Hierarchies of quark masses and the mixing matrix in the standard theory
TL;DR: In this paper, the authors studied the general dependence of mixing angles on heavy fermion masses when mass hierarchies exist among the fermions and showed that the Cabibbo-Kobayashi-Maskawa matrix is a good approximation to the CCA matrix.
Abstract: We study the general dependence of mixing angles on heavy fermion masses when mass hierarchies exist among the fermions. For two generations and small Cabibbo angle, this angle is directly shown to scale like $\mu_1/m_s \pm \mu_2/m_c$, where $|\mu_1| \ll m_s, |\mu_2| \ll m_c$ are independent mass scales. For $n=3$ generations, we extend to the Yukawa matrices of $u$- and $d$-type quarks the property that the $2\times 2$ upper-left sub-matrix of the Cabibbo-Kobayashi-Maskawa matrix $K$ is a good approximation to the Cabibbo matrix $C$. Then, without any additional Ansatz concerning the existence of mass hierarchies or the smallness of the mixing angles, the moduli of its entries $K_{13},K_{23},K_{31},K_{32}$ are shown to scale like $[\beta_{13},\beta_{23},\beta_{31},\beta_{32}] \sqrt{{m_c}/{m_t}} \pm [\delta_{13},\delta_{23},\delta_{31},\delta_{32}] \sqrt{{m_s}/{m_b}}$, where the $\beta$'s and the $\delta$'s are coefficients smaller than 10. This method, when used for two generations, gives a dependence on $m_s$ and $m_c$ ``weaker'' than the one obtained first, but which matches a well known behaviour for the Cabibbo angle: $\theta_c \approx \sqrt{\epsilon_d (m_d/m_s)} - \sqrt{\epsilon_u(m_u/m_c)}$, with $\epsilon_d,\epsilon_u \leq 1$. The asymptotic behaviour in the case of three generations can also be strengthened into a $1/m_{b,t}$ behaviour by incorporating our knowledge about the hierarchies of quark masses and the smallness of the mixing angles.
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TL;DR: In this article, the Kobayashi-Maskawa matrix is expanded in powers of a small parameter equal to $sin{\ensuremath{\theta}}_{c}=0.22$ and the term of order is determined from the measured $B$ lifetime.
Abstract: The quark mixing matrix (Kobayashi-Maskawa matrix) is expanded in powers of a small parameter $\ensuremath{\lambda}$ equal to $sin{\ensuremath{\theta}}_{c}=0.22$. The term of order ${\ensuremath{\lambda}}^{2}$ is determined from the recently measured $B$ lifetime. Two remaining parameters, including the $\mathrm{CP}$-nonconservation effects, enter only the term of order ${\ensuremath{\lambda}}^{3}$ and are poorly constrained. A significant reduction in the limit on $\frac{{\ensuremath{\epsilon}}^{\ensuremath{'}}}{\ensuremath{\epsilon}}$ possible in an ongoing experiment would tightly constrain the $\mathrm{CP}$-nonconservation parameter and could rule out the hypothesis that the only source of $\mathrm{CP}$ nonconservation is the Kobayashi-Maskawa mechanism.
1,568 citations
TL;DR: In this paper, the low-energy effective lagrangian produced by a heavy chiral fermion, in the limit M → ∞, was determined, and the effective interaction of vector fields, in anomalous and non-anomalous sectors, and couplings of the Higgs boson were considered.
28 citations