(r6) Formalism < UTfit

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The Cabibbo-Kobayashi-Maskawa ( CKM) matrix
$V_{CKM}=\left(\begin{array}{ccc} V_{ud} & V_{us} & V_{ub}\\ V_{cd} & V_{cs} & V_{cb}\\ V_{td} & V_{ts} & V_{tb}\end{array}\right)$
is a 3x3 unitary matrix which originates from the disalignment in flavour space of the up and down components of the quark doublet of the Standard Model ( SM). In the quark mass eigenstate basis, the CKM matrix appears in the SM charged-current interaction Lagrangian
${\cal L}^{cc}=\frac{g}{2\sqrt{2}} \sum_{i,j} \bar u_i \gamma_\mu (1-\gamma_5) (V_{CKM})_{ij} d_j\, W^\mu+ H.c.$
where the quark fields are and , while is the weak coupling constant and $W^\mu$ is the field which creates the vector boson. The CKM matrix elements are the only flavour- and CP-violating couplings present in the SM. In general, the CKM matrix can be parametrized using three rotation angles and one phase. The parametrization is however not unique. The standard parametrization, advocated by the PDG, uses $\theta_{12},\theta_{13},\theta_{23}$ in $[0,\pi/2]$ and $\delta$ in $(-\pi,\pi]$ defined so that
$V_{CKM}=\left(\begin{array}{ccc} \cos\theta_{12}\cos\theta_{13}& \sin\theta_{12}\cos\theta_{13}& \sin\theta_{13}\, e^{-i\delta}\\ -\sin\theta_{12}\cos\theta_{23}-\cos\theta_{12}\sin\theta_{13}\sin\theta_{23}\, e^{i\delta} & \cos\theta_{12}\cos\theta_{23}-\sin\theta_{12}\sin\theta_{13}\sin\theta_{23}\, e^{i\delta} & \cos\theta_{13}\sin\theta_{23}\\ \sin\theta_{12}\sin\theta_{23}-\cos\theta_{12}\sin\theta_{13}\cos\theta_{23}\, e^{i\delta} & -\cos\theta_{12}\sin\theta_{23}-\sin\theta_{12}\sin\theta_{13}\cos\theta_{23}\, e^{i\delta} & \cos\theta_{13}\cos\theta_{23}\end{array}\right)\,.$
We start extracting the CKM parameters from the measurements of $\vert V_{ud}\vert,\,\vert V_{cb}\vert,\,\vert V_{ub}\vert$ and $\gamma$ using
$\begin{array}{ll}\sin\theta_{13}=\vert V_{ub}\vert, & \cos\theta_{13}=\sqrt{1-\sin^2\theta_{13}},\\ \cos\theta_{12}=\frac{\vert V_{ud}\vert}{\cos\theta_{13}}, & \sin\theta_{12}=\sqrt{1-\cos^2\theta_{12}},\\ \sin\theta_{23}=\frac{\vert V_{cb}\vert}{\cos\theta_{13}}, & \cos\theta_{23}=\sqrt{1-\sin^2\theta_{23}},\end{array} ~~\delta=2\arctan\left(\frac{1\mp\sqrt{1-(a^2-1)\tan^2\gamma}}{(a-1)\tan\gamma}\right), ~a=\frac{\cos\theta_{12}\sin\theta_{13}\sin\theta_{23}}{\sin\theta_{12}\cos\theta_{23}}.$
The sign in the formula for $\delta$ corresponds to $\cos\gamma\leq0~(\cos\gamma\geq0)$ . Additional constraints are then applied using the method described in the section Statistical Method. Results are also given in the popular Wolfenstein parametrization which allows for a transparent expansion of the CKM matrix in terms of the small Cabibbo angle $\theta_{12}$ . The Wolfenstain parameters $\lambda,\,A,\,\rho,\,\eta$ are defined by the following equations
$\lambda=\sin\theta_{12},\quad A=\frac{\sin\theta_{23}}{\sin^2\theta_{12}},\quad \rho=\frac{\sin\theta_{13}\cos\delta}{\sin\theta_{12}\sin\theta_{23}},\quad\eta=\frac{\sin\theta_{13}\sin\delta}{\sin\theta_{12}\sin\theta_{23}}\,.$
The relations induced by the unitarity of the CKM matrix include six "triangular" relations, among which
$V_{ud}V_{ub}^*+V_{cd}V_{cb}^*+ V_{td}V_{tb}^*=0$
is referred to as the Unitarity Triangle ( UT). It can be rewritten as
$R_t\,e^{-i\beta}+R_u\,e^{i\gamma}=1\,,$
with
$R_t=\left|\frac{V_{td}V_{tb}^*}{V_{cd}V_{cb}^*}\right|,\quad R_u=\left|\frac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*}\right|,\quad\beta=\arg\left(-\frac{V_{cd}V_{cb}^*}{V_{td}V_{tb}^*}\right),\quad\gamma=\arg\left(-\frac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*}\right).$

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