11. THE CKM QUARK-MIXING MATRIX · 2006. 9. 6. · 2 11. CKM quark-mixing matrix Figure 11.1:...

20
11. CKM quark-mixing matrix 1 11. THE CKM QUARK-MIXING MATRIX Written January 2006 by A. Ceccucci (CERN), Z. Ligeti (LBNL), and Y. Sakai (KEK). 11.1. 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, L Y = Y d ij Q I Li φd I Rj Y u ij Q I Li φ u I Rj +h.c., (11.1) 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. Q I L are left handed quark doublets, and d I R and u I R are right handed down- and up-type quark singlets, respectively, in the weak-eigenstate basis. When φ acquires a vacuum expectation value, φ = (0,v/ 2), Eq. (11.1) yields mass terms for the quarks. The physical states are obtained by diagonalizing Y u,d by four unitary matrices, V u,d L,R , as M f diag = V f L Y f V f R (v/ 2), f = u, d. As a result, the charged current W ± interactions couple to the physical u Lj and d Lk quarks with couplings given by V CKM V u L V dL = V ud V us V ub V cd V cs V cb V td V ts V tb . (11.2) This Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2] is a 3 × 3 unitary matrix. It can be parameterized by three mixing angles and a CP -violating phase. Of the many possible parameterizations, a standard choice is [3] V = c 12 c 13 s 12 c 13 s 13 e s 12 c 23 c 12 s 23 s 13 e c 12 c 23 s 12 s 23 s 13 e s 23 c 13 s 12 s 23 c 12 c 23 s 13 e c 12 s 23 s 12 c 23 s 13 e c 23 c 13 , (11.3) where s ij = sin θ ij , c ij = cos θ ij , and δ is the KM phase [2] responsible for all CP -violating phenomena in flavor changing processes in the SM. The angles θ ij can be chosen to lie in the first quadrant, so s ij ,c ij 0. It is known experimentally that s 13 s 23 s 12 1, and it is convenient to exhibit this hierarchy using the Wolfenstein parameterization. We define [4–6] s 12 = λ = |V us | |V ud | 2 + |V us | 2 , s 23 = 2 = λ V cb V us , s 13 e = V ub = 3 (ρ + )= 3 ρ + i ¯ η) 1 A 2 λ 4 1 λ 2 [1 A 2 λ 4 ρ + i ¯ η)] . (11.4) These ensure that ¯ ρ + i ¯ η = (V ud V ub )/(V cd V cb ) is phase-convention independent and the CKM matrix written in terms of λ, A, ¯ ρ and ¯ η is unitary to all orders in λ. The definitions of ¯ ρ, ¯ η reproduce all approximate results in the literature. E.g.ρ = ρ(1 λ 2 /2+ ...) and we can write V CKM to O(λ 4 ) either in terms of ¯ ρ, ¯ η or, traditionally, V = 1 λ 2 /2 λ 3 (ρ ) λ 1 λ 2 /2 2 3 (1 ρ ) 2 1 + O(λ 4 ) . (11.5) CITATION: W.-M. Yao et al., Journal of Physics G 33, 1 (2006) available on the PDG WWW pages (URL: http://pdg.lbl.gov/) August 30, 2006 10:12

Transcript of 11. THE CKM QUARK-MIXING MATRIX · 2006. 9. 6. · 2 11. CKM quark-mixing matrix Figure 11.1:...

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11. CKM quark-mixing matrix 1

11. THE CKM QUARK-MIXING MATRIXWritten January 2006 by A. Ceccucci (CERN), Z. Ligeti (LBNL), and Y. Sakai (KEK).

11.1. 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,

LY = −Y dij QI

Li φdIRj − Y u

ij QILi ε φ∗uI

Rj + h.c., (11.1)

where Y u,d are 3× 3 complex matrices, φ is the Higgs field, i, j are generation labels, andε is the 2 × 2 antisymmetric tensor. QI

L are left handed quark doublets, and dIR and uI

Rare right handed down- and up-type quark singlets, respectively, in the weak-eigenstatebasis. When φ acquires a vacuum expectation value, 〈φ〉 = (0, v/

√2), Eq. (11.1) yields

mass terms for the quarks. The physical states are obtained by diagonalizing Y u,d by fourunitary matrices, V u,d

L,R, as Mfdiag = V f

L Y f V f†R (v/

√2), f = u, d. As a result, the charged

current W± interactions couple to the physical uLj and dLk quarks with couplings givenby

VCKM ≡ V uL V d†

L =

⎛⎝ Vud Vus Vub

Vcd Vcs VcbVtd Vts Vtb

⎞⎠ . (11.2)

This Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2] is a 3 × 3 unitary matrix. Itcan be parameterized by three mixing angles and a CP -violating phase. Of the manypossible parameterizations, a standard choice is [3]

V =

⎛⎝ c12c13 s12c13 s13e−iδ

−s12c23−c12s23s13eiδ c12c23−s12s23s13eiδ s23c13

s12s23−c12c23s13eiδ −c12s23−s12c23s13eiδ c23c13

⎞⎠ , (11.3)

where sij = sin θij , cij = cos θij , and δ is the KM phase [2] responsible for all CP -violatingphenomena in flavor changing processes in the SM. The angles θij can be chosen to lie inthe first quadrant, so sij , cij ≥ 0.

It is known experimentally that s13 � s23 � s12 � 1, and it is convenient to exhibitthis hierarchy using the Wolfenstein parameterization. We define [4–6]

s12 = λ =|Vus|√

|Vud|2 + |Vus|2, s23 = Aλ2 = λ

∣∣∣∣ Vcb

Vus

∣∣∣∣ ,

s13eiδ = V ∗

ub = Aλ3(ρ + iη) =Aλ3(ρ + iη)

√1 − A2λ4

√1 − λ2[1 − A2λ4(ρ + iη)]

. (11.4)

These ensure that ρ + iη = −(VudV ∗ub)/(VcdV ∗

cb) is phase-convention independent and theCKM matrix written in terms of λ, A, ρ and η is unitary to all orders in λ. The definitionsof ρ, η reproduce all approximate results in the literature. E.g., ρ = ρ(1− λ2/2 + . . .) andwe can write VCKM to O(λ4) either in terms of ρ, η or, traditionally,

V =

⎛⎝ 1 − λ2/2 λ Aλ3(ρ − iη)

−λ 1 − λ2/2 Aλ2

Aλ3(1 − ρ − iη) −Aλ2 1

⎞⎠ + O(λ4) . (11.5)

CITATION: W.-M. Yao et al., Journal of Physics G 33, 1 (2006)

available on the PDG WWW pages (URL: http://pdg.lbl.gov/) August 30, 2006 10:12

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2 11. CKM quark-mixing matrix

Figure 11.1: Sketch of the unitarity triangle.

The CKM matrix elements are fundamental parameters of the SM, so their precisedetermination is important. The unitarity of the CKM matrix imposes

∑i VijV

∗ik = δjk

and∑

j VijV∗kj = δik. The six vanishing combinations can be represented as triangles in

a complex plane, of which the ones obtained by taking scalar products of neighboringrows or columns are nearly degenerate. The areas of all triangles are the same, half ofthe Jarlskog invariant, J [7], which is a phase-convention independent measure of CPviolation, Im

[VijVklV

∗il V

∗kj

]= J

∑m,n εikmεjln.

The most commonly used unitarity triangle arises from

Vud V ∗ub + Vcd V ∗

cb + Vtd V ∗tb = 0 , (11.6)

by dividing each side by the best-known one, VcdV ∗cb (see Fig. 1). Its vertices are exactly

(0, 0), (1, 0) and, due to the definition in Eq. (11.4), (ρ, η). An important goal offlavor physics is to overconstrain the CKM elements, and many measurements can beconveniently displayed and compared in the ρ, η plane.

Processes dominated by loop contributions in the SM are sensitive to new physicsand can be used to extract CKM elements only if the SM is assumed. In Sec. 11.2 and11.3 we describe such measurements assuming the SM, and discuss implications for newphysics in Sec. 11.5.

11.2. Magnitudes of CKM elements

11.2.1. |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 ninemost precise determinations [8,9] yields [10]

|Vud| = 0.97377 ± 0.00027. (11.7)

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11. CKM quark-mixing matrix 3

The error is dominated by theoretical uncertainties stemming from nuclear Coulombdistortions and radiative corrections. A recent measurement of 46V performed usingan atomic Penning trap method [9] deviates by 2.7 standard deviations from theaverage and may indicate a need to remeasure superallowed 0+ → 0+ transitions withmodern atomic physics techniques. A precise determination of |Vud| is also obtainedfrom the measurement of the neutron lifetime. The theoretical uncertainties are verysmall but the determination is limited by the knowledge of the ratio of the axial-vector and vector couplings, gA = GA/GV [10]. The PIBETA experiment [11] hasimproved the measurement of the π+ → π0e+ν branching ratio to 0.6%, and quote|Vud| = 0.9728 ± 0.0030, in agreement with the more precise results listed above. Theinterest in this measurement is that the determination of |Vud| is very clean theoretically,because it is a pure vector transition and is free from nuclear structure uncertainties.

11.2.2. |Vus| :The magnitude of Vus has been extracted traditionally from semileptonic kaon decays.

Significant progress on the experimental side has taken place during the last two years.After a high statistics measurement of B(K+ → π0e+ν) [12], a suite of measurements ofneutral kaon branching ratios [13–15], form factors [16], and lifetime [17] followed. Anearly value of f+(0) = 0.961 ± 0.008 [18] is still broadly accepted [10,19]. It yields fromthe kaon semileptonic decay rate

|Vus| = 0.2257 ± 0.0021. (11.8)

A word of caution is in order, since other calculations of f+(0) [20,21] differ by as muchas 2%, with uncertainties around 1%.

Other determinations of |Vus| involve leptonic kaon decays, hyperon decays, andτ decays. The calculation of the ratio of the kaon and pion decay constants enablesone to extract |Vus/Vud| from K → µν(γ) and π → µν(γ), where (γ) indicates thatradiative decays are included [22]. The KLOE measurement of the K+ → µ+ν(γ)branching ratio [23], combined with the unquenched lattice QCD calculation,fK/fπ = 1.198 ± 0.003+0.016

−0.005 [24], leads to |Vus| = 0.2245+0.0012−0.0031, where the accuracy is

limited by the knowledge of the ratio of the decay constants. Hereafter, the first error isstatistical and the second is systematic, unless mentioned otherwise. The determinationfrom hyperon decays was recently updated [25]. These authors focus on the analysisof the vector form factor, protected from first order SU(3) breaking effects by theAdemollo-Gatto theorem [26], and treat the ratio between the axial and vector formfactors g1/f1 as experimental input, thus avoiding first order SU(3) breaking effects inthe axial-vector contribution. They find |Vus| = 0.2250 ± 0.0027, although this does notinclude an estimate of the theoretical uncertainty due to second order SU(3) breaking,contrary to Eq. (11.8). Concerning hadronic τ decays to strange particles, the latestdeterminations based on OPAL or ALEPH data also yields consistent results [27] withaccuracy around 0.003.

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11.2.3. |Vcd| :The magnitude of Vcd can be extracted from semileptonic charm decays if theoretical

knowledge of the form factors is available. Recently, the first three-flavor unquenchedlattice QCD calculations for D → K ν and D → π ν have been published [28]. Usingthese estimates and the isospin-averaged charm semileptonic width measured by CLEO-c,one obtains |Vcd| = 0.213± 0.008± 0.021 [29], where the first uncertainty is experimentaland the second is from the theoretical error of the form factor.

This determination is not yet as precise as the one based on neutrino and antineutrinointeractions. The difference of the ratio of double-muon to single-muon production byneutrino and antineutrino beams is proportional to the charm cross section off valenced-quarks, and therefore to |Vcd|2 times the average semileptonic branching ratio of charmmesons, Bµ. The method was used first by CDHS [30] and then by CCFR [31,32] andCHARM II [33]. Averaging these results is complicated, not only because it requiresassumptions about the scale of the QCD corrections, but also because Bµ is an effectivequantity, which depends on the specific neutrino beam characteristics. Given that no newexperimental input is available, we quote the average provided in the previous edition ofthis review, Bµ|Vcd|2 = (0.463± 0.034)× 10−2 [3]. Analysis cuts make these experimentsinsensitive to neutrino energies smaller than 30 GeV. Thus, Bµ should be computed usingonly neutrino interactions with visible energy larger than 30 GeV. A recent appraisal [34]based on charm production fractions measured in neutrino interactions [35,36] givesBµ = 0.088 ± 0.006. Data from the CHORUS experiment [37] are now sufficiently preciseto extract Bµ directly, by comparing the number of charm decays with a muon to thetotal number of charmed hadrons found in the nuclear emulsions. Requiring the visibleenergy to be larger than 30 GeV, CHORUS finds Bµ = 0.085 ± 0.009 ± 0.006. To extract|Vcd|, we use the average of these two determinations, Bµ = 0.0873 ± 0.0052, and obtain

|Vcd| = 0.230 ± 0.011. (11.9)

11.2.4. |Vcs| :The determination of |Vcs| from neutrino and antineutrino scattering suffers from the

uncertainty of the s-quark sea content. Measurements sensitive to |Vcs| from on-shell W±decays were performed at LEP-2. The branching ratios of the W depend on the six CKMmatrix elements involving quarks with masses smaller than MW . The W branching ratioto each lepton flavor is 1/B(W → ν�) = 3

[1 +

∑u,c,d,s,b |Vij |2 (1 + αs(mW )/π)

]. The

measurement assuming lepton universality, B(W → ν�) = (10.83 ± 0.07 ± 0.07) % [38],implies

∑u,c,d,s,b |Vij |2 = 2.002 ± 0.027. This is a precise test of unitarity, but only

flavor-tagged measurements determine |Vcs|. DELPHI measured tagged W+ → cs decays,obtaining |Vcs| = 0.94+0.32

−0.26 ± 0.13 [39].The direct determination of |Vcs| is possible from semileptonic D or leptonic Ds decays,

relying on the calculations of hadronic matrix elements. The measurement in D+s → +ν

requires the lattice QCD calculation of the decay constant, fDs , while in semileptonic Ddecays one needs the form factors which depend on the invariant mass of the lepton pair,q2. Unquenched lattice QCD calculations have predicted the normalization and the shape

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11. CKM quark-mixing matrix 5

of the form factor in D → K ν and D → π ν [28]. Using these theoretical results andthe isospin averaged semileptonic widths one obtains [29]

|Vcs| = 0.957 ± 0.017 ± 0.093, (11.10)

where the first error is experimental and the second one, which is dominant, is from thetheoretical error of the form factor.

11.2.5. |Vcb| :This matrix element can be determined from exclusive and inclusive semileptonic

decays of B mesons to charm. The inclusive determinations use the semileptonic decayrate measurement together with the leptonic energy and the hadronic invariant-massspectra. The theoretical foundation of the calculation is the operator product expansion(OPE) [40,41]. It expresses the total rate and moments of differential energy andinvariant-mass 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 differentfor different moments, and a large number of measured moments overconstrains all theparameters and tests the consistency of the determination. The precise extraction of |Vcb|requires using a “threshold” quark mass definition [42,43]. Inclusive measurements havebeen performed using B mesons from Z0 decays at LEP and at e+e− machines operatedat the Υ (4S). At LEP the large boost of B mesons from Z0 allows one to determinethe moments throughout phase space, which is not possible otherwise, but the largestatistics available at the B factories leads to more precise determinations. An average ofthe measurements and a compilation of the references are provided by Kowalewski andMannel [44]: |Vcb| = (41.7 ± 0.7) × 10−3.

Exclusive determinations are based on semileptonic B decays to D and D∗. In themb,c � ΛQCD limit, all form factors are given by a single Isgur-Wise function [45], whichdepends on the product of the four-velocities, w = v · v′, of the initial and final statehadrons. Heavy quark symmetry determines the normalization of the rate at w = 1, themaximum momentum transfer to the leptons, and |Vcb| is obtained from an extrapolationto w = 1. The exclusive determination, |Vcb| = (40.9 ± 1.8) × 10−3 [44], is less precisethan the inclusive one, because the theoretical uncertainty in the form factor and theexperimental uncertainty in the rate near w = 1 are both about 3%. The |Vcb| reviewquotes the average as [44]

|Vcb| = (41.6 ± 0.6) × 10−3. (11.11)

11.2.6. |Vub| :The determination of |Vub| from inclusive B → Xu ν decay suffers from large

B → Xc ν backgrounds. In most regions of phase space where the charm backgroundis kinematically forbidden the hadronic physics enters via unknown nonperturbativefunctions, so-called shape functions. (In contrast, the nonperturbative physics for |Vcb|is encoded in a few parameters.) At leading order in ΛQCD/mb there is only one shapefunction, which can be extracted from the photon energy spectrum in B → Xsγ [46,47]and applied to several spectra in B → Xu ν. The subleading shape functions are modeled

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6 11. CKM quark-mixing matrix

in the current calculations. Phase space cuts for which the rate does not depend on theshape function are also possible [48]. The measurements of both hadronic and leptonicsystem are important for effective choice of phase space. A different approach is toextend the measurements deeper into the B → Xc ν region to reduce the theoreticaluncertainties. Analyses of the electron-energy endpoint from CLEO [49], BABAR [50] andBelle [51] quote B → Xueν partial rates for |�pe| ≥ 2.0 GeV and 1.9 GeV, which are wellbelow the charm endpoint. The large and pure B–B samples at the B factories permitthe selection of B → Xu ν decays in events where the other B is fully reconstructed [52].With this full-reconstruction tag method the four-momenta of both the leptonic and thehadronic systems can be measured. It also gives access to a wider kinematic region dueto improved signal purity.

To extract |Vub| from an exclusive channel, the form factors have to be known.Experimentally, better signal-to-background ratios are offset by smaller yields. TheB → π ν branching ratio is now known to 8%. The first unquenched lattice QCDcalculations of the B → π ν form factor for q2 > 16 GeV2 appeared recently [53,54].Light-cone QCD sum rules are applicable for q2 < 14 GeV2 [55] and yield somewhatsmaller |Vub|, (3.3+0.6

−0.4) × 10−3. The theoretical uncertainties in extracting |Vub| frominclusive and exclusive decays are different. A combination of the determinations isquoted by the Vcb and Vub minireview as [44],

|Vub| = (4.31 ± 0.30) × 10−3, (11.12)

which is dominated by the inclusive measurement. In the previous edition of the RPP [56]the average was reported as |Vub| = (3.67±0.47)×10−3, with an uncertainty around 13%.The new average is 17% larger, somewhat above the range favored by the measurementof sin 2β discussed below. Given the rapid theoretical and experimental progress, it willbe interesting how the determination of |Vub| develops.

11.2.7. |Vtd| and |Vts| :The CKM elements |Vtd| and |Vts| cannot be measured from tree-level decays of the

top quark, so one has to rely on determinations from B–B oscillations mediated by boxdiagrams or loop-mediated rare K and B decays. Theoretical uncertainties in hadroniceffects limit the accuracy of the current determinations. These can be reduced by takingratios of processes that are equal in the flavor SU(3) limit to determine |Vtd/Vts|.

The mass difference of the two neutral B meson mass eigenstates is very well measured,∆md = 0.507± 0.004 ps−1 [57]. It is dominated by box diagrams with top quarks. Usingthe unquenched lattice QCD calculation of the Bd decay constant and bag parameter,

fBd

√BBd

= 244 ± 11 ± 24 MeV [58,59], and assuming |Vtb| = 1, one finds [60]

|Vtd| = (7.4 ± 0.8) × 10−3, (11.13)

where the uncertainty is dominated by that of fBd

√BBd

. Several uncertainties

are reduced in the lattice calculation of the ratio ξ = (fBs

√BBs)/(fBd

√BBd

) =

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11. CKM quark-mixing matrix 7

1.21 ± 0.04+0.04−0.01, and therefore the constraint on |Vtd/Vts| from ∆md and ∆ms would

be more reliable theoretically. So far only a lower bound is available, ∆ms > 16.6 ps−1

at 95% CL [57], which implies |Vtd/Vts| < 0.22 at 95% CL. After completion of thisreview, new DØ and CDF analyses appeared, obtaining 17 ps−1 < ∆ms < 21 ps−1 at90% CL [61], and ∆ms = (17.31+0.33

−0.18±0.07) ps−1 [62], respectively. These provide a newtheoretically clean and significantly improved constraint, |Vtd/Vts| = 0.208+0.008

+0.006 [62].

The inclusive rate B(B → Xsγ) = (3.55 ± 0.26) × 10−4 for Eγ > 1.6 GeV [57] issensitive to VtbV

∗ts via t-quark penguins. However, a large contribution to the rate comes

from the b → ccs four-quark operators mixing into the electromagnetic dipole operator.Since any CKM determination from loop processes necessarily assumes the SM, usingVcbV

∗cs ≈ −VtbV

∗ts, we obtain |Vts| = (40.6 ± 2.7) × 10−3 [63].

The corresponding exclusive decays suffer from larger theoretical uncertainties dueto unknown hadronic form factors, although the experimental accuracy is better.Recently, Belle observed exclusive B → (ρ/ω)γ signals [64] in b → dγ decays. Thetheoretically cleanest determination of |Vtd/Vts| in exclusive radiative B decays will comefrom B(B0 → ρ0γ)/B(B0 → K∗0γ), because the poorly known spectator interactioncontribution is expected to be smaller in this ratio than that in charged B decay(W -exchange vs. weak annihilation). For now, we average the neutral and charged modes,assuming isospin symmetry, which gives B(B → ργ)/B(B → K∗γ) = 0.017 ± 0.006 [57].This ratio is |Vtd/Vts|2/ξ2

γ , where ξγ contains the hadronic physics, and is unity inthe SU(3) limit. Using ξγ = 1.2 ± 0.2 [65], implies |Vtd/Vts| = 0.16 ± 0.04, where wecombined the experimental and theoretical errors in quadrature.

A theoretically clean determination of |VtdV ∗ts| is possible from K+ → π+νν decay [66].

Experimentally, only three events have been observed [67], and the rate is consistent withthe SM with large uncertainties. Much more data are needed for a precision measurement.

11.2.8. |Vtb| :The direct determination of |Vtb| from top decays uses the ratio of branching fractions

R = B(t → Wb)/B(t → Wq) = |Vtb|2/(∑

q |Vtq|2) = |Vtb|2, where q = b, s, d. The CDFand DØ measurements performed on data collected during Run II of the Tevatron giveR = 1.12+0.27

−0.23 [68] and R = 1.03+0.19−0.17 [69], respectively. Both measurements result in the

95% CL lower limit|Vtb| > 0.78 . (11.14)

An attempt at constraining |Vtb| from the precision electroweak data was madein [70]. The result, mostly driven by the top-loop contributions to Γ(Z → bb), gives|Vtb| = 0.77+0.18

−0.24.

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8 11. CKM quark-mixing matrix

11.3. Phases of CKM elements

As can be seen from Fig. 11.1, the angles of the unitarity triangle are

β = φ1 = arg(−VcdV ∗

cb

VtdV ∗tb

),

α = φ2 = arg(− VtdV ∗

tb

VudV ∗ub

),

γ = φ3 = arg(−VudV ∗

ub

VcdV ∗cb

). (11.15)

Since CP violation involves phases of CKM elements, many measurements of CP -violatingobservables can be used to constraint these angles and the ρ, η parameters.

11.3.1. ε and ε′ :The measurement of CP violation in K0–K0 mixing, |ε| = (2.233± 0.015)× 10−3 [71],

provides important information about the CKM matrix. In the SM [72]

|ε| =G2

F f2KmKm2

W

12√

2 π2∆mKBK

{ηcS(xc)Im[(VcsV

∗cd)2]

+ ηtS(xt)Im[(VtsV∗td)2] + 2ηctS(xc, xt)Im(VcsV

∗cdVtsV

∗td)

}, (11.16)

where S is an Inami-Lim function [73], xq = m2q/m2

W , and ηi are perturbativeQCD corrections. The constraint from ε in the ρ, η plane are bounded by hyperbolasapproximately. The dominant uncertainties are due to the bag parameter, for which weuse BK = 0.79 ± 0.04 ± 0.09 from lattice QCD [74], and the parametric uncertaintyproportional to σ(A4) [i.e., σ(|Vcb|4)].

The measurement of ε′ provides a qualitative test of the CKM mechanismbecause its nonzero experimental average, Re(ε′/ε) = (1.67 ± 0.23) × 10−3 [71],demonstrated the existence of direct CP violation, a prediction of the KM ansatz. WhileRe(ε′/ε) ∝ Im(VtdV ∗

ts), this quantity cannot easily be used to extract CKM parameters,because the electromagnetic penguin contributions tend to cancel the gluonic penguinsfor large mt [75], thereby significantly increasing the hadronic uncertainties. Mostestimates [76,77] are in agreement with the observed value, indicating a positive valuefor η. Progress in lattice QCD, in particular finite-volume calculations [78,79], mayeventually provide a determination of the K → ππ matrix elements.

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11. CKM quark-mixing matrix 9

11.3.2. β / φ1 :

11.3.2.1. Charmonium modes:CP violation measurements in B meson decays provide direct information on the

angles of the unitarity triangle, shown in Fig. 11.1. These overconstraining measurementsserve to improve the determination of the CKM elements or to reveal effects beyond theSM.

The time-dependent CP asymmetry of neutral B decays to a final state f common toB0 and B0 is given by [80,81]

Af =Γ(B0(t) → f) − Γ(B0(t) → f)Γ(B0(t) → f) + Γ(B0(t) → f)

= Sf sin(∆mt) − Cf cos(∆mt), (11.17)

where

Sf =2 Imλf

1 + |λf |2, Cf =

1 − |λf |21 + |λf |2

, λf =q

p

Af

Af. (11.18)

Here q/p describes B0–B0 mixing and, to a good approximation in the SM, q/p =V ∗

tbVtd/VtbV∗td = e−2iβ+O(λ4) in the usual phase convention. Af (Af ) is the amplitude of

B0 → f (B0 → f) decay. If f is a CP eigenstate and amplitudes with one CKM phasedominate a decay, then |Af | = |Af |, Cf = 0 and Sf = sin(arg λf ) = ηf sin 2φ, whereηf is the CP eigenvalue of f and 2φ is the phase difference between the B0 → f andB0 → B0 → f decay paths. A contribution of another amplitude to the decay with adifferent CKM phase makes the value of Sf sensitive to relative strong interaction phasesbetween the decay amplitudes (it also makes Cf �= 0 possible).

The b → ccs decays to CP eigenstates (B0 → charmonium K0S,L) are the theoretically

cleanest examples, measuring Sf = −ηf sin 2β. The b → sqq penguin amplitudes havedominantly the same weak phase as the b → ccs tree amplitude. Since only λ2-suppressedpenguin amplitudes introduce a new CP -violating phase, amplitudes with a singleweak phase dominate these decays, and we expect ||A/A| − 1| < 0.01. The e+e−asymmetric-energy B-factory experiments BABAR [82] and Belle [83] provide precisemeasurements. The world average is [57]

sin 2β = 0.687 ± 0.032 . (11.19)

This measurement has a four-fold ambiguity in β, which can be resolved by a global fitmentioned below. Experimentally, the two-fold ambiguity β → π/2−β (but not β → π+β)can be resolved by a time-dependent angular analysis of B0 → J/ψK∗0 [84,85] or a time-dependent Dalitz plot analysis of B0 → D0h0 (h0 = π0, η, ω) with D0 → K0

Sπ+π− [86].The latter gives the better sensitivity and disfavors the solutions with cos 2β < 0 at the97% CL, consistent with the global CKM fit result.

The b → ccd mediated transitions, such as B0 → J/ψπ0 and B0 → D(∗)+D(∗)−, alsomeasure approximately sin 2β. However, the dominant component of b → d penguin

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10 11. CKM quark-mixing matrix

amplitude has different CKM phase (V ∗tbVtd) than the tree amplitude (V ∗

cbVcd), andtheir magnitudes are of the same order in λ. Therefore, the effect of penguins couldbe large, resulting in Sf �= −ηf sin 2β and Cf �= 0. These decay modes have also beenmeasured by BABAR and Belle. The world averages [57], SJ/ψπ0 = −0.69 ± 0.25,SD+D− = −0.29±0.63, and SD∗+D∗− = −0.75±0.23, are consistent with sin 2β obtainedfrom B0 → charmonium K0 decays, and the Cf ’s are also consistent with zero, althoughthe uncertainties are sizable.

11.3.2.2. Penguin dominated modes:The b → sqq penguin dominated decays have the same CKM phase as the b → ccs

tree level decays, up to corrections suppressed by λ2, since V ∗tbVts = −V ∗

cbVcs[1 + O(λ2)].Therefore, decays such as B0 → φK0 and η′K0 provide sin 2β measurements in the SM.If new physics contributes to the b → s loop diagrams and has a different weak phase, itwould give rise to Sf �= −ηf sin 2β and possibly Cf �= 0. Therefore, the main interest inthese modes is not simply to measure sin 2β, but to search for new physics. Measurementsof many other decay modes in this category, such as B → π0K0

S , K0SK0

SK0S , etc.,

have also been performed by BABAR and Belle. The results and their uncertainties aresummarized in Fig. 12.3 and Table 12.1 of Ref. [81].

11.3.3. α / φ2 :Since α is the angle between V ∗

tbVtd and V ∗ubVud, only time-dependent CP asymmetries

in b → uud dominated modes can directly measure sin 2α, in contrast to sin 2β, whereseveral different transitions can be used. Since b → d penguin amplitudes have a differentCKM phase than b → uud tree amplitudes, and their magnitudes are of the same order inλ, the penguin contribution can be sizable and makes the α determination complicated.To date, α has been measured in B → ππ, ρπ and ρρ decay modes, among which B → ρρhas the best precision.

11.3.3.1. B → ππ:It is now experimentally well established that there is a sizable contribution of b → d

penguin amplitudes in B → ππ decays. Then Sπ+π− in the time-dependent B0 → π+π−analysis no longer measures sin 2α, but

Sπ+π− =√

1 − C2π+π− sin(2α + 2∆α), (11.20)

where 2∆α is the phase difference between e2iγAπ+π− and Aπ+π− . The value of∆α, hence α, can be extracted using the isospin relations among the amplitudes ofB0 → π+π−, B0 → π0π0, and B+ → π+π0 decays [87],

1√2

Aπ+π− + Aπ0π0 − Aπ+π0 = 0, (11.21)

and a similar one for the Aππ’s. This method utilizes the fact that a pair of pionsin B → ππ decay must be in a zero angular momentum state, and then, because of

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11. CKM quark-mixing matrix 11

Bose statistics, they must have even isospin. Consequently, π0π± is in a pure isospin-2state, while the penguin amplitudes only contribute to the isospin-0 final state. Thelatter does not hold for the electroweak penguin amplitudes, but their effect is expectedto be small. The isospin analysis uses the world averages [57] Sπ+π− = −0.50 ± 0.12,Cπ+π− = −0.37 ± 0.10, the branching fractions of all three modes, and the directCP asymmetry Cπ0π0 = −0.28+0.39

−0.40. This analysis leads to 16 mirror solutions for0 ≤ α < 2π. Because of this, and the sizable experimental error of the B0 → π0π0 rateand CP asymmetry, only a loose constraint on α can be obtained at present, 0◦ < α < 17◦and 73◦ < α < 180◦ at 68% CL.

11.3.3.2. B → ρρ:The decay B0 → ρ+ρ− contains two vector mesons in the final state, which in general

is a mixture of CP -even and CP -odd components. Therefore, it was thought thatextracting α from this mode would be complicated.

However, the longitudinal polarization fractions (fL) in B+ → ρ+ρ0 and B0 → ρ+ρ−decays were measured to be close to unity [88], which implies that the final states arealmost purely CP -even. Furthermore, B(B0 → ρ0ρ0) < 1.1×10−6 at 90% CL [89] is muchsmaller than B(B0 → ρ+ρ−) = (26.4+6.1

−6.4)×10−6 and B(B+ → ρ+ρ0) = (26.2+3.6−3.7)×10−6,

which implies that the effect of the penguin diagrams is small. The isospin analysis usingthe world averages [57], Sρ+ρ− = −0.22 ± 0.22 and Cρ+ρ− = −0.02 ± 0.17, and theabove-mentioned branching fractions of B → ρρ modes gives α = (96 ± 13)◦ [90] with amirror solution at 3π/2 − α. A possible small violation of Eq. (11.21)) due to the finitewidth of the ρ [91] is neglected.

11.3.3.3. B → ρπ:The final state in B0 → ρ+π− decay is not a CP eigenstate, but this decay proceeds

via the same quark level diagrams as B0 → π+π− and both B0 and B0 can decay toρ+π−. Consequently, mixing induced CP violations can occur in four decay amplitudes,B0 → ρ±π∓ and B0 → ρ±π∓. The measurements of CP violation parameters for thesedecays, where B0 → π+π−π0 decays are treated as quasi-two-body decays, have beenmade both by BABAR [92] and the Belle [93]. However, the isospin analysis is rathercomplicated and no significant model independent constraint on α has been obtained yet.

The time-dependent Dalitz plot analysis of B0 → π+π−π0 decays permits theextraction of α with a single discrete ambiguity, α → α + π, since one knows the variationof the strong phases in the interference regions of the ρ+π−, ρ−π+, and ρ0π0 amplitudesin the Dalitz plot [94]. The BABAR measurement gives α = (113+27

−17±6)◦ [95]. Althoughthis constraint is moderate so far, it favors one of the mirror solutions in B → ρ+ρ−.

Combining the above-mentioned three decay modes [90], α is constrained as

α = (99+13−8 )◦. (11.22)

A different statistical approach [96] gives similar constraint from the combination of thesemeasurements.

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12 11. CKM quark-mixing matrix

11.3.4. γ / φ3 :By virtue of Eq. (11.15), 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 themeasurements of α and β, and implies that the direct measurements of γ are unlikely tobe affected by physics beyond the SM.

11.3.4.1. B± → DK±:The interference of B− → D0K− (b → cus) and B− → D0K− (b → ucs) transitions

can be studied in final states accessible in both D0 and D0 decays [80]. In principle, itis possible to extract the B and D decay amplitudes, the relative strong phases, and theweak phase γ from the data.

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

rB =∣∣∣A(B− → D0K−)/A(B− → D0K−)

∣∣∣ , (11.23)

which is around 0.1 − 0.2. The original GLW method [97,98] considers D decays to CP

eigenstate final states, such as B± → D(∗)CP (→ π+π−)K±(∗). To alleviate the smallness

of rB and make the interfering amplitudes (which are products of the B and D decayamplitudes) comparable in magnitude, the ADS method [99] considers final states whereCabibbo-allowed D0 and doubly Cabibbo-suppressed D0 decays interfere. Extensivemeasurements have been made by the B factories using both methods [100], but theconstraints on γ are fairly weak so far.

It was realized that both D0 and D0 have large branching fractions to certainthree-body final states, such as KSπ+π−, and the analysis can be optimized by studyingthe Dalitz plot dependence of the interferences [101,102]. The best present determinationof γ comes from this method. Belle [103] and BABAR [104] obtained γ = 68+14

−15 ± 13± 11◦and γ = 67± 28± 13± 11◦, respectively, where the last uncertainty is due to the D decaymodeling. The error is sensitive to the central value of the amplitude ratio rB (and r∗B forthe D∗K mode), for which Belle found somewhat larger central values than BABAR. Thesame values of r

(∗)B enter the ADS analyses, and the data can be combined to fit for r

(∗)B

and γ. The possibility of D0–D0 mixing has been neglected in all measurements, but itseffect on γ is far below the present experimental accuracy [105], unless D0–D0 mixing isdue to CP -violating new physics, in which case it could be included in the analysis [106].

Combining the GLW, ADS, and Dalitz analyses [90], γ is constrained as

γ = (63+15−12)

◦. (11.24)

The likelihood function of γ is not Gaussian, and the 95% CL range is 40◦ < γ < 110◦.Similar results are found in [96]. New results [107] reported after the completion of thisreview indicate a decrease of r

(∗)B , which leads to an increased error on γ.

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11. CKM quark-mixing matrix 13

11.3.4.2. B → D(∗)±π∓:The interference of b → u and b → c transitions can be studied in B0 → D(∗)+π−

(b → cud) and B0 → B0 → D(∗)+π− (b → ucd) decays and their CP conjugates, sinceboth B0 and B0 decay to D(∗)±π∓ (or D±ρ∓, etc.). Since there are only tree and nopenguin contributions to these decays, in principle, it is possible to extract from the fourobservable time-dependent rates the magnitudes of the two hadronic amplitudes, theirrelative 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(B0 → D+π−)/A(B0 → D+π−) = O(0.01) (and similarly for rD∗π and rDρ),and therefore it has not been possible to measure it. To obtain 2β + γ, SU(3) flavorsymmetry and dynamical assumptions have been used to relate A(B0 → D−π+) toA(B0 → D−

s π+), so this measurement is not model independent at present. Theamplitude ratio is larger in the analogous Bs decays to D±

s K∓, where it will bepossible to measure it and model independently extract γ − 2βs at LHCb, as proposedin Ref. [108]. Combining the D±π∓ D∗±π∓ and D±ρ∓ measurements [109] givessin(2β + γ) = 0.8+0.18

−0.24, consistent with the previously discussed results for β and γ.

11.4. 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 =0.9992 ± 0.0011 (1st row), |Vcd|2 + |Vcs|2 + |Vcb|2 = 0.968 ± 0.181 (2nd row), and|Vud|2 + |Vcd|2 + |Vtd|2 = 1.001 ± 0.005 (1st column), respectively. For the second row, amore stringent check is obtained from the measurement of

∑u,c,d,s,b |Vij |2 in Sec.11.2.4

minus the sum in the first row above: |Vcd|2 + |Vcs|2 + |Vcb|2 = 1.003 ± 0.027. Theseprovide strong tests of the unitarity of the CKM matrix. The sum of the three angles ofthe unitarity triangle, α + β + γ = (184+20

−15)◦, is also consistent with the SM expectation.

The CKM matrix elements can be most precisely determined by a global fit that uses allavailable measurements and imposes the SM constraints (i.e., three generation unitarity).There are several approaches to combining the experimental data. CKMfitter [90,6] andRef. [110] (which develops [111,112] further) use frequentist statistics with differentpresentations of the theoretical errors, while UTfit [96,113] uses a Bayesian approach.These approaches provide similar results [114].

The constraints implied by the unitarity of the three generation CKM matrixsignificantly reduce the allowed range of some of the CKM elements. The fit for theWolfenstein parameters defined in Eq. (11.4) gives

λ = 0.2272 ± 0.0010 , A = 0.818+0.007−0.017 ,

ρ = 0.221+0.064−0.028 , η = 0.340+0.017

−0.045 , (11.25)

These values are obtained using the method of Refs. [6,90], while using the prescriptionof Refs. [96,113] gives λ = 0.2262 ± 0.0014, A = 0.815 ± 0.013, ρ = 0.235 ± 0.031, andη = 0.349 ± 0.020. The allowed ranges of the magnitudes of all nine CKM elements are

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14 11. CKM quark-mixing matrix

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1 1.5 2

sin 2β

sol. w/ cos 2β < 0(excl. at CL > 0.95)

excluded at CL > 0.95

γ

γ

α

α

∆md

∆ms & ∆md

εK

εK

|Vub/Vcb|

sin 2β

sol. w/ cos 2β < 0(excl. at CL > 0.95)

excluded at CL > 0.95

α

βγ

ρ

η

excluded area has CL > 0.95

Figure 11.2: Constraints on the ρ, η plane. The shaded areas have 95% CL. Seefull-color version on color pages at end of book.

VCKM =

⎛⎝ 0.97383+0.00024

−0.00023 0.2272+0.0010−0.0010 (3.96+0.09

−0.09) × 10−3

0.2271+0.0010−0.0010 0.97296+0.00024

−0.00024 (42.21+0.10−0.80) × 10−3

(8.14+0.32−0.64) × 10−3 (41.61+0.12

−0.78) × 10−3 0.999100+0.000034−0.000004

⎞⎠ , (11.26)

and the Jarlskog invariant is J = (3.08+0.16−0.18) × 10−5.

Fig. 11.2 illustrates the constraints on the ρ, η plane from various measurements andthe global fit result. The shaded 95% CL regions all overlap consistently around theglobal fit region, though the consistency of |Vub/Vcb| and sin 2β is not very good, asmentioned previously.

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11. CKM quark-mixing matrix 15

11.5. Implications beyond the SM

If the constraints of the SM are lifted, K, B and D decays and mixings are describedby many more parameters than just the four CKM parameters and the W , Z andquark masses. The most general effective Lagrangian at lowest order contains arounda hundred flavor changing operators, and the observable effects of interactions at theweak-scale or above are encoded in their coefficients. For example, ∆md, Γ(B → ργ), andΓ(B → Xd + −) are all proportional to |VtdV ∗

tb|2 in the SM, however, they may receiveunrelated contributions from new physics. Similar to the measurements of sin 2β fromtree- and loop-dominated modes, such overconstraining measurements of the magnitudesand phases of CKM elements provide excellent sensitivity to new physics. Anothervery clean test of the SM can come from future measurements of B(K0

L → π0νν) andB(K+ → π+νν). These loop induced decays are sensitive to new physics and will allow adetermination of β independent of its value measured in B decays [115].

Not all CP -violating measurements can be interpreted as constraints on the ρ, η plane.Besides the angles in Eq. (11.15), it is also useful to define βs = arg(−VtsV

∗tb/VcsV

∗cb),

which is the small, λ2-suppressed, angle of a “squashed” unitarity triangle obtained bytaking the scalar product of the second and third columns. It is the phase between Bs

mixing and b → ccs decays, and sin 2βs can be measured via time-dependent CP violationin Bs → J/ψφ, similar to sin 2β in B → J/ψK0. Checking if βs agrees with its SMprediction, sin 2βs = 0.036 ± 0.003 [116,90], is an equally important test of the theory.

In the kaon sector, both CP -violating observables ε and ε′ are tiny, so models in whichall sources of CP violation are small were viable before the B-factory measurements.Since the measurement of sin 2β we know that CP violation can be an O(1) effect, and itis only flavor mixing that is suppressed between the three quark generations. Thus, manymodels with spontaneous CP violation are excluded. Model independent statements forthe constraints imposed by the CKM measurements on new physics are hard to make,because most models that allow for new flavor physics contain a large number of newparameters. For example, the flavor sector of the MSSM contains 69 CP -conservingparameters and 41 CP -violating phases (i.e., 40 new ones) [117].

In a large class of models the unitarity of the CKM matrix is maintained, and thedominant new physics effect is a contribution to the B0–B0 mixing amplitude [118],which can be parameterized as M12 = MSM

12 (1+hdeiσd). While the constraints on hd andσd are significant (before the measurements of γ and α they were not), new physics witha generic weak phase may still contribute to M12 at order 20% of the SM. Measurementsunimportant for the SM CKM fit, such as the CP asymmetry in semileptonic decays, playa role in constraining such extensions of the SM [116]. Similar results for the constraintson new physics in K0 and B0

s mixing are discussed in Refs. [113,119].

The CKM elements are fundamental parameters, so they should be measured asprecisely as possible. The overconstraining measurements of CP asymmetries, mixing,semileptonic, and rare decays have started to severely constrain the magnitudes andphases of possible new physics contributions to flavor changing interactions. When newparticles are observed at the LHC, it will be important to know the flavor parameters asprecisely as possible to understand the underlying physics.

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16 11. CKM quark-mixing matrix

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