Narrow Results

A model of the 3 × 3 Cabibbo–Kobayashi–Maskawa matrix, V, is presented in which the parameters are the eigenvalues and the components of its eigenvectors. In this model, the small departure of the experimentally determined V from being moduli symmetric (i.e. |V_ij|=|V_ji|) is controlled by the small difference between two of the eigenvalues. In case any two eigenvalues are equal, one obtains a moduli symmetric V depending on only three parameters. Our model gives very good fits to the available data including CP-violation. Our value of sin 2β≈0.7 and other parameters associated with the ''unitarity triangle'' are in good agreement with data and other analyses.

A new approach to the parametrization of the CKM matrix, V, is considered in which V is written as a linear combination of the unit matrix I and a nondiagonal matrix U which causes inter-generational-mixing, that is V = cos θ I + i sin θ U. Such a V depends on three real parameters including the parameter θ. It is interesting that a value of θ = π/4 is required to fit the available data on the CKM-matrix including CP-violation. Predictions of this fit for the angles α, β and γ for the unitarity triangle corresponding to , are given. For θ = π/4, we obtain α = 88.46°, β = 45.046° and γ = 46.5°. These values are just about in agreement, within errors, with the present data. It is very interesting that the unitarity triangle is expected to be approximately a right-angle, isosceles triangle. Our prediction sin 2β = 1 is in excellent agreement with the value 0.99 ± 0.15 ± 0.05 reported by the Belle collaboration at the Lepton–Photon 2001 meeting.

If the unitary quark-mixing matrix, V, is moduli symmetric, it will depend on three real parameters. This means that there is a relation between the four parameters needed to parametrize a general V. It is shown that there exists a very simple relation involving |V₁₁|², |V₃₃|², and . This relation is compared with the present experimental data. It is concluded that a moduli symmetric V is not ruled out.

The ratios of B_d and B_s meson decay constant have been estimated using the variational method in conjunction with the relativistic Hamiltonian of the heavy meson in the framework of the statistical model. The ratio of CKM matrix elements has been extracted from mixing using the estimated decay constants and ratio of B_d and B_s mass differences. The results are found to be in good agreement with the existing theoretical and experimental predictions.

Semileptonic decays of the B mesons provide an excellent probe for the weak and strong interactions of the bottom quark. The large data samples collected at the B Factories have pushed the experimental studies of the semileptonic B decays to a new height and stimulated significant theoretical developments. We review recent progresses in this fast-evolving field, with an emphasis on the determination of the magnitude of the Cabibbo–Kobayashi–Maskawa matrix element |V_ub|.

It is shown that two specific properties of the unitary matrix V can be expressed directly in terms of the matrix elements and eigenvalues of the hermitian matrix M which is diagonalized by V. These are the asymmetry Δ(V) = |V₁₂|²-|V₂₁|², of V with respect to the main diagonal and the Jarlskog invariant . These expressions for Δ(V) and J(V) provide constraints on possible mass matrices from the available data on V.

Recent experimental measurements and lattice QCD calculations are now reaching the precision (and accuracy) needed to over-constrain the CKM parameters and . In this brief review, we discuss the current status of lattice QCD calculations needed to connect the experimental measurements of B meson properties to quark flavor-changing parameters. Special attention is given to B→πℓν, which is becoming a competitive way to determine |V_ub|, and to mixings, which now include reliable extrapolation to the physical light quark mass. The combination of the recent measurement of the B_s mass difference and current lattice calculations dramatically reduces the uncertainty in |V_td|. We present an outlook for reducing dominant lattice QCD uncertainties entering CKM fits and remark on lattice calculations for other decay channels.

It is shown how the two experimentally measurable properties of the mixing matrix V, the asymmetry Δ(V) = |V₁₂|² - |V₂₁|² of V with respect to the main diagonal and the Jarlskog invariant , can be exploited to obtain constraints on possible structures of mass matrices in the quark sector. Specific mass matrices are examined in detail as an illustration.

If the Standard Model is correct, and fundamental fermions exist only in the three generations, then the CKM matrix should be unitary. However, there remains a question over a deviation from unitarity from the value of the neutron lifetime. We discuss a simple space-based experiment that, at an orbit height of 500 km above Earth, would measure the kinetic-energy, solid-angle, flux spectrum of gravitationally bound neutrons (kinetic energy K< 0.606 eV at this altitude). The difference between the energy spectrum of neutrons that come up from the Earth's atmosphere and that of the undecayed neutrons that return back down to the Earth would yield a measurement of the neutron lifetime. This measurement would be free of the systematics of laboratory experiments. A package of mass < 25 kg could provide a 10^-3 precision in two years.

We review the renormalization group evolution of quark and lepton masses, mixing angles and phases both in the UED extension of the Standard Model (SM) and of the Minimal Supersymmetric Standard Model (MSSM). We consider two typical scenarios: all matter fields propagating in the bulk and matter fields constrained to the brane. The resulting renormalization group evolution equations in these scenarios are compared with the existing results in the literature, together with their implications.

We review the current status of |V_cb|, |V_ub| and |V_tb|, the absolute values of the matrix elements in the CKM third column.

The flavor mixing of the quarks is described by the CKM matrix, which is parametrized by three mixing angles and one phase parameter. We discuss a new texture for the two mass matrices of the six quarks. The three flavor mixing angles can be calculated — they are functions of the ratios of the quark masses. The third mixing angle is given by the CKM matrix element $\left|V_{cb}\right|$. We find: $\left|V_{cb}\right|\simeq 2(m_s/m_b)$. The calculated mixing angles agree with the mixing angles, measured in many experiments.

New developments in both the theories and experiments related to the extraction of the top-row Cabibbo–Kobayashi–Maskawa matrix elements $V_{ud}$ and $V_{us}$ led to a series of new anomalies, for instance, the apparent violation of the top-row unitarity relation. It is important to further reduce all the associated Standard Model theory uncertainties in order to better understand whether such observations point towards the possibility of physics beyond the Standard Model, or rather some unexpectedly large Standard Model effects. This requires improved studies of tree-level and higher-order Standard Model corrections that enter the beta decays of pions, neutron, nuclei and kaons. We will briefly review the recent progress along this direction and discuss possible improvements in the future.