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Overview of the Cabibbo–Kobayashi–Maskawa Matrix†∗ PRAMANA c Indian Academy of Sciences Vol. 79, No. 5 — journal of November 2012 physics pp. 1091–1108 Overview of the Cabibbo–Kobayashi–Maskawa matrix†∗ TIM GERSHON1,2 1Department of Physics, University of Warwick, Coventry, United Kingdom 2European Organization for Nuclear Research (CERN), Geneva, Switzerland E-mail: [email protected] Abstract. The current status of the determination of the elements of the Cabibbo–Kobayashi– Maskawa quark-mixing matrix is reviewed. Tensions in the global fits are highlighted. Particular attention is paid to the progress in, and prospects for, measurements of CP violation effects. Keywords. Quark flavour physics; quark mixing; CP violation. PACS Nos 11.30.Er; 11.30.Hv; 12.15.Hh; 13.20.–v; 13.25.–k; 14.65.–q 1. Introduction The Cabibbo–Kobayashi–Maskawa (CKM) matrix [1,2] describes the mixing between the three different families of quark in the Standard Model (SM) of particle physics. It is therefore a 3 × 3 unitary matrix, and can be written in terms of four real parameters. For example, in the Wolfenstein parametrization [3,4], it can be expressed as ⎛ ⎞ Vud Vus Vub ⎝ ⎠ VCKM = Vcd Vcs Vcb (1) ⎛ Vtd Vts Vtb ⎞ 1 − λ2/2 λ Aλ3(ρ − iη) = ⎝ −λ 1 − λ2/2 Aλ2 ⎠ + O λ4 , (2) Aλ3(1 − ρ − iη) −Aλ2 1 where the expansion parameter λ is the sine of the Cabibbo angle (λ = sin θC ≈ Vus). With four independent parameters, a 3 × 3 unitary matrix cannot be forced to be real- valued, and hence CP violation arises because the couplings for quarks and antiquarks = ∗ have different phases, i.e. VCKM VCKM. In the SM, all CP violations in the quark sector arise from this fact, which is encoded in the Wolfenstein parameter η. Moreover, all flavour-changing interactions of the quarks are described by the four parameters of the CKM matrix, which makes it a remarkably predictive paradigm, describing phenom- ena from the lowest energies (such as nuclear transitions and pion decays) to the highest ∗ †Copyright of this article remains with CERN under the Creative Commons License (CC-BY 3.0). DOI: 10.1007/s12043-012-0418-y; ePublication: 6 November 2012 1091 Tim Gershon (W boson and top quark decays) within the realm of accelerator-based particle physics. Inevitably, a very broad range of theoretical tools (such as chiral perturbation theory, lat- tice quantum chromodynamics, and heavy quark effective theories) is needed to relate such diverse processes to the underlying physical parameters. Only a brief summary of the current status is given here, with most of the attention given to experimental progress (discussions of progress in theory can be found in refs [5,6]). More detailed reviews can be found, for example, in refs [7–11], and in the summaries of CKM2010, the 6th International Workshop on the CKM Unitarity Triangle [12–17]. 2. CP-conserving parameters – magnitudes of CKM matrix elements 2.1 The Fermi constant As shown in figure 1, any absolute determination of the magnitude of a CKM matrix element requires knowledge of the Fermi constant, GF. The most precise information on GF is obtained from the measurement of the lifetime of the muon, τμ, since 2 5 1 G mμ = F (1 + q) , (3) τμ 192π 3 where mμ is the mass of the muon (known to better than 50 parts per billion [19]) and q accounts for phase-space, QED and hadronic radiative corrections (known to better than 1 part per million [20,21]). A recent measurement of the positive muon lifetime by the MuLan Collaboration [18], set in its historical context in figure 1,gives τμ+ = (2196980.3 ± 2.2) ps , (4) from which a determination of the Fermi constant to better than 1 part per million is obtained −5 −2 GF = (1.1663788 ± 7) × 10 GeV . (5) Balandin - 1974 Giovanetti - 1984 (*) W Bardin - 1984 Chitwood - 2007 i Barczyk - 2008 (u,c,t) j MuLan - R06 GFVij (d,s,b) MuLan - R07 2.19695 2.19700 2.19705 2.19710 2.19715 (a) (b) Lifetime (μs) Figure 1. (a) Diagram for a flavour-changing charged current interaction, the strength of which involves the Fermi constant GF and the CKM matrix element Vij.(b) Progress in the determination of the muon lifetime over the last 40 years (from ref. [18]). 1092 Pramana – J. Phys., Vol. 79, No. 5, November 2012 Overview of the Cabibbo–Kobayashi–Maskawa matrix 2.2 Determination of |Vud| + + The most precise determination of |Vud| is from super-allowed 0 → 0 nuclear beta decays. The relation K ft = , (6) 2 | |2 2GF Vud where the parameters f and t are obtained from measurements of the energy gap and from the half-life and branching fraction, respectively, is expected to be nucleus-independent 6 3 2 5 to first order (K is a known constant, K/(c) = 2π ln 2/(mec ) ). However, as shown in figure 2, the precision is such that second-order effects related to the nuclear medium (radiative and isospin-breaking corrections) need to be accounted for. This is achieved by obtaining a corrected quantity, labelled Ft [22], that is confirmed to be constant to −4 3 × 10 (see figure 2). This allows |Vud| to be extracted, |Vud| = 0.97425 ± 0.00022 . (7) Various alternative approaches allow determinations of |Vud|, with different merits. Nuclear mirror decays (transitions between states with nuclear isospin 1/2) and neutron lifetime measurements are sensitive to both vector and axial-vector couplings, and the latter does not require nucleus-dependent or isospin-breaking corrections to be known. The current experimental status of the neutron lifetime is controversial [7,23](seealso refs [24,25]), but future experiments should reduce its uncertainty. Determinations from 3090 3080 3070 (s) 3060 ft 3050 3040 3030 10C 22Mg 38Km 46V 14O 26Alm 34Ar 50Mn 62Ga 74Rb 34Cl 42Sc 54Co 3090 (s) 3080 t 3070 3060 0 10 20 30 40 Figure 2. Uncorrected ( ft) and corrected (Ft) values obtained from different 0+ → 0+ transitions (from ref. [22]). Pramana – J. Phys., Vol. 79, No. 5, November 2012 1093 Tim Gershon pion beta decay (which has only vector couplings) have the smallest theoretical uncer- tainty, though significantly increased data samples would be necessary to approach the current sensitivity on |Vud|. 2.3 Determination of |Vus| The last few years have seen significant progress in the determination of |Vus| from semileptonic kaon decays, as reviewed in refs [26,27]. Figure 3 summarizes the values of f+(0) |Vus| determined experimentally, using data from the BNL-E865, KLOE, KTeV, ISTRA+ and NA48 experiments (the latest preliminary results from the NA48 Collabo- ration [28] are not included), as well as the calculations of f+(0), mainly using lattice quantum chromodynamics (QCD) techniques. Using f+(0) = 0.959 ± 0.005 [29], the average value is obtained as |Vus| = 0.2254 ± 0.0013 . (8) A result with comparable precision on the ratio of CKM matrix elements is obtained from the widths of leptonic kaon and pion decays. The experimental data together with lattice QCD input, fK/fπ = 1.193 ± 0.006 [26], and accounting for isospin violation [30] gives |Vus/Vud| = 0.2316 ± 0.0012 , (9) where both experimental and theoretical uncertainties are essentially uncorrelated with those in the average for |Vus| given above. This then allows a comparison of the different determinations, as well as a test of the unitarity of the first row of the CKM matrix. As shown in figure 4, unitarity is found to hold to better than one part in 103. An alternative way of viewing this result is that the Fermi constant measured in the quark sector is consis- tent with the determination from the muon lifetime. This is thus a beautiful demonstration of the universality of the weak interaction. Κ0π+ f (0) 0.213 0.214 0.215 0.216 0.217 + Leutwyler & Roos 84 0.961(8) Quark M. KL e3 Q.M. Bijnens & Talavera 0.976(10) χPT + LR χ μ Jamin et al 0.974(11) PT + disp. KL 3 Cirigliano et al 0.984(12) χPT + 1/Nc PT+LECs- χ - Nf=0 SPQcdR 0.960(5)(7) Clover K e3 S N =2 f RBC 0.968(9)(6) DWF JLQCD* 0.967(6) Clover ± K e3 QCDSF* 0.9647(15) stat Clover ETMC-09 0.9560(84) TWMF - LATTICE - LATTICE - Nf=2+1 ± Stag K μ3 HPQCD-FNAL* 0.962(11) RBC-UKQCD-07 0.9644(49) +31 DWF RBC-UKQCD-10 0.9599(37) -43 0.213 0.214 0.215 0.216 0.217 (a) (b) 0.94 0.95 0.96 0.97 0.98 0.99 1.00 Figure 3. (a)Valuesof f+(0) |Vus| obtained from different semileptonic kaon decays, K 0π + giving an average f+(0) |Vus| = 0.2163±0.0005. (b) Calculations of f+ (0) (from ref. [26]). 1094 Pramana – J. Phys., Vol. 79, No. 5, November 2012 Overview of the Cabibbo–Kobayashi–Maskawa matrix 0.228 + → + Vud (0 0 ) Vus (Kl3) 0.226 us V fit fit with unitarity 0.224 ) (K μ2 /V ud unitarity V us 0.972 0.974 0.976 Vud Figure 4. Combination of constraints on the magnitudes of the elements of the first row of the CKM matrix (from ref. [26]). Alternative approaches to measure |Vus| are possible using hyperon decays or hadronic tau lepton decays. For the latter, the method relies on comparison of the inclusive strange and non-strange branching fractions. These are determined experimentally from sums of exclusive measurements, and since not all decays have yet been measured, rely somewhat on extrapolations (see ref. [31] for a detailed review). A recent study [32] estimates the value from hadronic tau decays to be |Vus| = 0.2166 ± 0.0019(exp.) ± 0.0005(th.) , (10) which is discrepant from the value from semileptonic kaon decays at the level of 3.7σ .
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