12. CKM Quark-Mixing Matrix

1

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). Theyarise from the Yukawa interactions with the Higgs condensate,
LY = −Yijd QILi φ dIRj − Yiju QILi  φ∗ uIRj + h.c.,(12.1)

where Y u,d are 3×3 complex matrices, φ is the Higgs field, i, j are generation labels, and  is the 2×2
antisymmetric tensor. QIL are left-handed quark doublets, and dIR and uIR are right-handed downand up-type quark singlets, respectively,in the weak-eigenstate basis. When φ acquires a vacuum√
expectation value, hφi = (0, v/ 2), Eq. (12.1) yields mass terms for the quarks. The physical states√u,df
are obtained by diagonalizing Y u,d by four unitary matrices, VL,R
, as Mdiag= VLf Y f VRf † (v/ 2),
f = u, d. As a result, the charged-current W ± interactions couple to the physical uLj and dLk
quarks with couplings given by




dL
−g

√ (uL , cL , tL )γ µ Wµ+ VCKM sL  + h.c.,
2
bL





Vud Vus Vub

VCKM ≡ VLu VLd † =  Vcd Vcs Vcb  .
Vtd Vts Vtb(12.2)

This Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2] is a 3 × 3 unitary matrix. It can beparameterized by three mixing angles and the CP -violating KM phase [2]. Of the many possible
conventions, a standard choice has become [3]10VCKM = 0 c23
0 −s23





0
c13
0 s13 e−iδ
c12


s23  01
0 −s12c23
0−s13 eiδ 0
c13

s12
c12
0



0

0
1





c12 c13s12 c13s13 e−iδiδ
iδ
c12 c23 − s12 s23 s13 e
s23 c13  ,= −s12 c23 − c12 s23 s13 e
s12 s23 − c12 c23 s13 eiδ −c12 s23 − s12 c23 s13 eiδ c23 c13(12.3)

where sij = sin θij , cij = cos θij , and δ is the phase responsible for all CP -violating phenomena inflavor-changing processes in the SM. The angles θij can be chosen to lie in the first quadrant, sosij , cij ≥ 0.
It is known experimentally that s13  s23  s12  1, and it is convenient to exhibit thishierarchy using the Wolfenstein parameterization. We define [4–6]|Vus |,|Vud |2 + |Vus |2

Vcb
s23 = Aλ2 = λ,Vus√Aλ3 (ρ̄ + iη̄) 1 − A2 λ4iδ∗3
s13 e = Vub = Aλ (ρ + iη) = √.
1 − λ2 [1 − A2 λ4 (ρ̄ + iη̄)]
s12 = λ = p(12.4)∗ )/(V V ∗ ) is phase convention independent, and theThese relations ensure that ρ̄ + iη̄ = −(Vud Vub
cd cb
CKM matrix written in terms of λ, A, ρ̄, and η̄ is unitary to all orders in λ. The definitions of ρ̄, η̄
reproduce all approximate results in the literature; i.e., ρ̄ = ρ(1−λ2 /2+. . .) and η̄ = η(1−λ2 /2+. . .),and one can write VCKM to O(λ4 ) either in terms of ρ̄, η̄ or, traditionally,

1 − λ2 /2λAλ3 (ρ − iη)4−λ
1 − λ2 /2
Aλ2
VCKM =  + O(λ ) .
3
2
Aλ (1 − ρ − iη)−Aλ1
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:15am(12.5)

2

12. CKM Quark-Mixing Matrix

Figure 12.1: Sketch of the unitarity triangle.
The CKM matrix elements are fundamental parameters of the SM, so their precise determinationPP∗ = δ . The
is important. The unitarity of the CKM matrix imposes i Vij Vik∗ = δjk and j Vij Vkj
ik
six vanishing combinations can be represented as triangles in a complex plane, of which thoseobtained 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∗ =JP
independent measure of CP violation, defined by Im Vij Vkl Vil∗ Vkj
m,n εikm εjln .
The most commonly used unitarity triangle arises from∗
Vud Vub+ Vcd Vcb∗ + Vtd Vtb∗ = 0 ,(12.6)by dividing each side by Vcd Vcb∗ (see Fig. 12.1). Its vertices are exactly (0, 0), (1, 0), and, due tothe definition in Eq. (12.4), (ρ̄, η̄). An important goal of flavor physics is to overconstrain the
CKM elements, and many measurements can be conveniently displayed and compared in the ρ̄, η̄
plane. While the Lagrangian in Eq. (12.1) is renormalized, and the CKM matrix has a well-known
scale dependence above the weak scale [8], below µ = mW the CKM elements can be treated as
constants, with all µ-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 givethe 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 |Vud |The most precise determination of |Vud | comes from the study of superallowed 0+ → 0+ nuclear
beta decays, which are pure vector transitions. Taking the average of the fifteen most precise
determinations [9] yields [10]|Vud | = 0.97367 ± 0.00032 .
(12.7)
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12. CKM Quark-Mixing Matrix

This uncertainty is slightly more than twice as large as that in the 2020 edition, due to a moreconservative estimate of the nuclear structure uncertainties. A less precise determination of |Vud |can be obtained from the measurement of the neutron lifetime. The theoretical uncertainties arevery small, but the determination is limited by the knowledge of the ratio of the axial-vector andvector couplings, gA = GA /GV [10]. The PIBETA experiment [11] has improved the measurementof the π + → π 0 e+ ν branching ratio to 0.6%, and Ref. [12] quotes |Vud | = 0.9739 ± 0.0027, inagreement with the more precise result listed above. The interest in this measurement is that thedetermination of |Vud | is very clean theoretically, because it is a pure vector transition and is freefrom nuclear-structure uncertainties.
12.2.2 |Vus |The product of |Vus | and the form factor at q 2 = 0, |Vus | f+ (0), has been extracted traditionallyfrom KL0 → πeν decays in order to avoid isospin-breaking corrections (π 0 − η mixing) that affect
K ± 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 tojustify 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 KL0 → πeν, KL0 → πµν, K ± → π 0 e± ν, K ± → π 0 µ± ν and KS0 → πeν. The
average of these five decay modes yields |Vus | f+ (0) = 0.21656 ± 0.00035. Results obtained fromeach decay mode, and exhaustive references to the experimental data, are listed for instance inRef. [10]. The form factor average f+ (0) = 0.9698 ± 0.0017 [14] from Nf = 2 + 1 + 1 lattice QCD
calculations gives |Vus | = 0.2233 ± 0.0005 [10].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 |Vus /Vud |from K → µν(γ) and π → µν(γ), where (γ) indicates that radiative decays are included [19].
The value of Γ (K → µν(γ)) [10] derived from the KLOE measurement of the corresponding
branching ratio [20], combined with the lattice QCD result, fK /fπ = 1.1932 ± 0.0021 [14], leads to|Vus | = 0.2250 ± 0.0004, where the accuracy is limited by the knowledge of the ratio of the decay
constants. The averagep of these two determinations, with the error scaled according to the PDG
prescription [21] by χ2 = 2.5, is quoted as [10]|Vus | = 0.22431 ± 0.00085 .

(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
g1 /f1 as experimental input, thus avoiding first order SU (3) breaking effects in the axial-vector
contribution. They find |Vus | = 0.2250 ± 0.0027, although this does not include an estimate of thetheoretical uncertainty due to second-order SU (3) breaking, contrary to Eq. (12.8). Concerning
hadronic τ decays to strange particles, averaging the inclusive decay and the exclusive τ → hν(h = π, K) measurements yields |Vus | = 0.2207 ± 0.0014 [24].
12.2.3 |Vcd |The magnitude of Vcd can be extracted from semileptonic charm decays, using theoretical knowledge of the form factors. In semileptonic D decays, lattice QCD calculations have predicted the nor1
For 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 Nf = 2 + 1 + 1 and 2 + 1 calculations are available, we use the former.

31st May, 2024

12. CKM Quark-Mixing Matrix

4

malization of the D → π`ν and D → K`ν form factors [14]. The dependence on the invariant massof the lepton pair, q 2 , is determined from lattice QCD and theoretical constraints from analyticity [15]. Using Nf = 2+1+1 lattice QCD calculations for D → π`ν, f+Dπ (0) = 0.612±0.035 [14], andthe average [24] of the measurements of D → π`ν decays by BABAR [25], BESIII [26], CLEO-c [27],and Belle [28], one obtains |Vcd | = 0.2330 ± 0.0029 ± 0.0133, where the first uncertainty is experimental, and the second is from the theoretical uncertainty of the form factor.
The determination of |Vcd | is also possible from the leptonic decay D+ → µ+ ν and τ + ν. The
experimental uncertainties have not decreased significantly recently. Averaging the BESIII [29] andearlier CLEO [30] for µ+ ν and BESIII [31] for τ + ν measurements, and using the Nf = 2 + 1 + 1
lattice QCD result, fD = 212.0 ± 0.7 MeV [14], yields |Vcd | = 0.2181 ± 0.0049 ± 0.0007 [24] 2
Earlier determinations of |Vcd | came from neutrino scattering data. The difference of the ratioof double-muon to single-muon production by neutrino and antineutrino beams is proportional tothe charm cross section off valence d quarks, and therefore to |Vcd |2 times the average semileptonic branching ratio of charm mesons, Bµ . The method was used first by CDHS [32] and thenby CCFR [33] and CHARM II [34]. Averaging these results is complicated, because it requires
assumptions about the scale of the QCD corrections, and because Bµ 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, Bµ |Vcd |2 = (0.463 ± 0.034) × 10−2 [35]. Analysis cuts make these experiments insensitive to neutrino energies smaller than 30 GeV. Thus, Bµshould 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
Bµ = 0.088 ± 0.006. Data from the CHORUS experiment [38] are sufficiently precise to extract Bµ
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 Bµ = 0.085 ± 0.009 ± 0.006. We use the average of these two determinations,
Bµ = 0.087 ± 0.005, and obtain |Vcd | = 0.230 ± 0.011. Averaging the three determinations above,
we find|Vcd | = 0.221 ± 0.004 .

(12.9)

12.2.4 |Vcs |The direct determination of |Vcs | is possible from semileptonic D or leptonic Ds decays, using
lattice QCD calculations of the semileptonic D form factor or the Ds decay constant. For muonic
decays, the average of Belle [39], CLEO-c [40], BABAR [41], and BESIII [42, 43] is B(Ds+ → µ+ ν) =(5.43 ± 0.16) × 10−3 [24]. For decays to τ leptons, the average of CLEO-c [40, 44], BABAR [41],Belle [39], and BESIII [42] gives B(Ds+ → τ + ν) = (5.40±0.23)×10−2 [24]. From each of these values,
determinations of |Vcs | can be obtained using the PDG values for the mass and lifetime of the Ds ,the masses of the leptons, and fDs = (249.9 ± 0.5) MeV [14]. The average of these determinations
gives |Vcs | = 0.984 ± 0.012, where the error is dominated by the experimental uncertainty. In
semileptonic D decays, lattice QCD calculations of the D → K`ν form factor are available [14].
Using f+DK (0) = 0.7385 ± 0.0044 and the average [24] of CLEO-c [27], Belle [28], BABAR [45], andrecent BESIII [26, 46] measurements of D → K`ν decays, one obtains |Vcs | = 0.972 ± 0.007, where
the dominant uncertainty is from the theoretical calculation of the form factor. Averaging thedeterminations from leptonic and semileptonic decays, we find|Vcs | = 0.975 ± 0.006 .
Measurements of on-shell W ± decays sensitive to |V(12.10)cs | were made by LEP-2. The W branchingratios depend on the six CKM elements involving only the quarks lighter than mW . The W
2

Hereafter the first error is statistical and the second is systematic, unless mentioned otherwise.
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512. CKM Quark-Mixing Matrix

branching ratio to each lepton flavor is 1/B(W → `ν̄` ) = 3 1+ u,c,d,s,b |Vij |2 (1+αs (mW )/π)+. . . .
Assuming lepton universality, the measurement B(W → `ν̄` ) = (10.83 ± 0.07 ± 0.07) % [47] implies
P2
u,c,d,s,b |Vij | = 2.002 ± 0.027. This is a precise test of unitarity; however, only flavor-tagged
W -decays determine |Vcs | directly, such as DELPHI’s tagged W + → cs̄ analysis, yielding |Vcs | =+0.32
0.94−0.26± 0.13 [48].
P12.2.5 |Vcb |This matrix element can be determined from exclusive and inclusive semileptonic decays ofB mesons to charm. The inclusive 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 decayrate and various spectra as expansions in αs and inverse powers of the heavy-quark mass. The
dependence on mb , mc , and the parameters that occur at subleading order is different for different
moments. The measurements of many moments overconstrain these parameters, and also test theconsistency of their determination. The precise extraction of |Vcb | requires using a “threshold”
quark mass definition [51, 52]. Inclusive measurements have been performed using B mesons fromZ 0 decays at LEP, and at e+ e− colliders operated at the Υ (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]: |Vcb | = (42.2 ± 0.5) × 10−3 .
Complementary determinations are based on exclusive semileptonic B decays to D and D∗ .
In the mb,c  ΛQCD 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 · v 0 . Heavy-quark
symmetry determines the rate (in the symmetry limit) at w = 1, the point in phase space atwhich the invariant mass of the `ν̄ pair is maximal; |Vcb | is obtained from extrapolating a fit to thespectrum to w = 1. The current update of the Vcb and Vub minireview quotes from exclusive decays|Vcb | = (39.8 ± 0.6) × 10−3 [15], based on the only unfolded measurement of B → D∗ semileptonic
decay distributions [54], and using
p a more general fit [55] than in earlier B factory measurements.
With the uncertainty scaled by χ2 = 3.0, this yields the combination [15],|Vcb | = (41.1 ± 1.2) × 10−3 .

(12.11)

Determinations of |Vcb | that are currently less precise and not included in this average, can be(∗)
obtained from the measurement of Bs → Ds µν̄ decays [56]. In addition, semileptonic decays toτ leptons measured in B → D(∗) τ ν̄ and related modes are also sensitive to |Vcb |. The most precise
data involving τ leptons are the |Vcb |-independent ratios, B(B → D(∗) τ ν̄)/B(B → D(∗) `ν̄) measured
by BaBar, Belle, and LHCb. If the current, approximately 3.3 σ [24] hint of lepton non-universality
prevails, the determination of |Vcb | becomes more complicated.
12.2.6 |Vub |The determination of |Vub | from inclusive B → Xu `ν̄ decay is complicated due to large B →Xc `ν̄ backgrounds. In most regions of phase space where the charm background is kinematicallyforbidden, the hadronic physics enters via unknown nonperturbative functions, so-called shapefunctions. (By contrast, the nonperturbative physics for |Vcb | is encoded in a few parameters.)
At leading order in ΛQCD /mb , there is only one shape function, which can be extracted fromthe photon energy spectrum in B → Xs γ [57, 58], and applied to several spectra in B → Xu `ν̄.
The subleading shape functions are modeled in the current determinations. Phase space cuts forwhich 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
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12. CKM Quark-Mixing Matrix

6of phase space. A different approach is to make the measurements more inclusive by extending
them deeper into the B → Xc `ν̄ region, and thus reduce the theoretical uncertainties. Analyses ofthe electron-energy endpoint from CLEO [60], BABAR [61], and Belle [62] quote B → Xu eν̄ partial
rates for |~pe | ≥ 2.0 GeV and 1.9 GeV, which are well below the charm endpoint. The large andpure BB samples at the B factories permit the selection of B → Xu `ν̄ decays in events where theother B is fully reconstructed [63]. With this full-reconstruction tag method, the four-momentaof both the leptonic and the hadronic final states can be measured. It also gives access to awider kinematic region, because of improved signal purity. Ref. [15] quotes the inclusive average,−3|Vub | = (4.13 ± 0.12 +0.13−0.14 ± 0.18) × 10 , 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 |Vub | from exclusive decays, the form factors have to be known. Experimentally,
better signal-to-background ratios are offset by smaller yields. The B → π`ν̄ branching ratio is
now known to 5%. Lattice QCD calculations of the B → π`ν̄ form factor are available [64] for thehigh q 2 region (q 2 > 16 or 18 GeV2 ). A fit to the experimental partial rates and lattice QCD results
versus q 2 yields |Vub | = (3.70 ± 0.10 ± 0.12) × 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,|Vub | = (3.67 ± 0.09 ± 0.12) × 10−3 [15, 24].
The uncertainties in extracting |Vub | from inclusive and exclusive decays p
are different to a large
extent. An average of these determinations, with the uncertainty scaled by χ2 = 1.4, is [15]|Vub | = (3.82 ± 0.20) × 10−3 .

(12.12)A determination of |Vub | not included in this average can be obtained from B(B → τ ν̄) =(1.09 ± 0.21) × 10−4 [24]. Using fB = (190.0 ± 1.3) MeV [14] and τB ± = (1.638 ± 0.004) ps [65],we find the remarkably consistent result, |Vub | = (4.11 ± 0.39) × 10−3 . This decay is sensitive, forexample, to tree-level charged Higgs contributions, and the measured rate is consistent with the SM
expectation. The LHCb measurement |Vub /Vcb | = 0.083 ± 0.004 [15] from the ratios of Λb → p+ µ− ν̄
−
0
− +
0
− +
2
and Λb → Λ+c µ ν̄ [66] and Bs → K µ ν and Bs → Ds µ ν [67] in different regions of q , provides
another complementary determination.
12.2.7 |Vtd | and |Vts |The CKM elements |Vtd | and |Vts | are not likely to be precisely measurable in tree-level processes
involving top quarks, so one has to rely on determinations from B 0 – B 0 mixing, dominated by box
diagrams with top quarks, or loop-mediated rare K and B decays. Theoretical uncertainties inhadronic 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 |Vtd /Vts |.
The phenomenon of B 0 – B 0 mixing was discovered by ARGUS [68], and the mass difference is
now precisely measured as ∆md = (0.5069 ± 0.0019) ps−1 [69]. In the Bs0 system, ∆ms was firstmeasured significantly by CDF [70] and the world average, dominated by an LHCb measurement−1[71], is ∆ms = (17.765 ± 0.006)
q ps [69]. Neglecting correctionsqsuppressed by |Vtb | − 1, and usingbB = (210.6 ± 5.5) MeV and fBBsdthe lattice QCD results fBd|Vtd | = (8.6 ± 0.2) × 10−3 ,bB = (256.1 ± 5.7) MeV [14],Bs|Vts | = (41.5 ± 0.9) × 10−3 .

(12.13)The uncertainties are dominatedby qlattice QCD. Several uncertainties are reduced in the calculationqbB / fB B
bB = 1.216 ± 0.016 [14] and therefore the constraint onof the ratio ξ = fBs B
sdd|Vtd /Vts | from ∆md /∆ms is more reliable theoretically. These provide a theoretically clean and31st May, 2024

712. CKM Quark-Mixing Matrix

significantly improved determination,

Vtd /Vts = 0.207 ± 0.001 ± 0.003 .

(12.14)The inclusive branching ratio B(B → Xs γ) = (3.49 ± 0.19) × 10−4 extrapolated to Eγ >E0 = 1.6 GeV [24] is also sensitive to |Vtb Vts |. In addition to t-quark penguins, a substantial partof the rate comes from charm contributions proportional to Vcb Vcs∗ via the application of 3 × 3
CKM unitarity (which is used here). With the NNLO calculation of B(B → Xs γ)Eγ >E0 /B(B →Xc eν̄) [72], we obtain |Vts /Vcb | = 1.00 ± 0.04. The Bs → µ+ µ− rate is also proportional to |Vtb Vts |2
in the SM, and the world average, B(Bs → µ+ µ− ) = (3.01 ± 0.35) × 10−9 [65], is consistent with
the SM, with sizable uncertainties.
A complementary determination of |Vtd /Vts | is possible from the ratio of B → ργ and K ∗ γ rates.
The ratio of the neutral modes is theoretically cleaner than that of the charged ones, because thepoorly known spectator-interaction contribution is expected to be smaller (W -exchange vs. weakannihilation). For now, because of low statistics, we average the charged and neutral rates assuming
the isospin symmetry and heavy-quark limit motivated relation, |Vtd /Vts |2 /ξγ2 = [Γ (B + → ρ+ γ) +2Γ (B 0 → ρ0 γ)]/[Γ (B + → K ∗+ γ) + Γ (B 0 → K ∗0 γ)] = (3.35 ± 0.48)% [24, 73]. Here ξγ contains thepoorly known hadronic physics. Using ξγ = 1.2 ± 0.2 [74] gives |Vtd /Vts | = 0.220 ± 0.016 ± 0.037,
where the first uncertainty is experimental and the second is theoretical.
A theoretically clean determination of |Vtd Vts∗ | is possible from K + → π + ν ν̄ 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 |Vtb |The determination of |Vtb | from top decays uses the ratio of branching fractions R = B(t →P
W b)/B(t → W q) = |Vtb |2 /( q |Vtq |2 ) = |Vtb |2 , where q = b, s, d. The CDF and DØ measurements
performed on data collected during Run II of the Tevatron give |Vtb | > 0.85 [78] and 0.99 >|Vtb | > 0.90 [79], respectively, at 95% CL. CMS measured the same quantity at 8 TeV and obtained|Vtb | > 0.975 [80] at 95% CL.
The direct determination of |Vtb |, 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 |Vtb | = 1.02+0.06.
The
LHC experiments, ATLAS and CMS, have measured−0.05
single top quark production cross sections (and extracted |Vtb |) in t-channel, W t-channel, and schannel at 7 TeV, 8 TeV, and 13 TeV [82]. The average of these |Vtb | values is calculated to be|Vtb | = 1.007 ± 0.030, where all systematic errors and theoretical errors are treated to be fully
correlated. The average of Tevatron and LHC values gives|Vtb | = 1.010 ± 0.027 .

(12.15)The experimental systematic uncertainties dominate, and a dedicated combination would be welcome.
A weak constraint on |Vtb | can be obtained from precision electroweak data, where top quarks+0.18
enter in loops. The sensitivity is best in Γ (Z → bb̄) and yields |Vtb | = 0.77−0.24[83].
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12. CKM Quark-Mixing Matrix

812.3 Phases of CKM elements
As can be seen from Fig. 12.1, the angles of the unitarity triangle are
Vcd Vcb∗
β = φ1 = arg −,Vtd Vtb∗
Vtd Vtb∗
α = φ2 = arg −∗ ,Vud Vub∗ 
Vud Vubγ = φ3 = arg −.
Vcd Vcb∗(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 ρ̄, η̄ parameters.
12.3.1  and 0
The measurement of CP violation in K 0 –K 0 mixing, || = (2.228 ± 0.011) × 10−3 [84], provides
important information about the CKM matrix. The phase of  is determined by long-distancephysics,  = 12 eiφ sin φ arg(−M12 /Γ12 ), where φ = arctan |2∆mK /∆ΓK | ' 43.5◦ . The SM
prediction can be written as = κ eiφn
G2F m2W mK
2 b
∗ 2
√
f
B
K K ηtt S(xt ) Im[(Vts Vtd ) ]12 2 π 2 ∆mKo∗∗ 2+ 2ηct S(xc , xt ) Im(Vcs Vcd
Vts Vtd∗ ) + ηcc xc Im[(Vcs Vcd) ] ,(12.17)

where κ ' 0.94 ± 0.02 [85] includes the effects of strangeness changing ∆s = 1 operators andadditional dependence on φ 6= π/4 (see also Ref. [86]). The displayed terms are the short-distance∆s = 2 contribution to ImM12 in the usual phase convention, S is an Inami-Lim function [87],xq = m2q /m2W , and ηij are perturbative QCD corrections. The constraint from  in the ρ̄, η̄ plane isbK = 0.717 ±bounded by approximate hyperbolas. Lattice QCD determined the bag parameter B∗
20.024 [14] and the main uncertainties are from (Vts Vtd ) (approximately given by that of |Vcb |4 or
A4 ), the ηij coefficients, and estimates of κ .
The measurement of 6 Re(0 /) = 1 − |η00 /η+− |2 , where each ηij = hπ i π j |H|KL i/hπ i π j |H|KS i
violates CP , provides a qualitative test of the CKM mechanism, and strong constraints on many
BSM scenarios. Its nonzero value, Re(0 /) = (1.67 ± 0.23) × 10−3 [84], demonstrated the existenceof direct CP violation, a prediction of the KM ansatz. While Re(0 /) ∝ Im(Vtd Vts∗ ), this quantity
cannot easily be used to extract CKM parameters, because cancellations between the electromagnetic and gluonic penguin contributions for large mt [88] enhance the hadronic uncertainties. Most
SM estimates [89] agree with the observed value, indicating that η̄ is positive. Progress in lattice
QCD [90] may yield a precise SM prediction in the future, and trigger new work on assessing theconsistency of the SM with the measured value [91, 92].
12.3.2 β / φ112.3.2.1 Charmonium modes
CP -violation measurements in B-meson decays provide direct information on the angles of theunitarity triangle, shown in Fig. 12.1. These overconstraining measurements serve to improve thedetermination 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
0
B is given by [93–95]
Af =Γ (B 0 (t) → f ) − Γ (B 0 (t) → f )= Sf sin(∆md t) − Cf cos(∆md t) ,
Γ (B 0 (t) → f ) + Γ (B 0 (t) → f )31st May, 2024

(12.18)

12. CKM Quark-Mixing Matrix

9
where
Sf =

2 Imλf,
1 + |λf |2

Cf =1 − |λf |2,
1 + |λf |2

λf =

q Āf
.
p Af

(12.19)Here, q/p describes B 0 –B 0 mixing and, to a good approximation in the SM, q/p = Vtb∗ Vtd /Vtb Vtd∗ =4e−2iβ+O(λ ) in the usual phase convention. Af (Āf ) is the amplitude of the B 0 → f (B 0 → f )decay. If f is a CP eigenstate, and amplitudes with one CKM phase dominate the decay, then|Af | = |Āf |, Cf = 0, and Sf = sin(arg λf ) = ηf sin 2φ, where ηf is the CP eigenvalue of f and2φ is the phase difference between the B 0 → f and B 0 → B 0 → f decay paths. A contributionof another amplitude to the decay with a different CKM phase makes the value of Sf sensitive torelative strong-interaction phases between the decay amplitudes (it also makes Cf 6= 0 possible).
0 ) give currently the mostThe b → cc̄s decays to CP eigenstates (B 0 → charmonium KS,L
precise measurements of Sf = −ηf sin 2β. The b → s penguin amplitudes have dominantly thesame weak phase as the b → cc̄s tree amplitude. Since only λ2 -suppressed penguin amplitudes
introduce a different CP -violating phase, amplitudes with a single weak phase dominate, and we
expect |ĀψK /AψK | − 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 someother measurements, is [24, 95, 99]
sin 2β = 0.709 ± 0.011 .

(12.20)This measurement has a four-fold ambiguity in β, which can be resolved by a global fit as
mentioned in Sec. 12.4. Experimentally, the two-fold ambiguity β → π/2−β (but not β → π+β) canbe resolved by a time-dependent angular analysis of B 0 → J/ψK ∗0 [100, 101], or a time-dependent
Dalitz plot analysis of B 0 → D0 h0 . The time-dependent Dalitz plot analysis of B 0 → D0 h0(h0 = π 0 , η, ω) with D0 → KS0 π + π − , jointly performed by Belle and BABAR, excludes the π/2 − βsolution with 7.3σ confidence level [102]. These results exclude the negative cos 2β solutions, inagreement with the global CKM fit, which is no longer shown in Fig. 12.2.
The b → cc̄d mediated transitions, such as B 0 → J/ψπ 0 and B 0 → D(∗)+ D(∗)− , also measureapproximately sin 2β. However, the dominant component of the b → d penguin amplitude has adifferent CKM phase (Vtb∗ Vtd ) than the tree amplitude (Vcb∗ Vcd ), and their magnitudes are of the sameorder in λ. Therefore, the effect of penguins could be large, resulting in Sf 6= −ηf sin 2β and Cf 6= 0.
Such decay modes have been measured by BABAR, Belle, and LHCb. The world averages [24],+0.16, SD+ D− = −0.84 ± 0.12, and SD∗+ D∗− = −0.71 ± 0.09
SJ/ψπ0 = −0.86 ± 0.14, SJ/ψρ0 = −0.66−0.120+(where ηf = +1 for the J/ψπ and D D− modes, while J/ψρ0 and D∗+ D∗− are mixtures of CP
even and odd states), are consistent with sin 2β obtained from B 0 → charmonium K 0 decays, andthe Cf ’s are consistent with zero, although the uncertainties are sizable.
The b → cūd decays B 0 → D0(∗) h0 , with D0 → CP eigenstates and D0 → KS0 π + π − with Dalitz
plot analysis, have no penguin contributions, and provide theoretically clean sin 2β measurements.
The average of joint analyses of BABAR and Belle data [102, 103] give sin 2β = 0.71 ± 0.09 [24].
12.3.2.2 Penguin-dominated modes
The b → sq̄q penguin-dominated decays have the same CKM phase as the b → cc̄s tree level
decays, up to corrections suppressed by λ2 , since Vtb∗ Vts = −Vcb∗ Vcs [1 + O(λ2 )]. Therefore, decays
such as B 0 → φK 0 and η 0 K 0 provide sin 2β measurements in the SM. Any BSM contribution to
the amplitude with a different weak phase would give rise to Sf 6= −ηf sin 2β, and possibly Cf 6= 0.
Therefore, the main interest in these modes is not simply to measure sin 2β, but to search for new
physics. Measurements of many other decay modes in this category, such as B → π 0 KS0 , KS0 KS0 KS0 ,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
31st May, 2024

1012. CKM Quark-Mixing Matrix

between tree-dominated and penguin-dominated modes in Bs0 decays provides similar sensitivity to
new physics.
12.3.3 α / φ2
∗ V , only time-dependent CP asymmetries in decay
Since α is the phase between Vtb∗ Vtd and Vub
udmodes dominated by b → uūd transitions can directly measure sin 2α, in contrast to sin 2β, where
several different quark-level transitions can be used. Since b → d penguin amplitudes have a
different CKM phase than b → uūd tree amplitudes, and their magnitudes are of the same order in
λ, the penguin contribution can be sizable, which makes the determination of α complicated. To
date, α has been measured in B → ππ, ρπ and ρρ decay modes.
12.3.3.1 B → ππ
It is well-established from the data that there is a sizable contribution of b → d penguin
amplitudes in B → ππ decays. Thus, Sπ+ π− in the time-dependent B 0 → π + π − analysis does notmeasure sin 2α, butqSπ+ π− =1 − Cπ2+ π− sin(2α + 2∆α) ,(12.21)

where 2∆α is the phase difference between e2iγ Āπ+ π− and Aπ+ π− . The value of ∆α, and hence α,
can be extracted using the isospin relation among the amplitudes of B 0 → π + π − , B 0 → π 0 π 0 , andB + → π + π 0 decays [104],1√ Aπ + π − + Aπ 0 π 0 − Aπ + π 0 = 0 ,(12.22)2
and a similar expression for the Āππ ’s. This method utilizes the fact that a pair of pions fromB → ππ decay must be in a zero angular momentum state, and, because of Bose statistics, they musthave even isospin. Consequently, π ± π 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 averagesof BABAR, Belle, and LHCb measurements, Sπ+ π− = −0.666 ± 0.029, Cπ+ π− = −0.311 ± 0.030, thedecay widths of all three modes, and the direct CP asymmetry Cπ0 π0 = −0.33 ± 0.22 [24]. This
analysis leads to 16 mirror solutions for 0 ≤ α < 2π. Because of this, and due to the experimental
uncertainties, some of these solutions are not well separated [95].
12.3.3.2 B → ρρThe decay B 0 → ρ+ ρ− 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 theanalysis that the longitudinal polarization fractions in B + → ρ+ ρ0 and B 0 → ρ+ ρ− decays were
measured to be close to unity [105], which implies that the final states are almost purely CP even. Furthermore, B(B 0 → ρ0 ρ0 ) = (0.96 ± 0.15) × 10−6 is much smaller than B(B 0 → ρ+ ρ− ) =(27.5±1.7)×10−6 and B(B + → ρ+ ρ0 ) = (23.8±1.7)×10−6 [24], which implies that the effect of the
penguin contributions is small. The isospin analysis using the world averages, Sρ+ ρ− = −0.14±0.13
and Cρ+ ρ− = 0.00 ± 0.09 [24], together with the time-dependent CP asymmetry, Sρ0 ρ0 = −0.3 ± 0.7
and Cρ0 ρ0 = −0.2 ± 0.9 [106], and the above mentioned branching fractions and longitudinalpolarization fractions, gives two solutions (with mirror solutions at 3π/2 − α) [95]. A possible
small violation of Eq. (12.22) due to the finite width of the ρ [107] is so far neglected.
12.3.3.3 B → ρπ
The final state in B 0 → ρ+ π − decay is not a CP eigenstate, but this decay proceeds via thesame quark-level diagrams as B 0 → π + π − , and both B 0 and B 0 can decay to ρ+ π − , while the final
state in B 0 → ρ0 π 0 is a CP eigenstate. Consequently, mixing-induced CP violation can occur in
B 0 and B 0 decays to ρ± π ∓ and ρ0 π 0 . The time-dependent Dalitz plot analysis of B 0 → π + π − π 0
31st May, 2024

1112. CKM Quark-Mixing Matrix

decays permits the extraction of α with a single discrete ambiguity, α → α + π, since one knows thevariation of the strong phases in the interference regions of the ρ+ π − , ρ− π + , and ρ0 π 0 amplitudes
in the Dalitz plot [108]. The combination of Belle [109] and BABAR [110] measurements gives only
moderate constraints [95].
Combining the B → ππ, ρπ, and ρρ decay modes [24, 95, 99], α is constrained as
α = 84.1+4.5
−3.8

◦

.

(12.23)

Similar results can be found in Refs. [111, 112].
12.3.4 γ / φ3
By virtue of Eq. (12.16), γ 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
α and β, and implies that measurements of γ are unlikely to be affected by physics beyond the SM.
12.3.4.1 B(s) → D(s) K (∗)
The interference of B − → D0 K − (b → cūs) and B − → D0 K − (b → uc̄s) transitions can bestudied in final states accessible in both D0 and D0 decays [93]. In principle, it is possible toextract the B and D decay amplitudes, the relative strong phases, and the weak phase γ from thedata [95].
A practical complication is that the precision depends sensitively on the ratio of the interfering
amplitudes
rB = A(B − → D0 K − ) A(B − → D0 K − ) ,(12.24)
which is around 0.1. The original GLW method [113, 114] considers D decays to CP eigenstates,(∗)
such as B ± → DCP (→ π + π − )K (∗)± . To alleviate the smallness of rB 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 D0 and doubly-Cabibbo-suppressed
D0 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 γ, while the ADS
method provides a moderate constraint.
The BPGGSZ method [116,117] utilizes the fact that both D0 and D0 can have large branching
fractions to CP self-conjugate three- and four-body final states, such as KS0 π + π − , and the analysis
can be optimized by studying the Dalitz plot dependence of the interferences. The best present
determination of γ 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,γ = (70.0 ± 4.0)◦ is obtained [95]. The uncertainty is sensitive to the central value of the amplitude∗ for the D ∗ K mode), for which Belle found somewhat larger central values thanratio rB (and rB(∗)
BABAR and LHCb. The same values of rB enter the ADS analyses, and the data (including 4-body(∗)
D decays [121]) can be combined to fit for rB and γ. The effect of D0 – D0 mixing on γ is either
below the present experimental accuracy or can be taken into account in the analysis [122] (even if
D0 – D0 mixing is due to CP -violating new physics [123]).
The amplitude ratio is much larger in the analogous Bs0 → Ds± K ∓ decays, which allows a modelindependent extraction of γ − 2βs [124] (here βs = arg(−Vts Vtb∗ /Vcs Vcb∗ ) is related to the phase ofBs mixing). Measurements by LHCb with Bs0 → Ds± K ∓ [125] and Bs0 → Ds± K ∓ π + π − [126] give◦γ = (79+19−21 ) using a constraint on 2βs (see Sec. 12.5).
Combining all the above measurements [24, 95, 99], γ is constrained as
γ = 65.7 ± 3.0
Similar results can be found in Refs. [111, 112].
31st May, 2024◦.

(12.25)

12. CKM Quark-Mixing Matrix

1212.3.4.2 B 0 → D(∗)± π ∓The interference of b → u and b → c transitions can be studied in B 0 → D(∗)+ π − (b → cūd) and0¯ decays and their CP conjugates, since both B 0 and B 0 decay to
B → B 0 → D(∗)+ π − (b̄ → ūcd)
D(∗)± π ∓ (or D± ρ∓ , 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 2β + γ.
A complication is that the ratio of the interfering amplitudes is very small, rDπ = A(B 0 →D+ π − )/A(B 0 → D+ π − ) = O(0.01) (and similarly for rD∗ π and rDρ ), and therefore it has notbeen possible to measure it. To obtain 2β + γ, SU (3) flavor symmetry and dynamical assumptions have been used to relate A(B 0 → D− π + ) to A(B 0 → Ds− π + ), so this measurement is notmodel independent at present. Combining the D± π ∓ , D∗± π ∓ and D± ρ∓ measurements [127] givessin(2β + γ) > 0.68 at 68% CL [111], consistent with the previously discussed results for β and γ.

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 |Vud |2 + |Vus |2 + |Vub |2 = 0.9984 ± 0.0007 (1strow), |Vcd |2 + |Vcs |2 + |Vcb |2 = 1.001 ± 0.012 (2nd row), |Vud |2 + |Vcd |2 + |Vtd |2 = 0.9971 ± 0.0020 (1stcolumn), and |Vus |2 +|Vcs |2 +|Vts |2 = 1.003±0.012 (2nd column), respectively. Due to the recent reduction of the value of |Vud |, there is a 2.3σ tension with unitarity in the 1st row, leading also to poorconsistency of the SM fit below. The uncertainties in the second row and column are dominated byP
that of |Vcs |. For the second row, another check is obtained from the measurement of u,c,d,s,b |Vij |2
in Sec. 12.2.4, minus the sum in the first row above: |Vcd |2 + |Vcs |2 + |Vcb |2 = 1.002 ± 0.027. These
provide strong tests of the unitarity of the CKM matrix. With the significantly improved direct
determination of |Vtb |, 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, α + β + γ = 172 ± 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 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
λ = 0.22501 ± 0.00068 ,+0.016
A = 0.826−0.015,

ρ̄ = 0.1591 ± 0.0094 ,+0.0073η̄ = 0.3523−0.0071.

(12.26)

These values are obtained using the method of Refs. [6, 111]. The prescription of Refs. [112, 131]gives λ = 0.22497 ± 0.00070, A = 0.839 ± 0.011, ρ̄ = 0.1581 ± 0.0092, and η̄ = 0.3548 ± 0.0072 [132];
these results are now very close to one another. The fit results for the magnitudes of all nine CKM
elements are
0.97435 ± 0.00016 0.22501 ± 0.00068 0.003732+0.000090−0.000085

VCKM = 0.22487 ± 0.00068 0.97349 ± 0.00016 0.04183+0.00079−0.00069  ,0.00858+0.00019
0.04111+0.00077
0.999118+0.000029−0.00017−0.00068−0.000034


31st May, 2024

(12.27)

12. CKM Quark-Mixing Matrix

13

1.5
excluded area has CL > 0.95

γ
1.0

∆md & ∆mssin 2β

0.5

∆md

η

εK

α
β

γ

0.0

α
Vub

α

-0.5

εK

-1.0

γ
-1.5
-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

ρ
Figure 12.2: Constraints on the ρ̄, η̄ plane. The shaded areas have 95% CL.
−5and the Jarlskog invariant is J = 3.12+0.13−0.12 × 10 . The parameters in Eq. (12.3) are

sin θ12 = 0.22501 ± 0.00068 ,sin θ13 = 0.003732+0.000090−0.000085 ,sin θ23 = 0.04183+0.00079−0.00069 ,δ = 1.147 ± 0.026 .

(12.28)Fig. 12.2 illustrates the constraints on the ρ̄, η̄ 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 γ), the resulting fit is almost identical for λ in Eq. (12.26), while the other parameters’ central values can change by about a sigma and their uncertainties double, yielding+0.024
λ = 0.22509 ± 0.00068, A = 0.811 ± 0.024, ρ̄ = 0.166+0.022−0.021 , and η̄ = 0.367−0.023 . 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, Bs , 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) × SU (2) × U (1) gauge symmetry. Flavor-changing neutral currents, suppressedin the SM, are especially sensitive to beyond SM contributions. Processes studied in great detail,31st May, 2024

1412. CKM Quark-Mixing Matrix

both experimentally and theoretically, include neutral meson mixings, B(s) → Xγ, X`+ `− , `+ `− ,K → πν ν̄, etc. The BSM contributions to these operators are suppressed by powers of the scale atwhich they are generated. Already at lowest order, there are many dimension-6 operators, and theobservable effects of BSM interactions are encoded in their coefficients. In the SM, these coefficientsare determined by just the four CKM parameters, and the W , Z, and quark masses. For example,∆md , Γ (B → ργ), Γ (B → π`+ `− ), and Γ (B → `+ `− ) are all proportional to |Vtd Vtb |2 in theSM, 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 themeasurements of sin 2β in tree- and loop-dominated decay modes, overconstraining measurementsof 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 theneutral meson mixing amplitudes [134] by (zij /Λ2 )(q i γ µ PL qj )2 , where zij is an unknown coefficientand Λ is the scale suppressing this BSM contribution (see, [135, 136]). It is only known since thefirst measurements of γ and α that the SM gives the leading contribution to B 0 – 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 Λ/|zij |1/2 has toexceed about 104 TeV for K 0 – K 0 mixing, 103 TeV for D0 – D0 mixing, 500 TeV for B 0 – B 0 mixing,and 100 TeV for Bs0 – B 0s 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, |zij |  1 is required. Even if |zij | aresuppressed by a loop factor and |Vti∗ Vtj |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, such0 decays, Ad,s , are important [140]. The current world
as the CP asymmetry in semileptonic Bd,sSL
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 Bs system is βs = arg(−Vts Vtb∗ /Vcs Vcb∗ ),which is the small, λ2 -suppressed, angle of a “squashed” unitarity triangle, obtained by taking thescalar product of the second and third columns of the CKM matrix. This angle can be measured
via time-dependent CP violation in Bs0 → J/ψ φ, similar to β in B 0 → J/ψK 0 . Since the J/ψ φ
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, theasymmetry for the CP -even part is 2βs , when one neglects subdominant amplitudes with a weakphase Vub . (Sometimes the notation φs = −2βs plus a possible BSM contribution to the Bs mixing
phase is used.) Testing if the data agree with the SM prediction, 2βs = 0.03726+0.00078−0.00077 [111], is
another sensitive probe of the SM. The current world average, dominated by LHC measurements
including the Bs → J/ψ K + K − and J/ψ π + π − decay modes, is 2βs = 0.040 ± 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 D0 – D0 mixing,
provide complementary sensitivity to BSM.
In the kaon sector, the CP -violating observables,  and 0 , are tiny, so models in which all sourcesof CP violation are small were viable before the B-factory measurements. Since the measurementof sin 2β, 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
31st May, 2024

15

12. CKM Quark-Mixing Matrix

excluded. In the kaon sector, clean tests of the SM can come from measurements of K + → π + ν ν̄ [77]and KL0 → π 0 ν ν̄ [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 flavorchanging 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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[2] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
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[5] A. J. Buras, M. E. Lautenbacher and G. Ostermaier, Phys. Rev. D50, 3433 (1994), [hepph/9403384].
[6] J. Charles et al. (CKMfitter Group), Eur. Phys. J. C41, 1, 1 (2005), [hep-ph/0406184].
[7] C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985).
[8] W. J. Marciano and A. Sirlin, Nucl. Phys. B93, 303 (1975); K. S. Babu, Z. Phys. C35, 69(1987).
[9] J. C. Hardy and I. S. Towner, Phys. Rev. C 102, 4, 045501 (2020).
[10] E. Blucher, G. D’Ambrosio, and W.J. Marciano, “Vud , Vus , the Cabibbo Angle and CKM
Unitarity,” in this Review.
[11] D. Pocanic et al., Phys. Rev. Lett. 93, 181803 (2004), [hep-ex/0312030].
[12] A. Czarnecki, W. J. Marciano and A. Sirlin, Phys. Rev. D 101, 9, 091301 (2020),[arXiv:1911.04685].
[13] M. Antonelli et al. (FlaviaNet Working Group on Kaon Decays), Eur. Phys. J. C69, 399(2010), [arXiv:1005.2323]; see also http://www.lnf.infn.it/wg/vus.
[14] S. Aoki et al. (Flavour Lattice Averaging Group) "FLAG Review 2021 (March 24, 2023 update)", http://flag.unibe.ch/2021; The original papers that led to the quoted averages
are cited in this reference.
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