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Revision 14

20 Sep 2011 - Main.MarcoCiuchini

Line: 1 to 1
	CKM formalism

Revision 13

20 Oct 2010 - Main.MarcoCiuchini

Line: 1 to 1
	CKM formalism
Line: 11 to 11
	${\cal L}^{cc}=\frac{g}{2\sqrt{2}} \sum_{i,j} \bar u_i \gamma_\mu (1-\gamma_5) (V_{CKM})_{ij} d_j\, W^\mu+ H.c.$
Changed:
< <	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
> >	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}\
Line: 54 to 54
	$\rho+i\,\eta=\sqrt{\frac{1-A^2\lambda^4}{1-\lambda^2}}\frac{\bar\rho+i\,\bar\eta}{1-A^2\lambda^4(\bar\rho+i\,\bar\eta)}\simeq \left(1+\frac{\lambda^2}{2}\right)\left(\bar\rho+i\,\bar\eta\right)+{\cal O}(\lambda^4)\,.$
Changed:
< <	At the lowest order in $\lambda$ , $\rho~$ and $\eta~$ coincide with $\bar\rho~$ and $\bar\eta~$ .
> >	At the lowest order in $\lambda$ , $\rho~$ and $\eta~$ coincide with $\bar\rho~$ and $\bar\eta~$ .

Revision 12

21 Jul 2010 - Main.VincenzoVagnoni

Line: 1 to 1
	CKM formalism

Revision 11

16 Jul 2010 - Main.VincenzoVagnoni

-<
<
+Under construction
The Cabibbo-Kobayashi-Maskawa ( CKM) matrix 
 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 
 where the quark fields are  and , while  is the weak coupling constant and  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  in  and  in  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}\
->
>
+  CKM formalism 

The Cabibbo-Kobayashi-Maskawa ( CKM) matrix



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



where the quark fields are  and , while  is the weak coupling constant and  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  in  and  in  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 
 is referred to as the Unitarity Triangle ( UT). It can be rewritten as 
 with 
 ,  and ,  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 . The UT is determined by one complex number 
 namely by the coordinates  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  and  using \begin{array}{ll}\sin\theta_{13}=\vert V_{ub}\vert, & \cos\theta_{13}=\sqrt{1-\sin^2\theta_{13}},\
->
>
+\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 



is referred to as the Unitarity Triangle ( UT). It can be rewritten as 



with 



,  and ,  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 . The UT is determined by one complex number



 namely by the coordinates  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  and  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  corresponds to . 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 . The Wolfenstain parameters  are defined by the following equations   
 The CKM matrix can be expanded as   
 The exact and expanded relations between the UT apex coordinates  and the Wolfenstein parameters are given by 
 At the lowest order in ,  and  coincide with  and .
->
>
+The sign  in the formula for  corresponds to . 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 . The Wolfenstain parameters  are defined by the following equations 

  

The CKM matrix can be expanded as 

  

The exact and expanded relations between the UT apex coordinates  and the Wolfenstein parameters are given by 



 At the lowest order in ,  and  coincide with  and .

Revision 10

29 Jun 2010 - Main.MarcoCiuchini

 Under construction
-<
<
+The Cabibbo-Kobayashi-Maskawa ( CKM) matrix 
 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 
 where the quark fields are  and , while  is the weak coupling constant and  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  in  and  in  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}\
->
>
+The Cabibbo-Kobayashi-Maskawa ( CKM) matrix 
 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 
 where the quark fields are  and , while  is the weak coupling constant and  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  in  and  in  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 
 is referred to as the Unitarity Triangle ( UT). It can be rewritten as 
 with 
 ,  and ,  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 . The UT is determined by one complex number 
 namely by the coordinates  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  and  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  corresponds to . 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 . The Wolfenstain parameters  are defined by the following equations   
 The CKM matrix can be expanded as   
 The exact and expanded relations between  and the Wolfenstein parameters are given by 
 At the lowest order in ,  and  coincide with the UT apex coordinates  and .
->
>
+ The sign  in the formula for  corresponds to . 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 . The Wolfenstain parameters  are defined by the following equations   
 The CKM matrix can be expanded as   
 The exact and expanded relations between the UT apex coordinates  and the Wolfenstein parameters are given by 
 At the lowest order in ,  and  coincide with  and .

Revision 9

29 Jun 2010 - Main.AdrianBevan

Line: 1 to 1
	Under construction
Changed:
< <	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}\
> >	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- 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}},\

Revision 8

28 Jun 2010 - Main.MarcoCiuchini

Line: 1 to 1
	Under construction 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}\
Line: 7 to 7
	\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}}.
Changed:
< <	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 small Cabibbo angle $\theta_{12}$ . The Wolfenstain parameters $\lambda,\,A,\,\rho,\,\eta~$ are defined by the following equations $\lambda=\sin\theta_{12},\quad A=\frac{\sin\theta_{23}}{\sin^2\theta_{12}},\quad \rho=\frac{\sin\theta_{13}\cos\delta}{\sin\theta_{12}\sin\theta_{23}},\quad\eta=\frac{\sin\theta_{13}\sin\delta}{\sin\theta_{12}\sin\theta_{23}}\,.$ At the first order, $\lambda~$ is the Cabibbo angle and $\rho~$ and $\eta~$ coincide with the UT coordinates $\bar\rho~$ and $\bar\eta~$ . The exact relation between $\rho,\,\eta~$ and $\bar\rho,\,\bar\eta~$ is given by $\rho+i\,\eta=\sqrt{\frac{1-A^2\lambda^4}{1-\lambda^2}}\frac{\bar\rho+i\,\bar\eta}{1-A^2\lambda^4(\bar\rho+i\,\bar\eta)}\simeq \left(1+\frac{\lambda^2}{2}\right)\left(\bar\rho+i\,\bar\eta\right)+{\cal O}(\lambda^4)\,.$ </latex></latex>
> >	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 $\bar\rho,\,\bar\eta~$ 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 the UT apex coordinates $\bar\rho~$ and $\bar\eta~$ .

Revision 7

28 Jun 2010 - Main.MarcoCiuchini

 Under construction
-<
<
+The Cabibbo-Kobayashi-Maskawa ( CKM) matrix 
 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 
 where the quark fields are  and , while  is the weak coupling constant and  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  in  and  in  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}\
->
>
+The Cabibbo-Kobayashi-Maskawa ( CKM) matrix 
 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 
 where the quark fields are  and , while  is the weak coupling constant and  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  in  and  in  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  and  using \begin{array}{ll}\sin\theta_{13}=\vert V_{ub}\vert, & \cos\theta_{13}=\sqrt{1-\sin^2\theta_{13}},\
->
>
+\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 
 is referred to as the Unitarity Triangle ( UT). It can be rewritten as 
 with 
 ,  and ,  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 . The UT is determined by one complex number 
 namely by the coordinates  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  and  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  corresponds to . 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 . The Wolfenstain parameters  are defined by the following equations   
 The relations induced by the unitarity of the CKM matrix include six "triangular" relations, among which 
 is referred to as the Unitarity Triangle ( UT). It can be rewritten as 
 with
->
>
+ The sign  in the formula for  corresponds to . 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 small Cabibbo angle . The Wolfenstain parameters  are defined by the following equations   
 At the first order,  is the Cabibbo angle and  and  coincide with the UT coordinates  and . The exact relation between  and  is given by 
  </latex></latex>

Revision 6

17 Jun 2010 - Main.AchilleStocchi

Line: 1 to 1
	Under construction 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}\

Revision 5

17 Jun 2010 - Main.MarcoCiuchini

Line: 1 to 1
	Under construction 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}\
Line: 7 to 7
	\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}}.
Changed:
< <	The sign in the formula for $\delta$ corresponds to $\cos\gamma<0~(\cos\gamma>0)$ . 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 $R_t=\left\|\frac{V_{td}V_{tb}^}{V_{cd}V_{cb}^}\right\|,\quad R_u=\left\|\frac{V_{ud}V_{ub}^}{V_{cd}V_{cb}^}\right\|,\quad\beta=\arg\left(-\frac{V_{cd}V_{cb}^}{V_{td}V_{tb}^}\right),\quad\gamma=\arg\left(-\frac{V_{ud}V_{ub}^}{V_{cd}V_{cb}^*}\right).$
> >	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 $R_t=\left\|\frac{V_{td}V_{tb}^}{V_{cd}V_{cb}^}\right\|,\quad R_u=\left\|\frac{V_{ud}V_{ub}^}{V_{cd}V_{cb}^}\right\|,\quad\beta=\arg\left(-\frac{V_{cd}V_{cb}^}{V_{td}V_{tb}^}\right),\quad\gamma=\arg\left(-\frac{V_{ud}V_{ub}^}{V_{cd}V_{cb}^*}\right).$

Revision 4

16 Jun 2010 - Main.MarcoCiuchini

 Under construction
-<
<
+The Cabibbo-Kobayashi-Maskawa ( CKM) matrix 
 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 mass eigenstate basis, the CKM matrix appears in the SM charged current interaction 
 The CKM matrix elements are the only flavour- and CP-violating couplings present in the SM. 
The CKM matrix can be parametrized using three rotation angles . in  and one phase  in . The standard parametrization reads 
 We extract the CKM parameters from the measurements of  and  using \begin{array}{ll}\sin\theta_{13}=\vert V_{ub}\vert, & \cos\theta_{13}=\sqrt{1-\sin^2\theta_{13}},\
->
>
+The Cabibbo-Kobayashi-Maskawa ( CKM) matrix 
 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 
 where the quark fields are  and , while  is the weak coupling constant and  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  in  and  in  defined so that 
 We start extracting the CKM parameters from the measurements of  and  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  corresponds to .
->
>
+ The sign  in the formula for  corresponds to . 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 . The Wolfenstain parameters  are defined by the following equations   
 The relations induced by the unitarity of the CKM matrix include six "triangular" relations, among which 
 is referred to as the Unitarity Triangle ( UT). It can be rewritten as 
 with

Revision 3

11 Jun 2010 - Main.MarcoCiuchini

Line: 1 to 1
	Under construction
Changed:
< <	$V_{CKM}=\left(\begin{array}{ccc} V_{ud} & V_{us} & V_{ub}\\ V_{cd} & V_{cs} & V_{cb}\\ V_{td} & V_{ts} & V_{tb}\end{array}\right)$
> >	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 mass eigenstate basis, the CKM matrix appears in the SM charged current interaction ${\cal L}^{cc}=\frac{g}{2\sqrt{2}} \sum_{i,j} \bar u_i \gamma_\mu (1-\gamma_5) (V_{CKM})_{ij} d_j+ H.c.$ The CKM matrix elements are the only flavour- and CP-violating couplings present in the SM. The CKM matrix can be parametrized using three rotation angles . $\theta_{12},\theta_{13},\theta_{22}$ in $[0,\pi/2]$ and one phase $\delta$ in $(-\pi,\pi]$ . The standard parametrization reads $V_{CKM}=\left(\begin{array}{ccc} \cos\theta_{12}\cos\theta_{13}& \sin\theta_{12}\cos\theta_{13}& \sin\theta_{13}\, e^{-i\delta}\\ V_{cd} & V_{cs} & \sin\theta_{23}\cos\theta_{13}\\ V_{td} & V_{ts} & V_{tb}\end{array}\right)$ We extract 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 $\cos\gamma<0~(\cos\gamma>0)$ .

Revision 2

18 May 2010 - Main.MarcoCiuchini

Line: 1 to 1
	Under construction
Added:
> >	$V_{CKM}=\left(\begin{array}{ccc} V_{ud} & V_{us} & V_{ub}\\ V_{cd} & V_{cs} & V_{cb}\\ V_{td} & V_{ts} & V_{tb}\end{array}\right)$

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