(iii) the search for decays of the observed CP-even Higgs h boson into a pair of a bosons;

Radiative decays $\Upsilon \to a\gamma$ have been searched for in various colliders, the most recent results are searches for radiative decays of the $\Upsilon(1s)$ to $a\gamma$ with a subsequent decay of the $a$ boson to a pair of taus at CLEO [556], and the radiative decays of the $\Upsilon(1s, 2s, 3s)$ to $a\gamma$ with subsequent decays to a pair of muons or taus by BaBar [519, 520].

Direct inclusive searches for the light pseudo scalar $a$ boson were performed in the $a \to \mu\mu$ channel at the Tevatron by D0 [514] and by ATLAS [506], CMS [507], and LHCb [164] at the LHC.

Finally, searches for the decays of the Higgs boson to a pair of $a$ bosons were performed with subsequent decays to four photons, in the four muons final state, in the two muons and two taus final state, and in the four taus final state.

No significant excess in the searches for a light CP-odd $a$ boson was found and limits on the production times branching fractions of the $a$ boson have been set.

References for all these searches are summarised in Table 11.11.

### 11.6.7.8 Searches in Higgs decays

Dark-photons or axion-like particles searches can also be carried out within the decays of the Higgs boson to four leptons through dilepton pair resonances. Such search has been carried out by ATLAS and CMS [557]. Alternatively signatures of dark photons escaping detection with mising transverse momentum signatures have also been searched for by ATLAS [558].

### 11.6.7.9 Searches for doubly charged Higgs bosons $H^{\pm\pm}$

As discussed in Section 11.6.5, the generation of small neutrino masses via the standard EWSB mechanism described in Section 11.2 requires unnaturally small Yukawa couplings, provided that neutrinos are Dirac-type fermions. A Majorana mass term with a see-saw mechanism for neutrinos, would allow for naturally small masses and would also yield a framework for the appealing scenario of leptogenesis. However, within the SM, Majorana mass terms correspond to (non-renormalizable) dimension-5 operators. Such effective interactions can be generated via renormalisable interactions with an electroweak triplet of complex scalar fields (corresponding to a type-II see-saw mechanism). Other models such as the Zee-Babu model, with the introduction of two $SU(2)_L$ singlets, also generate Majorana mass terms. The signature of such models would be the presence of doubly charged Higgs bosons $H^{\pm\pm}$.

The main production mechanisms of $H^{\pm\pm}$ bosons at hadron colliders are the pair production in the s-channel through the exchange of a Z boson or a photon and the associated production with a charged Higgs boson through the exchange of a W boson. Various searches for doubly charged Higgs bosons have been performed by ATLAS and CMS at Run 1 [559] and Run 2 [521, 522]. Typically, these searches aim at low values of the Higgs triplet vacuum expectation for which the doubly charged Higgs boson will decay mostly to leptons (for high values, the decay to W bosons will become predominant). These searches assume that the coupling to W bosons is negligible and that the main production mode is through the Drell-Yan process.

### 11.6.7.10 Searches for non-standard production processes of the Higgs boson

The discovery of the Higgs boson has also allowed for searches of BSM processes involving standard decays of the Higgs boson. One example directly pertaining to the search for additional states of the EWSB sector is the search for Higgs bosons in the cascade decay of a heavy CP-even Higgs boson decaying to charged Higgs boson and a W boson, and the charged Higgs boson subsequently decaying to H and another W boson. This search has been performed by ATLAS in $b\bar{b}$ decays of the 125 GeV Higgs boson [560].

### 11.6.7.11 Outlook on searches for additional states

The LHC program of searches for additional states covers a large variety of decay and production channels. Since the Higgs boson discovery, many new channels have been explored at the LHC, e.g., the searches for additional states decaying into $hh$ or $Vh$ or $ZA$. The search for charged Higgs bosons has been extended to include the WZ, WA and the very difficult $t\bar{b}$ decay channel.

## 11.7 Summary and outlook

**Summary**– The discovery of the Higgs boson is a major milestone in the history of particle physics as well as an extraordinary achievement of the LHC machine and the ATLAS and CMS experiments. Eleven years after the discovery, substantial progress in the field of Higgs boson physics has been accomplished and a significant number of measurements probing the nature of this unique particle have been made. They are revealing an increasingly precise profile of the Higgs boson.

The LHC has concluded its Run 2 in 2018, delivering a dataset of 13 TeV pp collisions corresponding to an integrated luminosity of more than $150\,\text{fb}^{-1}$ of data collected by ATLAS and CMS. The Run 3 started in 2022 at 13.6 TeV and more than $70\,\text{fb}^{-1}$ of additional data were delivered. First cross section measurements at this new centre-of-mass energy have been presented and are reported herein. With the substantial increase in production rates at the higher center-of-mass energy (mostly from Run 1 to Run 2) and the larger datasets, new landmark results in Higgs physics have been achieved. They are highlighted in the 10th anniversary papers of the Higgs boson discovery [183, 184].

Three new results of fundamental importance have been achieved with Run-2 datasets by ATLAS and CMS independently: (i) the clear and unambiguous observation of the Higgs boson decay to taus; (ii) the clear and unambiguous observation of the Higgs boson decay to a pair of b quarks; (iii) the clear and unambiguous observation of the production of the Higgs boson through the $t\bar{t}H$ process. These results provide direct evidence for the Yukawa coupling of the Higgs boson to fermions of the third generation: taus, bottom quarks and top quarks, at rates compatible with those expected in the SM. In addition, the evidence for the decay of the Higgs boson to muons is being consolidated. These, and all other experimental measurements, are consistent with the EWSB mechanism of the SM.

New theoretical calculations and developments in Monte-Carlo simulations pertaining to Higgs physics are still occurring at a rapid pace. For example, the theoretical prediction for the dominant gluon fusion production mode now includes the latest N3LO result at the inclusive and differential levels, which is twice as precise as previous N2LO calculations. With these improvements in the state-of-the-art theory predictions and the increase in luminosity and center-of-mass energy, Higgs physics has definitively entered a precision era. Its impact can already be seen on the latest Run-2 combined measurements of the Higgs boson couplings (see Section 11.5).

Since the discovery of the Higgs boson, new ideas have emerged to probe its rare decays and production modes, as well as to indirectly measure the Higgs boson width through the study of its off-shell couplings, or via on-shell interference effects. The Higgs boson has now become part of the standard toolkit in searches for new physics, with Higgs measurements being used in SMEFT global fits.

Many extensions of the SM at higher energies call for an enlargement of the EWSB sector. Hence, direct searches for additional scalar states can provide valuable insights on the dynamics of the EWSB mechanism. The ATLAS and CMS experiments have searched for additional Higgs bosons in the Run-2 data, and have imposed constraints in broad ranges of mass and couplings for various scenarios with an extended Higgs sector.

The landscape of Higgs physics has been scrutinized extensively since its discovery. The current precisions on the measurements of the couplings of the Higgs boson to gauge bosons and third generation fermions are typically of the order of 10–20%. The uncertainty on the Higgs boson coupling to the muon shrank below 50%, and the upper limits on the branching fraction to new invisible particles are approximately 10%. Sensitivity to the Higgs boson self-coupling has not reached the SM value yet, and there is no information on how the Higgs field acquired its VEV in the early times of the Universe. However, significant progress in being made through exploration of di-Higgs production, including new analyses based on machine learning algorithms. This situation allows for new challenges to ultimately increase further the reach in
precision and it also widens the possibilities of unveiling the true nature and the dynamics of the electroweak symmetry breaking mechanism.

**Outlook-** The unitarisation of the vector boson scattering (VBS) amplitudes, dominated at high energies by their longitudinal polarisation, has been the basis of the *no lose* theorem at the LHC, and was a determining consideration in the building of the accelerator and detectors. It motivated the existence of a Higgs boson or the observability of manifestations of strong dynamics at the TeV scale. Now that a Higgs boson has been found and its couplings to gauge bosons are - within current accuracy - consistent with the SM predictions, perturbative unitarity is preserved to a large extent with the sole exchange of the Higgs boson, and without the need for any additional states. VBS remains still an important channel to further investigate in order to better understand the nature of the Higgs sector and the possible completion of the SM at the TeV scale. In association with the double Higgs boson production channel by vector boson fusion, VBS could, for instance, confirm that the Higgs boson is part of a weak doublet and also establish whether it is an elementary object or a composite state that could emerge as a pseudo-Nambu-Goldstone boson from a new underlying broken symmetry.

The fermion-Higgs boson couplings are not governed by local gauge symmetry. Thus, in addition to a new particle, the LHC has also discovered a new force, different in nature from the other fundamental interactions since it is non-universal and distinguishes between the three families of quarks and leptons. The existence of the Higgs boson embodies the problem of an unnatural cancellation among the quantum corrections to its mass if new physics is present at scales significantly higher than the EW scale. The non-observation of additional states which could stabilise the Higgs boson mass is a challenge for natural scenarios like Supersymmetry or models with a new strong interaction in which the Higgs boson is not a fundamental particle. This increasingly pressing paradox starts questioning the principle of naturalness and it also inspires new solutions tied to the cosmological evolution of the Universe.

The search for the Higgs boson has occupied the particle physics community for the last 50 years. Its discovery has shaped and sharpened the physics programs of the LHC and of prospective future accelerators [126,561]. With respect to current data, at the HL-LHC the precision will improve by a factor of approximately 5–10 depending on the observables considered. Table 11.8 displays the expected sensitivities in the characterization of the Higgs boson at HL-LHC: in this table, the parameters $\kappa_i$ specify by how much the coupling of the Higgs boson to a given particle $i$ deviates from the SM expectation. The only channels which are expected to be limited by data statistics are the rare decays to muons and $Z\gamma$. In all other cases, the experimental systematic uncertainties are similar to the statistical uncertainties, but the dominant source of uncertainty arises from theory, and this remains the case even after assuming that, by the end of the HL-LHC run, the theory uncertainties can be reduced by a factor two compared to the current uncertainties, a hypothesis that appears realistic but still requires dedicated and concerted work [118]. For both hadron and lepton colliders, some theoretical progress is crucial to fully exploit and capitalise on the experimental data. In particular, the expected HL-LHC data together with rapid ongoing progress in theoretical calculations are defining a new era of precision Higgs boson measurements.

## Acknowledgements

We would like to thank many of our colleagues for proofreading this review, for useful criticism and for their input in general: W. Altmannshofer, J. de Blas, G. Branco, G. Buchalla, J. Campbell, F. Caola, F. Cerutti, C. Csáki, R. Contino, J. Conway, N. Craig, A. David, S. Dawson, J.B. De Vivie, J. D'Hondt, G. Durieux, C. Englert, J.R. Espinosa, A. Falkowski, L. Fayard, W. Fischer, S. Forte, S. Gori, M. Grazzini, J. Gu, H. Haber, B. Heinemann, S. Heinemeyer, J. Hubisz, A. Korytov, B. Jäger, H. Ji, T. Junk, P. Langacker, J. Lykken, F. Maltoni, M.L. Mangano, B. Mansoulié, M. McCullough, R. Mishra, M. Mühlleitner, B. Murray, M. Neubert, A. Nisati, Y. Paul,

G. Perez, G. Petrucciani, A. Pomarol, E. Pontón, R. Rattazzi, D. Rebuzzi, L. Reina, F. Riva, R. Salerno, J. Santiago, E. Salvioni, N. Shah, G. Shaughnessy, M. Spira, A. Strumia, K. Tackmann, R. Tanaka, J. Terning, A. Vartak, C. Wagner, N. Wardle, A. Weiler, A. Wulzer, and G. Zanderighi. We are also most grateful to the ATLAS, CDF, CMS and D0 collaborations for their help with this review.

M.C. is supported by Fermilab, that is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy. C.G. is supported by the Helmholtz Association and by the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy - EXC 2121 "Quantum Universe" - 390833306. V.S. is supported by the grant DE-SC0009919 of the United States Department of Energy.

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12. CKM Quark-Mixing Matrix

Revised April 2024 by A. Ceccucci (CERN), Z. Ligeti (LBNL) and Y. Sakai (KEK).

12.1 Introduction

The masses and mixings of quarks have a common origin in the Standard Model (SM). They arise from the Yukawa interactions with the Higgs condensate,

$$
\mathcal{L}_Y = -Y_{ij}^d \overline{Q}_{Li} \phi d_{Rj}^I - Y_{ij}^u \overline{Q}_{Li} \epsilon \phi^* u_{Rj}^I + \text{h.c.}, \quad (12.1)
$$

where $Y^{u,d}$ are $3 \times 3$ complex matrices, $\phi$ is the Higgs field, $i, j$ are generation labels, and $\epsilon$ is the $2 \times 2$ antisymmetric tensor. $Q_L^I$ are left-handed quark doublets, and $d_R^I$ and $u_R^I$ are right-handed down- and up-type quark singlets, respectively, in the weak-eigenstate basis. When $\phi$ acquires a vacuum expectation value, $\langle\phi\rangle = (0, v/\sqrt{2})$, Eq. (12.1) yields mass terms for the quarks. The physical states are obtained by diagonalizing $Y^{u,d}$

by four unitary matrices, $V_{L,R}^{u,d}$, as $M_{\text{diag}}^f = V_L^f Y^f V_R^{f\dagger} (v/\sqrt{2})$, $f=u,d$. As a result, the charged-current $W^\pm$ interactions couple to the physical $u_{Lj}$ and $d_{Lk}$ quarks with couplings given by

$$
\begin{equation}
\begin{split}
& \frac{-g}{\sqrt{2}} (\overline{u_L}, \overline{c_L}, \overline{t_L}) \gamma^\mu W_\mu^+ V_{\text{CKM}} \begin{pmatrix} d_L \\ s_L \\ b_L \end{pmatrix} + \text{h.c.}, \\
& V_{\text{CKM}} \equiv V_L^u V_L^{d\dagger} = \begin{pmatrix} V_{\text{ud}} & V_{\text{us}} & V_{\text{ub}} \\ V_{\text{cd}} & V_{\text{cs}} & V_{\text{cb}} \\ V_{\text{td}} & V_{\text{ts}} & V_{\text{tb}} \end{pmatrix}.
\end{split}
\tag{12.2}
\end{equation}
$$

This Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2] is a 3×3 unitary matrix. It can be parameterized by three mixing angles and the CP-violating KM phase [2]. Of the many possible conventions, a standard choice has become [3]

$$
\begin{align}
V_{\text{CKM}} &= \begin{pmatrix} 1 & 0 & 0 \\ 0 & c_{23} & s_{23} \\ 0 & -s_{23} & c_{23} \end{pmatrix} \begin{pmatrix} c_{13} & 0 & s_{13} e^{-i\delta} \\ 0 & 1 & 0 \\ -s_{13} e^{i\delta} & 0 & c_{13} \end{pmatrix} \begin{pmatrix} c_{12} & s_{12} & 0 \\ -s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \end{pmatrix} \nonumber \\
&= \begin{pmatrix} c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i\delta} \\ -s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i\delta} & c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i\delta} & s_{23} c_{13} \\ s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i\delta} & -c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i\delta} & c_{23} c_{13} \end{pmatrix}, \tag{12.3}
\end{align}
$$

where $s_{ij} = \sin \theta_{ij}$, $c_{ij} = \cos \theta_{ij}$, and $\delta$ is the phase responsible for all CP-violating phenomena in flavor-changing processes in the SM. The angles $\theta_{ij}$ can be chosen to lie in the first quadrant, so $s_{ij}, c_{ij} \ge 0$.

It is known experimentally that $s_{13} \ll s_{23} \ll s_{12} \ll 1$, and it is convenient to exhibit this hierarchy using the Wolfenstein parameterization. We define [4–6]

$$
\begin{align}
s_{12} &= \lambda = \frac{|V_{us}|}{\sqrt{|V_{ud}|^2 + |V_{us}|^2}}, & s_{23} &= A\lambda^2 = \lambda \left|\frac{V_{cb}}{V_{us}}\right|, \nonumber \\
s_{13} e^{i\delta} &= V_{ub}^* = A\lambda^3(\rho + i\eta) = \frac{A\lambda^3(\bar{\rho} + i\bar{\eta})\sqrt{1-A^2\lambda^4}}{\sqrt{1-\lambda^2}[1-A^2\lambda^4(\bar{\rho} + i\bar{\eta})]}. \tag{12.4}
\end{align}
$$

These relations ensure that $\bar{\rho}+i\bar{\eta} = -(V_{ud}V_{ub}^*)/(V_{cd}V_{cb}^*)$ is phase
convention independent, and the CKM matrix written in terms
of $\lambda, A, \bar{\rho}$, and $\bar{\eta}$ is unitary to all orders in $\lambda$. The definitions
of $\bar{\rho}, \bar{\eta}$ reproduce all approximate results in the literature; i.e.,
$\bar{\rho} = \rho(1-\lambda^2/2+\dots)$ and $\bar{\eta} = \eta(1-\lambda^2/2+\dots)$, and one can write
$V_{\text{CKM}}$ to $\mathcal{O}(\lambda^4)$ either in terms of $\bar{\rho}, \bar{\eta}$ or, traditionally,

$$
V_{\text{CKM}} = \begin{pmatrix} 1 - \lambda^2/2 & \lambda & A\lambda^3(\rho - i\eta) \\ -\lambda & 1 - \lambda^2/2 & A\lambda^2 \\ A\lambda^3(1 - \rho - i\eta) & -A\lambda^2 & 1 \end{pmatrix} + \mathcal{O}(\lambda^4). \quad (12.5)
$$

The CKM matrix elements are fundamental parameters of the SM, so their precise determination is important. The unitarity of the CKM matrix imposes $\sum_i V_{ij} V_{ik}^* = \delta_{jk}$ and $\sum_j V_{ij} V_{kj}^* = \delta_{ik}$. The six vanishing combinations can be represented as triangles in a complex plane, of which those obtained by taking scalar products of neighboring rows or columns are nearly degenerate. The areas of all triangles are the same, half of the Jarlskog invariant, $J$ [7], which is a phase-convention-independent measure of CP violation, defined by $\text{Im}[V_{ij}V_{kl}V_{il}^*V_{kj}^*] = J \sum_{m,n} \epsilon_{ikm}\epsilon_{jln}$.

The most commonly used unitarity triangle arises from

$$
V_{ud} V_{ub}^* + V_{cd} V_{cb}^* + V_{td} V_{tb}^* = 0, \quad (12.6)
$$

by dividing each side by $V_{cd}V_{cb}^*$ (see Fig. 12.1). Its vertices are
exactly (0,0), (1,0), and, due to the definition in Eq. (12.4),
($\bar{\rho}, \bar{\eta}$). An important goal of flavor physics is to overconstrain
the CKM elements, and many measurements can be conveniently
displayed and compared in the $\bar{\rho}, \bar{\eta}$ plane. While the Lagrangian in
Eq. (12.1) is renormalized, and the CKM matrix has a well-known

<figure><img src="image_46.png" /><figcaption>Figure 12.1: Sketch of the unitarity triangle.</figcaption></figure>

scale dependence above the weak scale [8], below $\mu = m_W$ the CKM elements can be treated as constants, with all $\mu$-dependence contained in the running of quark masses and higher-dimension operators.

Unless explicitly stated otherwise, we describe all measurements assuming the SM, to extract magnitudes and phases of CKM elements in Sec. 12.2 and 12.3. Processes dominated by loop-level contributions in the SM are particularly sensitive to new physics beyond the SM (BSM). We give the global fit results for the CKM elements in Sec. 12.4, and discuss some implications for beyond standard model physics in Sec. 12.5.

12.2 Magnitudes of CKM elements

12.2.1 $|V_{ud}|$

The most precise determination of $|V_{ud}|$ comes from the study of superallowed $0^+ \to 0^+$ nuclear beta decays, which are pure vector transitions. Taking the average of the fifteen most precise determinations [9] yields [10]

$$
|V_{ud}| = 0.97367 \pm 0.00032. \qquad (12.7)
$$

This uncertainty is slightly more than twice as large as that in the
2020 edition, due to a more conservative estimate of the nuclear
structure uncertainties. A less precise determination of |*V*<sub>ud</sub>| can
be obtained from the measurement of the neutron lifetime. The
theoretical uncertainties are very small, but the determination
is limited by the knowledge of the ratio of the axial-vector and
vector couplings, g<sub>A</sub> = G<sub>A</sub>/G<sub>V</sub> [10]. The PIBETA experiment
[11] has improved the measurement of the π<sup>+</sup> → π<sup>0</sup>e<sup>+</sup>ν branching
ratio to 0.6%, and Ref. [12] quotes |*V*<sub>ud</sub>| = 0.9739 ± 0.0027, in
agreement with the more precise result listed above. The interest in this measurement is that the determination of $|V_{ud}|$ is very clean theoretically, because it is a pure vector transition and is free from nuclear-structure uncertainties.

### 12.2.2 $|V_{us}|$

The product of $|V_{us}|$ and the form factor at $q^2 = 0$, $|V_{us}|f_+(0)$, has been extracted traditionally from $K_L^0 \to \pi e \nu$ decays in order to avoid isospin-breaking corrections ($\pi^0 - \eta$ mixing) that affect $K^\pm$ semileptonic decay, and the complications induced by a second (scalar) form factor present in the muonic decays. The last round of measurements has led to enough experimental constraints to justify the comparison between different decay modes. Systematic errors related to the experimental quantities, e.g., the lifetime of neutral or charged kaons, and the form factor determinations for electron and muonic decays, differ among decay modes, and the consistency between different determinations enhances the confidence in the final result. For this reason, we follow the prescription [13] to average $K_L^0 \to \pi e \nu$, $K_L^0 \to \pi \mu \nu$, $K^\pm \to \pi^0 e^\pm \nu$, $K^\pm \to \pi^0 \mu^\pm \nu$ and $K_S^0 \to \pi e \nu$. The average of these five decay modes yields $|V_{us}|f_+(0) = 0.21656 \pm 0.00035$. Results obtained from each decay mode, and exhaustive references to the experimental data, are listed for instance in Ref. [10]. The form factor average $f_+(0) = 0.9698 \pm 0.0017$ [14] from $N_f = 2 + 1 + 1$ lattice QCD calculations gives $|V_{us}| = 0.2233 \pm 0.0005$ [10].<fnref n="1" /> The broadly used classic calculation of $f_+(0)$ [16] is in good agreement with this value, while other calculations [18] differ by as much as 2%.

The calculation of the ratio of the kaon and pion decay constants enables one to extract $|V_{us}/V_{ud}|$ from $K \to \mu\nu(\gamma)$ and $\pi \to \mu\nu(\gamma)$, where $(\gamma)$ indicates that radiative decays are included [19]. The value of $\Gamma(K \to \mu\nu(\gamma))$ [10] derived from the KLOE measurement of the corresponding branching ratio [20], combined with the lattice QCD result, $f_K/f_\pi = 1.1932 \pm 0.0021$ [14], leads to $|V_{us}| = 0.2250 \pm 0.0004$, where the accuracy is limited by the knowledge of the ratio of the decay constants. The average of these two determinations, with the error scaled according to the PDG prescription [21] by $\sqrt{\chi^2} = 2.5$, is quoted as [10]

$$|V_{us}| = 0.22431 \pm 0.00085. \qquad (12.8)$$

It is important to include both QED and QCD sources of isospin violations in the lattice QCD calculations.

The latest determination from hyperon decays can be found in Ref. [22]. The authors focus on the analysis of the vector form factor, protected from first order flavor SU(3) breaking effects by the Ademollo-Gatto theorem [23], and treat the ratio between the axial and vector form factors $g_1/f_1$ as experimental input, thus avoiding first order SU(3) breaking effects in the axial-vector contribution. They find $|V_{us}| = 0.2250 \pm 0.0027$, although this does not include an estimate of the theoretical uncertainty due to second-order SU(3) breaking, contrary to Eq. (12.8). Concerning hadronic $\tau$ decays to strange particles, averaging the inclusive decay and the exclusive $\tau \to h\nu$ ($h = \pi, K$) measurements yields $|V_{us}| = 0.2207 \pm 0.0014$ [24].

### 12.2.3 $|V_{cd}|$

The magnitude of $V_{cd}$ can be extracted from semileptonic charm decays, using theoretical knowledge of the form factors. In semileptonic D decays, lattice QCD calculations have predicted the normalization of the $D \to \pi \ell \nu$ and $D \to K \ell \nu$ form factors [14]. The dependence on the invariant mass of the lepton pair, $q^2$, is determined from lattice QCD and theoretical constraints from analyticity [15]. Using $N_f = 2 + 1 + 1$ lattice QCD calculations for $D \to \pi \ell \nu$, $f_D^{D\pi}(0) = 0.612 \pm 0.035$ [14], and the average [24] of the measurements of $D \to \pi \ell \nu$ decays by BABAR [25], BESIII [26], CLEO-c [27], and Belle [28], one obtains $|V_{cd}| = 0.2330 \pm 0.0029 \pm 0.0133$, where the first uncertainty is experimental, and the second is from the theoretical uncertainty of the form factor.

The determination of $|V_{cd}|$ is also possible from the leptonic decay $D^+ \to \mu^+ \nu$ and $\tau^+ \nu$. The experimental uncertainties have not decreased significantly recently. Averaging the BESIII [29] and earlier CLEO [30] for $\mu^+ \nu$ and BESIII [31] for $\tau^+ \nu$ measurements, and using the $N_f = 2 + 1 + 1$ lattice QCD result, $f_D = 212.0 \pm 0.7$ MeV [14], yields $|V_{cd}| = 0.2181 \pm 0.0049 \pm 0.0007$ [24]<fnref n="2" />

Earlier determinations of $|V_{cd}|$ came from neutrino scattering data. The difference of the ratio of double-muon to single-muon production by neutrino and antineutrino beams is proportional to the charm cross section off valence d quarks, and therefore to $|V_{cd}|^2$ times the average semileptonic branching ratio of charm mesons, $B_\mu$. The method was used first by CDHS [32] and then by CCFR [33] and CHARM II [34]. Averaging these results is complicated, because it requires assumptions about the scale of the QCD corrections, and because $B_\mu$ is an effective quantity, which depends on the specific neutrino beam characteristics. With no recent experimental input available, we quote the average from a past review, $B_\mu|V_{cd}|^2 = (0.463 \pm 0.034) \times 10^{-2}$ [35]. Analysis cuts make these experiments insensitive to neutrino energies smaller than 30 GeV. Thus, $B_\mu$ should be computed using only neutrino interactions with visible energy larger than 30 GeV. An appraisal [36] based on charm-production fractions measured in neutrino interactions [37] gives $B_\mu = 0.088 \pm 0.006$. Data from the CHORUS experiment [38] are sufficiently precise to extract $B_\mu$ directly, by comparing the number of charm decays with a muon to the total number of charmed hadrons found in the nuclear emulsions. Requiring the visible energy to be larger than 30 GeV, CHORUS found $B_\mu = 0.085 \pm 0.009 \pm 0.006$. We use the average of these two determinations, $B_\mu = 0.087 \pm 0.005$, and obtain $|V_{cd}| = 0.230 \pm 0.011$. Averaging the three determinations above, we find

$$|V_{cd}| = 0.221 \pm 0.004. \qquad (12.9)$$

### 12.2.4 $|V_{cs}|$

The direct determination of $|V_{cs}|$ is possible from semileptonic D or leptonic $D_s$ decays, using lattice QCD calculations of the semileptonic D form factor or the $D_s$ decay constant. For muonic decays, the average of Belle [39], CLEO-c [40], BABAR [41], and BESIII [42, 43] is $B(D_s^+ \to \mu^+ \nu) = (5.43 \pm 0.16) \times 10^{-3}$ [24]. For decays to $\tau$ leptons, the average of CLEO-c [40, 44], BABAR [41], Belle [39], and BESIII [42] gives $B(D_s^+ \to \tau^+ \nu) = (5.40 \pm 0.23) \times 10^{-2}$ [24]. From each of these values, determinations of $|V_{cs}|$ can be obtained using the PDG values for the mass and lifetime of the $D_s$, the masses of the leptons, and $f_{D_s} = (249.9 \pm 0.5)$ MeV [14]. The average of these determinations gives $|V_{cs}| = 0.984 \pm 0.012$, where the error is dominated by the experimental uncertainty. In semileptonic D decays, lattice QCD calculations of the $D \to K \ell \nu$ form factor are available [14]. Using $f_D^{D\kappa}(0) = 0.7385 \pm 0.0044$ and the average [24] of CLEO-c [27], Belle [28], BABAR [45], and recent BESIII [26, 46] measurements of $D \to K \ell \nu$ decays, one obtains $|V_{cs}| = 0.972 \pm 0.007$, where the dominant uncertainty is from the theoretical calculation of the form factor. Averaging the determinations from leptonic and semileptonic decays, we find

$$|V_{cs}| = 0.975 \pm 0.006. \qquad (12.10)$$

Measurements of on-shell $W^\pm$ decays sensitive to $|V_{cs}|$ were made by LEP-2. The W branching ratios depend on the six CKM elements involving only the quarks lighter than $m_W$. The W branching ratio to each lepton flavor is $1/B(W \to \ell \bar{\nu}_\ell) = 3[1 + \sum_{u,c,d,s,b} |V_{ij}|^2 (1 + \alpha_s(m_W)/\pi) + \dots]$. Assuming lepton universality, the measurement $B(W \to \ell \bar{\nu}_\ell) = (10.83 \pm 0.07 \pm 0.07)\%$ [47] implies $\sum_{u,c,d,s,b} |V_{ij}|^2 = 2.002 \pm 0.027$. This is a precise test of unitarity; however, only flavor-tagged W-decays determine $|V_{cs}|$ directly, such as DELPHI's tagged $W^+ \to c\bar{s}$ analysis, yielding $|V_{cs}| = 0.94^{+0.32}_{-0.26} \pm 0.13$ [48].

### 12.2.5 $|V_{cb}|$

This matrix element can be determined from exclusive and inclusive semileptonic decays of B mesons to charm. The in-

<footnote n="1">¹For lattice QCD inputs, we use the averages from Ref. [14], unless the minireviews [10, 15] choose different values. We only use unquenched results, and if both $N_f = 2 + 1 + 1$ and $2 + 1$ calculations are available, we use the former.</footnote>

<footnote n="2">²Hereafter the first error is statistical and the second is systematic, unless mentioned otherwise.</footnote>
clusive determinations use the semileptonic decay rate measurement, together with (certain moments of) the lepton energy and the hadronic invariant-mass spectra. The theoretical basis is the operator product expansion [49, 50], which allows calculation of the decay rate and various spectra as expansions in $\alpha_s$ and inverse powers of the heavy-quark mass. The dependence on $m_b$, $m_c$, and the parameters that occur at subleading order is different for different moments. The measurements of many moments overconstrain these parameters, and also test the consistency of their determination. The precise extraction of $|V_{cb}|$ requires using a “threshold” quark mass definition [51, 52]. Inclusive measurements have been performed using B mesons from $Z^0$ decays at LEP, and at $e^+e^-$ colliders operated at the $\Upsilon(4S)$. At LEP, the large boost of B mesons from the $Z^0$ decay allows the determination of the moments throughout phase space, which is not possible otherwise, but the large statistics available at the B factories lead to more precise determinations. An average of the measurements and a compilation of the references are provided in Ref. [15]: $|V_{cb}| = (42.2 \pm 0.5) \times 10^{-3}$.

Complementary determinations are based on exclusive semileptonic B decays to $D$ and $D^*$. In the $m_{b,c} \gg \Lambda_{QCD}$ limit, all form factors are given by a single Isgur-Wise function [53], which depends on the product of the four-velocities of the B and $D^{(*)}$ mesons, $w = v \cdot v^t$. Heavy-quark symmetry determines the rate (in the symmetry limit) at $w = 1$, the point in phase space at which the invariant mass of the $\ell\bar{\nu}$ pair is maximal; $|V_{cb}|$ is obtained from extrapolating a fit to the spectrum to $w = 1$. The current update of the $V_{cb}$ and $V_{ub}$ minireview quotes from exclusive decays $|V_{cb}| = (39.8 \pm 0.6) \times 10^{-3}$ [15], based on the only unfolded measurement of $B \to D^*$ semileptonic decay distributions [54], and using a more general fit [55] than in earlier B factory measurements. With the uncertainty scaled by $\sqrt{\chi^2} = 3.0$, this yields the combination [15],

$$|V_{cb}| = (41.1 \pm 1.2) \times 10^{-3}. \qquad (12.11)$$

Determinations of $|V_{cb}|$ that are currently less precise and not included in this average, can be obtained from the measurement of $B_s \to D_s^{(*)} \mu\bar{\nu}$ decays [56]. In addition, semileptonic decays to $\tau$ leptons measured in $B \to D^{(*)} \tau\bar{\nu}$ and related modes are also sensitive to $|V_{cb}|$. The most precise data involving $\tau$ leptons are the $|V_{cb}|$-independent ratios, $B(B \to D^{(*)} \tau\bar{\nu})/B(B \to D^{(*)} \ell\bar{\nu})$ measured by BaBar, Belle, and LHCb. If the current, approximately 3.3 $\sigma$ [24] hint of lepton non-universality prevails, the determination of $|V_{cb}|$ becomes more complicated.

### 12.2.6 $|V_{ub}|$

The determination of $|V_{ub}|$ from inclusive $B \to X_u \ell\bar{\nu}$ decay is complicated due to large $B \to X_c \ell\bar{\nu}$ backgrounds. In most regions of phase space where the charm background is kinematically forbidden, the hadronic physics enters via unknown nonperturbative functions, so-called shape functions. (By contrast, the nonperturbative physics for $|V_{cb}|$ is encoded in a few parameters.) At leading order in $\Lambda_{QCD}/m_b$, there is only one shape function, which can be extracted from the photon energy spectrum in $B \to X_s \gamma$ [57, 58], and applied to several spectra in $B \to X_u \ell\bar{\nu}$. The subleading shape functions are modeled in the current determinations. Phase space cuts for which the rate has only subleading dependence on the shape function are also possible [59]. The measurements of both the hadronic and the leptonic systems are important for an optimal choice of phase space. A different approach is to make the measurements more inclusive by extending them deeper into the $B \to X_c \ell\bar{\nu}$ region, and thus reduce the theoretical uncertainties. Analyses of the electron-energy endpoint from CLEO [60], BABAR [61], and Belle [62] quote $B \to X_u e\bar{\nu}$ partial rates for $|\bar{p}_e| \ge 2.0$ GeV and 1.9 GeV, which are well below the charm endpoint. The large and pure $B\bar{B}$ samples at the B factories permit the selection of $B \to X_u \ell\bar{\nu}$ decays in events where the other B is fully reconstructed [63]. With this full-reconstruction tag method, the four-momenta of both the leptonic and the hadronic final states can be measured. It also gives access to a wider kinematic region, because of improved signal purity. Ref. [15] quotes the inclusive average, $|V_{ub}| = (4.13 \pm 0.12)_{-0.14}^{+0.13} \pm 0.18) \times 10^{-3}$, where the first error is experimental, the second arises from the

model dependence quoted by the individual measurements, and the third is an additional one estimated in Ref. [15].

To extract $|V_{ub}|$ from exclusive decays, the form factors have to be known. Experimentally, better signal-to-background ratios are offset by smaller yields. The $B \to \pi\ell\bar{\nu}$ branching ratio is now known to 5%. Lattice QCD calculations of the $B \to \pi\ell\bar{\nu}$ form factor are available [64] for the high $q^2$ region ($q^2 > 16$ or $18$ GeV$^2$). A fit to the experimental partial rates and lattice QCD results versus $q^2$ yields $|V_{ub}| = (3.70 \pm 0.10 \pm 0.12) \times 10^{-3}$ [24]. Using additional input from light-cone QCD sum rules (which are thought to be reliable in the small $q^2$ region), yield a combination, $|V_{ub}| = (3.67 \pm 0.09 \pm 0.12) \times 10^{-3}$ [15, 24].

The uncertainties in extracting $|V_{ub}|$ from inclusive and exclusive decays are different to a large extent. An average of these determinations, with the uncertainty scaled by $\sqrt{\chi^2} = 1.4$, is [15]

$$|V_{ub}| = (3.82 \pm 0.20) \times 10^{-3}. \qquad (12.12)$$

A determination of $|V_{ub}|$ not included in this average can be obtained from $B(B \to \tau\bar{\nu}) = (1.09 \pm 0.21) \times 10^{-4}$ [24]. Using $f_B = (190.0 \pm 1.3)$ MeV [14] and $\tau_{B\pm} = (1.638 \pm 0.004)$ ps [65], we find the remarkably consistent result, $|V_{ub}| = (4.11 \pm 0.39) \times 10^{-3}$. This decay is sensitive, for example, to tree-level charged Higgs contributions, and the measured rate is consistent with the SM expectation. The LHCb measurement $|V_{ub}/V_{cb}| = 0.083 \pm 0.004$ [15] from the ratios of $\Lambda_b \to p^+\mu^-\bar{\nu}$ and $\Lambda_b \to \Lambda_c^+\mu^-\bar{\nu}$ [66] and $B_s^0 \to K^+\mu^+\nu$ and $B_s^0 \to D_s^-\mu^+\nu$ [67] in different regions of $q^2$, provides another complementary determination.

### 12.2.7 $|V_{td}|$ and $|V_{ts}|$

The CKM elements $|V_{td}|$ and $|V_{ts}|$ are not likely to be precisely measurable in tree-level processes involving top quarks, so one has to rely on determinations from $B^0 - \bar{B}^0$ mixing, dominated by box diagrams with top quarks, or loop-mediated rare K and B decays. Theoretical uncertainties in hadronic effects limit the accuracy of the current determinations. These can be reduced by taking ratios of processes that are equal in the flavor SU(3) limit to determine $|V_{td}/V_{ts}|$.

The phenomenon of $B^0 - \bar{B}^0$ mixing was discovered by ARGUS [68], and the mass difference is now precisely measured as $\Delta m_d = (0.5069 \pm 0.0019)$ ps$^{-1}$ [69]. In the $B_s^0$ system, $\Delta m_s$ was first measured significantly by CDF [70] and the world average, dominated by an LHCb measurement [71], is $\Delta m_s = (17.765 \pm 0.006)$ ps$^{-1}$ [69]. Neglecting corrections suppressed by $|V_{tb}| - 1$, and using the lattice QCD results $f_{B_d} \sqrt{\hat{B}_{B_d}} = (210.6 \pm 5.5)$ MeV and $f_{B_s} \sqrt{\hat{B}_{B_s}} = (256.1 \pm 5.7)$ MeV [14],

$$|V_{td}| = (8.6 \pm 0.2) \times 10^{-3}, \qquad |V_{ts}| = (41.5 \pm 0.9) \times 10^{-3}. \qquad (12.13)$$

The uncertainties are dominated by lattice QCD. Several uncertainties are reduced in the calculation of the ratio $\xi = (f_{B_s} \sqrt{\hat{B}_{B_s}})/(f_{B_d} \sqrt{\hat{B}_{B_d}}) = 1.216 \pm 0.016$ [14] and therefore the constraint on $|V_{td}/V_{ts}|$ from $\Delta m_d/\Delta m_s$ is more reliable theoretically. These provide a theoretically clean and significantly improved determination,

$$|V_{td}/V_{ts}| = 0.207 \pm 0.001 \pm 0.003. \qquad (12.14)$$

The inclusive branching ratio $B(B \to X_s \gamma) = (3.49 \pm 0.19) \times 10^{-4}$ extrapolated to $E_\gamma > E_0 = 1.6$ GeV [24] is also sensitive to $|V_{tb}V_{ts}|$. In addition to t-quark penguins, a substantial part of the rate comes from charm contributions proportional to $V_{cb}V_{cs}^*$ via the application of 3 × 3 CKM unitarity (which is used here). With the NNLO calculation of $B(B \to X_s \gamma)_{E_\gamma > E_0}/B(B \to X_c e\bar{\nu})$ [72], we obtain $|V_{ts}/V_{cb}| = 1.00 \pm 0.04$. The $B_s \to \mu^+\mu^-$ rate is also proportional to $|V_{tb}V_{ts}|^2$ in the SM, and the world average, $B(B_s \to \mu^+\mu^-) = (3.01 \pm 0.35) \times 10^{-9}$ [65], is consistent with the SM, with sizable uncertainties.

A complementary determination of $|V_{td}/V_{ts}|$ is possible from the ratio of $B \to \rho\gamma$ and $K^*\gamma$ rates. The ratio of the neutral modes is theoretically cleaner than that of the charged ones, because the poorly known spectator-interaction contribution is expected to be
smaller (W-exchange vs. weak annihilation). For now, because of low statistics, we average the charged and neutral rates assuming the isospin symmetry and heavy-quark limit motivated relation, $|V_{td}/V_{ts}|^2/\xi_\gamma^2 = [\Gamma(B^+ \to \rho^+\gamma) + 2\Gamma(B^0 \to \rho^0\gamma)]/[\Gamma(B^+ \to K^{*+}\gamma) + \Gamma(B^0 \to K^{*0}\gamma)] = (3.35 \pm 0.48)\%$ [24, 73]. Here $\xi_\gamma$ contains the poorly known hadronic physics. Using $\xi_\gamma = 1.2 \pm 0.2$ [74] gives $|V_{td}/V_{ts}| = 0.220 \pm 0.016 \pm 0.037$, where the first uncertainty is experimental and the second is theoretical.

A theoretically clean determination of $|V_{td}V_{ts}^*|$ is possible from $K^+ \to \pi^+\nu\bar{\nu}$ decay [75]. Experimentally, more than 20 candidates have been observed [76, 77] and the rate is consistent with the SM within errors. Much more data are needed for a precision measurement.

### 12.2.8 $|V_{tb}|$

The determination of $|V_{tb}|$ from top decays uses the ratio of branching fractions $R = \mathcal{B}(t \to Wb)/\mathcal{B}(t \to Wq) = |V_{tb}|^2/(\sum_q |V_{tq}|^2) = |V_{tb}|^2$, where $q = b, s, d$. The CDF and DØ measurements performed on data collected during Run II of the Tevatron give $|V_{tb}| > 0.85$ [78] and $0.99 > |V_{tb}| > 0.90$ [79], respectively, at 95% CL. CMS measured the same quantity at 8 TeV and obtained $|V_{tb}| > 0.975$ [80] at 95% CL.

The direct determination of $|V_{tb}|$, without assuming unitarity, is possible from the single top quark production cross section. The $(3.30^{+0.52}_{-0.40})$ pb combined cross section [81] of DØ and CDF measurements implies $|V_{tb}| = 1.02^{+0.06}_{-0.05}$. The LHC experiments, ATLAS and CMS, have measured single top quark production cross sections (and extracted $|V_{tb}|$) in t-channel, Wt-channel, and s-channel at 7 TeV, 8 TeV, and 13 TeV [82]. The average of these $|V_{tb}|$ values is calculated to be $|V_{tb}| = 1.007 \pm 0.030$, where all systematic errors and theoretical errors are treated to be fully correlated. The average of Tevatron and LHC values gives

$$|V_{tb}| = 1.010 \pm 0.027. \qquad (12.15)$$

The experimental systematic uncertainties dominate, and a dedicated combination would be welcome.

A weak constraint on $|V_{tb}|$ can be obtained from precision electroweak data, where top quarks enter in loops. The sensitivity is best in $\Gamma(Z \to b\bar{b})$ and yields $|V_{tb}| = 0.77^{+0.18}_{-0.24}$ [83].

## 12.3 Phases of CKM elements

As can be seen from Fig. 12.1, the angles of the unitarity triangle are

$$ \begin{aligned} \beta &= \phi_1 = \arg \left( - \frac{V_{cd}V_{cb}^*}{V_{td}V_{tb}^*} \right), \\ \alpha &= \phi_2 = \arg \left( - \frac{V_{td}V_{tb}^*}{V_{ud}V_{ub}^*} \right), \\ \gamma &= \phi_3 = \arg \left( - \frac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*} \right). \end{aligned} \qquad (12.16) $$

Since CP violation involves phases of CKM elements, many measurements of CP-violating observables can be used to constrain these angles and the $\bar{\rho}, \bar{\eta}$ parameters.

### 12.3.1 $\epsilon$ and $\epsilon'$

The measurement of CP violation in $K^0-\bar{K}^0$ mixing, $|\epsilon| = (2.228 \pm 0.011) \times 10^{-3}$ [84], provides important information about the CKM matrix. The phase of $\epsilon$ is determined by long-distance physics, $\epsilon = \frac{1}{2}e^{i\phi_e}\sin\phi_e \arg(-M_{12}/\Gamma_{12})$, where $\phi_e = \arctan|2\Delta m_K/\Delta\Gamma_K| \approx 43.5^\circ$. The SM prediction can be written as

$$ \epsilon = \kappa_e e^{i\phi_e} \frac{G_F^2 m_W^2 m_K}{12\sqrt{2} \pi^2 \Delta m_K} f_K^2 \hat{B}_K \left\{ \eta_{tt} S(x_t) \operatorname{Im}[(V_{ts}V_{td}^*)^2] + 2\eta_{ct} S(x_c, x_t) \operatorname{Im}(V_{cs}V_{cd}^*V_{ts}V_{td}^*) + \eta_{cc} x_c \operatorname{Im}[(V_{cs}V_{cd}^*)^2] \right\}, \qquad (12.17) $$

where $\kappa_e \approx 0.94 \pm 0.02$ [85] includes the effects of strangeness changing $\Delta s = 1$ operators and additional dependence on $\phi_e \neq 0$.

$\pi/4$ (see also Ref. [86]). The displayed terms are the short-distance $\Delta s = 2$ contribution to $\operatorname{Im}M_{12}$ in the usual phase convention, $S$ is an Inami-Lim function [87], $x_q = m_q^2/m_W^2$, and $\eta_{ij}$ are perturbative QCD corrections. The constraint from $\epsilon$ in the $\bar{\rho}, \bar{\eta}$ plane is bounded by approximate hyperbolas. Lattice QCD determined the bag parameter $\hat{B}_K = 0.717 \pm 0.024$ [14] and the main uncertainties are from $(V_{ts}V_{td}^*)^2$ (approximately given by that of $|V_{cb}|^4$ or $A^4$), the $\eta_{ij}$ coefficients, and estimates of $\kappa_e$.

The measurement of $6 \operatorname{Re}(\epsilon'/\epsilon) = 1 - |\eta_{00}/\eta_{+-}|^2$, where each $\eta_{ij} = \langle\pi^i\pi^j|\mathcal{H}|K_L\rangle/\langle\pi^i\pi^j|\mathcal{H}|K_S\rangle$ violates CP, provides a qualitative test of the CKM mechanism, and strong constraints on many BSM scenarios. Its nonzero value, $\operatorname{Re}(\epsilon'/\epsilon) = (1.67 \pm 0.23) \times 10^{-3}$ [84], demonstrated the existence of direct CP violation, a prediction of the KM ansatz. While $\operatorname{Re}(\epsilon'/\epsilon) \propto \operatorname{Im}(V_{td}V_{ts}^*)$, this quantity cannot easily be used to extract CKM parameters, because cancellations between the electromagnetic and gluonic penguin contributions for large $m_t$ [88] enhance the hadronic uncertainties. Most SM estimates [89] agree with the observed value, indicating that $\bar{\eta}$ is positive. Progress in lattice QCD [90] may yield a precise SM prediction in the future, and trigger new work on assessing the consistency of the SM with the measured value [91, 92].

### 12.3.2 $\beta/\phi_1$

#### 12.3.2.1 Charmonium modes

CP-violation measurements in B-meson decays provide direct information on the angles of the unitarity triangle, shown in Fig. 12.1. These overconstraining measurements serve to improve the determination of the CKM elements, and to reveal possible effects beyond the SM.

The time-dependent CP asymmetry of neutral B decays to a final state $f$ common to $B^0$ and $\bar{B}^0$ is given by [93–95]

$$ \begin{aligned} A_f &= \frac{\Gamma(\bar{B}^0(t) \to f) - \Gamma(B^0(t) \to f)}{\Gamma(\bar{B}^0(t) \to f) + \Gamma(B^0(t) \to f)}, \\ &= S_f \sin(\Delta m_d t) - C_f \cos(\Delta m_d t), \end{aligned} \qquad (12.18) $$

where

$$ S_f = \frac{2 \operatorname{Im}\lambda_f}{1 + |\lambda_f|^2}, \qquad C_f = \frac{1 - |\lambda_f|^2}{1 + |\lambda_f|^2}, \qquad \lambda_f = \frac{q}{p} \frac{\bar{A}_f}{A_f}. \qquad (12.19) $$

Here, $q/p$ describes $B^0-\bar{B}^0$ mixing and, to a good approximation in the SM, $q/p = V_{tb}^*V_{td}/V_{tb}V_{td}^* = e^{-2i\beta+O(\lambda^4)}$ in the usual phase convention. $A_f$ ($\bar{A}_f$) is the amplitude of the $B^0 \to f$ ($\bar{B}^0 \to f$) decay. If $f$ is a CP eigenstate, and amplitudes with one CKM phase dominate the decay, then $|A_f| = |\bar{A}_f|$, $C_f = 0$, and $S_f = \sin(\operatorname{arg}\lambda_f) = \eta_f \sin 2\phi$, where $\eta_f$ is the CP eigenvalue of $f$ and $2\phi$ is the phase difference between the $B^0 \to f$ and $B^0 \to \bar{B}^0 \to f$ decay paths. A contribution of another amplitude to the decay with a different CKM phase makes the value of $S_f$ sensitive to relative strong-interaction phases between the decay amplitudes (it also makes $C_f \neq 0$ possible).

The $b \to c\bar{c}s$ decays to CP eigenstates ($B^0 \to \text{charmonium } K_{S,L}^0$) give currently the most precise measurements of $S_f = -\eta_f \sin 2\beta$. The $b \to s$ penguin amplitudes have dominantly the same weak phase as the $b \to c\bar{c}s$ tree amplitude. Since only $\lambda^2$-suppressed penguin amplitudes introduce a different CP-violating phase, amplitudes with a single weak phase dominate, and we expect $|\bar{A}_{\psi K}/A_{\psi K}| - 1| < 0.01$. The $e^+e^-$ asymmetric-energy B-factory experiments, BABAR [96] and Belle [97], and LHCb [98] provided precise measurements. The world average, including some other measurements, is [24, 95, 99]

$$ \sin 2\beta = 0.709 \pm 0.011. \qquad (12.20) $$

This measurement has a four-fold ambiguity in $\beta$, which can be resolved by a global fit as mentioned in Sec. 12.4. Experimentally, the two-fold ambiguity $\beta \to \pi/2 - \beta$ (but not $\beta \to \pi + \beta$) can be resolved by a time-dependent angular analysis of $B^0 \to J/\psi K^{*0}$ [100, 101], or a time-dependent Dalitz plot analysis of $B^0 \to \bar{D}^0 h^0$. The time-dependent Dalitz plot analysis of $B^0 \to \bar{D}^0 h^0$ ($h^0 = \pi^0, \eta, \omega$) with $\bar{D}^0 \to K_S^0\pi^+\pi^-$, jointly performed by Belle and BABAR, excludes the $\pi/2 - \beta$ solution with
7.3σ confidence level [102]. These results exclude the negative cos 2β solutions, in agreement with the global CKM fit, which is no longer shown in Fig. 12.2.

The $b \to c\bar{c}d$ mediated transitions, such as $B^0 \to J/\psi\pi^0$ and $B^0 \to D^*(+)D^*(-)$, also measure approximately $\sin 2\beta$. However, the dominant component of the $b \to d$ penguin amplitude has a different CKM phase ($V_{tb}^*V_{td}$) than the tree amplitude ($V_{cb}^*V_{cd}$), and their magnitudes are of the same order in $\lambda$. Therefore, the effect of penguins could be large, resulting in $S_f \neq -\eta_f \sin 2\beta$ and $C_f \neq 0$. Such decay modes have been measured by BABAR, Belle, and LHCb. The world averages [24], $S_{J/\psi\pi^0} = -0.86 \pm 0.14$, $S_{J/\psi\rho^0} = -0.66^{+0.16}_{-0.12}$, $S_{D^+D^-} = -0.84 \pm 0.12$, and $S_{D^{*+}D^{*-}} = -0.71 \pm 0.09$ (where $\eta_f = +1$ for the $J/\psi\pi^0$ and $D^+D^-$ modes, while $J/\psi\rho^0$ and $D^{*+}D^{*-}$ are mixtures of CP even and odd states), are consistent with $\sin 2\beta$ obtained from $B^0 \to$ charmonium $K^0$ decays, and the $C_f$'s are consistent with zero, although the uncertainties are sizable.

The $b \to c\bar{u}d$ decays $B^0 \to \bar{D}^{0(*)}h^0$, with $\bar{D}^0 \to CP$ eigenstates and $\bar{D}^0 \to K_S^0\pi^+\pi^-$ with Dalitz plot analysis, have no penguin contributions, and provide theoretically clean $\sin 2\beta$ measurements. The average of joint analyses of BABAR and Belle data [102, 103] give $\sin 2\beta = 0.71 \pm 0.09$ [24].

### 12.3.2.2 Penguin-dominated modes

The $b \to s\bar{q}q$ penguin-dominated decays have the same CKM phase as the $b \to c\bar{c}s$ tree level decays, up to corrections suppressed by $\lambda^2$, since $V_{tb}^*V_{ts} = -V_{cb}^*V_{cs}[1 + \mathcal{O}(\lambda^2)]$. Therefore, decays such as $B^0 \to \phi K^0$ and $\eta^* K^0$ provide $\sin 2\beta$ measurements in the SM. Any BSM contribution to the amplitude with a different weak phase would give rise to $S_f \neq -\eta_f \sin 2\beta$, and possibly $C_f \neq 0$. Therefore, the main interest in these modes is not simply to measure $\sin 2\beta$, but to search for new physics. Measurements of many other decay modes in this category, such as $B \to \pi^0 K_S^0$, $K_S^0 K_S^0 K_S^0$, etc., have also been performed by BABAR and Belle. The results and their uncertainties are summarized in Fig. 13.3 and Table 13.1 of Ref. [94]. The comparison of CP violation measurements between tree-dominated and penguin-dominated modes in $B_s^0$ decays provides similar sensitivity to new physics.

### 12.3.3 $\alpha / \phi_2$

Since $\alpha$ is the phase between $V_{tb}^*V_{td}$ and $V_{ub}^*V_{ud}$, only time-dependent CP asymmetries in decay modes dominated by $b \to u\bar{u}d$ transitions can directly measure $\sin 2\alpha$, in contrast to $\sin 2\beta$, where several different quark-level transitions can be used. Since $b \to d$ penguin amplitudes have a different CKM phase than $b \to u\bar{u}d$ tree amplitudes, and their magnitudes are of the same order in $\lambda$, the penguin contribution can be sizable, which makes the determination of $\alpha$ complicated. To date, $\alpha$ has been measured in $B \to \pi\pi$, $\rho\pi$ and $\rho\rho$ decay modes.

#### 12.3.3.1 $B \to \pi\pi$

It is well-established from the data that there is a sizable contribution of $b \to d$ penguin amplitudes in $B \to \pi\pi$ decays. Thus, $S_{\pi^+\pi^-}$ in the time-dependent $B^0 \to \pi^+\pi^-$ analysis does not measure $\sin 2\alpha$, but

$$S_{\pi^+\pi^-} = \sqrt{1 - C_{\pi^+\pi^-}^2} \sin(2\alpha + 2\Delta\alpha), \quad (12.21)$$

where $2\Delta\alpha$ is the phase difference between $e^{2i\gamma}\bar{A}_{\pi^+\pi^-}$ and $A_{\pi^+\pi^-}$. The value of $\Delta\alpha$, and hence $\alpha$, can be extracted using the isospin relation among the amplitudes of $B^0 \to \pi^+\pi^-$, $B^0 \to \pi^0\pi^0$, and $B^+ \to \pi^+\pi^0$ decays [104],

$$\frac{1}{\sqrt{2}} A_{\pi^+\pi^-} + A_{\pi^0\pi^0} - A_{\pi^+\pi^0} = 0, \quad (12.22)$$

and a similar expression for the $\bar{A}_{\pi\pi'}$'s. This method utilizes the fact that a pair of pions from $B \to \pi\pi$ decay must be in a zero angular momentum state, and, because of Bose statistics, they must have even isospin. Consequently, $\pi^\pm\pi^0$ is in a pure isospin-2 state, while the penguin amplitudes only contribute to the isospin-0 final state. The latter does not hold for the electroweak penguin amplitudes, but their effect is expected to be small. The isospin

analysis uses the world averages of BABAR, Belle, and LHCb measurements, $S_{\pi^+\pi^-} = -0.666 \pm 0.029$, $C_{\pi^+\pi^-} = -0.311 \pm 0.030$, the decay widths of all three modes, and the direct CP asymmetry $C_{\pi^0\pi^0} = -0.33 \pm 0.22$ [24]. This analysis leads to 16 mirror solutions for $0 \le \alpha < 2\pi$. Because of this, and due to the experimental uncertainties, some of these solutions are not well separated [95].

#### 12.3.3.2 $B \to \rho\rho$

The decay $B^0 \to \rho^+\rho^-$ contains two vector mesons in the final state, and so in general is a mixture of CP-even and CP-odd components. At the current level of precision, it simplifies the analysis that the longitudinal polarization fractions in $B^+ \to \rho^+\rho^0$ and $B^0 \to \rho^+\rho^-$ decays were measured to be close to unity [105], which implies that the final states are almost purely CP-even. Furthermore, $B(B^0 \to \rho^0\rho^0) = (0.96 \pm 0.15) \times 10^{-6}$ is much smaller than $B(B^0 \to \rho^+\rho^-) = (27.5 \pm 1.7) \times 10^{-6}$ and $B(B^+ \to \rho^+\rho^0) = (23.8 \pm 1.7) \times 10^{-6}$ [24], which implies that the effect of the penguin contributions is small. The isospin analysis using the world averages, $S_{\rho^+\rho^-} = -0.14 \pm 0.13$ and $C_{\rho^+\rho^-} = 0.00 \pm 0.09$ [24], together with the time-dependent CP asymmetry, $S_{\rho^0\rho^0} = -0.3 \pm 0.7$ and $C_{\rho^0\rho^0} = -0.2 \pm 0.9$ [106], and the above mentioned branching fractions and longitudinal polarization fractions, gives two solutions (with mirror solutions at $3\pi/2 - \alpha$) [95]. A possible small violation of Eq. (12.22) due to the finite width of the $\rho$ [107] is so far neglected.

#### 12.3.3.3 $B \to \rho\pi$

The final state in $B^0 \to \rho^+\pi^-$ decay is not a CP eigenstate, but this decay proceeds via the same quark-level diagrams as $B^0 \to \pi^+\pi^-$, and both $B^0$ and $\bar{B}^0$ can decay to $\rho^+\pi^-$, while the final state in $B^0 \to \rho^0\pi^0$ is a CP eigenstate. Consequently, mixing-induced CP violation can occur in $B^0$ and $\bar{B}^0$ decays to $\rho^{\pm}\pi^{\mp}$ and $\rho^0\pi^0$. The time-dependent Dalitz plot analysis of $B^0 \to \pi^+\pi^-\pi^0$ decays permits the extraction of $\alpha$ with a single discrete ambiguity, $\alpha \to \alpha + \pi$, since one knows the variation of the strong phases in the interference regions of the $\rho^+\pi^-$, $\rho^-\pi^+$, and $\rho^0\pi^0$ amplitudes in the Dalitz plot [108]. The combination of Belle [109] and BABAR [110] measurements gives only moderate constraints [95].

Combining the $B \to \pi\pi$, $\rho\pi$, and $\rho\rho$ decay modes [24, 95, 99], $\alpha$ is constrained as

$$\alpha = (84.1^{+4.5}_{-3.8})^{\circ}. \quad (12.23)$$

Similar results can be found in Refs. [111, 112].

### 12.3.4 $\gamma / \phi_3$

By virtue of Eq. (12.16), $\gamma$ does not depend on CKM elements involving the top quark, so it can be measured in tree-level $B$ decays. This is an important distinction from the measurements of $\alpha$ and $\beta$, and implies that measurements of $\gamma$ are unlikely to be affected by physics beyond the SM.

#### 12.3.4.1 $B_{(s)} \to D_{(s)} K^{(*)}$

The interference of $B^- \to D^0 K^-$ ($b \to c\bar{u}s$) and $B^- \to \bar{D}^0 K^-$ ($b \to u\bar{c}s$) transitions can be studied in final states accessible in both $D^0$ and $\bar{D}^0$ decays [93]. In principle, it is possible to extract the $B$ and $D$ decay amplitudes, the relative strong phases, and the weak phase $\gamma$ from the data [95].

A practical complication is that the precision depends sensitively on the ratio of the interfering amplitudes

$$r_B = \left| A(B^- \to \bar{D}^0 K^-) / A(B^- \to D^0 K^-) \right|, \quad (12.24)$$

which is around 0.1. The original GLW method [113, 114] considers $D$ decays to CP eigenstates, such as $B^\pm \to D_{CP}^{(*)}(\to \pi^+\pi^-)K^{(*)\pm}$. To alleviate the smallness of $r_B$ and make the interfering amplitudes (which are products of the $B$ and $D$ decay amplitudes) comparable in magnitude, the ADS method [115] considers final states where Cabibbo-allowed $\bar{D}^0$ and doubly-Cabibbo-suppressed $D^0$ decays interfere. Measurements have been made by the $B$ factories, CDF, and LHCb, using both methods [24]. The GLW method currently gives only a loose constraint on $\gamma$, while the ADS method provides a moderate constraint.

The BPGGSZ method [116, 117] utilizes the fact that both $D^0$ and $\bar{D}^0$ can have large branching fractions to CP self-conjugate
three- and four-body final states, such as $K_S^0\pi^+\pi^-$, and the analysis can be optimized by studying the Dalitz plot dependence of the interferences. The best present determination of $\gamma$ comes from this method, dominated by 3-body $D$ decay modes. Combining results in 3-body decay modes from Belle [118], BABAR [119], and the most precise LHCb [120] one, $\gamma = (70.0\pm4.0)°$ is obtained [95]. The uncertainty is sensitive to the central value of the amplitude ratio $r_B$ (and $r_B^*$ for the $D^*K$ mode), for which Belle found somewhat larger central values than BABAR and LHCb. The same values of $r_B^{(*)}$ enter the ADS analyses, and the data (including 4-body $D$ decays [121]) can be combined to fit for $r_B^{(*)}$ and $\gamma$. The effect of $D^0-\bar{D}^0$ mixing on $\gamma$ is either below the present experimental accuracy or can be taken into account in the analysis [122] (even if $D^0-\bar{D}^0$ mixing is due to CP-violating new physics [123]).

The amplitude ratio is much larger in the analogous $B_s^0 \to D_s^\pm K^\mp$ decays, which allows a model-independent extraction of $\gamma - 2\beta_s$ [124] (here $\beta_s = \arg(-V_{ts}V_{tb}^*/V_{cs}V_{cb}^*)$ is related to the phase of $B_s$ mixing). Measurements by LHCb with $B_s^0 \to D_s^\pm K^\mp$ [125] and $B_s^0 \to D_s^\pm K^\mp\pi^+\pi^-$ [126] give $\gamma = (79^{+1}_{-21})°$ using a constraint on $2\beta_s$ (see Sec. 12.5).

Combining all the above measurements [24, 95, 99], $\gamma$ is constrained as

$$ \gamma = (65.7 \pm 3.0)^{\circ}. \qquad (12.25) $$

Similar results can be found in Refs. [111, 112].

$$ 12.3.4.2 \quad B^0 \to D^{(*)\pm}\pi^{\mp} $$

The interference of $b \to u$ and $b \to c$ transitions can be studied in $\bar{B}^0 \to D^{(*)+}\pi^{-}$ ($b \to c\bar{u}d$) and $\bar{B}^0 \to B^0 \to D^{(*)+}\pi^{-}$ ($\bar{b} \to \bar{u}cd$) decays and their CP conjugates, since both $B^0$ and $\bar{B}^0$ decay to $D^{(*)\pm}\pi^{\mp}$ (or $D^{\pm}\rho^{\mp}$, etc.). Since there are only tree and no penguin contributions to these decays, in principle, it is possible to extract from the four time-dependent rates the magnitudes of the two hadronic amplitudes, their relative strong phase, and the weak phase between the two decay paths, which is $2\beta + \gamma$.

A complication is that the ratio of the interfering amplitudes is very small, $r_{D\pi} = A(B^0 \to D^+\pi^-)/A(\bar{B}^0 \to D^+\pi^-) = O(0.01)$ (and similarly for $r_{D^*\pi}$ and $r_{D\rho}$), and therefore it has not been possible to measure it. To obtain $2\beta + \gamma$, SU(3) flavor symmetry and dynamical assumptions have been used to relate $A(\bar{B}^0 \to D^-\pi^+) \to A(\bar{B}^0 \to D_s^-\pi^+)$, so this measurement is not model independent at present. Combining the $D^{\pm}\pi^{\mp}$, $D^{*\pm}\pi^{\mp}$ and $D^{\pm}\rho^{\mp}$ measurements [127] gives $\sin(2\beta + \gamma) > 0.68$ at 68% CL [111], consistent with the previously discussed results for $\beta$ and $\gamma$.

## 12.4 Global fit in the Standard Model

Using the independently measured CKM elements mentioned in the previous sections, the unitarity of the CKM matrix can be checked. We obtain $|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 0.9984 \pm 0.0007$ (1st row), $|V_{cd}|^2 + |V_{cs}|^2 + |V_{cb}|^2 = 1.001 \pm 0.012$ (2nd row), $|V_{ud}|^2 + |V_{cd}|^2 + |V_{td}|^2 = 0.9971 \pm 0.0020$ (1st column), and $|V_{us}|^2 + |V_{cs}|^2 + |V_{ts}|^2 = 1.003 \pm 0.012$ (2nd column), respectively. Due to the recent reduction of the value of $|V_{ud}|$, there is a $2.3\sigma$ tension with unitarity in the 1st row, leading also to poor consistency of the SM fit below. The uncertainties in the second row and column are dominated by that of $|V_{cs}|$. For the second row, another check is obtained from the measurement of $\sum_{u,c,d,s,b} |V_{ij}|^2$ in Sec. 12.2.4, minus the sum in the first row above: $|V_{cd}|^2 + |V_{cs}|^2 + |V_{cb}|^2 = 1.002 \pm 0.027$. These provide strong tests of the unitarity of the CKM matrix. With the significantly improved direct determination of $|V_{tb}|$, the unitarity checks for the third row and column have also become fairly precise, leaving decreasing room for mixing with other states. The sum of the three angles of the unitarity triangle, $\alpha + \beta + \gamma = (172 \pm 5)°$, is also consistent with the SM expectation.

The CKM matrix elements can be most precisely determined using a global fit to all available measurements and imposing the SM constraints (i.e., three generation unitarity). The fit must also use theory predictions for hadronic matrix elements, which sometimes have significant uncertainties. There are several approaches to combining the experimental data. CKMfitter [6, 111] and Ref. [128] (which develops [129, 130] further) use frequentist

<figure><img src="image_47.png" /><figcaption>Figure 12.2: Constraints on the $\bar{\rho}, \bar{\eta}$ plane. The shaded areas have 95% CL.</figcaption></figure>

statistics, while UTfit [112, 131] uses a Bayesian approach. These approaches provide similar results.

The constraints implied by the unitarity of the three generation CKM matrix significantly reduce the allowed range of some of the CKM elements. The fit for the Wolfenstein parameters defined in Eq. (12.4) gives

$$ \begin{aligned} \lambda &= 0.22501 \pm 0.00068, & A &= 0.826^{+0.016}_{-0.015}, \\ \bar{\rho} &= 0.1591 \pm 0.0094, & \bar{\eta} &= 0.3523^{+0.0073}_{-0.0071}. \end{aligned} \qquad (12.26) $$

These values are obtained using the method of Refs. [6, 111]. The prescription of Refs. [112, 131] gives $\lambda = 0.22497 \pm 0.00070$, $A = 0.839 \pm 0.011$, $\bar{\rho} = 0.1581 \pm 0.0092$, and $\bar{\eta} = 0.3548 \pm 0.0072$ [132]; these results are now very close to one another. The fit results for the magnitudes of all nine CKM elements are

$$ |V_{CKM}| = \begin{pmatrix} 0.97435 \pm 0.00016 & 0.22501 \pm 0.00068 & 0.003732^{+0.000090}_{-0.000085} \\ 0.22487 \pm 0.00068 & 0.97349 \pm 0.00016 & 0.04183^{+0.000079}_{-0.000069} \\ 0.00858^{+0.00019}_{-0.00017} & 0.04111^{+0.00077}_{-0.00068} & 0.999118^{+0.000029}_{-0.000034} \end{pmatrix}, \qquad (12.27) $$

and the Jarlskog invariant is $J = (3.12^{+0.13}_{-0.12}) \times 10^{-5}$. The parameters in Eq. (12.3) are

$$ \begin{aligned} \sin\theta_{12} &= 0.22501 \pm 0.00068, & \sin\theta_{13} &= 0.003732^{+0.000090}_{-0.000085}, \\ \sin\theta_{23} &= 0.04183^{+0.00079}_{-0.00069}, & \delta &= 1.147 \pm 0.026. \end{aligned} \qquad (12.28) $$

Fig. 12.2 illustrates the constraints on the $\bar{\rho}, \bar{\eta}$ plane from various measurements, and the global fit result. The shaded 95% CL regions all overlap consistently around the global fit region.

If one uses only tree-level inputs (magnitudes of CKM elements not coupling to the top quark and the angle $\gamma$), the resulting fit is almost identical for $\lambda$ in Eq. (12.26), while the other parameters' central values can change by about a sigma and their uncertainties double, yielding $\lambda = 0.22509 \pm 0.00068$, $A = 0.811 \pm 0.024$, $\bar{\rho} = 0.166^{+0.022}_{-0.021}$, and $\bar{\eta} = 0.367^{+0.024}_{-0.023}$. This illustrates how the constraints can be less tight in the presence of BSM physics.

## 12.5 Implications beyond the SM

The effects in $B, B_s, K$, and $D$ decays and mixings due to high-scale physics ($W, Z, t, H$ in the SM, and unknown heavier particles) can be parameterized by operators composed of SM
fields, obeying the $SU(3) \times SU(2) \times U(1)$ gauge symmetry. Flavor-changing neutral currents, suppressed in the SM, are especially sensitive to beyond SM contributions. Processes studied in great detail, both experimentally and theoretically, include neutral meson mixings, $B_{(s)} \to X\gamma$, $X\ell^+\ell^-$, $\ell^+\ell^-$, $K \to \pi\nu\bar{\nu}$, etc. The BSM contributions to these operators are suppressed by powers of the scale at which they are generated. Already at lowest order, there are many dimension-6 operators, and the observable effects of BSM interactions are encoded in their coefficients. In the SM, these coefficients are determined by just the four CKM parameters, and the $W$, $Z$, and quark masses. For example, $\Delta m_d$, $\Gamma(B \to \rho\gamma)$, $\Gamma(B \to \pi\ell^+\ell^-)$, and $\Gamma(B \to \ell^+\ell^-)$ are all proportional to $|V_{td}V_{tb}|^2$ in the SM, however, they may receive unrelated BSM contributions. These BSM contributions may or may not obey the SM relations. (For example, the flavor sector of the MSSM contains 69 CP-conserving parameters and 41 CP-violating phases, i.e., 40 new ones [133]). Thus, similar to the measurements of $\sin 2\beta$ in tree- and loop-dominated decay modes, overconstraining measurements of the magnitudes and phases of flavor-changing neutral-current amplitudes gives good sensitivity to BSM.

To illustrate the level of suppression required for BSM contributions, consider a class of models in which the unitarity of the CKM matrix is maintained, and the dominant BSM effects modify the neutral meson mixing amplitudes [134] by $(z_{ij}/\Lambda^2)(\bar{q}_i\gamma^\mu P_L q_j)^2$, where $z_{ij}$ is an unknown coefficient and $\Lambda$ is the scale suppressing this BSM contribution (see, [135, 136]). It is only known since the first measurements of $\gamma$ and $\alpha$ that the SM gives the leading contribution to $B^0 - \bar{B}^0$ mixing [6,137]. Nevertheless, new physics with a generic weak phase may still contribute to neutral meson mixings at a significant fraction of the SM [131, 138, 139]. The existing data imply that $\Lambda/|z_{ij}|^{1/2}$ has to exceed about $10^4$ TeV for $K^0 - \bar{K}^0$ mixing, $10^3$ TeV for $D^0 - \bar{D}^0$ mixing, 500 TeV for $B^0 - \bar{B}^0$ mixing, and 100 TeV for $B_s^0 - \bar{B}_s^0$ mixing [131, 136]. (Some other operators are even better constrained [131].) The constraints are the strongest in the kaon sector, because the CKM suppression is the most severe. Thus, if there is new physics at the TeV scale, $|z_{ij}| \ll 1$ is required. Even if $|z_{ij}|$ are suppressed by a loop factor and $|V_{ti}^*V_{tj}|^2$ (in the down quark sector), similar to the SM, one expects percent-level effects, which may be observable in forthcoming flavor physics experiments. To constrain such extensions of the SM, many measurements irrelevant for the SM-CKM fit, such as the CP asymmetry in semileptonic $B_{d,s}^0$ decays, $A_{SL}^{d,s}$, are important [140]. The current world averages [24] are consistent with the SM, with experimental uncertainties far greater than those of the theory predictions.

There are many key measurements sensitive to BSM physics, which do not constrain the unitarity triangle in Fig. 12.1. For example, a key quantity in the $B_s$ system is $\beta_s = \arg(-V_{ts}V_{tb}^*/V_{cs}V_{cb}^*)$, which is the small, $\lambda^2$-suppressed, angle of a “squashed” unitarity triangle, obtained by taking the scalar product of the second and third columns of the CKM matrix. This angle can be measured via time-dependent CP violation in $B_s^0 \to J/\psi\phi$, similar to $\beta$ in $B^0 \to J/\psi K^0$. Since the $J/\psi\phi$ final state is not a CP eigenstate, an angular analysis of the decay products is needed to separate the CP-even and CP-odd components, which give opposite asymmetries. In the SM, the asymmetry for the CP-even part is $2\beta_s$, when one neglects subdominant amplitudes with a weak phase $V_{ub}$. (Sometimes the notation $\phi_s = -2\beta_s$ plus a possible BSM contribution to the $B_s$ mixing phase is used.) Testing if the data agree with the SM prediction, $2\beta_s = 0.03726^{+0.00078}_{-0.00077}$ [111], is another sensitive probe of the SM. The current world average, dominated by LHC measurements including the $B_s \to J/\psi K^+K^-$ and $J/\psi \pi^+\pi^-$ decay modes, is $2\beta_s = 0.040 \pm 0.016$ [69]. Since the uncertainty is much larger than that in the SM, a lot will be learned from more precise future measurements. Searches for CP violation in the charm sector, in particular in $D^0 - \bar{D}^0$ mixing, provide complementary sensitivity to BSM.

In the kaon sector, the CP-violating observables, $\epsilon$ and $\epsilon'$, are tiny, so models in which all sources of CP violation are small were viable before the B-factory measurements. Since the measurement of $\sin 2\beta$, we know that CP violation can be an $O(1)$ effect,

and only flavor mixing is suppressed between the three quark generations. Thus, many models with spontaneous CP violation were excluded. In the kaon sector, clean tests of the SM can come from measurements of $K^+ \to \pi^+\nu\bar{\nu}$ [77] and $K_L^0 \to \pi^0\nu\bar{\nu}$ [141]. These loop-induced rare decays are sensitive to BSM, and will allow precise tests [142] of the CKM paradigm, independent of B decays.

The CKM elements are fundamental parameters, so they should be measured as precisely as possible. The overconstraining measurements of CP asymmetries, mixing, semileptonic, and rare decays severely constrain the magnitudes and phases of possible BSM contributions to flavor-changing interactions. If new particles are observed at the LHC, it will be important to explore their flavor parameters as precisely as possible to understand the underlying physics.

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13. CP Violation in the Quark Sector

Revised August 2023 by T. Gershon (Warwick U.) and Y. Nir (Weizmann Inst.).

The CP transformation combines charge conjugation *C* with parity *P*. Under *C*, particles and antiparticles are interchanged, by conjugating all internal quantum numbers, e.g., $Q \to -Q$ for electromagnetic charge. Under *P*, the handedness of space is reversed, $\vec{x} \to -\vec{x}$. Thus, for example, a left-handed electron $e_L^{-}$ is transformed under CP into a right-handed positron, $e_R^+$.

If CP were an exact symmetry, the laws of nature would be the same for matter and for antimatter. We observe that most phenomena are *C*- and *P*-symmetric, and therefore, also *CP*-symmetric. In particular, these symmetries are respected by the electromagnetic and strong interactions. The weak interactions, on the other hand, violate *C* and *P* in the strongest possible way. For example, the W bosons couple to left-handed electrons, $e_L$, and to their *CP*-conjugate right-handed positrons, $e_R^+$, but to neither their *C*-conjugate left-handed positrons, $e_L^-$, nor their *P*-conjugate right-handed electrons, $e_R^-$. While weak interactions violate *C* and *P* separately, CP is still preserved in most weak interaction processes. The CP symmetry is, however, violated in certain processes involving interference effects, as discovered in neutral *K* decays in 1964 [1], and established later in *B* (2001) and *D* (2019) decays. For example, as discovered in 1967, a $K_L$ meson decays more often to $\pi^-e^+\nu_e$ than to $\pi^+e^-\bar{\nu}_e$, thus allowing electrons and positrons to be unambiguously distinguished, but the decay-rate asymmetry is only at the 0.003 level. The CP-violating effects observed in the *B* system are larger: the parameter describing the CP asymmetry in the decay time distribution of $B^0/\bar{B}^0$ meson transitions to CP eigenstates like $J/\psi K_S$ is about 0.7 [2,3]. These effects are related to $K^0-\bar{K}^0$ and $B^0-\bar{B}^0$ mixing, but CP violation arising solely from decay amplitudes has also been observed, first in $K \to \pi\pi$ decays [4–6], subsequently in $B^0$ [7, 8], $B^+$ [9–11], and $B_s^0$ [12] decays, and most recently in charm decays [13]. All of these observed CP asymmetries are consistent with the Standard Model predictions. Similar effects could also occur in decays of baryons, but have not yet been observed. Given that neutrino masses and lepton mixing have been established, it is expected that CP is violated also in the lepton sector [14]. Discovering CP violation in the lepton sector is one of the main goals of current and near-future experiments. CP violation has not yet been observed in processes involving the top quark, nor in flavor-conserving processes such as electric dipole moments; for these, any significant observation would be a clear indication of physics beyond the Standard Model.

In addition to parity and to continuous Lorentz transformations, there is one other spacetime operation that could be a symmetry of the interactions: time reversal $T$, $t \to -t$. Violations of $T$ symmetry have been observed in neutral $K$ decays [15]. More recently, $T$ violation has been observed between states that are not CP-conjugate [16], exploiting the fact that for neutral $B$ mesons both flavor tagging and CP tagging can be used [17]. Moreover, $T$ violation is expected as a corollary of CP violation if the combined CPT transformation is a fundamental symmetry of nature [18]. All observations indicate that CPT is indeed a symmetry of nature [15]. Furthermore, one cannot build a locally Lorentz-invariant quantum field theory with a Hermitian Hamiltonian that violates CPT. (At several points in our discussion, we avoid assumptions about CPT, in order to identify cases where evidence for CP violation relies on assumptions about CPT.)

Within the Standard Model, CP symmetry is broken by complex phases in the Yukawa couplings (that is, the couplings of the Higgs scalar to quarks). When all transformations to remove unphysical phases in this model are exhausted, a single CP-violating parameter remains [19]. In the basis of mass eigenstates, this single phase appears in the $3 \times 3$ unitary matrix that gives the W-boson couplings to an up-type antiquark and a down-type quark. (If the Standard Model is supplemented with Majorana mass terms for the neutrinos, the analogous mixing matrix for leptons has three CP-violating phases.) The beautifully consistent and economical Standard-Model description of CP violation in terms of Yukawa couplings, known as the Kobayashi-Maskawa (KM) mechanism [19], agrees with all measurements to date. Fur-

thermore, one can fit the data allowing contributions from beyond the Standard Model (referred to subsequently as new physics) to loop processes to compete with, or even dominate over, the Standard Model amplitudes [20,21]. Such an analysis provides model-independent proof that the KM phase is different from zero, and that the matrix of three-generation quark mixing is the dominant source of CP violation in meson decays.

The current level of experimental accuracy and the theoretical uncertainties involved in the interpretation of the various observations leave room, however, for additional subdominant sources of CP violation from new physics. Indeed, almost all extensions of the Standard Model imply that there are such additional sources. Moreover, CP violation is a necessary condition for baryogenesis, the process of dynamically generating the matter-antimatter asymmetry of the Universe [22]. Despite the phenomenological success of the KM mechanism, it fails (by several orders of magnitude) to accommodate the observed asymmetry [23]. This discrepancy strongly suggests that nature provides additional sources of CP violation beyond the KM mechanism. The evidence for neutrino masses implies that CP can be violated also in the lepton sector. This situation makes leptogenesis [24,25], a scenario where CP-violating phases in the Yukawa couplings of the neutrinos play a crucial role in the generation of the baryon asymmetry, a very attractive possibility. The expectation of new sources motivates the large ongoing experimental effort to find deviations from the predictions of the KM mechanism.

CP violation can be experimentally searched for in a variety of processes, such as hadron decays, electric dipole moments of neutrons, electrons and nuclei, and neutrino oscillations. Hadron decays via the weak interaction probe flavor-changing CP violation. The search for electric dipole moments may find (or constrain) sources of CP violation that, unlike the KM phase, are not related to flavor-changing couplings. Following the discovery of the Higgs boson [26,27], searches for CP violation in the Higgs sector are becoming feasible. Future searches for CP violation in neutrino oscillations might provide further input on leptogenesis.

The present measurements of CP asymmetries provide some of the strongest constraints on the weak couplings of quarks. Future measurements of CP violation in $K, D, B$, and $B_s^0$ meson decays will provide additional constraints on the flavor parameters of the Standard Model, and can probe new physics. In this review, we give the formalism and basic physics motivations that are relevant to present and near future measurements of CP violation in the quark sector.

13.1 Formalism

The phenomenology of CP violation for neutral flavored mesons is particularly interesting, since many of the observables can be cleanly interpreted. Although the phenomenology is superficially different for $K^0, D^0, B^0$, and $B_s^0$ decays, this is primarily because each of these systems is governed by a different balance between decay rates, oscillations, and lifetime splitting. However, the general considerations presented in this section are identical for all flavored neutral pseudoscalar mesons. The phenomenology of CP violation for neutral mesons that do not carry flavor quantum numbers (such as the $\eta^{(l)}$ state) is quite different: such states are their own antiparticles and have definite CP eigenvalues, so the signature of CP violation is simply the decay to a final state with the opposite CP. Such decays are mediated by the electromagnetic or (OZI-suppressed [28–30]) strong interaction, where CP violation is not expected and has not yet been observed. In the remainder of this review, we restrict ourselves to considerations of weakly decaying hadrons.

In this section, we present a general formalism for, and classification of, CP violation in the decay of a weakly decaying hadron, denoted $M$. We pay particular attention to the case that $M$ is a $K^0, D^0, B^0$, or $B_s^0$ meson. Subsequent sections describe the CP-violating phenomenology, approximations, and alternative formalisms that are specific to each system.

13.1.1 Charged- and neutral-hadron decays

We define decay amplitudes of $M$ (which could be charged or neutral) and its CP conjugate $\bar{M}$ to a multi-particle final state $f$
and its CP conjugate $\bar{f}$ as

$$A_f = \langle f | \mathcal{H} | M \rangle, \quad \bar{A}_f = \langle f | \mathcal{H} | \bar{M} \rangle, \quad (13.1a)$$

$$A_{\bar{f}} = \langle \bar{f} | \mathcal{H} | M \rangle, \quad \bar{A}_{\bar{f}} = \langle \bar{f} | \mathcal{H} | \bar{M} \rangle, \quad (13.1b)$$

where $\mathcal{H}$ is the Hamiltonian governing weak interactions. The action of CP on these states introduces phases $\xi_M$ and $\xi_f$ that depend on their flavor content, according to

$$CP|M\rangle = e^{+i\xi_M} |\bar{M}\rangle, \quad CP|f\rangle = e^{+i\xi_f} |\bar{f}\rangle, \quad (13.2a)$$

$$CP|\bar{M}\rangle = e^{-i\xi_M} |M\rangle, \quad CP|\bar{f}\rangle = e^{-i\xi_f} |\bar{f}\rangle, \quad (13.2b)$$

so that $(CP)^2 = 1$. The phases $\xi_M$ and $\xi_f$ are arbitrary and unobservable because of the flavor symmetry of the strong interaction. If CP is conserved by the dynamics, $[CP, \mathcal{H}] = 0$, then $A_f$ and $\bar{A}_{\bar{f}}$ have the same magnitude and an arbitrary unphysical relative phase

$$\bar{A}_{\bar{f}} = e^{i(\xi_f - \xi_M)} A_f. \quad (13.3)$$

### 13.1.2 Neutral-meson mixing

A state that is initially a superposition of $M^0$ and $\bar{M}^0$, say

$$|\psi(0)\rangle = a(0)|M^0\rangle + b(0)|\bar{M}^0\rangle, \quad (13.4)$$

will evolve in time acquiring components that describe all possible decay final states $\{f_1, f_2, \dots\}$, that is,

$$|\psi(t)\rangle = a(t)|M^0\rangle + b(t)|\bar{M}^0\rangle + c_1(t)|f_1\rangle + c_2(t)|f_2\rangle + \dots. \quad (13.5)$$

If we are interested in computing only the values of $a(t)$ and $b(t)$ (and not the values of all $c_i(t)$), and if the times $t$ under study are much larger than the typical strong interaction scale, then we can use a much simplified formalism [31]. The simplified time evolution is determined by a $2 \times 2$ effective Hamiltonian $\mathbf{H}$ that is not Hermitian, since otherwise the mesons would only oscillate and not decay. Any complex matrix, such as $\mathbf{H}$, can be written in terms of Hermitian matrices $\mathbf{M}$ and $\mathbf{\Gamma}$ as

$$\mathbf{H} = \mathbf{M} - \frac{i}{2} \mathbf{\Gamma}. \quad (13.6)$$

$\mathbf{M}$ and $\mathbf{\Gamma}$ are associated with $(M^0, \bar{M}^0) \leftrightarrow (M^0, \bar{M}^0)$ transitions via off-shell (dispersive), and on-shell (absorptive) intermediate states, respectively. Diagonal elements of $\mathbf{M}$ and $\mathbf{\Gamma}$ are associated with the flavor-conserving transitions $M^0 \to M^0$ and $\bar{M}^0 \to \bar{M}^0$, while off-diagonal elements are associated with flavor-changing transitions $M^0 \leftrightarrow \bar{M}^0$.

The eigenvectors of $\mathbf{H}$ have well-defined masses and decay widths. To specify the components of the strong interaction eigenstates, $M^0$ and $\bar{M}^0$, in the light $(M_L)$ and heavy $(M_H)$ mass eigenstates, we introduce three complex parameters: $p, q$, and, for the case that both CP and CPT are violated in mixing, $z$. Then

$$|M_L\rangle \propto p\sqrt{1-z} |M^0\rangle + q\sqrt{1+z} |\bar{M}^0\rangle, \quad (13.7a)$$

$$|M_H\rangle \propto p\sqrt{1+z} |M^0\rangle - q\sqrt{1-z} |\bar{M}^0\rangle, \quad (13.7b)$$

with the normalization $|q|^2 + |p|^2 = 1$ when $z = 0$. (Another possible choice of labeling, which is in standard usage for $K$ mesons, defines the mass eigenstates according to their lifetimes: $K_S$ for the short-lived and $K_L$ for the long-lived state. The $K_L$ is experimentally found to be the heavier state. Yet another choice is often used for the $D$ mesons [32]: the eigenstates are labeled according to their dominant CP content.)

The real and imaginary parts of the eigenvalues $\omega_{L,H}$ corresponding to $|M_{L,H}\rangle$ represent their masses and decay widths, respectively. The mass and width splittings are

$$\Delta m \equiv m_H - m_L = \mathcal{R}\epsilon(\omega_H - \omega_L), \quad (13.8a)$$

$$\Delta\Gamma \equiv \Gamma_H - \Gamma_L = -2\mathcal{I}m(\omega_H - \omega_L). \quad (13.8b)$$

Note that here $\Delta m$ is positive by definition, while the sign of $\Delta\Gamma$ must be experimentally determined. The sign of $\Delta\Gamma$ has not yet been established for $B^0$ mesons, while $\Delta\Gamma < 0$ is established for $K$

and $B_s^0$ mesons. The Standard Model predicts $\Gamma_L > \Gamma_H$ for $B_{(s)}^0$ mesons; for this reason, $\Delta\Gamma = \Gamma_L - \Gamma_H$, which is still a signed quantity, is often used in the $B_{(s)}^0$ literature and is the convention used in the PDG experimental summaries.

Solving the eigenvalue problem for $\mathbf{H}$ yields

$$\left(\frac{q}{p}\right)^2 = \frac{\mathbf{M}_{12}^* - (i/2)\mathbf{\Gamma}_{12}^*}{\mathbf{M}_{12} - (i/2)\mathbf{\Gamma}_{12}}, \quad (13.9)$$

and

$$z \equiv \frac{\delta m - (i/2)\delta \Gamma}{\Delta m - (i/2)\Delta \Gamma}, \quad (13.10)$$

where

$$\delta m \equiv \mathbf{M}_{11} - \mathbf{M}_{22}, \quad \delta \Gamma \equiv \mathbf{\Gamma}_{11} - \mathbf{\Gamma}_{22} \quad (13.11)$$

are the differences in effective mass and decay-rate expectation values for the strong interaction states $M^0$ and $\bar{M}^0$.

If either CP or CPT is a symmetry of $\mathbf{H}$ (independently of whether $T$ is conserved or violated), then the values of $\delta m$ and $\delta \Gamma$ are both zero, and hence $z = 0$. We also find that

$$\omega_H - \omega_L = 2\sqrt{\left(\mathbf{M}_{12} - \frac{i}{2}\mathbf{\Gamma}_{12}\right)\left(\mathbf{M}_{12}^* - \frac{i}{2}\mathbf{\Gamma}_{12}^*\right)}. \quad (13.12)$$

If either CP or $T$ is a symmetry of $\mathbf{H}$ (independently of whether CPT is conserved or violated), then $\mathbf{\Gamma}_{12}/\mathbf{M}_{12}$ is real, leading to

$$\left(\frac{q}{p}\right)^2 = e^{2i\xi_M} \quad \Rightarrow \quad \left|\frac{q}{p}\right| = 1, \quad (13.13)$$

where $\xi_M$ is the arbitrary unphysical phase introduced in Eq. (13.2). If, and only if, CP is a symmetry of $\mathbf{H}$ (independently of CPT and $T$), then both of the above conditions hold, with the result that the mass eigenstates are orthogonal

$$\langle M_H | M_L \rangle = |p|^2 - |q|^2 = 0. \quad (13.14)$$

### 13.1.3 CP-violating observables

All CP-violating observables in $M$ and $\bar{M}$ decays to final states $f$ and $\bar{f}$ can be expressed in terms of phase-convention-independent combinations of $A_f, \bar{A}_f, A_{\bar{f}},$ and $\bar{A}_{\bar{f}}$, together with, for neutral meson decays only, $q/p$. CP violation in charged meson and all baryon decays depends only on the combination $|\bar{A}_{\bar{f}}/A_f|$, while CP violation in flavored neutral meson decays is enriched by $M^0 \leftrightarrow \bar{M}^0$ oscillations, and depends, additionally, on $|q/p|$ and on $\lambda_f \equiv (q/p)(\bar{A}_f/A_f)$.

The decay rates of the two neutral kaon mass eigenstates, $K_S$ and $K_L$, are different enough $(\Gamma_S/\Gamma_L \sim 500)$ that one can, in most cases, actually study their decays independently. For $D^0, B^0$, and $B_s^0$ mesons, however, values of $\Delta\Gamma/\Gamma$ (where $\Gamma \equiv (\Gamma_H + \Gamma_L)/2$) are relatively small, and so both mass eigenstates must be considered in their evolution. We denote the state of an initially pure $|M^0\rangle$ or $|\bar{M}^0\rangle$ after an elapsed proper time $t$ as $|M_{\text{phys}}^0(t)\rangle$ or $|\bar{M}_{\text{phys}}^0(t)\rangle$, respectively. Using the effective Hamiltonian approximation, but not assuming CPT to be a good symmetry, we obtain

$$|M_{\text{phys}}^0(t)\rangle = (g_+(t) + z g_-(t)) |M^0\rangle - \sqrt{1-z^2} \frac{q}{p} g_-(t) |\bar{M}^0\rangle, \quad (13.15a)$$

$$|\bar{M}_{\text{phys}}^0(t)\rangle = (g_+(t) - z g_-(t)) |\bar{M}^0\rangle - \sqrt{1-z^2} \frac{p}{q} g_-(t) |M^0\rangle, \quad (13.15b)$$

where

$$g_{\pm}(t) \equiv \frac{1}{2} \left[ \exp\left(-im_H t - \frac{1}{2}\Gamma_H t\right) \pm \exp\left(-im_L t - \frac{1}{2}\Gamma_L t\right) \right] \quad (13.16)$$

and $z = 0$ if either CPT or CP is conserved.
Defining $x \equiv \Delta m/\Gamma$ and $y \equiv \Delta\Gamma/(2\Gamma)$, and assuming $z = 0$, one obtains the following time-dependent decay rates:

$$ \frac{d\Gamma[M_{\text{phys}}^0(t) \to f]/dt}{e^{-\Gamma t} N_f} = (|A_f|^2 + |(q/p)\bar{A}_f|^2) \cosh(y\Gamma t) \\ + (|A_f|^2 - |(q/p)\bar{A}_f|^2) \cos(x\Gamma t) \\ + 2 \mathcal{R}e((q/p)A_f^*\bar{A}_f) \sinh(y\Gamma t) \\ - 2 \mathcal{I}m((q/p)A_f^*\bar{A}_f) \sin(x\Gamma t), \qquad (13.17a) $$

$$ \frac{d\Gamma[\overline{M}_{\text{phys}}^0(t) \to f]/dt}{e^{-\Gamma t} N_f} = (|(p/q)A_f|^2 + |\overline{A}_f|^2) \cosh(y\Gamma t) \\ - (|(p/q)A_f|^2 - |\overline{A}_f|^2) \cos(x\Gamma t) \\ + 2 \mathcal{R}e((p/q)A_f \overline{A}_f^*) \sinh(y\Gamma t) \\ - 2 \mathcal{I}m((p/q)A_f \overline{A}_f^*) \sin(x\Gamma t), \qquad (13.17b) $$

where $N_f$ is a common, time-independent, normalization factor that can be determined bearing in mind that the range of $t$ is $0 < t < \infty$. Decay rates to the CP-conjugate final state $\bar{f}$ are obtained analogously, with $N_f = N_{\bar{f}}$ and the substitutions $A_f \to A_{\bar{f}}$ and $\bar{A}_f \to \bar{A}_{\bar{f}}$ in Eqs. (13.17a) and (13.17b). Terms proportional to $|A_f|^2$ or $|\bar{A}_f|^2$ are associated with decays that occur without any net $M^0 \leftrightarrow \overline{M}^0$ oscillation, while terms proportional to $|(q/p)\bar{A}_f|^2$ or $|(p/q)A_f|^2$ are associated with decays following a net oscillation. The sinh($y\Gamma t$) and sinh($x\Gamma t$) terms of Eqs. (13.17a) and (13.17b) are associated with the interference between these two cases. Note that, in multi-body decays such as $D^0 \to K_S\pi^+\pi^-$ or $B^0 \to \pi^+\pi^-\pi^+\pi^-$, amplitudes are functions of variables that describe the phase-space of the final state. Interference may be present in some regions but not others, and is strongly influenced by resonant substructure.

When neutral pseudoscalar mesons are produced coherently in pairs from the decay of a vector resonance, $V \to M^0 \overline{M}^0$ (for example, $\Upsilon(4S) \to B^0 \overline{D}^0$, $\psi(3770) \to D^0 \overline{D}^0$ or $\phi \to K^0 \overline{K}^0$), the time-dependence of their subsequent decays to final states $f_1$ and $f_2$ has a similar form to Eqs. (13.17a) and (13.17b):

$$ \frac{d\Gamma[V_{\text{phys}}(t_1, t_2) \to f_1 f_2]/d(\Delta t)}{e^{-\Gamma |\Delta t|} N_{f_1 f_2}} = (|a_+|^2 + |a_-|^2) \cosh(y\Gamma\Delta t) \\ + (|a_+|^2 - |a_-|^2) \cos(x\Gamma\Delta t) \\ - 2 \mathcal{R}e(a_+^* a_-) \sinh(y\Gamma\Delta t) \\ + 2 \mathcal{I}m(a_+^* a_-) \sin(x\Gamma\Delta t), \qquad (13.18) $$

where $\Delta t \equiv t_2 - t_1$ is the difference in the production times, $t_1$ and $t_2$, of $f_1$ and $f_2$, respectively, and the dependence on the average decay time and on decay angles has been integrated out. The normalization factor $N_{f_1 f_2}$ can be evaluated, noting that the range of $\Delta t$ is $-\infty < \Delta t < \infty$. The coefficients in Eq. (13.18) are determined by the amplitudes for no net oscillation from $t_1 \to t_2$, $\bar{A}_{f_1} A_{f_2}$, and $A_{f_1} \bar{A}_{f_2}$, and for a net oscillation, $(q/p)\bar{A}_{f_1} \bar{A}_{f_2}$ and $(p/q)A_{f_1} A_{f_2}$, via

$$ a_+ \equiv \bar{A}_{f_1} A_{f_2} - A_{f_1} \bar{A}_{f_2}, \qquad (13.19a) $$

$$ a_- \equiv -\sqrt{1-z^2} \left( \frac{q}{p} \bar{A}_{f_1} \bar{A}_{f_2} - \frac{p}{q} A_{f_1} A_{f_2} \right) \\ + z (\bar{A}_{f_1} A_{f_2} + A_{f_1} \bar{A}_{f_2}) . \qquad (13.19b) $$

Assuming CPT conservation, $z = 0$, and identifying $\Delta t \to t$ and $f_2 \to f$, we find that Eqs. (13.18) and (13.19) reduce to Eq. (13.17a) with $A_{f_1} = 0$, $\bar{A}_{f_1} = 1$, or to Eq. (13.17b) with $\bar{A}_{f_1} = 0$, $A_{f_1} = 1$. Indeed, this plays an important role in experiments that exploit the coherence of $V \to M^0 \overline{M}^0$ production. Final states $f_1$ with $A_{f_1} = 0$ or $\bar{A}_{f_1} = 0$ are called tagging states,

because they identify the decaying pseudoscalar meson as, respectively, $\overline{M}^0$ or $M^0$. Before one of $M^0$ or $\overline{M}^0$ decays, they evolve in phase, so that there is always one $M^0$ and one $\overline{M}^0$ present. A tagging decay of one meson sets the clock for the time evolution of the other: it starts at $t_1$ as purely $M^0$ or $\overline{M}^0$, with time evolution that depends only on $t_2 - t_1$.

When $f_1$ is a state that both $M^0$ and $\overline{M}^0$ can decay into, then Eq. (13.18) contains interference terms proportional to $A_{f_1} \bar{A}_{f_1} \neq 0$ that are not present in Eqs. (13.17a) and (13.17b). Even when $f_1$ is dominantly produced by $M^0$ decays rather than $\overline{M}^0$ decays, or vice versa, $A_{f_1} \bar{A}_{f_1}$ can be non-zero owing to doubly-CKM-suppressed decays (with amplitudes suppressed by at least two powers of $\lambda$ relative to the dominant amplitude, in the language of Section 13.3), and these terms should be considered for precision studies of CP violation in coherent $V \to M^0 \overline{M}^0$ decays [33]. The correlations in $V \to M^0 \overline{M}^0$ decays can also be exploited to determine phase differences between favored and suppressed decay amplitudes [34, 35].

### 13.1.4 Classification of CP-violating effects

We distinguish three types of CP-violating effects that can occur in the quark sector:

#### I. CP violation in decay is defined by

$$ |\bar{A}_{\bar{f}}/A_f| \neq 1. \qquad (13.20) $$

In charged meson (and all baryon) decays, where mixing effects are absent, this is the only possible source of CP asymmetries:

$$ A_{f\pm} \equiv \frac{\Gamma(M^{-} \to f^{-}) - \Gamma(M^{+} \to f^{+})}{\Gamma(M^{-} \to f^{-}) + \Gamma(M^{+} \to f^{+})} = \frac{|\bar{A}_{f^{-}}/A_{f^{+}}|^2 - 1}{|\bar{A}_{f^{-}}/A_{f^{+}}|^2 + 1}. \qquad (13.21) $$

Note that the usual sign convention for CP asymmetries of hadrons is for the difference between the rate involving the particle that contains a heavy quark and that which contains an antiquark. Hence, Eq. (13.21) corresponds to the definition for $B^\pm$ mesons, but the opposite sign is used for $D_{(s)}^\pm$ decays.

#### II. CP (and T) violation in mixing is defined by

$$ |q/p| \neq 1. \qquad (13.22) $$

In charged-current semileptonic neutral meson decays $M, \bar{M} \to \ell^\pm X^\mp$ (taking $|A_{\ell+X^-}| = |\bar{A}_{\ell-X^+}|$ and $A_{\ell-X^+} = \bar{A}_{\ell-X^-} = 0$, as is the case in the Standard Model, to lowest order in $G_F$, and in most of its extensions), this is the only source of CP violation, and can be measured via the asymmetry of “wrong-sign” decays induced by oscillations:

$$ A_{SL}(t) \equiv \frac{d\Gamma/dt[\overline{M}_{\text{phys}}^0(t) \to \ell^+ X^-] - d\Gamma/dt[M_{\text{phys}}^0(t) \to \ell^- X^+]}{d\Gamma/dt[\overline{M}_{\text{phys}}^0(t) \to \ell^+ X^-] + d\Gamma/dt[M_{\text{phys}}^0(t) \to \ell^- X^+]}, \qquad (13.23a) $$

$$ = \frac{1 - |q/p|^4}{1 + |q/p|^4}. \qquad (13.23b) $$

Note that this asymmetry of time-dependent decay rates is actually time-independent.

#### III. CP violation in interference between a decay without mixing, $M^0 \to f$, and a decay with mixing, $M^0 \to \overline{M}^0 \to f$ (such an effect occurs only in decays to final states that are common to $M^0$ and $\overline{M}^0$, including all CP eigenstates), is defined by

$$ \arg(\lambda_f) + \arg(\lambda_{\bar{f}}) \neq 0, \quad \text{with} \quad \lambda_f \equiv \frac{q}{p} \frac{\bar{A}_f}{A_f}. \qquad (13.24) $$

For final CP eigenstates, $f_{CP}$, the condition Eq. (13.24) simplifies to

$$ \mathcal{I}m(\lambda_{f_{CP}}) \neq 0, \qquad (13.25) $$
This form of CP violation can be observed, for example, using the asymmetry of neutral meson decay rates into CP eigenstates

$$ \begin{aligned} \mathcal{A}_{f_{CP}}(t) &\equiv \frac{d\Gamma/dt[\bar{M}_{\text{phys}}^0(t) \rightarrow f_{CP}]}{d\Gamma/dt[\bar{M}_{\text{phys}}^0(t) \rightarrow f_{CP}]} - \frac{d\Gamma/dt[\bar{M}_{\text{phys}}^0(t) \rightarrow f_{CP}]}{d\Gamma/dt[\bar{M}_{\text{phys}}^0(t) \rightarrow f_{CP}]} \\ &\equiv \frac{d\Gamma/dt[\bar{M}_{\text{phys}}^0(t) \rightarrow f_{CP}]}{d\Gamma/dt[\bar{M}_{\text{phys}}^0(t) \rightarrow f_{CP}]} + \frac{d\Gamma/dt[\bar{M}_{\text{phys}}^0(t) \rightarrow f_{CP}]}{d\Gamma/dt[\bar{M}_{\text{phys}}^0(t) \rightarrow f_{CP}]} . \end{aligned} \quad (13.26) $$

If $\Delta\Gamma = 0$, as expected to a good approximation for $B^0$ mesons but not for $K^0$ and $B_s^0$ mesons, and $|q/p| = 1$, then $\mathcal{A}_{f_{CP}}$ has a particularly simple form (see Eq. (13.75), below). If, in addition, the decay amplitudes fulfill $|\bar{A}_{f_{CP}}| = |A_{f_{CP}}|$, the interference between decays with and without mixing is the only source of asymmetry and $\mathcal{A}_{f_{CP}}(t) = \text{Im}(\lambda_{f_{CP}}) \sin(x\Gamma t)$.

Examples of these three types of CP violation will be given in Sections 13.4, 13.5, and 13.6.

## 13.2 Theoretical Interpretation: General Considerations

Consider the $M \to f$ decay amplitude $A_f$, and the CP conjugate process, $\bar{M} \to \bar{f}$, with decay amplitude $\bar{A}_{\bar{f}}$. There are two types of phases that may appear in these decay amplitudes. Complex parameters in any Lagrangian term that contributes to the amplitude will appear in complex conjugate form in the CP-conjugate amplitude. Thus, their phases appear in $A_f$ and $\bar{A}_{\bar{f}}$ with opposite signs. In the Standard Model, these phases occur only in the couplings of the $W^\pm$ bosons, and hence, are often called “weak phases.” The weak phase of any single term is convention-dependent. However, the difference between the weak phases in two different terms in $A_f$ is convention-independent. A second type of phase can appear in scattering or decay amplitudes, even when the Lagrangian is real. This phase originates from the possible contribution from intermediate on-shell states in the decay process. Since such phases are generated by CP-invariant interactions, they are the same in $A_f$ and $\bar{A}_{\bar{f}}$. Usually the dominant rescattering is due to strong interactions; hence the designation “strong phases” for the phase shifts so induced. Again, only the relative strong phases between different terms in the amplitude are physically meaningful.

The “weak” and “strong” phases discussed here appear in addition to the spurious CP-transformation phases of Eq. (13.3). Those spurious phases are due to an arbitrary choice of phase convention, and do not originate from any dynamics or induce any CP violation. For simplicity, we set them to zero from here on.

It is useful to write each contribution $a_i$ to $A_f$ in three parts: its magnitude $|a_i|$, its weak phase $\phi_i$, and its strong phase $\delta_i$. If, for example, there are two such contributions, $A_f = a_1 + a_2$, we have

$$ A_f = |a_1|e^{i(\delta_1 + \phi_1)} + |a_2|e^{i(\delta_2 + \phi_2)}, \quad (13.27a) $$

$$ \bar{A}_{\bar{f}} = |a_1|e^{i(\delta_1 - \phi_1)} + |a_2|e^{i(\delta_2 - \phi_2)}. \quad (13.27b) $$

Similarly, for neutral mesons, it is useful to write

$$ \mathbf{M}_{12} = |\mathbf{M}_{12}|e^{i\phi_M}, \quad \mathbf{\Gamma}_{12} = |\mathbf{\Gamma}_{12}|e^{i\phi_\Gamma}. \quad (13.28) $$

Each of the phases appearing in Eqs. (13.27) and (13.28) is convention-dependent, but combinations such as $\delta_1 - \delta_2$, $\phi_1 - \phi_2$, $\phi_M - \phi_\Gamma$, and $\phi_M + \phi_1 - \bar{\phi}_1$ (where $\bar{\phi}_1$ is a weak phase contributing to $\bar{A}_f$) are physical.

It is now straightforward to evaluate the various asymmetries in terms of the theoretical parameters introduced here. We will do so with approximations that are often relevant to the most interesting measured asymmetries.

1. The CP asymmetry in charged meson and all baryon decays

[Eq. (13.21)] is given by

$$ \mathcal{A}_f = - \frac{2|a_1 a_2| \sin(\delta_2 - \delta_1) \sin(\phi_2 - \phi_1)}{|a_1|^2 + |a_2|^2 + 2|a_1 a_2| \cos(\delta_2 - \delta_1) \cos(\phi_2 - \phi_1)}. \quad (13.29) $$

The quantity of most interest to theory is the weak phase difference $\phi_2 - \phi_1$. Its extraction from the asymmetry requires, however, that the amplitude ratio $|a_2/a_1|$ and the strong phase difference $\delta_2 - \delta_1$ are known. Both quantities depend on non-perturbative hadronic parameters that are difficult to calculate, but in some cases can be obtained from experiment.

2. In the approximation that $|\Gamma_{12}/\mathbf{M}_{12}| \ll 1$ (valid for $B^0$ and $B_s^0$ mesons), the CP asymmetry in semileptonic neutral-meson decays [Eq. (13.23)] is given by

$$ \mathcal{A}_{\text{SL}} = - \left| \frac{\Gamma_{12}}{\mathbf{M}_{12}} \right| \sin(\phi_M - \phi_\Gamma). \quad (13.30) $$

The quantity of most interest to theory is the weak phase $\phi_M - \phi_\Gamma$. Its extraction from the asymmetry requires, however, that $|\Gamma_{12}/\mathbf{M}_{12}|$ is known. State of the art calculations of this quantity for the $B^0$ and $B_s^0$ mesons have uncertainties of around 10% [36].

3. In the approximations that only a single weak phase contributes to decay, $A_f = |a_f|e^{i(\delta_f + \phi_f)}$, and that $|\Gamma_{12}/\mathbf{M}_{12}| = 0$, we obtain $|\lambda_f| = 1$, and the CP asymmetries in decays to a final CP eigenstate $f$ [Eq. (13.26)] with eigenvalue $\eta_f = \pm 1$ are given by

$$ \mathcal{A}_{f_{CP}}(t) = \mathrm{Im}(\lambda_f) \sin(\Delta mt) \quad \text{with} \quad \mathrm{Im}(\lambda_f) = \eta_f \sin(\phi_M + 2\phi_f). \quad (13.31) $$

Note that the phase measured is purely a weak phase, and no hadronic parameters are involved in the extraction of its value from $\mathrm{Im}(\lambda_f)$.

The discussion above allows us to introduce another classification of CP-violating effects:

1. Indirect CP violation is consistent with taking $\phi_M \neq 0$ and setting all other CP violating phases to zero. CP violation in mixing (type II) belongs to this class.

2. Direct CP violation cannot be accounted for by just $\phi_M \neq 0$. CP violation in decay (type I) belongs to this class.

The historical significance of this classification is related to theory. In superweak models [37], CP violation appears only in diagrams that contribute to $\mathbf{M}_{12}$, hence predicting no direct CP violation. In most models and, in particular, in the Standard Model, CP violation is both direct and indirect. As concerns type III CP violation, a single observation of such an effect would be consistent with indirect CP violation, but observing $\eta_{f_1} \mathrm{Im}(\lambda_{f_1}) \neq \eta_{f_2} \mathrm{Im}(\lambda_{f_2})$ (for the same decaying meson and two different final CP eigenstates $f_1$ and $f_2$) would establish direct CP violation. The experimental observation of $\epsilon' \neq 0$, which was achieved by establishing that $\mathrm{Im}(\lambda_{\pi^+\pi^-}) \neq \mathrm{Im}(\lambda_{\pi^0\pi^0})$ (see Section 13.4), excluded the superweak scenario.

## 13.3 Theoretical Interpretation: The KM Mechanism

Of all the Standard Model quark parameters, only the Kobayashi-Maskawa (KM) phase is CP-violating. Having a single source of CP violation, the Standard Model is very predictive for CP asymmetries: some vanish, and those that do not are correlated.

To be precise, CP could be violated also by strong interactions. The experimental upper bound on the electric-dipole moment of the neutron [38] implies, however, that $\theta_{\text{QCD}}$, the non-perturbative parameter that determines the strength of this type of CP violation, is tiny, if not zero [39]. The smallness of $\theta_{\text{QCD}}$ constitutes a theoretical puzzle, known as “the strong CP problem.” This, however, is irrelevant to our discussion of hadron decays.

The charged current interactions (that is, the $W^\pm$ interactions) for quarks are given by

$$ -\mathcal{L}_{W^\pm} = \frac{g}{\sqrt{2}} \bar{u}_{Li} \gamma^\mu (V_{\text{CKM}})_{ij} d_{Lj} W_\mu^+ + \text{h.c.} \quad (13.32) $$
Here $i, j = 1, 2, 3$ are generation numbers. The Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix for quarks is a $3 \times 3$ unitary matrix [40]. Ordering the quarks by their masses, i.e., $(u_1, u_2, u_3) \to (u, c, t)$ and $(d_1, d_2, d_3) \to (d, s, b)$, the elements of $V_{CKM}$ are written as follows:

$$V_{CKM} = \begin{pmatrix} V_{ud} & V_{us} & V_{ub} \\ V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb} \end{pmatrix} . \quad (13.33)$$

While a general $3 \times 3$ unitary matrix depends on three real angles

and six phases, the freedom to redefine the phases of the quark mass eigenstates can be used to remove five of the phases, leaving a single physical phase, the Kobayashi-Maskawa phase, that is responsible for all CP violation in the Standard Model.

The fact that one can parameterize $V_{CKM}$ by three real and only one imaginary physical parameters can be made manifest by choosing an explicit parametrization. The Wolfenstein parametrization [41, 42] is particularly useful:

$$V_{CKM} = \begin{pmatrix} 1 - \frac{1}{2}\lambda^2 - \frac{1}{8}\lambda^4 & \lambda & A\lambda^3(\rho - i\eta) \\ -\lambda + \frac{1}{2}A^2\lambda^5[1 - 2(\rho + i\eta)] & 1 - \frac{1}{2}\lambda^2 - \frac{1}{8}\lambda^4(1 + 4A^2) & A\lambda^2 \\ A\lambda^3[1 - (1 - \frac{1}{2}\lambda^2)(\rho + i\eta)] & -A\lambda^2 + \frac{1}{2}A\lambda^4[1 - 2(\rho + i\eta)] & 1 - \frac{1}{2}A^2\lambda^4 \end{pmatrix} . \quad (13.34)$$

Here $\lambda \approx 0.23$ (not to be confused with $\lambda_f$), the sine of the Cabibbo angle, plays the role of an expansion parameter, and $\eta$ represents the CP-violating phase. Terms of $O(\lambda^6)$ have been neglected.

The unitarity of the CKM matrix, $(VV^\dagger)_{ij} = (V^\dagger V)_{ij} = \delta_{ij}$, leads to twelve distinct complex relations among the matrix elements. The six relations with $i \neq j$ can be represented geometrically as triangles in the complex plane. Two of these,

$$V_{ud}V_{ub}^* + V_{cd}V_{cb}^* + V_{td}V_{tb}^* = 0, \quad (13.35a)$$

$$V_{td}V_{ud}^* + V_{ts}V_{us}^* + V_{tb}V_{ub}^* = 0, \quad (13.35b)$$

have terms of equal order, $O(A\lambda^3)$, and so have corresponding triangles whose interior angles are all $O(1)$ physical quantities that can be independently measured. The angles of the first triangle (see Fig. 13.1) are given by

$$\alpha \equiv \varphi_2 \equiv \arg \left( - \frac{V_{td} V_{tb}^*}{V_{ud} V_{ub}^*} \right) \approx \arg \left( - \frac{1 - \rho - i\eta}{\rho + i\eta} \right), \quad (13.36a)$$

$$\beta \equiv \varphi_1 \equiv \arg \left( - \frac{V_{cd} V_{cb}^*}{V_{td} V_{tb}^*} \right) \approx \arg \left( \frac{1}{1 - \rho - i\eta} \right), \quad (13.36b)$$

$$\gamma \equiv \varphi_3 \equiv \arg \left( - \frac{V_{ud} V_{ub}^*}{V_{cd} V_{cb}^*} \right) \approx \arg (\rho + i\eta) . \quad (13.36c)$$

The angles of the second triangle are equal to $(\alpha, \beta, \gamma)$ up to corrections of $O(\lambda^2)$. The notations $(\alpha, \beta, \gamma)$ and $(\varphi_1, \varphi_2, \varphi_3)$ are both in common usage but, for convenience, we only use the first convention in the following.

<figure><img src="image_48.png" /><figcaption>Figure 13.1: Graphical representation of the unitarity constraint $V_{ud}V_{ub}^* + V_{cd}V_{cb}^* + V_{td}V_{tb}^* = 0$ as a triangle in the complex plane.</figcaption></figure>

Another relation that can be represented as a triangle,

$$V_{us}V_{ub}^* + V_{cs}V_{cb}^* + V_{ts}V_{tb}^* = 0, \quad (13.37)$$

and, in particular, its small angle, of $O(\lambda^2)$,

$$\beta_s \equiv \arg \left( - \frac{V_{ts} V_{tb}^*}{V_{cs} V_{cb}^*} \right), \quad (13.38)$$

is convenient for analyzing CP violation in the $B_s^0$ sector.

All unitarity triangles have the same area, commonly denoted by $J/2$ [43]. If CP is violated, $J$ is different from zero and can be taken as the single CP-violating parameter. In the Wolfenstein parametrization of Eq. (13.34), $J \simeq \lambda^6 A^2 \eta$.

## 13.4 Kaons

CP violation was discovered in $K \to \pi\pi$ decays in 1964 [1]. The same mode provided the first observation of direct CP violation [4–6].

The decay amplitudes actually measured in neutral $K$ decays refer to the mass eigenstates $K_L$ and $K_S$, rather than to the $K$ and $\bar{K}$ states referred to in Eq. (13.1). The final $\pi^+\pi^-$ and $\pi^0\pi^0$ states are CP-even. In the CP conservation limit, $K_S(K_L)$ would be CP-even (odd), and therefore would (would not) decay to two pions. We define CP-violating amplitude ratios for two-pion final states,

$$\eta_{00} \equiv \frac{\langle \pi^0 \pi^0 | \mathcal{H} | K_L \rangle}{\langle \pi^0 \pi^0 | \mathcal{H} | K_S \rangle}, \quad \eta_{+-} \equiv \frac{\langle \pi^+ \pi^- | \mathcal{H} | K_L \rangle}{\langle \pi^+ \pi^- | \mathcal{H} | K_S \rangle} . \quad (13.39)$$

Another important observable is the asymmetry of time-integrated semileptonic decay rates:

$$\delta_L \equiv \frac{\Gamma(K_L \to \ell^+ \nu_\ell \pi^-) - \Gamma(K_L \to \ell^- \bar{\nu}_\ell \pi^+)}{\Gamma(K_L \to \ell^+ \nu_\ell \pi^-) + \Gamma(K_L \to \ell^- \bar{\nu}_\ell \pi^+)} . \quad (13.40)$$

CP violation has been observed as an appearance of $K_L$ decays to two-pion final states [44],

$$|\eta_{00}| = (2.220 \pm 0.011) \times 10^{-3}, \quad (13.41a)$$

$$|\eta_{+-}| = (2.232 \pm 0.011) \times 10^{-3}, \quad (13.41b)$$

$$|\eta_{00}/\eta_{+-}| = 0.9950 \pm 0.0007, \quad (13.41c)$$

where the CP-conserving phases $\phi_{ij}$ of the amplitude ratios $\eta_{ij}$ have been determined both assuming CPT invariance:

$$\phi_{00} = (43.52 \pm 0.05)^\circ, \quad \phi_{+-} = (43.51 \pm 0.05)^\circ, \quad (13.42)$$

and without assuming CPT invariance:

$$\phi_{00} = (43.7 \pm 0.6)^\circ, \quad \phi_{+-} = (43.4 \pm 0.5)^\circ. \quad (13.43)$$

CP violation has also been observed in semileptonic $K_L$ decays [44]

$$\delta_L = (3.32 \pm 0.06) \times 10^{-3}, \quad (13.44)$$

where $\delta_L$ is a weighted average of muon and electron measurements, as well as in $K_L$ decays to $\pi^+\pi^-\gamma$ and $\pi^+\pi^-e^+e^-$ [44]. CP violation in $K \to 3\pi$ decays has not yet been observed [44, 45].

Historically, CP violation in neutral $K$ decays has been described in terms of the complex parameters $\epsilon$ and $\epsilon'$. The observables $\eta_{00}$, $\eta_{+-}$, and $\delta_L$ are related to these parameters, and to
those of Section 13.1, by

$$
\eta_{00} = \frac{1 - \lambda_{\pi^0 \pi^0}}{1 + \lambda_{\pi^0 \pi^0}} = \epsilon - 2\epsilon', \quad (13.45a)
$$

$$
\eta_{+-} = \frac{1 - \lambda_{\pi^+ \pi^-}}{1 + \lambda_{\pi^+ \pi^-}} = \epsilon + \epsilon', \quad (13.45b)
$$

$$
\delta_L = \frac{1 - |q/p|^2}{1 + |q/p|^2} = \frac{2\mathrm{Re}(\epsilon)}{1 + |\epsilon|^2}, \quad (13.45c)
$$

where, in the last line, we have assumed that $|A_{\ell+\nu_{\ell}\pi^-}| = |
\bar{A}_{\ell-\bar{\nu}_{\ell}\pi^+}|$ and $|A_{\ell-\bar{\nu}_{\ell}\pi^+}| = |
\bar{A}_{\ell+\nu_{\ell}\pi^-}| = 0$. (The convention-
dependent parameter $\tilde{\epsilon} \equiv (1-q/p)/(1+q/p)$, sometimes used in
the literature, is, in general, different from $\epsilon$ but yields a similar
expression, $\delta_L = 2\mathcal{R}e(\tilde{\epsilon})/(1+|\tilde{\epsilon}|^2)$.) A fit to the $K \to \pi\pi$
data yields [44]

$$
|\epsilon| = (2.228 \pm 0.011) \times 10^{-3}, \quad (13.46a)
$$

$$
\mathcal{R}e(\epsilon'/\epsilon) = (1.66 \pm 0.23) \times 10^{-3}. \quad (13.46b)
$$

In discussing two-pion final states, it is useful to express the am-
plitudes $A_{\pi^0\pi^0}$ and $A_{\pi^+\pi^-}$ in terms of their isospin components
via

$$
A_{\pi^0 \pi^0} = \sqrt{\frac{1}{3}} |A_0| e^{i(\delta_0 + \phi_0)} - \sqrt{\frac{2}{3}} |A_2| e^{i(\delta_2 + \phi_2)}, \quad (13.47a)
$$

$$
A_{\pi^+\pi^-} = \sqrt{\frac{2}{3}} |A_0| e^{i(\delta_0 + \phi_0)} + \sqrt{\frac{1}{3}} |A_2| e^{i(\delta_2 + \phi_2)}, \quad (13.47b)
$$

where we parameterize the amplitude $A_I(\bar{A}_I)$ for $K^0(\bar{K}^0)$ decay
into two pions with total isospin $I=0$ or 2 as

$$
A_I \equiv (\langle \pi\pi \rangle_I |\mathcal{H}| K^0) = |A_I| e^{i(\delta_I + \phi_I)}, \quad (13.48a)
$$

$$
\bar{A}_I \equiv (\langle \pi\pi \rangle_I |\mathcal{H}| \bar{K}^0) = |A_I| e^{i(\delta_I - \phi_I)}. \quad (13.48b)
$$

The smallness of $|\eta_{00}|$ and $|\eta_{+-}|$ allows us to approximate

$$
\epsilon \approx \frac{1}{2} (1 - \lambda_{(\pi\pi)I=0}), \quad \epsilon' \approx \frac{1}{6} (\lambda_{\pi^0\pi^0} - \lambda_{\pi^+\pi^-}). \quad (13.49)
$$

The parameter $\epsilon$ represents indirect CP violation, while $\epsilon'$ pa-
rameterizes direct CP violation: $\mathcal{R}e(\epsilon')$ measures CP violation in
decay (type I), $\mathcal{R}e(\epsilon)$ measures CP violation in mixing (type II),
and $\mathcal{I}m(\epsilon)$ and $\mathcal{I}m(\epsilon')$ measure the interference between decays
with and without mixing (type III).

The following expressions for $\epsilon$ and $\epsilon'$ are useful for theoretical
evaluations:

$$
\epsilon \approx \frac{e^{i\pi/4}}{\sqrt{2}} \frac{\mathcal{I}m(\mathbf{M}_{12})}{\Delta m}, \quad \epsilon' \approx \frac{i}{\sqrt{2}} \left|\frac{A_2}{A_0}\right| e^{i(\delta_2 - \delta_0)} \sin(\phi_2 - \phi_0). \quad (13.50)
$$

The expression for $\epsilon$ is only valid in a phase convention where
$\phi_2 = 0$, corresponding to a real $V_{ud}V_{us}^*$, and in the approximation
that also $\phi_0 = 0$. The phase of $\epsilon$, $\arg(\epsilon) \approx \arctan(-2\Delta m/\Delta\Gamma)$,
is determined by non-perturbative QCD dynamics and is experi-
mentally determined to be about $\pi/4$. The calculation of $\epsilon$ bene-
fits from the fact that $\mathcal{I}m(\mathbf{M}_{12})$ is dominated by short distance
physics. Consequently, the main sources of uncertainty in theo-
retical interpretations of $\epsilon$ are the values of matrix elements, such
as $\langle K^0 | (\bar{s}d)_{V-A} (\bar{s}d)_{V-A} | \bar{K}^0 \rangle$. The expression for $\epsilon'$ is valid to
first order in $|A_2/A_0| \sim 1/20$. The phase of $\epsilon'$ is experimentally
determined, $\pi/2 + \delta_2 - \delta_0 \approx \pi/4$, and is independent of the model
of electroweak interactions. Note that, accidentally, $\epsilon'/\epsilon$ is real to
a good approximation. Determination of weak phase information
from the measurement of $\mathcal{R}e(\epsilon'/\epsilon)$ given in Eq. (13.46) has until
now been precluded by uncertainties in the hadronic parameters,
but recent advances in lattice QCD calculations and other theo-
retical approaches [46–48] suggest that it may become possible.

A future measurement of much interest is that of CP violation
in the rare $K \to \pi \nu \bar{\nu}$ decays. The signal for CP violation is
simply observing the $K_L \to \pi^0 \nu \bar{\nu}$ decay. The effect here is that

of interference between decays with and without mixing (type
III) [49]:

$$
\begin{equation} \tag{13.51}
\begin{split}
\frac{\Gamma(K_L \rightarrow \pi^0 \nu \bar{\nu})}{\Gamma(K^+ \rightarrow \pi^+ \nu \bar{\nu})} &= \frac{1}{2} [1 + |\lambda_{\pi \nu \bar{\nu}}|^2 - 2 \mathcal{R} e(\lambda_{\pi \nu \bar{\nu}})] \\
&\approx 1 - \mathcal{R} e(\lambda_{\pi \nu \bar{\nu}}),
\end{split}
\end{equation}
$$

where in the last equation we neglect CP violation in decay and
in mixing (expected, model-independently, to be of order 10⁻⁵
and 10⁻³, respectively). Such a measurement is experimentally
very challenging but would be theoretically very rewarding [50].
Similar to the CP asymmetry in B⁰ → J/ψK_S, the CP violation
in K → πνν̄ decay is predicted to be large (that is, the ratio in
Eq. (13.51) is neither CKM- nor loop-suppressed) and can be very
cleanly interpreted. In particular, the independent determinations
of the CKM parameters via B-meson and K-meson decays and
mixing will over-constrain the unitarity triangle and provide a
stringent text of the KM mechanism.

Within the Standard Model, the $K_L \to \pi^0 \nu \bar{\nu}$ decay is dom-
inated by an intermediate top quark contribution and, conse-
quently, can be interpreted in terms of CKM parameters [51]. (For
the charged mode, $K^+ \to \pi^+ \nu \bar{\nu}$, the contribution from an inter-
mediate charm quark is not negligible, and constitutes a source
of hadronic uncertainty.) In particular, $\mathcal{B}(K_L \to \pi^0 \nu \bar{\nu})$ provides
a theoretically clean way to determine the Wolfenstein parameter
$\eta$ [52]:

$$
\mathcal{B}(K_L \rightarrow \pi^0 \nu \bar{\nu}) = \kappa_L [X(m_t^2/m_W^2)]^2 A^4 \eta^2, \quad (13.52)
$$

where the hadronic parameter $\kappa_L \sim 2 \times 10^{-10}$ incorporates the
value of the four-fermion matrix element which is deduced, using
isospin relations, from $\mathcal{B}(K^+ \to \pi^0 e^+ \nu_e)$, and $X(m_t^2/m_W^2)$ is a
known function of the top mass. An explicit calculation gives
$\mathcal{B}(K_L \to \pi^0 \nu \bar{\nu}) = (3.00 \pm 0.30) \times 10^{-11}$ [53].

Currently the most stringent experimental limit is $\mathcal{B}(K_L \to \pi^0 \nu \bar{\nu}) < 3.0 \times 10^{-9}$ [54, 55], which does not yet reach the bound that can be derived from Eq. (13.51), $\mathcal{B}(K_L \to \pi^0 \nu \bar{\nu}) < 4.4 \times \mathcal{B}(K^+ \to \pi^+ \nu \bar{\nu})$ [49], with the most precise result for the charged kaon decay being $\mathcal{B}(K^+ \to \pi^+ \nu \bar{\nu}) = (10.6^{+4.0}_{-3.4} \pm 0.9) \times 10^{-11}$ [56]. Significant further progress is anticipated from experiments searching for $K \to \pi \nu \bar{\nu}$ decays in the next few years.

13.5 Charm

The existence of $D^0-\bar{D}^0$ mixing is well established [57–61], with the latest experimental constraints giving [62, 63] $x \equiv \Delta m/\Gamma = (0.407 \pm 0.044) \times 10^{-2}$ and $y \equiv \Delta \Gamma/(2\Gamma) = (0.647 \pm 0.024) \times 10^{-2}$. Long-distance contributions make it difficult to calculate Standard Model predictions for the $D^0-\bar{D}^0$ mixing parameters. Therefore, the goal of the search for $D^0-\bar{D}^0$ mixing is not to constrain the CKM parameters, but rather to probe new physics. Here CP violation plays an important role. Within the Standard Model, the CP-violating effects are predicted to be small, since the mixing and the relevant decays are described, to an excellent approximation, by the physics of the first two generations only. The expectation is that the Standard Model size of CP violation in D decays is $\mathcal{O}(10^{-3})$ or less. At present, the most sensitive searches involve the $D^0 \to K^+K^-$, $D^0 \to \pi^+\pi^-$ and $D^0 \to K^\pm\pi^\mp$ modes.

The neutral D mesons decay via a singly-Cabibbo-suppressed
transition to the CP eigenstates $K^+K^-$ and $\pi^+\pi^-$. These decays
are dominated by Standard-Model tree diagrams. Thus, we can
write, for $f = K^+K^-$ or $\pi^+\pi^-$,

$$
A_f = A_f^T e^{\pm i \phi_f^T} [1 + r_f e^{i(\delta_f + \phi_f)}], \quad (13.53\text{a})
$$

$$
\bar{A}_f = A_f^T e^{-i\phi_f^T} [1 + r_f e^{i(\delta_f - \phi_f)}], \quad (13.53\text{b})
$$

where $A_f^T e^{\pm i \phi_f^T}$ is the Standard Model tree-level contribution,
$\phi_f^T$ and $\phi_f$ are weak, CP violating phases, $\delta_f$ is a strong phase
difference, and $r_f$ is the ratio between a subleading ($r_f \ll 1$) con-
tribution with a weak phase different from $\phi_f^T$ and the Standard