Formalism < UTfit

CKM formalism

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 misalignment 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-non-diagonal 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
$V_{CKM}=\left(\begin{array}{ccc} \cos\theta_{12}\cos\theta_{13}& \sin\theta_{12}\cos\theta_{13}& \sin\theta_{13}\, e^{-i\delta}\\ -\sin\theta_{12}\cos\theta_{23}-\cos\theta_{12}\sin\theta_{13}\sin\theta_{23}\, e^{i\delta} & \cos\theta_{12}\cos\theta_{23}-\sin\theta_{12}\sin\theta_{13}\sin\theta_{23}\, e^{i\delta} & \cos\theta_{13}\sin\theta_{23}\\ \sin\theta_{12}\sin\theta_{23}-\cos\theta_{12}\sin\theta_{13}\cos\theta_{23}\, e^{i\delta} & -\cos\theta_{12}\sin\theta_{23}-\sin\theta_{12}\sin\theta_{13}\cos\theta_{23}\, e^{i\delta} & \cos\theta_{13}\cos\theta_{23}\end{array}\right)\,.$
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
$R_t=\left|\frac{V_{td}V_{tb}^*}{V_{cd}V_{cb}^*}\right|,\quad R_u=\left|\frac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*}\right|,\quad\beta=\arg\left(-\frac{V_{cd}V_{cb}^*}{V_{td}V_{tb}^*}\right),\quad\gamma=\arg\left(-\frac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*}\right).$
, and $\beta$ , $\gamma~$ are the non-trivial sides and angles of the normalized UT. The third side is the unit vector and the third angle is given by $\alpha=\pi-\beta-\gamma=\arg(-V_{td}V_{tb}^*/(V_{ud}V_{ub}^*))$ . The UT is determined by one complex number
$\bar\rho+i\,\bar\eta=R_u\, e^{i\gamma}\,,$
namely by the coordinates $(\bar\rho,\,\bar\eta)~$ in the complex plane of its only non-trivial apex (the others being (0,0) and (1,0)). Several experimental constraints can be conveniently represented on this plane and used to determine the UT, as shown in section Fit Results. 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 $\cos\gamma\leq 0~(\cos\gamma\geq 0)$ . Additional constraints, discussed in section Constraints, are then applied using the method described in section Statistical Method. Fit Results are also given in the popular Wolfenstein parametrization which allows for a transparent expansion of the CKM matrix in terms of the sine 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 CKM matrix can be expanded as
$V_{CKM}=\left(\begin{array}{ccc} 1-\frac{\lambda^2}{2} & \lambda & A\lambda^3(\rho-i\,\eta)\\ -\lambda & 1-\frac{\lambda^2}{2} & A\lambda^2\\ A\lambda^3(1-\rho-i\,\eta) & -A\lambda^2 & 1\end{array}\right)+{\cal O}(\lambda^4)$
The exact and expanded relations between the UT apex coordinates $\bar\rho,\,\bar\eta~$ and the Wolfenstein parameters are given by
$\rho+i\,\eta=\sqrt{\frac{1-A^2\lambda^4}{1-\lambda^2}}\frac{\bar\rho+i\,\bar\eta}{1-A^2\lambda^4(\bar\rho+i\,\bar\eta)}\simeq \left(1+\frac{\lambda^2}{2}\right)\left(\bar\rho+i\,\bar\eta\right)+{\cal O}(\lambda^4)\,.$
At the lowest order in $\lambda$ , $\rho~$ and $\eta~$ coincide with $\bar\rho~$ and $\bar\eta~$ .

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