CP Violation in the Quark Sector: Mixing Matrix Unitarity

Abstract

Since its discovery in the 1960s, the violation of CP symmetry has intrigued scientists and stimulated the advancement of knowledge in particle physics. Numerous experiments were designed and built to study it in increasingly deeper detail. Nowadays, the phenomenon is well framed within the Standard Model of Particle Physics. Nevertheless, new results are being produced by modern experiments at colliders that challenge the current understanding of the model. In this article, the current status of CP violation studies and the role of CP violation in the search for effects beyond the Standard Model are described together with the prospects for ongoing and future experiments.

1. Introduction

The Standard Model of Particle Physics (SM) describes the interaction of quarks and leptons with utmost precision. Developed over the course of the 20th century, the SM has been tested in a wide range of experiments showing remarkable predictive power. While extremely successful in describing the interaction of elementary particles, the SM is not able to explain some of the most fundamental questions in physics. For instance, limiting ourselves to the energy scale for which the SM is best suited, it does not explain the size of the asymmetry between matter and antimatter in the Universe, why there are exactly three generations of fermions, and the origin of the masses of neutrinos.

These compelling questions indicate that the SM is not the ultimate theory of particle physics, rather an excellent approximation of a more fundamental theory. The SM is, therefore, considered as an effective theory, valid up to a certain energy scale, $\Lambda$, where new physics is expected to appear. Probing higher energy scales is, therefore, a key aspect in the search for new physics. There are two main approaches to this problem: direct and indirect searches. The former consists in looking for new particles directly by producing them in a particle collider, while the latter consists in looking for deviations from the SM predictions in precision measurements. At the Large Hadron Collider (LHC) both these approaches are pursued thanks to the unprecedented energy and luminosity of the proton–proton collisions.

In this review, we will focus on the indirect searches for new physics in the flavor sector, in particular, those concerning the study of the matter–antimatter (CP) asymmetry in hadron decays.

2. CP Violation in the Standard Model

The charge–parity (CP) symmetry is a fundamental symmetry of the SM of particle physics, being the combination of two fundamental symmetry operations, charge-conjugation (C) and parity (P). Before the discovery of CP violation in the neutral kaon system [1], it was believed that the CP symmetry was an exact symmetry of nature. The necessity to describe this phenomenon in the SM led to the introduction of an irreducible complex phase in the Cabibbo–Kobayashi–Maskawa (CKM) quark mixing matrix [2], a complex matrix representing the probability of each up-type quark (u, c, t) to interact with a down-type quark (d, s, b) by means of a W boson emission or absorption.

This phase appears in the CKM matrix predominantly in the elements $V_{ub}^{\phantom{\ast }}$ and $V_{td}^{\phantom{\ast }}$ that in the Wolfenstein parametrization [3] are expressed as

$$\begin{array}{cc}\hfill V_{ub}^{\phantom{\ast }}& \approx A{\lambda }^3(\rho -i\eta ),\hfill \\ \hfill V_{td}^{\phantom{\ast }}& \approx A{\lambda }^3(1-\rho -i\eta ),\hfill \end{array}$$

(1)

which leads to an intuitive way to show CP violation by drawing unitarity triangles in the complex plane. The unitarity ($V{V}^{†}=I$) of the CKM matrix implies column and row orthogonality

$$\begin{array}{c}\hfill \sum _i{V}_{ij}{V}_{ik}^{\ast }={\delta }_{jk}\mathrm{and}\sum _j{V}_{ij}{V}_{kj}^{\ast }={\delta }_{ik},\end{array}$$

and each of these conditions can be represented as a triangle in the complex plane. The most common triangle is built from the unitarity relation

$$\begin{array}{c}\hfill V_{ud}^{\phantom{\ast }}{V}_{ub}^{\ast }+V_{cd}^{\phantom{\ast }}{V}_{cb}^{\ast }+V_{td}^{\phantom{\ast }}{V}_{tb}^{\ast }=0\end{array}$$

(2)

that can be rewritten as

$$\begin{array}{c}\hfill 1-{\displaystyle \frac{|V_{ud}^{\phantom{\ast }}|}{|V_{cd}^{\phantom{\ast }}\left|\right|V_{cb}^{\ast }|}}{V}_{ub}^{\ast }-{\displaystyle \frac{|V_{tb}^{\ast }|}{|V_{cd}^{\phantom{\ast }}\left|\right|V_{cb}^{\ast }|}}{V}_{td}^{\phantom{\ast }}=0\end{array}$$

(3)

when isolating the complex terms. The sum of three complex numbers can be drawn as a triangle in the complex plane with the three sides of the triangle corresponding to the three terms in the sum, as shown in Figure 1. The angles of the triangle are related to the CP-violating complex phase. In this case, they are defined as

$$\begin{array}{c}\hfill \alpha =arg\left(-{\displaystyle \frac{V_{td}^{\phantom{\ast }}{V}_{td}^{\ast }}{V_{ud}^{\phantom{\ast }}{V}_{ub}^{\ast }}}\right)\beta =arg\left(-{\displaystyle \frac{V_{cd}^{\phantom{\ast }}{V}_{cb}^{\ast }}{V_{td}^{\phantom{\ast }}{V}_{td}^{\ast }}}\right)\gamma =arg\left(-{\displaystyle \frac{V_{ud}^{\phantom{\ast }}{V}_{ub}^{\ast }}{V_{cd}^{\phantom{\ast }}{V}_{cb}^{\ast }}}\right).\end{array}$$

(4)

A fundamental ingredient for observing CP violation is the presence of at least two amplitudes that can interfere. These amplitudes can be of a different nature; for example, in the neutral meson system, the two amplitudes are the direct decay and the decay after oscillation. With charged particles, these amplitudes are represented by the different resonant decay paths that lead to the same final state. Different amplitudes show not only a difference in the weak (CP-violating) phase, but also in the strong (CP-conserving) phase. The non-zero difference between these two phases is what allows for the interference between the two amplitudes and the observation of CP violation [5,6].

CP violation first manifested itself in the decays of neutral kaons, but has become widely studied in the decays of B mesons. In Figure 1, there are many labels superimposed on the plot, each corresponding to a different measurement of CP violation. ${ϵ}_K$ is the sole constraint imposed by measurements in the neutral kaons system, while all the other measurements are related to B mesons decays.

The aforementioned unitarity triangle is the most commonly drawn since it is the one that shows the largest CP violation effects and is also the most constrained one. Alternative triangles can be drawn when focusing on the transitions involving $B_S^0$ and D decays but they are not as clear as the B one, since they are characterized by smaller CP violation effects that lead to triangles flattened on the real axis. Nevertheless, observations of CP violation in the $B_S^0$ and D systems have been made. In the upcoming section, we will discuss the measurements of CP violation in the B and D systems.

3. Experimental Status of CP Violation

Measurements of CP violation were first made in the 1960s by studying neutral kaon decays [1], but it was not until the discovery of the b quark that an extensive experimental program could start. Once B mesons were discovered and the technology was ready to produce them in large quantities, an extensive research program was undertaken by two competing experiments, BaBar [7] and Belle [8], that ran in the early 2000s at the asymmetric $e^{+}{e}^{-}$ colliders PEP-II and KEKB, respectively.

3.1. CP Violation in $B^0$ Decays

The first observation of CP violation in the B meson system was made by two collaborations in 2001 [9,10] by measuring

$$\begin{array}{c}\hfill sin\left(2\beta \right)=0.59\pm 0.14\pm 0.05\mathrm{and}sin\left(2\beta \right)=0.99\pm 0.14\pm 0.06,\end{array}$$

(5)

respectively. They combined the analyses of $B^0\to \left(c\overline{c}\right)K_{\mathsf{S}}^0$ ($c\overline{c}$ = $J/\psi$, $\psi \left(2S\right)$, ${\chi }_{c1}$) and $B^0\to J/\psi K_{\mathsf{L}}^0$ decays in this result.

CP violation is observed as an asymmetry in the decay time distribution of the $B^0$ mesons decaying to $J/\psi$$K_{\mathsf{S}}^0$ and $J/\psi$$K_{\mathsf{L}}^0$, as shown in Figure 2

$$\begin{array}{cc}\hfill A_{CP}(\Delta t)& \equiv {\displaystyle \frac{N_{{\overline{B}}^0}(\Delta t)-N_{B^0}(\Delta t)}{N_{{\overline{B}}^0}(\Delta t)B{N}_{B^0}(\Delta t)}}\hfill \\ \hfill & ={\eta }_{f}sin\left(2\beta \right)sin(\Delta m_d\Delta t),\hfill \end{array}$$

(6)

with ${\eta }_f$ being the CP eigenvalue of the final state and $\Delta t=t_{CP}-t_{\mathrm{tag}}$ the decay time difference between the reconstructed and tagging B mesons. It is interesting to note that CP violation at the B Factories is measured through the measurement of the difference in the decay time of the two B mesons in the event. Such a measurement is possible thanks to the entanglement of the B mesons, i.e., being produced from the same $e^{+}{e}^{-}\to b\overline{b}$ interaction, the large acceptance of the detectors covering almost the whole solid angle, and to the spatial separation between the two B mesons decay vertices, which is achieved by the asymmetric energy of the beams that boosts the B mesons in the laboratory frame.

At hadron colliders, such as the LHC, the B mesons are still produced in pairs with a large boost, but most of the time, one of them is not reconstructed due to the detector’s acceptances. In this case, CP violation is measured by studying the decay time distribution of the B meson that is reconstructed, and the rest of the event is used to infer the flavor of the B meson (tagging [11]). Larger yields have been collected since the first observation of CP violation in B⁰ decays, improving the precision of the measurements and requiring Equation (6) to be extended to account for second-order effects

$$\begin{array}{cc}\hfill A^{CP}\left(t\right)& \equiv {\displaystyle \frac{\Gamma ({\overline{B}}^0\left(t\right)\to f)-\Gamma (B^0\left(t\right)\to f)}{\Gamma ({\overline{B}}^0\left(t\right)\to f)+\Gamma (B^0\left(t\right)\to f)}}\hfill \\ \hfill & ={\displaystyle \frac{Ssin\left(\Delta m_{d}t\right)-Ccos\left(\Delta m_{d}t\right)}{cosh\left(\frac{1}{2}\Delta {\Gamma }_{d}t\right)+{\mathcal{A}}_{\Delta \Gamma }sinh\left(\frac{1}{2}\Delta {\Gamma }_{d}t\right)}}.\hfill \end{array}$$

(7)

Here S, C, and ${\mathcal{A}}_{\Delta \Gamma }$ are the CP violation parameters, $\Delta m_d$ and $\Delta {\Gamma }_d$ are the difference between the mass and the decay width of the two $B^0$ mass eigenstates, respectively. It is useful to note that the parameter S is related to the CP violation parameter $sin\left(2\beta \right)$ by the relation $S=sin(2\beta +\Delta {\Phi }_d+\Delta {\Phi }_d^{NP})$, where $\Delta {\Phi }_d$ is a contribution from loop (or penguin [12], see Appendix A) diagrams (suppressed in the SM), and $\Delta {\Phi }_d^{NP}$ is a contribution of the same type arising from phenomena beyond the SM. The B Factories have measured the CP violation parameters S and C in various $B^0\to c\overline{c}{K}^0$ decays, obtaining

$$\begin{array}{cccc}\hfill S& =0.687\pm 0.028\left(\mathit{stat}\right)\pm 0.012\left(\mathit{syst}\right)\hfill & \hfill C& =0.024\pm 0.020\left(\mathit{stat}\right)\pm 0.016\left(\mathit{syst}\right),\hfill \end{array}$$

(8)

$$\begin{array}{cccc}\hfill S& =0.667\pm 0.023\left(\mathit{stat}\right)\pm 0.012\left(\mathit{syst}\right)\hfill & \hfill C& =0.006\pm 0.016\left(\mathit{stat}\right)\pm 0.012\left(\mathit{syst}\right),\hfill \end{array}$$

(9)

where the first set of values (Equation (8)) are measured by the BaBar experiment [13], and the second (Equation (9)) by Belle [14]. These results represents the legacy of the B Factories and were quite recently superseded by the LHCb experiment [15], whose latest measurement of this parameter is

$$\begin{array}{cc}\hfill S\left(\psi K_{\mathsf{S}}^0\right)& =0.717\pm 0.013\left(stat\right)\pm 0.008\left(syst\right),\hfill \\ \hfill C\left(\psi K_{\mathsf{S}}^0\right)& =0.008\pm 0.012\left(stat\right)\pm 0.003\left(syst\right),\hfill \end{array}$$

(10)

which is the most precise measurement of CP violation in $B^0$ decays to date [16].

Despite being the most precise, the measurement of CP violation in $B^0$ decays through the angle $\beta$ is not the only one. The other two angles of the unitarity triangle, $\alpha$ and $\gamma$, can be measured through the study of other $B^0$ decays.

In particular, the angle $\alpha$ can be measured through the study of $b\to u$ transitions, like $B^0\to \pi \pi$ and $B^0\to \rho \rho$ decays. In these decays, the interference between the tree-level $b\to u$ and the box diagram of the $B^0$–${\overline{B}}^0$ mixing result in the CP asymmetry parameters $S=sin2\alpha$ and $C=0$. In reality, the penguin $b\to d$ diagrams also contribute and introduce theoretical uncertainties that need to be corrected (penguin pollution). A seminal paper by Gronau and London [17] provided the strategy to measure $\alpha$ in a model-independent way by combining measurements of CP asymmetries in the isospin-related $B^0\to {\pi }^{+}{\pi }^{-}$, $B^0\to {\pi }^0{\pi }^0$, and $B^{+}\to {\pi }^{+}{\pi }^0$ decays. In this context, the experiments measure the CP asymmetry and the combination of the measurements allows to extract the angle $\alpha$. The most precise measurement of CP violation in $B^0$ decays to ${\pi }^{+}{\pi }^{-}$, performed by the LHCb experiment [18] with a dataset of 4.7 fb⁻¹ luminosity, is

$$\begin{array}{cc}\hfill C_{\pi \pi }& =-0.311\pm 0.045,\hfill \\ \hfill S_{\pi \pi }& =-0.706\pm 0.042,\hfill \end{array}$$

(11)

while BaBar and Belle studied the decays $B^0\to {\pi }^0{\pi }^0$ and $B^{+}\to {\pi }^{+}{\pi }^0$ with a sensitivity smaller than anticipated [19,20]. These results led the B Factories to pursue alternative avenues to measure the angle $\alpha$ by performing a time-dependent CP violation measurement in $B^0\to \rho \rho$ and $B^0\to {\rho }^0(\to {\pi }^{+}{\pi }^{-}){\pi }^0$ decays [21,22,23,24,25,26]. It is interesting to note that nowadays the most stringent constraints to the determination of $\alpha$ come from the measurements of the $B^0\to \rho \rho$ decays, as shown in Figure 3, which were not considered at the beginning of the B Factories program. It was at the time thought that the $B^0\to \rho \rho$ were theoretically challenging due to the need of performing three isospin analyses for longitudinal and transverse polarizations of the $\rho$ mesons. In reality, the longitudinal polarization dominates the decay [21,22,27]. Together with the relatively small branching ratio of the penguin-dominated $B^0\to {\rho }^0{\rho }^0$ decay, the $B^0\to {\rho }^{+}{\rho }^{-}$ decays provide a theoretically clean determination of the angle $\alpha$.

The last angle of the unitarity triangle, $\gamma$, can be measured through the study of $B\to DX$ decays, in which the interference between the $b\to c$ and the $b\to u$ transitions gives access to $\gamma \equiv arg\left[-V_{ud}^{\phantom{\ast }}{V}_{ub}^{\ast }/V_{cd}^{\phantom{\ast }}{V}_{cb}^{\ast }\right]$. Since it is measured through the study of tree-level decays, it has a very small irreducible theoretical uncertainty ${\delta }_{\gamma }/\gamma \le {10}^{-7}$ [28], which also makes it a promising ground for searches of physics beyond the SM effects. The experimental uncertainty on $\gamma$ saw a significant reduction in recent years thanks to the efforts of the LHCb collaboration. Typically, measurements of $\gamma$ are made by studying the $B^{\pm }\to \overline{D}{K}^{\pm }$ decays. They proceed through the $b\to c\overline{u}c$ and $b\to u\overline{c}c$ transitions, whose ratio is $r_B{e}^{i({\delta }_B\pm \gamma )}$, where $r_B$ is the absolute ratio of the two amplitudes and ${\delta }_B$ is the strong phase difference between the two amplitudes. This ratio can be measured experimentally by studying the rate of B⁺ mesons

$$\begin{array}{cc}\hfill \Gamma \left(B^{\pm }\to \stackrel{(-)}{D}[\to f]K^{\pm }\right)& ={\left|r_D{e}^{-i{\delta }_D}+r_B{e}^{i({\delta }_B\pm \gamma )}\right|}^2\hfill \\ \hfill & =r_D^2+r_B^2+2\kappa r_D{r}_{B}cos({\delta }_D-{\delta }_B\pm \gamma ),\hfill \end{array}$$

(12)

where no CP violation is assumed in the D meson decays; therefore, the only nuisance parameters are the absolute ratio ($r_D$) and the strong phase difference (${\delta }_D$) between the $D\to f$ and the $D\to \overline{f}$ decays. To account for the dilution of the interference in multibody B and D decays, the “coherence factor” $\kappa$ is present in the formula, being 1 for two-body decays and $\kappa <1$ for multibody decays. Since the measurement of $\gamma$ reduces to the measurement of the decay rates of charged-conjugate decays, it is important to have a good control of the systematic uncertainties, especially the charged-particles detection asymmetries.

The simplest topology for the measurement of $\gamma$ is the $B^{\pm }\to D^0{K}^{\pm }$ decay, where the $D^0$ meson is reconstructed in two-body final states. Two methods are used to measure $\gamma$ in these decays: the GLW method [29] and the ADS method [30]. In the former method, $D^0\to {\pi }^{+}{\pi }^{-},K^{+}{K}^{-}$ decays are used, offering $r_D^{hh}=1$ and ${\delta }_D^{hh}=0$, while in the latter, $D^0\to K^{+}{\pi }^{-},K^{-}{\pi }^{+}$ decays are used, which have $r_D^{K\pi }=0.06$, and make best use of the similarity of $r_D^{K\pi }$ and $r_B^{D^{0}K}$($r_B^{D^{0}K}\approx c_f|V_{cc}^{\phantom{\ast }}{V}_{ub}^{\ast }/V_{uc}^{\phantom{\ast }}{V}_{cb}^{\ast }|\approx 0.1$, where $c_f\approx 0.3$ is a color suppression factor), giving large interference effects and high sensitivity to the phase information.

Another class of measurements involves multibody $D^0$ decays, such as $D^0\to K_{\mathsf{S}}^0{\pi }^{+}{\pi }^{-}$, $D^0\to K_{\mathsf{S}}^0{K}^{+}{K}^{-}$, and $D^0\to K_{\mathsf{S}}^0{\pi }^{+}{\pi }^{-}{\pi }^0$ decays. The technique used in this case is commonly referred to as GGSZ [31], and takes advantage of the resonant structure of the $D^0$ decay to acquire sensitivity to $\gamma$. As shown before, the strong phase difference between the $D^0$ and ${\overline{D}}^0$ decays to the same final state is an important ingredient to the sensitivity on $\gamma$. By studying multibody $D^0$ decays, one gets access to regions in which the strong phase difference is large, therefore enhancing the sensitivity. The price to pay is the need of a model to describe the resonant structure of the $D^0$ decays, which introduces a theoretical uncertainty in the measurement. Alternatively, external input is needed to constrain the strong phase difference in the regions of the Dalitz plot. This model-independent approach is becoming more and more popular, thanks to the synergy between the LHCb and the BESIII experiments. The large sample of quantum-correlated $D^0$–${\overline{D}}^0$ pairs produced at the $e^{+}{e}^{-}$ colliders allows to measure the strong phase difference in the $D^0$ decays, and detailed measurements across the Dalitz plot of various $D^0$ decays are performed [32].

Finally, the angle $\gamma$ can also be measured through the study of $B^0\to D^{(\ast )\mp }{\pi }^{\pm }$ decays. In the SM, these decays proceed through the $\overline{b}\to \overline{c}u\overline{d}$ and $b\to u\overline{c}d$ transitions, and the interference between the two amplitudes gives access to the angle $\gamma$. The same final states can also be reached after B⁰–${\overline{B}}^0$ mixing though, and the asymmetry between the decay rates gives access to $2\beta +\gamma$. The measurement of $\gamma$ in these decays, therefore, relies on the knowledge of $\beta$ from the $B^0\to c\overline{c}{K}_{\mathsf{S}}^0$ decays. In this case, a time-dependent measurement of the CP asymmetries allows to measure the angle $\gamma$ [33,34,35]. Mesons initially produced as B⁰ decay to the final states $f=D^{(\ast )-}{\pi }^{+}$ and $\overline{f}=D^{(\ast )+}{\pi }^{-}$ as

$$\begin{array}{cc}\hfill \Gamma (B^0\left(t\right)\to f)& =e^{-{\Gamma }_{d}t}\left[1+C_{f}cos(\Delta m_{d}t)-S_{f}sin(\Delta m_{d}t)\right],\hfill \\ \hfill \Gamma (B^0\left(t\right)\to \overline{f})& =e^{-{\Gamma }_{d}t}\left[1+C_{\overline{f}}cos(\Delta m_{d}t)-S_{\overline{f}}sin(\Delta m_{d}t)\right].\hfill \end{array}$$

(13)

The time evolution of initially produced ${\overline{B}}^0$ mesons is the same except for the flipped sign of the C and S coefficients. These coefficients are related to the theoretical observables $r_{D\pi }$, $\delta$, $\beta$, and $\gamma$ by the relations

$$\begin{array}{cc}\hfill C_f& ={\displaystyle \frac{1-r_{D\pi }^2}{1+r_{D\pi }^2}}=-C_{\overline{f}},\hfill \\ \hfill S_f& =-{\displaystyle \frac{2r_{D\pi }sin[\delta -(2\beta +\gamma )]}{1+r_{D\pi }^2}},\hfill \\ \hfill S_{\overline{f}}& ={\displaystyle \frac{2r_{D\pi }sin[\delta +(2\beta +\gamma )]}{1+r_{D\pi }^2}},\hfill \end{array}$$

(14)

where $r_{D\pi }$ is the ratio of the $B^0\to D^{(\ast )-}{\pi }^{+}$ and $B^0\to D^{(\ast )+}{\pi }^{-}$ decay amplitudes and $\delta$ their strong phase difference. By constraining the values of $r_{D\pi }$ and $\beta$ from external measurements, the angle $\gamma$ can be extracted from the measurement of the S coefficients, since they only differ by a phase $2(2\beta +\gamma )$. Measurements of $\gamma$ with this approach were made by the B Factories using $B^0\to D^{(\ast )\mp }{\pi }^{\pm }$ and $B^0\to D^{\mp }{\rho }^{\pm }$ decays [36,37,38,39]. The first measurement of this kind at a hadron collider was made by LHCb using $B^0\to D^{\mp }{\pi }^{\pm }$ decays [40].

The LHCb and BaBar experiments have produced a compendium of all their measurements of $\gamma$ and provided their own averages. Up to 2013, BaBar was the leader in the measurement of $\gamma$ [41], with an average of $\gamma ={\left({69}_{-16}^{+17}\right)}^{\circ }$. This was outclassed by the LHCb experiment, whose latest average is $\gamma ={(67\pm 4)}^{\circ }$ [42], clearly dominating the world average of $\gamma ={(66.2_{-3.6}^{+3.4})}^{\circ }$ [16]. A summary of the sensitivity to $\gamma$ from the various measurements is shown in Figure 4. So far, the average is mostly constrained by the measurement of $\gamma$ using $B^{+}\to D^0{K}^{+}$ decays, with the GGSZ method.

To summarize the status of CP violation in B⁰ decays, the combination of all the measurements of the angles $\alpha$, $\beta$, and $\gamma$ gives [16]

$$\begin{array}{cc}\hfill \alpha & ={(85.2_{-4.3}^{+4.8})}^{\circ },\hfill \\ \hfill \beta & ={(22.2\pm 0.7)}^{\circ },\hfill \\ \hfill \gamma & ={(66.2_{-3.6}^{+3.4})}^{\circ }.\hfill \end{array}$$

(15)

When summing the angles of the unitarity triangle, the sum is [6]

$$\begin{array}{c}\hfill \alpha +\beta +\gamma ={(173\pm 6)}^{\circ },\end{array}$$

(16)

consistent with the SM expectations.

3.2. CP Violation in $B_S^0$ Decays

When discussing the measurement of $\gamma$, the possibility was omitted of measuring the angle by studying $B_S^0\to D_S^{\pm }{K}^{\mp }$ [43] and $B_S^0\to D_S^{\pm }{K}^{\mp }{\pi }^{+}{\pi }^{-}$ [44] decays.

In these decays, the sensitivity to CP violation arises from the interference of the mixing and decay amplitudes [33,34,35,45], and the CP-violating parameters are a function of $\gamma$ and ${\beta }_S\equiv arg[-\left(V_{\mathrm{tc}}^{\phantom{\ast }}{V}_{\mathrm{tb}}^{\ast }\right)/\left(V_{cc}^{\phantom{\ast }}{V}_{cb}^{\ast }\right)]$, the weak phase of the $B_S^0$–${\overline{B}}_S^0$ mixing. Equations (13) and (14) can be adapted to the $B_S^0$ system by replacing the B⁰ with $B_S^0$, the $D^{(\ast )}$ with $D_c^{+}$ mesons, $\Delta m$ with $\Delta m_c$, and the angle $\beta$ with ${\beta }_c$.

The weak phase ${\beta }_S$ is of particular interest since it is a sensitive probe of physics beyond the SM [46]. It is usually measured as ${\varphi }_S=-2{\beta }_S$ and its value is predicted to be ${\varphi }_S=-0.{0368}_{-0.0006}^{+0.0009}$ [47] using the known values of the CKM matrix elements. The most sensitive measurement of CP violation in $B_S^0$ decays is obtained through the study of $B_S^0\to \psi hh$ decays ($h=\pi ,K$). Since their final state particles may exhibit various polarizations depending on the $hh$ resonances involved (e.g., $\varphi \left(1020\right)$, $\rho \left(770\right)$), the decay amplitudes are studied on the transversity basis, where they are decomposed in terms of the helicity amplitudes, to disentangle the CP-odd and the CP-even contributions [48,49,50].

The specific topology of the $B_S^0\to J/\psi \varphi$ decay allows to measure ${\varphi }_S$ with good precision at hadron colliders. The $J/\psi$ and $\varphi$ resonances have very small widths, which makes their identification easier even without a specific particle identification sub-system. Therefore, many experiments have measured the ${\varphi }_S$ parameter using this decay mode [51,52,53,54,55]. The latest combination of these measurements is shown in Figure 5. As shown in the figure, the combination of the independent measurements from different experiments is consistent with the SM prediction.

3.3. CP Violation in ${\mathit{\Lambda }}_b^0$ Decays

Since the ${\mathit{\Lambda }}_b^0$ baryon is the lightest baryon containing a b quark, it is a good candidate to study CP violation in the baryon sector. Similar approaches as those developed for charged B mesons can be used to study CP violation in ${\mathit{\Lambda }}_b^0$ decays. In particular, the angle $\alpha$ can be measured by studying charmless ${\mathit{\Lambda }}_b^0$ decays, and the angle $\gamma$ with final states involving charm mesons. Mixing does not happen in baryon decays; therefore, measurements of CP violation in the interference between decay and mixing ($\beta$) are not possible.

No measurements of CP violation in ${\mathit{\Lambda }}_b^0$ decays have been made so far, but the LHCb experiment is aiming at it with many searches on various decay modes. Evidence of CP violation in the decays of multibody ${\mathit{\Lambda }}_b^0\to p{\pi }^{-}{\pi }^{+}{\pi }^{-}\left(K^{+}{K}^{-}\right)$ decays was reported by the LHCb experiment [57], but further data collected in Run2 of the LHC have not led to observation of the effect [58] yet. The technique used in this analysis searches for CP violation by measuring triple-product asymmetries [59,60,61]. In this case, asymmetries of kinematical distributions are measured separately on particle and anti-particle decays. These distributions are built to be odd under CP transformation, but may exhibit an asymmetry due to strong-phase differences. These strong-phase effects are canceled out by measuring the difference between the asymmetries of the two charged-conjugate states, which is a direct probe of CP violation.

Another class of measurements of CP violation in ${\mathit{\Lambda }}_b^0$ decays is the study of the ${\mathit{\Lambda }}_b^0\to p{K}^{-}$ and ${\mathit{\Lambda }}_b^0\to p{\pi }^{-}$ decays. These decays follow the same quark-level transitions of charmless B⁰ and $B_S^0$ two-body decays, in which CP violation is established. Nevertheless, the LHCb experiment has not yet reported any measurement of CP violation in these decays with sensitivities as low as 2% [62]. A challenging aspect of this analysis is the effect of the ${\mathit{\Lambda }}_b^0$ production asymmetry in the LHC, which is known with comparable precision to the statistical precision of the CP asymmetry measurement [63].

3.4. CP Violation in Charm Decays

The study of CP violation in charm decays is a challenging task, since the CP violation effects in the charm sector are expected to be very small in the SM. Nevertheless the charm sector is unique in the SM, as it is the only up-type quark allowing a thorough study of flavor-changing neutral currents (FCNC) and searches for physics beyond the SM (top quarks undergo decay before they change to hadronize [64,65], and the lighter hadrons built with u and $\overline{u}$ are their own antiparticle).

Since the quarks involved in the box diagram for $D^0$ mixing are $m_u,m_c,m_b\ll m_W$, the process is highly suppressed in the SM. This results in a similar size of the difference of the mass and width eigenvalues [66].

Direct CP violation in charm decays arises from the interference of the tree-level and penguin $c\to u\overline{c}c$ diagrams, and is expected to be of the order of ${10}^{-3}$ or less [67,68,69]. Only recently, the LHCb experiment reported the first observation of CP violation in the decays of $D^0$ mesons [70]. This measurement showed a difference in the CP asymmetries of $D^0\to K^{+}{K}^{-}$ and $D^0\to {\pi }^{+}{\pi }^{-}$ decays of $(-15.4\pm 2.9)\times {10}^{-4}$. Since $D^0$ mesons can mix into ${\overline{D}}^0$ before decaying, and given the size of expected CP violation and mixing, the experimental asymmetry is a combination of the direct CP violation ($a_{CP}^{\mathrm{dir}}\left(f\right)$) and the mixing-induced CP violation

$$\begin{array}{c}\hfill A_{CP}\left(f\right)\approx a_{CP}^{\mathrm{dir}}\left(f\right)-{\displaystyle \frac{\langle t\left(f\right)\rangle }{\tau \left(D^0\right)}}{A}_{\Gamma }\left(f\right),\end{array}$$

(17)

where $\langle t\left(f\right)\rangle$ is the mean decay time of $D^0\to f$ decays in the reconstructed sample, $\tau \left(D^0\right)$ is the lifetime of the $D^0$ meson, and $A_{\Gamma }\left(f\right)$ is the the asymmetry between the effective $D^0$ and ${\overline{D}}^0$ decay widths [71]. In the limit of U-spin symmetry $a_{CP}^{\mathrm{dir}}\left(K^{+}{K}^{-}\right)=-a_{CP}^{\mathrm{dir}}\left({\pi }^{+}{\pi }^{-}\right)$, and by assuming $A_{\Gamma }$ to be the same for the two decays, the difference between the CP asymmetries in the two decays is

$$\begin{array}{c}\hfill \Delta A_{CP}\equiv A_{CP}\left(K^{+}{K}^{-}\right)-A_{CP}\left({\pi }^{+}{\pi }^{-}\right)\approx \Delta a_{CP}^{\mathrm{dir}}-{\displaystyle \frac{\Delta \langle t\rangle }{\tau \left(D^0\right)}}{A}_{\Gamma },\end{array}$$

(18)

where $\Delta \langle t\rangle$ is the difference in the mean decay times of the two decays. The experimental advantage of measuring the difference in the CP asymmetries is the cancellation of experimental asymmetries arising from the production and detection of the $D^0$ mesons. Furthermore, while $A_{\Gamma }$ is of the same order of magnitude as $\Delta A_{CP}$, the correction factor $\Delta \langle t\rangle /\tau \left(D^0\right)<0.1$ since the experimental acceptance is typically similar between the two decay modes. Studies are ongoing to measure CP violation separately for $D^0\to K^{+}{K}^{-}$ and $D^0\to {\pi }^{+}{\pi }^{-}$ decays, which involves removal of the nuisance production and detection asymmetries by means of control samples. The latest result, based on 4.7 fb⁻¹ data collected by the LHCb experiment [72], reported

$$\begin{array}{cc}\hfill a_{CP}^{\mathrm{dir}}\left(K^{+}{K}^{-}\right)& =(7.7\pm 5.7)\times {10}^{-4},\hfill \\ \hfill a_{CP}^{\mathrm{dir}}\left({\pi }^{+}{\pi }^{-}\right)& =(23.2\pm 6.1)\times {10}^{-4}.\hfill \end{array}$$

While the quest to measure direct CP violation from a single decay mode is ongoing, it is not the only challenge in the study of CP violation in charm decays. The mixing-induced CP violation in the charm sector is expected to be even smaller than the direct CP violation, and still escapes being measured. In the literature, CP-violating observables in charm decays are expressed in terms of the mixing parameters x and y, which are defined as

$$\begin{array}{cc}\hfill x& ={\displaystyle \frac{(m_1-m_2)}{\Gamma }},\hfill \\ \hfill y& ={\displaystyle \frac{{\Gamma }_1-{\Gamma }_2}{\Gamma }},\hfill \end{array}$$

(19)

where $m_{1,2}$ and ${\Gamma }_{1,2}$ are the mass and decay width of the two mass eigenstates $D_{1,2}$, respectively, and $\Gamma$ is the average decay width. The two mass eigenstates can be written as a linear combination of the flavor eigenstates

$$\begin{array}{c}\hfill |D_{1,2}\rangle =p|D^0\rangle \pm q|{\overline{D}}^0\rangle ,\end{array}$$

(20)

with the complex coefficients satisfying the condition ${\left|p\right|}^2+{\left|q\right|}^2=1$. In this formalism, CP violation in mixing can manifest itself as a deviation of $|q/p|$ from unity, while interference of mixing and decay can give rise to a non-zero phase difference ${\varphi }_f\equiv arg(q{\overline{A}}_f/p{A}_f)$ between the $D^0$ ($A_f$) and ${\overline{D}}^0$ (${\overline{A}}_f$) decay amplitudes. In the case of decays to the same final state, ${\varphi }_f=\varphi$. The time evolution of the decay rates of $D^0$ and ${\overline{D}}^0$ mesons can, therefore, be studied in terms of the mixing parameters x and y measured separately in $D^0$ and ${\overline{D}}^0$ decays. For convenience, these parameters are expressed in terms of the CP-averaged mixing parameters

$$\begin{array}{cc}\hfill x_{CP}& ={\displaystyle \frac{1}{2}}\left[xcos\varphi \left(\left|{\displaystyle \frac{q}{p}}\right|+\left|{\displaystyle \frac{p}{q}}\right|\right)+ysin\varphi \left(\left|{\displaystyle \frac{q}{p}}\right|-\left|{\displaystyle \frac{p}{q}}\right|\right)\right],\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill y_{CP}& ={\displaystyle \frac{1}{2}}\left[ycos\varphi \left(\left|{\displaystyle \frac{q}{p}}\right|+\left|{\displaystyle \frac{p}{q}}\right|\right)-xsin\varphi \left(\left|{\displaystyle \frac{q}{p}}\right|-\left|{\displaystyle \frac{p}{q}}\right|\right)\right],\hfill \end{array}$$

(22)

and the CP-violating differences

$$\begin{array}{cc}\hfill \Delta x& ={\displaystyle \frac{1}{2}}\left[xcos\varphi \left(\left|{\displaystyle \frac{q}{p}}\right|-\left|{\displaystyle \frac{p}{q}}\right|\right)+ysin\varphi \left(\left|{\displaystyle \frac{q}{p}}\right|+\left|{\displaystyle \frac{p}{q}}\right|\right)\right],\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \Delta y& ={\displaystyle \frac{1}{2}}\left[ycos\varphi \left(\left|{\displaystyle \frac{q}{p}}\right|-\left|{\displaystyle \frac{p}{q}}\right|\right)-xsin\varphi \left(\left|{\displaystyle \frac{q}{p}}\right|+\left|{\displaystyle \frac{p}{q}}\right|\right)\right].\hfill \end{array}$$

(24)

In the absence of CP violation ($\varphi =0$, $|q/p|=1$), the mixing parameters $x_{CP}=x$ and $y_{CP}=y$ and the CP-violating differences $\Delta x$ and $\Delta y$ are zero. The latest average of the mixing and CP violation parameters in the charm sector is [16]

$$\begin{array}{cc}\hfill x& =(4.07\pm 0.44)\times {10}^{-3},\hfill \\ \hfill y& =(6.{45}_{-0.23}^{+0.24})\times {10}^{-3},\hfill \\ \hfill |q/p|& =0.{994}_{-0.015}^{+0.016},\hfill \\ \hfill \varphi & =-2.6_{-1.2}^{+1.1},\hfill \end{array}$$

and the CP violation parameters are graphically shown in Figure 6. The data are so far compatible with the absence of CP violation in the charm sector up to 2.1 $\sigma$.

The golden channel to measure mixing-induced CP violation in the charm sector is the $D^0\to K_{\mathsf{S}}^0{\pi }^{+}{\pi }^{-}$ decay. It gives access to all the aforementioned observables at once, and has a relatively large branching ratio. Seminal studies performed at the B Factories were made through a time-dependent amplitude analysis of the decay [73,74]. Such analyses require an excellent understanding of the decay amplitude of the $D^0$ meson and especially of the time-dependent reconstruction efficiency of the experiment. This has not been possible at LHCb so far, given the limited amount of simulated data available. Therefore, the LHCb experiment has pursued an alternative method [75] to study $D^0\to K_{\mathsf{S}}^0{\pi }^{+}{\pi }^{-}$ decays, which relies on the measurement of the strong phase differences from BESIII [32]. This model-independent approach avoids the need of a time-dependent amplitude analysis, at the cost of limited sensitivity to the parameters associated to the width difference ($y_{CP}$ and $\Delta y$). By applying this technique, the LHCb experiment obtained the first observation of the mixing parameter x and the most precise determination of CP violation parameters in mixing at the time [76].

Complementarily to $D^0\to K_{\mathsf{S}}^0{\pi }^{+}{\pi }^{-}$ decays, there are ways of measuring the difference in the decay widths of $D^0$ and ${\overline{D}}^0$ mesons. By studying the evolution of the ratio of $D^0\to K^{+}{K}^{-}\left({\pi }^{+}{\pi }^{-}\right)$ over $D^0\to K^{-}{\pi }^{+}$ decays over time, it is possible to measure the mixing-induced CP violation parameter $y_{CP}$

$$\begin{array}{c}\hfill {\displaystyle \frac{\widehat{\Gamma }(D^0\to f)+\widehat{\Gamma }({\overline{D}}^0\to f)}{\widehat{\Gamma }(D^0\to K^{-}{\pi }^{+})+\widehat{\Gamma }({\overline{D}}^0\to K^{+}{\pi }^{-})}}-1=y_{CP}^f-y_{CP}^{K\pi }\end{array}$$

(25)

The best measurement to date of this parameter is performed by the LHCb experiment [77] and gives

$$\begin{array}{cc}\hfill y_{CP}^{\pi \pi }-y_{CP}^{K\pi }& =(6.57\pm 0.53\pm 0.16)\times {10}^{-3},\hfill \\ \hfill y_{CP}^{KK}-y_{CP}^{K\pi }& =(7.08\pm 0.30\pm 0.14)\times {10}^{-3},\hfill \end{array}$$

which is consistent with the world average of y reported above; therefore, no evidence of CP violation in mixing is found.

4. Experimental Status of Quark-Mixing Matrix Unitarity

An important aspect of the CKM matrix is that it is unitary, i.e., the sum of the squares of the elements in each row and column is equal to one. This indicates that there are only three families of quarks, and the total strength of the charged current couplings between each up-type (down-type) quark and all down-type (up-type) quarks is universally consistent. Within the SM, this is a consequence of the universality of non-abelian gauge couplings [5]. As such, it deserves experimental investigation that is achieved by measuring the strength of the couplings as the magnitude of the CKM matrix elements. The latest averages of the experimental measurements of the magnitudes of the CKM matrix elements are summarized in

Table 1. The latest averages of the magnitudes of the CKM elements [6]. Each element is given as $|V_{ij}|$ with $i,j$ being the row and column indices of the table.

	d	s	b
u	$0.97373\pm 0.00031$	$0.2243\pm 0.0008$	$(3.82\pm 0.20)\times {10}^{-3}$
c	$0.221\pm 0.004$	$0.975\pm 0.006$	$(40.8\pm 1.4)\times {10}^{-3}$
t	$(8.6\pm 0.2)\times {10}^{-3}$	$(41.5\pm 0.9)\times {10}^{-3}$	$1.014\pm 0.029$

.

A brief description of the most sensitive measurements of the CKM matrix elements is given below.

The most precise determination of $|V_{ud}^{\phantom{\ast }}|$ comes from the superallowed $0^{+}\to 0^{+}$ nuclear $\beta$-decay transitions, which are mediated by the weak interaction. A complete review of the experimental and theoretical aspects of superallowed $\beta$-decays can be found in Ref. [78].

The value of $|V_{uc}^{\phantom{\ast }}|$ is obtained from the study of semileptonic kaon decays of the type $K\to \pi {\ell }^{+}\nu$ ($\ell =e,\mu$), whose amplitude can be expressed as

$$\begin{array}{cc}\hfill \mathcal{M}& =-i{\displaystyle \frac{G_F}{\sqrt{2}}}{V}_{us}^{\phantom{\ast }}{L}^{\mu }{H}_{\mu }\hfill \end{array}$$

(26)

where $G_F$ is the Fermi constant, $L^{\mu }$ is the leptonic current, and $H_{\mu }$ is the hadronic current:

$$\begin{array}{cc}\hfill L^{\mu }& ={\ell }^{+}{\gamma }^{\mu }(1-{\gamma }^5){\nu }_{\ell },\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill H_{\mu }& =\langle \pi \left(p^{\prime }\right)\left|\overline{u}{\gamma }_{\mu }(1-{\gamma }^5)S\right|K\left(p\right)\rangle ,\hfill \end{array}$$

(28)

where p and $p^{\prime }$ are the momenta of K and $\pi$, respectively. Their product leads to an effective Hamiltonian that can be expressed as

$$\begin{array}{c}\hfill {\mathcal{H}}_{\mathrm{eff}}={\displaystyle \frac{G_F}{\sqrt{2}}}{V}_{us}^{\phantom{\ast }}\left[\overline{u}{\gamma }_{\mu }s-\overline{u}{\gamma }_{\mu }{\gamma }_{5}s\right]{\ell }^{+}{\gamma }^{\mu }(1-{\gamma }^5){\nu }_{\ell }.\end{array}$$

(29)

Since $K\to \pi {\ell }^{+}\nu$ is a pseudoscalar meson transition (K and $\pi$ have $J^P=0^{-}$), the axial-vector component of $H_{\mu }$ is zero due to constraints on the spin of the outgoing u quark. The vector component of $H_{\mu }$ can be expressed in terms of the form factors $f_{+}\left(q^2\right)$ and $f_0\left(q^2\right)$ as

$$\begin{array}{c}\hfill \langle \pi \left(p^{\prime }\right)|\overline{u}{\gamma }_{\mu }s|K\left(p\right)\rangle =f_{+}\left(q^2\right)\left(p_{\mu }+p_{\mu }^{\prime }-{\displaystyle \frac{m_K^2-m_{\pi }^2}{q^2}}{q}_{\mu }\right)+f_0\left(q^2\right){\displaystyle \frac{m_K^2-m_{\pi }^2}{q^2}}{q}^{\mu },\end{array}$$

(30)

where $q=p-p^{\prime }$ is the momentum transfer. Analyses of $K\to \pi {\ell }^{+}\nu$ decays often assume a linear dependence of the form factors $f_{+,0}\left(q^2\right)=f_{+}\left(0\right)\left[1+{\lambda }_{+,0}(q^2/m_{\pi }^2)\right]$ [6,79] and the decay rate can be expressed in terms of $|V_{uc}^{\phantom{\ast }}|f_{+}\left(0\right)$. By averaging the results of $K_{\mathsf{L}}^0\to \pi e\nu$, $K_{\mathsf{L}}^0\to \pi \mu \nu$, $K^{\pm }\to {\pi }^0{\ell }^{\pm }\nu$, $K^{\pm }\to {\pi }^0{\mu }^{\pm }\nu$, and $K_{\mathsf{S}}^0\to \pi e\nu$, the Particle Data Group (PDG) obtained the value of $|V_{us}^{\phantom{\ast }}|f_{+}\left(0\right)=0.21635\pm 0.00038$ [6]. For the form factor, the PDG used the value $f_{+}\left(0\right)=0.9698\pm 0.0017$ [80], to obtain the value of $|V_{uc}^{\phantom{\ast }}|$ reported in Table 1.

To complete the first row of the CKM matrix, the value of $|V_{ub}^{\phantom{\ast }}|$ is obtained from the study of the inclusive and exclusive semileptonic decays of the type $b\to u{\ell }^{+}\nu$ ($\ell =e,\mu$). The inclusive determination of $|V_{ub}^{\phantom{\ast }}|$ is based on the measurement of the total rate of $B\to X_u{\ell }^{+}\nu$ decays, where $X_u$ is a hadronic system that contains a u quark. This measurement is challenging due to the presence of the large background from the $B\to X_c{\ell }^{+}\nu$ decays, where $X_c$ are charm hadrons. The theoretical estimate of this effect is crucial to extract the value of $|V_{ub}^{\phantom{\ast }}|$ from the data. The B Factories and CLEO made significant advancements in the inclusive determination of $|V_{ub}^{\phantom{\ast }}|$ by using two different approaches. Initially they studied the inclusive electron momentum [81,82,83] to determine a partial decay rate near the kinematic endpoint. Once the number of B$\overline{B}$ pairs became large enough, they also developed a technique based on the full reconstruction of a (tagging) B meson and of the recoiling B decaying semileptonically [84,85,86].

Exclusive measurements of $|V_{ub}^{\phantom{\ast }}|$ are possible by studying various decay modes, $B\to \pi {\ell }^{+}\nu$ and $B\to \rho {\ell }^{+}\nu$ [87,88,89,90,91], ${\overline{\mathit{\Lambda }}}_b^0\to \overline{p}{\ell }^{+}\nu$ [92], and ${\overline{B}}_S^0\to K^{-}{\ell }^{+}\nu$ [93]. In the case of the B decays, the transition is described in terms of the form factors $f_{+}\left(q^2\right)$ and $f_0\left(q^2\right)$ as in Equation (30). For the ${\mathit{\Lambda }}_b^0$ decays, four additional form factors are needed to account for the polarization of the baryons [94]. Since determining the absolute branching fraction of a decay at LHCb (and at hadron colliders in general) is quite challenging, measurements of $|V_{ub}^{\phantom{\ast }}/V_{cb}^{\phantom{\ast }}|$ are rather made by studying the ratio of branching fractions with respect to the ${\overline{B}}_S^0\to D_S^{-}{\ell }^{+}\nu$ (${\overline{\mathit{\Lambda }}}_b^0\to {\overline{\mathit{\Lambda }}}_c^{-}{\ell }^{+}\nu$).

The averages of the inclusive and exclusive measurements of $|V_{ub}^{\phantom{\ast }}|$ are [6]:

$$\begin{array}{cc}\hfill |V_{ub}^{\phantom{\ast }}{|}_{\mathrm{incl}}& =(4.13\pm 0.12{\left(\exp \right)}_{-0.14}^{+0.13}\left(\mathrm{theo}\right)\pm 0.18(\Delta \mathrm{model}))\times {10}^{-3},\hfill \\ \hfill |V_{ub}^{\phantom{\ast }}{|}_{\mathrm{excl}}& =(3.70\pm 0.10\pm 0.12)\times {10}^{-3}.\hfill \end{array}$$

A tension between the two averages is observed. This is a matter of debate within the community; nevertheless, the two averages are combined after scaling the uncertainties to account for the tension [6], giving the result reported in Table 1.

Moving to the second row, $|V_{cd}^{\phantom{\ast }}|$ can be determined from $D\to \pi {\ell }^{+}\nu$ decays. Experimental measurements from BaBar [95], BESIII [96,97], CLEO [98], and Belle [99] have been combined in conjunction with input from lattice QCD calculations (needed to estimate the form factor $f_{+}^{D\pi }\left(0\right)=0.612\pm 0.035$ [80]) to extract the value of $|V_{cd}^{\phantom{\ast }}|=0.2330\pm 0.0029\pm 0.0133$, where the first uncertainty is experimental and the second theoretical from the form factor determination. Alternative ways of determining $|V_{cd}^{\phantom{\ast }}|$ are from the study of $D^{+}\to {\mu }^{+}\nu$ and $D^{+}\to {\tau }^{+}\nu$ decays [100,101,102], and neutrino scattering data [103,104,105], yielding $|V_{cd}^{\phantom{\ast }}|=0.2181\pm 0.0049\pm 0.0007$ [16] and $0.230\pm 0.011$ [6], respectively.

The value of $|V_{cs}^{\phantom{\ast }}|$ is obtained directly from the branching fraction of $D_s^{+}\to {\mu }^{+}\nu$ and $D_s^{+}\to {\tau }^{+}\nu$ decays, using the lattice QCD calculation of the semileptonic $D_s^{+}$ decay constant [80], giving $|V_{cs}^{\phantom{\ast }}|=0.984\pm 0.012$ [6]. Another approach relies on lattice QCD calculations of the $D\to K{\ell }^{+}\nu$ form factors [80] and the experimental measurement of the branching fraction of $D\to K{\ell }^{+}\nu$ decays to obtain $|V_{cs}^{\phantom{\ast }}|=0.972\pm 0.007$ [6]. The average of these two values is reported in Table 1.

The magnitude of $|V_{cb}^{\phantom{\ast }}|$ is obtained from the study of inclusive and exclusive semileptonic decays of the type $B\to X_c{\ell }^{+}\nu$ ($\ell =e,\mu$). The form factors for the B decays are calculated with lattice QCD methods by various collaborations [106,107,108,109,110,111]. Exclusive determinations make use of the decays of B mesons to the ground states of D and $D^{\ast }$ charm mesons. The most recent analyses of $B\to D^{\ast }{\ell }^{+}\nu$ decays have been performed by BaBar [112], Belle [113,114], and Belle II [115], and they all study the kinematic distribution of the decay products in a four-dimensional space to extract the value of $|V_{cb}^{\phantom{\ast }}|$. In the analysis of $B\to D{\ell }^{+}\nu$ decays, only the product of the four momenta of the initial- and final-state hadrons is studied to extract $|V_{cb}^{\phantom{\ast }}|$. BaBar [116] and Belle [117] obtained results compatible with the $B\to D^{\ast }{\ell }^{+}\nu$ decays.

Not only B^± mesons can be used to determine $|V_{cb}^{\phantom{\ast }}|$, but also $B_s^0$, $B_c^{+}$, and ${\mathit{\Lambda }}_b^0$ hadrons. The LHCb collaboration studied $B_S^0\to D_S^{(\ast )}{\ell }^{+}\nu$ [118] decays, measuring $|V_{cb}^{\phantom{\ast }}|$ with a precision comparable to the theoretical uncertainties, even though not competitive yet with the B measurements from the B Factories. In this perspective, $B_c^{+}\to {\tau }^{+}\nu$ decays can also be used to determine the value of $|V_{cb}^{\phantom{\ast }}|$ with small theoretical uncertainties, but they are difficult to reconstruct at a hadron collider and they will be studied in a future $e^{+}{e}^{-}$ facility [119].

The inclusive determination of $|V_{cb}^{\phantom{\ast }}|$ has been investigated by multiple experiments through the measurements of moments as a function of either the minimum lepton momentum [120,121,122,123,124,125,126,127,128], or the squared lepton invariant mass [129,130].

The averages of the exclusive and inclusive measurements of $|V_{cb}^{\phantom{\ast }}|$ are [6]:

$$\begin{array}{cc}\hfill |V_{cb}^{\phantom{\ast }}{|}_{\mathrm{excl}}& =(42.2\pm 0.5)\times {10}^{-3},\hfill \\ \hfill |V_{cb}^{\phantom{\ast }}{|}_{\mathrm{incl}}& =(39.8\pm 0.6)\times {10}^{-3}.\hfill \end{array}$$

Marginal consistency between the two averages is observed, and the uncertainties are scaled to account for this before combining the two values [6], resulting in the reported values in Table 1.

The measurements of the matrix elements involving the t quark is challenging due to its large mass. Even when boosted to the energies of the LHC, the t quark does not hadronize or form bound states because its decay length is shorter than the typical scale of the hadronization process. Therefore, $|V_{td}^{\phantom{\ast }}|$ and $|V_{tc}^{\phantom{\ast }}|$ are not likely to be measured in tree-level decay processes, rather they are determined from B–$\overline{B}$ mixing processes, where the t quark is involved in the box diagram (see Appendix A). In particular, the mass differences between the two mass eigenstates of the B mesons, $\Delta m_d$ and $\Delta m_s$, are related to the $|V_{td}^{\phantom{\ast }}|$ and $|V_{tc}^{\phantom{\ast }}|$ matrix elements, respectively, enabling their determination. Many experiments have measured $\Delta m_d$, whose average $0.5065\pm 0.0019 \mathsf{p}{\mathsf{s}}^{-1}$ [6] is dominated by the latest LHCb measurement using $B^0\to D^{(\ast )-}{\mu }^{+}{\nu }_{\mu }X$ decays [131]. The average of the measurements of $\Delta m_s$ is $17.765\pm 0.006 \mathsf{p}{\mathsf{s}}^{-1}$ [6], with the most precise measurement provided by the LHCb collaboration using $B_S^0\to D_S^{-}{\pi }^{+}$ decays [132].

The value of $|V_{tb}^{\phantom{\ast }}|$ can be determined either by assuming the unitarity of the CKM matrix or without making this assumption. In the first case, the ratio of branching fractions $R=\mathcal{B}(t\to Wb)/\mathcal{B}(t\to Wq)=|V_{tb}^{\phantom{\ast }}{|}^2/({\sum }_q|V_{tq}{|}^2)=|V_{tb}^{\phantom{\ast }}{|}^2$, where $q=b,s,d$. This measurement was made during Run II of the Tevatron by CDF [133] and D0 [134], and by CMS [135] at LHC obtaining $|V_{tb}^{\phantom{\ast }}|>0.975$ at the 95% confidence level. In the second case, $|V_{tb}^{\phantom{\ast }}|$ can be measured from the single top quark production cross-section. Measurements of this cross-section have been made at Tevatron by CDF and D0 [136], and at LHC by ATLAS and CMS [137]. The value reported in Table 1 for $|V_{tb}^{\phantom{\ast }}|$ is the average of this second set of measurements.

Tests of the unitarity of the CKM matrix are made by verifying the equality to 1 of the sum of the squared of the matrix elements along each row and column:

$$\begin{array}{cc}\hfill |V_{ud}^{\phantom{\ast }}{|}^2+|V_{us}^{\phantom{\ast }}{|}^2+{\left|V_{ub}^{\phantom{\ast }}\right|}^2& =0.9985\pm 0.0007,\hfill \\ \hfill |V_{cd}^{\phantom{\ast }}{|}^2+|V_{cs}^{\phantom{\ast }}{|}^2+{\left|V_{cb}^{\phantom{\ast }}\right|}^2& =1.001\pm 0.012,\hfill \\ \hfill |V_{td}^{\phantom{\ast }}{|}^2+|V_{ts}^{\phantom{\ast }}{|}^2+{\left|V_{tb}^{\phantom{\ast }}\right|}^2& =1.03\pm 0.06,\hfill \\ \hfill |V_{ud}^{\phantom{\ast }}{|}^2+|V_{cd}^{\phantom{\ast }}{|}^2+{\left|V_{td}^{\phantom{\ast }}\right|}^2& =0.9971\pm 0.0019,\hfill \\ \hfill |V_{us}^{\phantom{\ast }}{|}^2+|V_{cs}^{\phantom{\ast }}{|}^2+{\left|V_{ts}^{\phantom{\ast }}\right|}^2& =1.003\pm 0.012,\hfill \\ \hfill |V_{tb}^{\phantom{\ast }}{|}^2+|V_{cb}^{\phantom{\ast }}{|}^2+{\left|V_{tb}^{\phantom{\ast }}\right|}^2& =1.03\pm 0.06.\hfill \end{array}$$

All of the unitarity relations are verified within the uncertainties, except the first row that shows a tension of 2.2 standard deviations. A smaller tension is also observed in the first column, but it is not statistically significant. In both cases the tension is driven by the determination of $|V_{ud}^{\phantom{\ast }}|$, and is generally called the Cabibbo angle anomaly.

5. The High Intensity Frontier

A tremendous improvement in the precision of the measurements of CP violation in the B and D meson systems has been achieved in the last decade. Figure 7 shows how much the constraints on the CKM unitarity triangle have improved since 2012. In particular, the uncertainty on the angle $\gamma$ is reduced by more than a factor of 2 with the constraints from the analysis of LHCb data.

The quest for ultimate precision in flavor physics studies is not over yet, and the high intensity frontier is the next step in this direction. The LHCb experiment is already running its first upgrade [138] and the Belle II experiment is in the process of taking data at its design capabilities [139]. The two experiments are expected to provide a significant improvement in the precision of the measurements of CP violation and the CKM matrix elements in the B and D meson systems. In particular, the LHCb upgrade should further reduce the uncertainty on the angle $\gamma$ down to less than $1^{\circ }$, measure ${\varphi }_c$ with a precision of less than 20% of the Standard Model, and have sensitivity to evidence of CP violation in the interference between decay and mixing of $D^0$ decays [140].

Similarly, Belle II should improve the precision on the angles $\alpha$ and $\beta$ by a factor of 2 [141]. Improved determinations of the CKM matrix elements will provide stringent tests of the CKM paradigm and will be sensitive to new physics effects at the loop level. Most importantly, the physics capabilities of the two experiments are complementary: LHCb excels in high efficiency for charged final states and benefits from larger cross-sections, while Belle II achieves high efficiency for neutral final states and offers a larger acceptance. Therefore, the combination of the two experiments will provide a complete picture.

Finally, the LHCb collaboration is proposing a second upgrade of the experiment, called LHCb Upgrade II [142], which will collect an integrated luminosity of 300 fb⁻¹ to test the CKM paradigm with unprecedented precision by the end of 2041, when CERN will stop the LHC operations.

6. Conclusions

Flavor physics is a fundamental part of the Standard Model of particle physics, and the study of CP violation in the B and D meson systems is a key ingredient to test the CKM paradigm. In the last two decades CP violation has been established in the B meson system, and it has been observed recently also in the D meson system by the LHCb experiment.

Despite the tremendous improvement in the precision of the measurements of CP violation in the B and D meson systems, effects beyond the Standard Model have not been observed yet. Small inconsistencies in the measurements of CP violation in the B meson system are present, but they are not statistically significant. The quest for ultimate precision in CP violation studies is not over yet, and the high intensity frontier is the next step in this direction. The next decade will be crucial to test the CKM paradigm with unprecedented precision, and the LHCb and Belle II experiments are expected to play a key role in this quest, urging an update to this review. Possibly the last word will be given by the LHCb Upgrade II that will reach ultimate precision in flavor physics for our generation.