# Difference: Formalism (1 vs. 14)

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

Changed:
<
<
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
>
>
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}\
Line: 54 to 54

Changed:
<
<
At the lowest order in , and coincide with and .
>
>
At the lowest order in , and coincide with and .
Revision 12
21 Jul 2010 - Main.VincenzoVagnoni
Line: 1 to 1

# CKM formalism

Revision 11
16 Jul 2010 - Main.VincenzoVagnoni
Line: 1 to 1
Changed:
<
<
Under construction
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

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}\
Changed:
<
<
\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}}.
Changed:
<
<
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
Line: 1 to 1
Under construction
Changed:
<
<
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}\
>
>
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}}.
Changed:
<
<
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
Line: 1 to 1
Under construction
Changed:
<
<
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}\
>
>
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}\
-\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}},\
Revision 8
28 Jun 2010 - Main.MarcoCiuchini
Line: 1 to 1
Under construction
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}\
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 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>
>
>
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 .
Revision 7
28 Jun 2010 - Main.MarcoCiuchini
Line: 1 to 1
Under construction
Changed:
<
<
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}\
>
>
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}\
Changed:
<
<
\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}}.
Changed:
<
<
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
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}\
Revision 5
17 Jun 2010 - Main.MarcoCiuchini
Line: 1 to 1
Under construction
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}\
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 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 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 4
16 Jun 2010 - Main.MarcoCiuchini
Line: 1 to 1
Under construction
Changed:
<
<
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}},\
>
>
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}}.
Changed:
<
<
>
>
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:
<
<
>
>
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