of a selection of key flavour observables after LHCb upgrades are listed in Tab.9 and illustrated in Fig.37, mostly from Ref. [664] with minor updates when available. Note that Belle II will have completed data taking when LHCb collects 23 fb⁻¹ data. Expected projection from Belle II, ATLAS and CMS are listed for comparison when applicable. A few highlights will be briefly mentioned here.

Rare decays. The decay of B^0\to {\mu }^{+}{\mu }^{-} is not very far from being observed with imminent Upgrade I data, and its branching fraction with respect to B_s^0\to {\mu }^{+} {\mu }^{-} will be measured with 10% uncertainty giving a powerful test of minimal flavour violation. A wide range of studies will be performed in b\to s{\ell }^{+}{\ell }^{-} or b \to d{\ell }^{+}{\ell }^{-} decays with improved precision, so the current hint of discrepancy in R_{K^{(\ast )}} with SM predictions will be confirmed or excluded with confidence. A series of tests on lepton flavour universality can be carried out in b \to c{\ell }^{-}\nu decays. The precision of R_{D^{\ast }} will reach per mille level.

CKM tests. The CKM unitarity triangle will be so precisely determined in the future that discrepancies between various measurement caused by physics beyond Standard Model will be extremely difficult to hide, as shown in Fig.38. The angle \gamma, currently still the least well-known, will be determined with an uncertainty of {1.5}^{\circ } after Run 3, similar as the precision expected from Belle II; the uncertainty will be further reduced to {0.35}^{\circ } after Upgrade II. The expected precision on B_s weak mixing angle {\varphi }_s will be pushed to a few mrad, the same level as the current precision determined indirectly from CKM fit using tree-level measurements.

CP violation in charm. After Upgrade II LHCb will be able to probe CP violation in charm with a sensitivity of \mathcal{O}({10}^{-5}), the only future facility promising in observing indirect CP violation in charm which is predicted to be \mathcal{O}({10}^{-4}) or less in the SM.

Heavy-flavour spectroscopy. LHCb has demonstrated its capability as a general purpose detector in the forward region, and it will be better equipped in this respect after Upgrades. With data sample over an order of magnitude large than currently available, LHCb will systematically update our knowledge on heavy hadron spectroscopy, including less-studied conventional hadrons, such as doubly heavy baryons, and those labelled as exotic nowadays, such as pentaquark and tetraquark states.

Beyond flavour physics. In the high p_{\mathrm{T}} range LHCb could also make contribution complementary to ATLAS and CMS, for instance to the precision determination of the effective weak mixing angle \sin {\theta }_W^2 and the W mass. Measurements of top pair and gauge boson production at LHCb are also crucial to study the poorly known gluon parton distribution functions at high-x range. This is an important study in the QCD, which help to understand the ubiquitous background for any new high-mass states in ATLAS or CMS. LHCb will also push sensitivity in search for dark-photon and long-lived particles predicted in several NP scenarios.

6.3 Summary

This manuscript briefly reviews the recent experimental highlights using data collected with the LHCb detector in its first 10-year operation. They are not only from the flavour physics benchmarks that the experiment was designed for, but also include unexpected discoveries revealing LHCb’s capability as a general purpose detector in the forward region:

● A large variety of new particles are discovered, either filling gaps in conventional heavy hadron spectroscopy or establishing new types of their own, like pentaquarks or tetraquarks;

● Some processes predicted to be extremely rare in the SM are observed, such as the B_s^0\to {\mu }^{+} {\mu }^{-} decay. Precise measurements are performed on semileptonic and radiative FCNC beauty decays, where NP at high energy scale can be probed with promising sensitivity, and tensions with the SM are found in some cases;

● The CKM parameters are determined precisely using multiple approaches in a wide range of final states. The angle \gamma, which was the least known in the unitarity triangle, has been determined with an unprecedented precision of about 4^{\circ };

● Heavy flavour study in the charm sector witnesses a couple of milestones, such as the observation of D^0−{\overline{D}}^0 mixing in a single measurement and observation of non-zero mass difference between the two mass eigenstates in the D^0- {\overline{D}}^0 system. Precision in probing the CP violation in charm keeps pushing forward.

This is by far not a complete list [4], and many interesting topics studied at LHCb are not covered due to limited space. With the Upgrade I detector in place, the LHCb experiment has resumed operation in 2022 and will continue to take data at a higher luminosity while preparing for the future Upgrade II. New exciting physics results are expected, which will continue to shape the landscape of heavy flavour physics and beyond.

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