A flexible parameterization of the CKM matrix via the singular-value-decomposition method

We investigate a flexible method by which we can test the unitarity of the quark flavor-mixing matrix step by step. Singular-value-decomposition (SVD) techniques are used in analyzing the mixing matrix over a broader parameter region than the unitarity region. Unitary constraints let us extract CP-violating properties without any specific parameterization when the magnitudes of at least three mixing-matrix elements in three-generation quark mixing are given. This method can al so be applied to the analysis of lepton flavor mixing, in which only a few moduli are presently measured. Received: 27 November 2002 / Published online: 7 March 2003 RID="a" ID="a" e-mail: cskim@yonsei.ac.kr RID="b" ID="b" e-mail: jdkim@cskim.yonsei.ac.kr     

The β angle as the CP violating phase in the CKM matrix

The CKM matrix describing quark mixing with three generations can be parameterized by three Euler mixing angles and one CP violating phase. In most of the parameterizations, the CP violating phase chosen is not a directly measurable quantity and is parameterization dependent. In this work, we propose to use the most accurately measured CP violating angle β in the unitarity triangle as the phase in the CKM matrix, and construct an explicit β parameterization. We also derive an approximate Wolfenstein-like expression for this parameterization.     

Unitarity boomerangs of quark and lepton mixing matrices

The most popular way to present mixing matrices of quarks (CKM) and leptons (PMNS) is the parametrization with three mixing angles and one CP-violating phase. There are two major options in this kind of parametrizations, one is the original Kobayashi–Maskawa (KM) matrix, and the other is the Chau–Keung (CK) matrix. In a new proposal by Frampton and He, a unitarity boomerang is introduced to combine two unitarity triangles, and this new presentation displays all four independent parameters of the KM parametrization in the quark sector simultaneously. In this Letter, we study the relations between KM and CK parametrizations, and also consider the quark–lepton complementarity (QLC) in the KM parametrization. The unitarity boomerang is discussed in the situation of the CK parametrization for comparison with that in the KM parametrization in the quark sector. Then we extend the idea of unitarity boomerang to the lepton sector, and check the corresponding unitarity boomerangs in the two cases of parametrizations.     

Unitarity polygons and CP violation areas and phases in the standard electroweak model

A geometrical interpretation of CP violation in terms of areas of unitarity triangles is presented, in the standard electroweak model with three families of quarks. The extension of the results to the case of four families is investigated. A special case is worked out in detail where it is shown how one determines the CP violating invariant phases and the areas of the CP violation triangles as functions of the moduli of the elements of the quark mixing matrix. For the most general four-family case, which is found to be rather involved, the necessary formulae, for determining the invariant phases and the CP violation areas, are given.     

Experimental analysis of weak mixing angles between three or four quark generations

New data on weak quark decays and on weak heavy quark production are used to obtain the allowed ranges of elements of the quark mixing matrix for three or four generations of sequential quarks. The analysis yields allowed ranges for the three mixing angles in the six-quark case and for the six mixing angles in the eight-quark case.     

First measurement of the ratio B(t --> Wb)/B(t --> Wq) and associated limit on the Cabibbo-Kobayashi-Maskawa element /V(tb)/

We present the first measurement of the ratio of branching fractions R identical withB(t-->Wb)/B(t-->Wq) from p_p collisions at sqrt[s] = 1.8 TeV. The data set corresponds to 109 pb(-1) of data recorded by the Collider Detector at Fermilab during the 1992-95 Tevatron run. We measure R = 0.94(+0.31)(-0.24)(stat+syst) or R>0.61 (0.56) at 90% (95)% C.L., in agreement with the standard model predictions. This measurement yields a limit on the Cabibbo-Kobayashi-Maskawa quark mixing matrix element /V(tb)/ under the assumption of three generations and unitarity.     

Phenomenological relations for quark and neutrino mixing angles

The most recent experimental data on quark and neutrino mixing angles are discussed. It is indicated that the results of the latest kaon-decay experiments are consistent with the unitarity condition for the first row of the Cabibbo-Kobayashi-Maskawa matrix if the currently available world-average value of the neutron lifetime is used to determine the element V _ud of this matrix. The quark mixing angles are calculated within the Fritzsch-Scadron-Delbourgo-Rupp phenomenological approach on the basis of values of the masses of light and heavy constituent quarks. The neutrino mixing angles are calculated to a high precision with the aid of the hypothesis that the quark and neutrino mixing angles are complementary. The results are compatible with experimental data.     

Flavour mixing and quark masses in sequential flavour dynamics

We discuss the mixing of quark flavours in sequential flavour dynamics in the case of an arbitrary number of quark generations. For n generations the dominant weak interaction mixing is described by n ? 1 mixing angles, which are computed in terms of the quark masses.     

Unitarity boomerang

For the three family quark flavor mixing, the best parametrization is the original Kobayashi–Maskawa matrix, VKM$V_{\mathrm{KM}}$, with four real parameters: three rotation angles θ_1,2,3${\theta }_{1,2,3}$ and one phase δ. A popular way of presentation is by the unitarity triangle which, however, explicitly displays only three, not four, independent parameters. Here we propose an alternative presentation which displays simultaneously all four parameters: the unitarity boomerang.     

Coherent mixing in three and four quark generations

New dynamical mechanism of quark mass generations and mixing is demonstrated in the examples of three and four generations. In the framework of the new mixing pattern, called the coherent mixing, the CKM elements are predicted compatible with experimental data for three generations, and are strongly constrained for four generations.     

Quark mixing in weak interactions

The status of the quark mixing in weak interaction is reviewed. The 3×3 quark mixing matrix for the three left-handed doublet model is analyzed using various experimental information involving strange, charmed, and b-flavored particles. Its interplay with nonleptonic decays, implication on neutral particle-antiparticle mixing and CP violation in heavy quark systems, and the possible origin of the quark mixing from quark mass matrix are discussed. Finally we briefly review the status of alternative sources for CP violation, and alternative models to the three left-hand quark doublet model.     

A novel formulation of Cabibbo–Kobayashi–Maskawa matrix renormalization

We present a gauge-independent quark mass counterterm for the on-shell renormalization of the Cabibbo–Kobayashi–Maskawa (CKM) matrix in the Standard Model that is directly expressed in terms of the Lorentz-invariant self-energy functions, and automatically satisfies the hermiticity constraints of the mass matrix. It is very convenient for practical applications and leads to a gauge-independent CKM counterterm matrix that preserves unitarity and satisfies other highly desirable theoretical properties, such as flavor democracy.     

Determination of |Vud| from Fermi decays and the unitarity of the KM-mixing matrix

We re-evaluate the ƒt-values and their radiative corrections for the eight superallowed β⁺ decays for which very accurate data are available and also include the results for ¹⁰C, for which the measurements are approaching a similar accuracy. The main new feature is a recalculation of the nuclear-structure dependent part C_NS of the O() radiative correction using wave functions more realistic than those used previously. This determines the value of the element V_ud of the quark mixing matrix. Moreover, we recalculate V_us in the framework of a quark model. The resulting values of these two matrix elements indicate a violation of unitarity of the mixing matrix. We discuss implications of this regarding possible extensions of the minimal version of the standard model of electroweek interactions.     

A soft origin for CKM-type CP violation

We present a two-Higgs-doublet model, with a Z₃ symmetry, in which CP violation originates solely in a soft (dimension-2) coupling in the scalar potential, and reveals itself solely in the CKM (quark mixing) matrix. In particular, in the mass basis the Yukawa interactions of the neutral scalars are all real. The model has only eleven parameters to fit the six quark masses and the four independent CKM-matrix observables. We find regions of parameter space in which the flavour-changing neutral couplings are so suppressed that they allow the scalars to be no heavier than a few hundred GeV.     

Role of kaon physics,present–future

The main issue for kaon physics in the next decade is a systematic study of the rare decay modes to test the unitarity of the quark flavour mixing matrix, the global symmetries of the Standard Model and new physics with the highest possible precision.     

Testing the symmetric Cabibbo-Kobayashi-Maskawa (CKM) matrix with lattice QCD calculations

We examine quantitatively the expectation of a symmetric quark mixing using recent lattice QCD calculations on the various hadronic matrix elements. We present a general parametrization of the symmetric CKM matrix which has directly measurable parameters. The direct measurements of the CKM matrix elements, unitarity and CP violation are used to constrain the parameters. The symmetric CKM matrix is also confronted with ε in the kaon system,x _d of \(B_d^0 - \bar B_d^0 \) mixing, and ε′/ε of ΔS=1 CP-violation in the kaon system. Our analyses reveal that the symmetric CKM matrix is consistent with the present experiments, providedm _t>130 GeV. The factorf _Bd ² B _Bd plays an important role in reaching the conclusions.     

Neutrino mixing and leptonic CP phase in neutrino oscillations

Oscillations of the Dirac neutrinos of three generations in vacuum are considered with allowance made for the effect of the CP-violating leptonic phase (analogue of the quark CP phase) in the lepton mixing matrix. The general formulas for the probabilities of neutrino transition from one sort to another in oscillations are obtained as functions of three mixing angles and the CP phase. It is found that the leptonic CP phase can, in principle, be reconstructed by measuring the oscillation-averaged probabilities of neutrino transition from one sort to another. The manifestation of the CP phase as a deviation of the probabilities of direct processes from those of inverse processes is an effect that is practically unobservable as yet.     

Analysis of horizontal symmetries in a minimal SU(2)L×SU(2)R×U(1) gauge theory

A left-right symmetric gauge theory of electroweak interactions is considered in which only one Higgs field and its charge conjugate couple to quarks. Among all possible parity transformations those allowing for phenomenologically acceptable mass matrices are singled out. In such a theory arbitrary horizontal symmetry groups with unitary representations are investigated. Both for two and three quark generations essentially only one model emerges which is compatible with empirical constraints on quark masses and mixing angles.     

Implications of $B \to \rho \gamma$ measurements in the standard model and supersymmetric theories

We study the implications of the recently improved upper limits on the branching ratios for the decays , expressed as . We work out the constraints that the current bound on implies on the parameters of the quark mixing matrix in the standard model (SM). Using the present profile of the unitarity triangle, we predict this ratio to lie in the range (at 95% C.L.), with the central value . We also work out the correlations involving , the isospin-violating ratio , and the direct CP-violating asymmetry in decays in the SM, in the minimal supersymmetric extension of the SM (MSSM), and in an extension of the MSSM involving an additional flavor-changing structure in transitions. Received: 19 August 2002 / Published online: 18 October 2002 RID="a" ID="a" e-mail: ali@mail.desy.de RID="b" ID="b" e-mail: lunghi@mail.desy.de     

General scan in flavor parameter space in models with vector quark doublets and an enhancement in the B→X_sγ process

In models with vector-like quark doublets, the mass matrices of up and down type quarks are related.Precise diagonalization of the mass matrices has become an obstacle in numerical studies. In this work we first propose a diagonalization method. As its application, in the Standard Model with one vector-like quark doublet we present the quark mass spectrum and Feynman rules for the calculation of B → X_sγ. We find that i) under the constraints of the CKM matrix measurements, the mass parameters in the bilinear term are constrained to a small value by the small deviation from unitarity; ii) compared with the fourth generation extension of the Standard Model, there is an enhancement to the B → X_sγ process in the contribution of vector-like quarks, resulting in a non-decoupling effect in such models.