Experimental determination of the CKM matrix
In the development of particle physics describing matter at the smallest distances, it has proved possible not only to understand the structure of the proton and the neutron in nuclei as made of two types, or flavours, of quarks (\(u\) and \(d\)) but also to produce heavier flavours of quarks (\(s\), \(c\), \(b\) and finally the heaviest quark \(t\)) in more and more energetic collisions. The latter quarks could not be identified easily in the “ordinary” matter around us as they decay very quickly in lighter quarks (leading back to \(u\) and \(d\) quarks).
The study of these quarks showed the existence of three fundamental interactions among them at the subatomic level. Electromagnetism has been known since the nineteenth century, but two subatomic interactions play also an important role : the strong interaction binds the quarks into the observed hadrons (like the proton and the neutron) whereas the weak interaction is responsible for the decay of a quark of a given flavour into a quark of another flavour (leading to the observed \(\beta\) decays of radioactive nuclei). These interactions are mediated through the exchange of different gauge bosons, namely the photon (electromagnetism), the gluons (strong interaction) and \(W^\pm\) and \(Z^0\) bosons (weak interaction).
The weak interaction does not only interact with quarks but also with leptons: the electron \(e\) and two other, heavier, flavours called muon (\(\mu\)) and tau (\(\tau\)), as well as their partners the neutrinos (\(\nu_e\), \(\nu_\mu\) and \(\nu_\tau\)). It turns out that all the quarks and leptons discovered can be organised in three families or generations. The first generation contains the \(u\) and \(d\) quarks as well as the electron and the neutrino \(\nu_e\). The second and third generations can be seen as heavier copies of the first generation, with the same sensitivities to the three fundamental interactions, but different masses. Finally, each particle is associated with an antiparticle, with the same mass but opposite charges dictating their sensitivity to the three interactions (for instance, an antiparticle has an opposite electric charge to its partner). The resulting theory of the three interactions and generations (which also includes the Higgs boson) is the Standard Model (SM) of particle physics.
The detailed study of the weak interaction has shown that the decay from one quark flavour into another one involves quantum states that are superpositions of the three quark flavours sharing the same electric charge (\(u,c,t\) or \(d,s,b\)). This mixing, which is possible thanks to the quantum nature of the particles involved, is described through the Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix (Cabbibo, N. (1963), Kobayashi, M. and Maskawa. T (1973)). Moreover it has been realised that this matrix contained the single source of observed differences between particles and antiparticles, namely CP violation, in the quark sector. The determination of CKM mixing matrix is therefore a central question to validate important aspects of the SM (dynamics of quark transitions, amount of matterantimatter asymmetry), but also a tool to identify potential inconsistencies in this picture which could be the footprints of New Physics, beyond the Standard Model, occurring at higher energies which have not been explored yet.
Contents 
History
The SM has been built progressively through the identification of the elementary fermions (quarks and leptons) and their interactions (electromagnetic, strong and weak), which are mediated by the gauge bosons (photon, gluon and \(Z^0\) and \(W^{\pm}\), respectively). In the 1960s, only three types of quarks (\(u, d, s\)) were known. Rates of weak transitions involving the strange \(s\) quark were observed to be different from the ones involving \(u\) and \(d\) only, for instance in the decays of charged kaons (\(u\)\(\overline(s)\) or \(\overline(u)\)\(s\)) and pions (\(u\)\(\overline(d)\) or \(\overline(u)\)\(d\)) to a muon and a neutrino. In order to account for this, it was assumed that the interaction eigenstates relevant for the weak interactions would be combinations of the d and \(s\) mass eigenstates (Cabbibo, N. (1963)). This effect is known as quark or flavour mixing and it could indeed happen between the \(d\) and \(s\) states since they have the same electrical charges, and more generally the same quantum numbers. Quark mixing can actually be described by the following unitary mixing matrix : \[\left(\begin{array}{c} d'\\ s'\\ \end{array} \right)=\left( \begin{array}{lcr} \cos\theta & \sin\theta \\ \sin\theta & \cos\theta \end{array} \right)\left(\begin{array}{c} d\\ s\\ \end{array} \right)\] where \(d\) and \(s\) denote the mass eigenstates, \(d'\) and \(s'\) the weak interaction eigenstates and the Cabibbo angle \(\theta\) is introduced. This \(2\times 2\) matrix can in principle be complex but not all the parameters might carry physical information, since global phases of the quark wave functions are not observable.
Phase redefinitions 
In quantum mechanics, the global phase of the wave function of a particle does not carry physical information. This result holds also in quantum field theory, which is the based for the current description of particles as excitations of quantum fields. One can thus perform global phase rotations, for instance of the field of the \(d\) quark: \(d \to d e^{i\theta} \) (referred to as phase redefinition) without modifying the physics involved. In the case of the mixing matrix, one can use both these phase redefinition and the unitary condition (\(UU^{\dagger}=U^{\dagger}U=1\), where \(U^\dagger\) indicates the Hermitian conjugate) to count the actual physical parameters describing this matrix. 
After performing phase redefinitions removing these spurious phases, this matrix becomes real and has a single free (real) parameter, the Cabibbo angle. With such a real mixing matrix, \(CP\) is conserved in weak interactions (although charge conjugation and parity were each maximally broken), i.e. the amplitude of a generic process \(a \to b\) and of its \(CP\)conjugate \(\overline{a}\to\overline{b}\) mediated by the weak interaction involve the same CKM matrix element. See A. Buras’ article for a detailed review of CP violation in electroweak interactions.
Mass eigenstates and interaction eigenstates 
The particles described in the Standard Model are quantum objects. Their propagation over time in the vacuum (without interaction) is governed by a Hamiltonian (“free propagation”). The states with a simple evolution over time are obtained by diagonalising the Hamiltonian and identifying eigenstates and eigenvalues corresponding to the particles propagating freely with welldefined masses. These mass eigenstates are appropriate to describe particles propagating freely without interactions.
The situation can change in the presence of interactions involving these particles. These interactions can be expressed in terms of the mass eigenstates. However, due to the quantum nature of the problem, one expects in general that these interactions involve actually superpositions of the mass eigenstates. These linear combinations of mass eigenstates, which are called interaction eigenstates, provide a very simple description of the interactions. The two types of eigenstates can be seen as two different bases to describe the same set of particles, related by a (unitary) rotation. 
The decay of neutral kaons \(K^0\) (\(d\)\(\overline(s)\) or \(\overline(d)\)\(s\)) to a pair of muons can not occur at tree level in the Standard Model, they can only occur via more complex processes such as the box diagrams shown in Figure 1 and Figure 2. They are also known in the litterature as flavour changing neutral currents (FNCN). This decay was still observed to be much smaller than predicted by this framework. This was explained by the prediction of the existence of a fourth quark (\(c\)). In this way two Feynman diagrams could connect the initial and final dimuon state, involving the exchange of \(W\) boson together with either an intermediate u quark or a \(c\) quark, as shown in Figure 1 and Figure 2. Taking into account the mixing described by the Cabibbo matrix, the two amplitudes cancel largely when computing the total decay probability, which explains the small observed rate. This is known as the GlashowIliopoulosMaiani mechanism after the physicists who proposed it, and represented a confirmation of the quark mixing mechanism.
A kaon experiment at Brookhaven National Laboratory observed in 1964 the decay of \(K^0_L\) mesons into a pair of pions (Christenson, J H et al. (1964)). This was not possible if \(CP\) was obeyed by weak interactions, as the initial \(K^0_L\) and final states would have opposite \(CP\) (since \(CP(K^0_L)=1\) or odd and \(CP(\pi\pi)=+1\) or even) and this quantity would not be conserved in the decay. This observation was thus interpreted as the proof of \(CP\) violation in weak processes. This could be accommodated using Kobayahi and Maskawa’s work on the implications of the existence of a third family of quarks. The mixing between mass states of the three quark generations yielding the weak interaction eigenstates is now described by a \(3\times3\) unitary matrix, leading to the following Lagrangian: \[\tag{1} \mathcal{L}_{W^{\pm}} =  \frac{g}{\sqrt{2}}\left[\overline{U}_{i}\gamma^{\mu}\frac{1\gamma^5}{2}\left(V_{\rm CKM} \right )_{ij} D_{j} W_{\mu}^{+} + \overline{D}_{j} \gamma^{\mu}\frac{1\gamma^5}{2}\left(V^{*}_{\rm CKM} \right )_{ij} {U}_{i} W_{\mu}^{}\right]\] where \(g\) is the electroweak coupling constant, \(i,j\) are generation indices, \(U=(u\,c\,t)\), \(D=(d\,s\,b)\) and \(V_{\rm CKM}\) is the unitary CabibboKobayashiMaskawa (CKM) matrix: \[\tag{2} V_{\rm CKM} = \left( \begin{array}{lcr} V_{ud} & V_{us} & V_{ub} \\ V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb} \end{array} \right).\] Taking into account again phase redefinitions, this unitary matrix can be described in terms of the four Wolfenstein parameters \(A,\lambda,\bar\rho,\bar\eta\). The latter parameter amounts to a phase that is responsible for all \(CP\)violating processes in the quark sector within the SM. Indeed, in the case of three families, the phase redefinitions do not allow to remove all the phases involved. As shown in equation (1), the amplitude of a process \(a \to b\) and of its \(CP\)conjugate \(\overline{a}\to\overline{b}\), are driven by a CKM matrix element and its complex conjugate, which are different if \(\overline\eta\neq 0\), leading to \(CP\) violation.
Expanding the CKM matrix in powers of \(\sin\theta\simeq\lambda\simeq 0.22\) up to \(\lambda^6\), one obtains: \[\tag{3} V_{\rm CKM}= \left( \begin{array}{ccc} 1\frac{1}{2}\lambda^{2}\frac{1}{8}\lambda^{4} & \lambda & A\lambda^{3}\left(\bar\rho i\bar\eta\right) \\ \lambda +\frac{1}{2}A^{2}\lambda^{5}\left[12(\bar\rho +i\bar\eta)\right] & 1\frac{1}{2}\lambda^{2}\frac{1}{8}\lambda^{4}(1+4A^{2}) & A\lambda^{2} \\ A\lambda^{3}\left[1(\bar\rho +i\bar\eta)\right]~ & ~A\lambda^{2}+\frac{1}{2}A\lambda^{4}\left[12(\bar\rho +i\bar\eta)\right]~ & ~1\frac{1}{2}A^{2}\lambda^{4} \end{array} \right),\] where the diagonal terms describing the transitions between quarks of the same generation are \(\mathcal{O}(1)\) and the offdiagonal elements related to transitions between different generations get smaller for higher mass states. This highly predictive framework motivated many experiments to try to measure all the parameters of the CKM matrix using different techniques and decay modes.
For what concerns the terms involving the first two quark families, many dedicated experiments constrained the CKM parameters with a very good accuracy from the 1970s till now as shown in section [2]. Most of the experimental activities in the past decades concerned the \(b\) sector, since the third generation was the least explored. The following table presents a summary of their main characteristics. We will discuss in the following the various sources of information on the CKM matrix obtained from these experiments (DescotesGenon, S and Koppenburg, P (2017)).
Experiment  Collisions  Energy  Period 

SLD  \(e^{+}e^{}\)  \(Z^0\) mass  1990s 
LEP  \(e^{+}e^{}\)  \(Z^0\) mass  1990s 
ARGUS  \(e^{+}e^{}\)  \(\Upsilon(4S)\) mass  1980s 
CLEO  \(e^{+}e^{}\)  \(\Upsilon(4S)\) mass  1980–2000 
Belle  \(e^{+}e^{}\)  \(\Upsilon(4S)\) mass  2000s 
BaBar  \(e^{+}e^{}\)  \(\Upsilon(4S)\) mass  2000s 
CDF  \(p\overline{p}\)  2  1985–2011 
LHCb  \(pp\)  7–13  2010–2018 
Measurement of \(V_{ud}, V_{cs}, V_{us}\) and \(V_{cd} \)
These terms describe the transitions between the quarks of the two lightest families and form the original \(2 \times 2\) Cabibbo matrix. As explained in section [3], in the complete \(3 \times 3\) CKM matrix, a \(CP\) violating phase appears, which can contribute also to these terms as shown in equation (3). Experimentally, it is however observed that \(CP\) violation is very small in these CKM matrix elements compared to the \(CP\)conserving part. Leptonic and semileptonic transitions between hadrons containing \(u\), \(d\), \(s\) and/or \(c\) quarks are exploited to measure these elements. The amplitudes for the corresponding branching ratios are generally expressed as the product of a CKM matrix element with a hadronic parameter describing the hadronisation of the initial and final quarks into hadrons (decay constants and form factors for leptonic and semileptonic decays, respectively). The latter are generally obtained from lattice QCD simulations. Additional electromagnetic corrections are added, when known (mostly for leptonic decays of light mesons).
Nonleptonic, semileptonic and leptonic decays 
The strong interaction that binds quarks into hadrons prevents them from ever escaping and being observed isolated. They are always observed inside hadrons, either baryons (typically 3 quarks) or mesons (typically a quark and an antiquark). On the contrary, leptons (electron, \(\tau\), \(\mu\) and their neutrino counterparts) can be produced and observed isolated.
It is customary to separate weak decays of quarks in terms of the presence (or the absence) of hadrons and leptons in the final state of the reaction:

Decay constants, form factors, bag parameters 
The equations of the strong interaction given by Quantum Chromodynamics (QCD) have not been solved analytically for energies below a few GeV.
Indeed, the coupling constant of the strong interaction between quarks and gluons becomes so large that perturbative approach does not work well (see also the article on asymptotic freedom). This means that the hadronisation process that leads from the quarks of the SM to the hadrons observed experimentally is very difficult to assess quantitatively. One can however isolate this part of the process when evaluating the decay probability of a hadron due to a specific quark transition. The resulting quantity, describing the lowenergy QCD part of the process, has different names and structures depending on the process. For instance, leptonic decays lead to decay constants, semileptonic decays to form factors, whereas the transition of a neutral meson into its antimeson is described through socalled bag parameters. All these quantities can be evaluated through a numerical resolution of QCD equations putting spacetime on a finite grid, following a method called lattice QCD requiring a lot of computing power. 
The most precise measurements of \( V_{ud} \) are obtained from the decay rates of nuclei experiencing superallowed \(\beta\) decays. In these favoured nuclear transitions, the wave function of the entire nucleus is left unchanged since these decays involve no change in angular momentum nor parity. These processes provide thus clean theoretical predictions, can be evaluated by precise calculations that do not require strong assumptions or approximations, allowing for a very precise determination of \(V_{ud}\). To date, the halflives of 14 superallowed \(\beta\) decays have been measured and the average value of \(V_{ud}\) is found to be (Particle Data Group, Tanabashi, M et al. (2018)): \[V_{ud} = 0.97420 \pm 0.00021,\] where the error is dominated by theoretical uncertainties from nuclear Coulomb distortions and radiative corrections. This CKM matrix element is also accessible from the measurement of the neutron lifetime, although this determination is limited by the knowledge of the ratio of the axial vector and vector (both states with spin 1 but even and odd parity, respectively) couplings and exhibits inconsistent values from different experiments. The analogous transition to \(\beta\) decays in the pion sector, known as the pion \(\beta\) decay, \(\pi^{+} \to \pi^{0} e^{+} \nu_{e}\), is free of nuclearstructure corrections and provides a stringent test of weak decays. The PIBETA experiment used this transition to extract a measurement of \(V_{ud}\) with a precision of \(0.3\%\) (Počanić, D et al. (2004)).
Semileptonic and leptonic kaon decays are used to measure \(V_{us}\), including charged and neutral \(K \to \pi \mathcal{l} \nu_{\mathcal{l}}\) decays and the leptonic \(K^{+} \to \mu^{+} \nu_{\mu}\)decay, with the main limitation arising from the knowledge on the form factors and decay constant, respectively. Including the latest results from the KLOE experiment, the average from these measurements provides (Particle Data Group, Tanabashi, M et al. (2018)): \[V_{us} = 0.2243 \pm 0.0005.\] This element can be also extracted from hyperon decays and hadronic \(\tau\) decays like \(\tau^{} \to K^{} \nu_{\tau}\) measured at LEP as well as by the Belle and BaBar collaborations, but with a larger uncertainty. In all the cases theoretical input on the relevant hadronic quantities (decay constants or form factors) is needed and generally taken from lattice QCD simulations to extract \(V_{us}\).
Analogously, the determination of \(V_{cd}\), is currently based on leptonic and semileptonic charm decays, namely \(D \to \pi e^{} \overline{\nu_{e}}\) and \(D^{+} \to \mu^{+} \nu_{\mu}\), explored by the CLEOc, Belle, BaBar and BESIII collaborations, and the relevant lattice QCD form factors. Earlier measurements were obtained from neutrino scattering data, from the difference in the ratio of doublemuon production (\(\nu_{\mu} + N \rightarrow \mu + c \rightarrow \mu^{+}\mu^{} + X\)), which proceeds through charm production, and singlemuon production (\(\nu_{\mu} + N \rightarrow \mu + X\)) in neutrino and antineutrino beams, at the CDSHS, CCFR and CHARM II experiments. The current world average gives (Particle Data Group, Tanabashi, M et al. (2018)) \[V_{cd} = 0.218 \pm 0.004,\] dominated by the measurements from semileptonic decays, whose precision is limited by the theoretical uncertainty of the form factors.
Also \(V_{cs}\) can be obtained from semileptonic \(D\) decays like \(D \to K \mathcal{l} \nu_{\mathcal{l}}\)and leptonic \(D_{s}^{+}\)decays like \(D_{s}^{+} \to \mu^{+} \nu_{\mu}\) or \(D_{s}^{+} \to \tau^{+} \nu_{\tau}\) and lattice QCD form factors or decay constants. The Belle, CLEOc, BaBar and BESIII experiments have measured these decays with precision, leading to the average (Particle Data Group, Tanabashi, M et al. (2018)) \[V_{cs} = 0.997 \pm 0.017,\] where the uncertainty is dominated by the experimental precision for leptonic decays and by the theoretical knowledge of the form factors for semileptonic decays. The tagged measurement of \(W^{+} \to c\overline{s}\) from the DELPHI experiment gives also a direct determination of \(V_{cs}\), far less precise than leptonic and semileptonic decays of charm hadrons.
Measurement of \( V_{ub}, V_{cb}\) and \(V_{tb}\)
Similarly to the matrix elements of the first two generations, the moduli of \(V_{cb}\) and \(V_{ub}\) can be accessed through the semileptonic \(b \to (u,c)\mathcal{l}\nu_{\mathcal{l}}\) decays \((\mathcal{l}=e,\mu)\). A longstanding discrepancy exists between the determinations obtained from exclusive decays and from inclusive modes, which are treated with different approaches.
In the case of \(V_{cb}\), one can first use the inclusive decay \(B\to X_c\mathcal{l}\nu_\mathcal{l}\) (\(X_c\) denoting all final states with a charm quark). Using the tool of Operator product expansion (OPE), one can express the decay rate as the product of \(V_{cb}\) by a series in \(1/m_b\) and \(1/m_c\), with coefficients that can be determined experimentally. This is obtained by considering moments of the differential branching ratio of \(B\to X_c\mathcal{l}\nu_\mathcal{l}\) with respect to the leptonic or the hadronic invariant mass. These coefficients can be determined together with \(V_{cb}\) through a global fit to experimental measurements, yielding the socalled inclusive value of \(V_{cb}\).
One can also consider exclusive decays. There are determinations of the \(B\to D\mathcal{l}\nu\) form factors based on lattice QCD that provide the normalisation at momentum transfer \(q^2=0\) (where the momentum transfer \(q=p_Bp_D\) is the diffrence of the \(B\) and \(D\) 4momenta). This normalisation is needed to analyse the experimental measurements which yield the product of the vector form factor at \(q^2=0\) by \(V_{cb}\). The situation is less satisfying for \(B\to D^*\mathcal{l}\nu_\mathcal{l}\). On the experimental side, one of the main issues comes from the existence of a background \(B\to D^{**}\mathcal{l}\nu_\mathcal{l}\) of wide charm resonances which is not very well understood currently. On the theoretical side, due to the lack of a complete lattice QCD determination of the form factors involved, heavyquark effective theory (HQET) is used to simplify the expression of the form factors and to constrain their dependence on the lepton energy. The HQET approach starts from the limit where both the \(b\) and the \(c\) quarks are considered as very heavy and expands the form factors in powers of \(1/m_b\) and \(1/m_c\). The resulting parametrisation (called CLN parametrisation) of the form factors depends only on a few coefficients that can be estimated using dedicated theoretical methods (e.g. sum rules). The values of \(V_{cb}\) extracted from data from \(B\) factories using the CLN parametrisation tend to disagree with the inclusive determination described above. The accuracy of the CLN parametrisation has been questioned recently: it is possible to resort to a more general parametrisation of the form factors (called BGL parametrisation) and fit this expression to the decay rate obtained from \(B\)factories. However, the fits of Babar and Belle data on exclusive decays to CLN and BGL parametrisations turn out to provide similar values for \(V_{cb}\). The agreement between the various extractions remain thus still under debate, with the current world averages (Particle Data Group, Zyla, P. A. et al. (2020)) \[\begin{aligned} V_{cb} &=& (42.2 \pm 0.8) \times 10^{3}\ ({\rm inclusive}) \\ V_{cb} &=& (39.5 \pm 0.9) \times 10^{3}\ ({\rm exclusive}) \end{aligned}\]
In the case of \(V_{ub}\), one can also use either exclusive or inclusive measurements to extract the CKM matrix element. The exclusive determination benefits from lattice QCD computations for the vector form factor of the decay \(B\to \pi\mathcal{l}\nu\), which can be combined with measurements of the differential decay rate.
The inclusive determination is more challenging. The full decay rate cannot be accessed, because a cut in the lepton energy must be performed to eliminate the huge \(b\to c\mathcal{l}\nu_{\overline{\mathcal{l}}}\) background. The OPE expansion in \(1/m_b\) must be modified, introducing poorly known shape functions describing the \(b\) quark dynamics in the \(B\) meson. They can be constrained partly from \(B \to X_s\gamma\), with some questions concerning the convergence rate of the series in \(1/m_b\). The current world averages are: \[\begin{aligned} V_{ub} &=& (4.49 \pm 0.15^{+0.16}_{0.17} \pm 0.17) \times 10^{3}\ {\rm (inclusive}) \\ V_{ub} &=& (3.70 \pm 0.10 \pm 0.12) \times 10^{3}\qquad ({\rm exclusive}) \end{aligned}\] The element \(V_{ub}\) can also be determined from the leptonic decay \(B^\to\tau^\nu_{\overline{\tau}}\) which has been studied at Bfactories, favouring values in agreement with the average of inclusive and exclusive determinations. The measurement of this leptonic decay is rather challenging, as it requires a very good understanding of \(\tau\) decays for their reconstruction and the elimination of a large set of significant backgrounds.
These determinations, which are essentially dominated by systematic uncertainties related to hadronic inputs, have thus led to a longstanding discrepancy between inclusive and exclusive determinations for \(V_{ub}\) and \(V_{cb}\). Currently, global fits use averages of both kinds of determination as inputs, and their outcome favours exclusive measurements for \(V_{ub}\) and inclusive measurements for \(V_{cb}\).
In addition, the LHCb experiment recently used baryon decays for the first time (LHCb Collaboration, Aaij, R et al. (2015)). The decay rates of \(\Lambda_b \to p\mu^{}\overline{\nu_{\mu}}\) and \(\Lambda_b \to \Lambda_c\mu^{}\overline{\nu_{\mu}}\) are compared to determine the ratio \(V_{ub}/V_{cb}\), using the available lattice QCD estimates of the different form factors involved. This determination is rather difficult. On one hand, it requires a good experimental knowledge of \(\Lambda_b\) templates, which describe the shape of the kinematics of the decay and are determined from simulations. On the other hand, it needs an accurate theoretical knowledge of a large number of form factors and their correlations from lattice QCD. This explains the rather large uncertainties attached to this determination. However, it provides an interesting alternative determination of the CKM matrix elements \(V_{ub}\) and \(V_{cb}\), affected by different types of uncertainties compared to the determinations described above. Figure 3 depicts the overall situation, including the constraints from inclusive and exclusive determinations of \(V_{ub}\), \(V_{cb}\), and \(V_{ub}/V_{cb}\).
Finally, the CKM element \(V_{tb}\) can be obtained via a direct measurement from the cross section for single top quark production. The combination of Tevatron and LHC data yields \(V_{tb}=1.019±0.025 \)(Particle Data Group, Tanabashi, M et al. (2018)), which is not competitive with the very accurate indirect determination of this element within the SM, where the rest of the constraints on the CKM parameters are combined with the unitarity of the CKM matrix. Less stringent constraints on \(V_{tb}\) can be obtained from the ratio of branching ratios \({\rm Br}(t\to W b)/{\rm Br}(t\to W q)\) and from LEP electroweak precision measurements.
Measurement of neutral meson mixing: \(\epsilon_K, V_{td}, V_{ts}\)
In the Standard Model, neutral mesons with a given flavour content can mix with their antiparticles through \(\Delta F=2\) box diagrams, where \(F\) is the flavour of the heavier quark, with two \(W\) bosons being exchanged, involving therefore products of the CKM matrix (see Figure 4 for an example in the case of \(B^0\) and \(B_s^0\)). It turns out that the mixing of charmed meson \(D^0\) (\(c \overline{u}\)) into its antiparticle cannot be exploited to set constraints on the CKM matrix due to large and poorly known effects from the strong interaction at low energy for example, but other neutral mesons can provide interesting constraints.
Concerning the neutral kaon system, indirect \(CP\) violation was observed in the decay of particles to a pair of pions. The \(\pi\pi\) final state system has a \(CP = +1\), therefore it was shown that this transition is possible because the meson (which decays to the CP=1 final state \(\pi\pi\pi\) in the majority of the cases) contains also a \(CP = +1 \) component. The complex observable \(\epsilon_K\) defined as the ratio of decay amplitudes : \[\epsilon_K = \frac{{\cal A } [K_L \to (\pi\pi) ]_{I =0} }{{\cal A } [K_S \to (\pi\pi) ]_{I =0}},\] where the final states have a null value of isospin, encodes the neutral kaon mixing. It is possible to relate \(\epsilon_K\) with the parameters from the formalism which describes the \(K^{0}\bar{K}^0\) oscillations where \(K^0\) is (\(\overline{s}d\)) and \(\bar{K}^0\) is (\(\overline{d}s\)).
Experimentally, the kaon mixing observables were measured in dedicated fixedtarget experiments such as NA48 (NA48, Batley, J R et al. (2002)) at CERN and KTeV (KTeV, Abouzaid, E et al. (2002)) at Fermilab. The current experimental values which are derived from the measurements of the decay amplitudes for \(K\to \pi^+\pi^\) and \(K\to \pi^0\pi^0\) are the modulus and phase of \(\epsilon_K\): \[\tag{4} \epsilon_K = 2.228(11) \times 10^{3},\] \[\phi_{\epsilon_K} = 42.52(5)^{\circ},\] leading to a combined constraint on \(V_{td}V_{ts}^*\) and \(V_{cd}V_{cs}^*\). More information can be found in A. Buras’ article on CP violation in electroweak interactions.
We turn now to the mixing of neutral \(b\)mesons. Because top quarks decay very quickly into jets that cannot be tagged easily according to their content in light quarks, the only way of measuring \( V_{tb} \) is through the measurement of the mixing of \(B^{0}\bar{B^{0}}\) and \(B_s^{0}\bar{B_s^{0}}\) mesons, as these processes are dominated by topquark boxes in the SM as shown in Figure 4. For each of these mesons, the two mass eigenstates resulting from mixing have different masses, and their difference of masses \(\Delta m_d\) and \(\Delta m_s\) (related to and respectively) can be accessed by looking at the time evolution of \(B^{0}\) and \(B_s^{0}\) mesons, using their decays in order to determine the frequency with which they evolve into their antiparticles.
Historically, the ARGUS experiment running at a symmetric \(\Upsilon(4S)\) machine, where B and \(\bar{B}\) mesons are produced at equal rates, was the first one to observe the oscillation in the \(B^0\bar{B}^0\) system in 1987. Experimentally, the observation was made through the measurement of samesign fast dilepton pairs, which implied that one of the B mesons had oscillated, since B and \(\bar{B}\) particles decay to leptons of opposite sign. Later on many experiments located on both highenergy colliders and \(\Upsilon(4S)\) asymmetric machines also measured this observable. High precision on the frequency of oscillation was achieved with timedependent measurements of \(b\)meson decays, where the frequency is extracted from the decay time distribution. Experimentally, three key ingredients are essential to measure oscillations:
 The identification of the flavour of the \(B\) meson when it is produced. In \(B\) factories, intricated or quantum entangled \(B^0\)\(\bar{B}^0\) or \(B^0_s\)\(\bar{B}^0_s\) pairs are produced, so that the decay of one \(b\)meson provides direct information on the flavour of the other \(b\)meson on the other side, whose evolution and decay can then be studied. In highenergy colliders, socalled flavour tagging algorithms which explore the nature of the particles produced together with the \(B\) meson of interest are used to find out if it contained a \(b\) or a \(\bar{b}\) quark initially.
 The identification of the flavour of the \(B\) meson when it decays. One can use flavour specific decays for example \(B^0_s\to D_s^+\pi^{}\) where the charge of the final state mesons are correlated with the flavour of the \(b\)quark when it decays.
 The measurement of the decay time defined as \(t= \frac{m_B}{p}L\) where \(m_B\) is the reconstructed invariant mass of the \(B\) meson, \(p\) the momentum and \(L\) the distance between the production and decay vertices.
The current world average, dominated by \(B\) factories and the LHCb experiment, is: \[\Delta m_d = (0.5064 \pm 0.0019) \mbox{ ps}^{1}.\] For what concerns the \(B^0_s\bar{B^0_s}\) mixing, the first observation was done by the CDF collaboration (CDF, Abulencia, A et al. (2006)) and later on by the LHCb collaboration (LHCb Collaboration, Aaij, R et al. (2013)). Figure 5 shows the decay time distribution for events tagged as mixed (different flavour at decay and production) or unmixed (same flavour at decay and production) in data. The current world average is: \[\Delta m_s = (17.757 \pm 0.021)\mbox{ ps}^{1}\] It is interesting to notice that even though the \(B^0\) and \(B^0_s\) mesons differ only in the flavour of their spectator quark, because of the different factors from the CKM matrix, the frequency of the \(B^0\) mixing is significantly higher than the \(B^0_s\) one. Once the frequencies are measured and using lattice QCD input for the so called bag factors describing the hadronisation of the quarklevel box diagrams into the oscillation between \(B\)mesons, the CKM parameters can be extracted: \[\tag{5} V_{td} = (8.1\pm 0.5) \times 10^{3}, \qquad\qquad V_{ts} = (39.4\pm 2.3) \times 10^{3},\] The current accuracy of these values is limited by theoretical uncertainties on hadronic effects (encoded in bag parameters), which cancel largely in the following ratio thanks to \(SU(3)\) flavour symmetry. \[\tag{6} V_{td}/V_{ts} = 0.210 \pm 0.001 \pm 0.008\]
Measurement of CKM angles
The unitarity of the CKM matrix can be exploited to provide graphical representations of our knowledge on the four Wolfenstein parameters. Indeed, orthogonality relations among rows or columns of the CKM matrix yield constraints among complex products of CKM matrix elements, which can be represented as triangles. One of the triangles is described by the angles \(\alpha\), \(\beta\) and \(\gamma\), related to the dynamics of the \( B^0 \) meson. Another angle called \(\beta_s\) can be constructed using the \( B^0_s \) meson dynamics. As shown in Figure 6, more triangles can be constructed, but the large differences between the size of their sides makes them difficult to explore experimentally, and they are therefore not used.
The \( B^0 \) triangle has small sides but is not squashed (compared to other triangles), which amounts to the possibility of observing large effects of \(CP\) violation in the dynamics of this meson. This explains why this triangle, generally called “the” unitarity triangle, is often used to summarise the current knowledge of the CKM matrix parameters, once it has been rescaled to have sides of order 1 (see Figure 7). These effects have been studied extensively in the \(B\)factories BaBar and Belle, working at an energy chosen to produce copious amounts of the \(\Upsilon(4S)\) resonance that decays into pairs of \(B\) and \(\bar{B}\) mesons (see Table).
The angles \(\alpha\), \(\beta\), \(\gamma\) and \(\beta_s\) describing these triangles provide information on \(CP\) violation and can be accessed through \(CP\)asymmetries for nonleptonic twobodymeson decays, out of which most of the hadronic quantities cancel. Depending on the final states considered, these CP asymmetries correspond to the interference of amplitudes with different weak phases leading to one of the angles \(\alpha\), \(\beta\), \(\gamma\) and \(\beta_s\).
The charmless decays of \(B\) mesons to \(\pi\pi, \rho\pi\) and \(\rho\rho\) have been used to measure the angle \(\alpha\) which is defined as: \[\alpha = \textrm{arg} (V_{td}V_{tb}^*/V_{ud}V^*_{ub})\] Experimentally the measurement of \(\alpha\) requires timedependent analyses. Such analyses consider the neutral meson \(B^0_d\) which can decay either directly in a given final state, or through a twostep process, consisting in the evolution of the meson into its antimeson \(\bar{B}^0_d\) (due to the neutral meson mixing described in the previous section) followed by the decay into the same final state. The analysis requires to study the time dependence of CP asymmetries such as: \[A_{CP}(t) = \frac{ \Gamma ( \bar{B}^0_d(t) \to \pi^{+} \pi^{})  \Gamma ( B^0_d(t) \to \pi^{+} \pi^{}) }{ \Gamma ( \bar{B}^0_d(t) \to \pi^{+} \pi^{}) + \Gamma ( B^0_d(t) \to \pi^{+} \pi^{}) },\] where \(B^0_d(t)\) denotes a state known to have been produced as a \(B^0\) meson at \(t=0\), that has evolved through mixing until its decay at time t (\(\bar{B}^0_d(t)\) means the same thing for \(\bar{B}^0\) ). This requires the experimental determination of the nature of the neutral meson initially produced and the study of its evolution over time. In the case of \(\rho(\to \pi\pi)\pi\), it also involves the analysis of the resonant structures contributing to the final states, called Dalitz plot analyses, which explore the kinematics of three body decays. To date, the angle \(\alpha\) has been measured only at \(B\)factories, the combination of all the modes gives: \[\alpha = (84.5^{+5.9}_{5.2})^{\circ},\] dominated by statistical uncertainties.
The golden mode \(B^0\to J/\psi K_S\) was first used by the \(B\)factories which by means of time dependent measurements were able to determine experimentally the angle \(\beta\) defined as: \[\beta = \textrm{arg} (V_{cd}V_{cb}^*/V_{td}V^*_{tb})\], based on the time dependence of CPasymmetries such as \[A_{CP}(t) = \frac{ \Gamma ( \bar{B}^0_d(t) \to J/\psi K_S)  \Gamma ( B^0_d(t) \to J/\psi K_S) }{ \Gamma ( \bar{B}^0_d(t) \to J/\psi K_S) + \Gamma ( B^0_d(t) \to J/\psi K_S) }.\] Later on, various other channels with higher charmonia resonances such as the \(\psi(2S)\) were also analysed. The current world average is \[\sin2\beta = 0.691 \pm 0.017\] which is statistically limited. The degeneracy among the resulting solutions for \(\beta\) can be lifted by constraints on \(\cos 2\beta\) obtained from the same channels, in agreement with Standard Model expectations.
The angle \(\gamma\) is defined as: \[\gamma = \textrm{arg} (V_{ud}V_{ub}^*/V_{cd}V^*_{cb}).\] There are many different experimental techniques to measure this angle. Most of them rely on the interference in \(B\) meson decays via \(b\to c\) and \(b\to u\) transitions, for instance \(B\to D^{(*)}K^{(*)}\) followed by a \(D^{(*)}\) decay chosen to allow for interferences. The angle can then be extracted by considering CP asymmetries such as: \[A_{CP} = \frac{ \Gamma ( B^{} \to D K^{})  \Gamma ( B^{+} \to D K^{+}) }{ \Gamma ( B^{} \to D K^{}) + \Gamma ( B^{+} \to D K^{+}) },\]. The combination of such measurements for various final states gives: \[\gamma = (73.5^{+4.2}_{5.1})^{\circ},\] dominated by statistical uncertainties.
In the Standard Model, the phase associated with \(B_s\) mixing is \[\beta_s = \textrm{arg}(V_{ts}V_{tb}^*/V_{cs}V^*_{cb}),\] which can be predicted with a very good accuracy as \(\beta_s = −0.0365^{+0.0013}_{0.0012}\) from a global fit involving various constraints on the CKM matrix (CKMfitter Group, Charles, J et al. (2005) and [4]). Many decay channels governed by \(b\to c\bar{c}s\) contribute to the determination of \(\phi_s\), as long as one can neglect subleading penguin contributions, which are decays occurring through Feynman diagrams which involve a loop allowing a transition from a \(b\) quark to an \(s\) quark. Most of the statistical power comes from the decay \(B_s\to J/\psi\phi\). The measurement of this weak phase requires an angular analysis to be able to separate the \(CP\)even and \(CP\)odd components in the decay (this is not needed in the case of the decay \(B^0\to J/\psi K_S\), where the final state is not a mixture of different \(CP\) states). A timedependent treatment to extract the observables related to the lifetime of the \(B_s\) mass eigenstates (average lifetime \(\Gamma_s\) and difference of lifetime \(\Delta \Gamma_s\)) is also needed.
Further information on the current experimental knowledge of these angles can be found in the [Article on weak angles in Scholarpedia]
.
Outlook
The CKM matrix is able to describe a wide range of quark transitions, caused by the weak interaction in the Standard Model. It can thus be constrained by many different processes, which have to be measured experimentally with high precision and computed with good theoretical control. The combination of these various measurements overconstrains the CKM matrix and sharpens our knowledge of its parameters (DescotesGenon, S and Koppenburg, P (2017)). This should be performed in an appropriate statistical framework in order to take into account the uncertainties attached to the measurements as well as the theoretical (hadronic) inputs, and the correlations among them (several analyses of this kind are available and regularly updated [5] and [6]). The resulting four parameters describing the CKM matrix are thus determined to a high accuracy (Particle Data Group, Tanabashi, M et al. (2018)): \[\begin{aligned} \lambda &=& 0.22453 \pm 0.00044, \qquad\qquad \bar\rho = 0.122^{+0.018}_{0.017},\\ A &=& 0.836 \pm 0.015, \qquad\qquad\qquad \bar\eta = 0.355^{+0.012}_{0.011}.\end{aligned}\] The above constraints show a very good degree of consistency, as can be illustrated for instance in the \((\bar\rho,\bar\eta)\) plane (see Figure 8). In particular, the fact that \(\bar\eta\neq 0\) is the indication that \(CP\) is not a valid symmetry of the weak interaction and thus of the SM, which features an asymmetry between matter and antimatter. However, this asymmetry is much smaller than the one observed in astrophysics and cosmology, indicating that processes beyond the SM ones must have occurred earlier in the evolution of the Universe, at higher energies.
It should be noted that many other processes are not considered yet in global fits of the CKM parameters, due to the difficulty to understand their hadronic dynamics to extract information on the CKM matrix. Among these very interesting processes that are observed experimentally but remain challenging from the theoretical point of view, one may mention a vast part of the nonleptonic twobody and threebody \(B\) decays, the measurement of direct CPviolation in \(K\) decays (as measured by \(\epsilon'/\epsilon\)), or the recent observation of CPviolation in \(D\) decays into two light mesons. Theoretical effort is currently devoted to improve the understanding of these modes so that they could be exploited to determine CKM parameters.
Our current knowledge is based on data gathered mostly from the B factories as well as LHC results, although other experiments are also involved in some inputs. The full exploitation of LHC Run 2 data (20152018) will be performed by the LHCb experiment, the ATLAS experiment and the CMS experiment in the coming years. Important changes are also expected after the year 2020, when both the BelleII experiment and the upgraded LHCb experiment will collect data at much higher luminosities. The target is a multiplication of the data sets by up to two orders of magnitude (Aushev, T et al. (2010), Cerri, A et al. (2018)). For instance, a reduction of experimental uncertainties by factors of around 10 on the angles \(\beta\), \(\gamma\), and \(\beta_s\) is to be expected. One may also expect improvements in the experimental measurements of the observables related to the angle \(\alpha\) and the matrix elements \(V_{ub}\) and \(V_{cb}\). Measurements involving \(B_s\), \(B_c\) and \(\Lambda_b\) decays should also be improved, as well as comparisons between semileptonic modes involving different leptons (electron, muons, taus). Moreover, the BESIII experiment as well as the NA62 and KOTO experiments should provide important constraints on semileptonic charm decays and rare strange decays that would help to constrain further \(V_{cd},V_{cs}\) and \(\bar\eta\) respectively.
The interpretation of these improved measurements will depend on developments in theoretical calculations. The computation using lattice QCD simulations has already reached a very mature stage for decay constants and form factors. At the accuracy obtained, some issues become relevant, such as the estimation of electromagnetic corrections, the extrapolation of the lattice QCD simulations up to the physical values of the quark masses, and the kinematic range available for heavytolight form factors. The experimental accuracy obtained for the individual constraints requires one to reassess some of the theoretical hypotheses commonly used to extract these quantities and add systematics that have been neglected up to now (isospin breaking, QED corrections…). Other improvements can be expected concerning more exploratory domains, such as the description of unstable mesons or the hadronic quantities related to baryon dynamics.
In the coming years, the addition of new experimental measurements should thus help to determine the CKM matrix in a much more detailed way, increasing the accuracy on their parameters. One may hope that they will ultimately identify discrepancies among the various constraints, hinting at physics beyond the Standard Model.
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