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
 

{\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 u_i=(u,c,t)~ and d_i=(d,s,b), while g~ is the weak coupling constant and W^\mu~ is the field which creates the W^-~ 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 u_i=(u,c,t)~ and d_i=(d,s,b), while g~ is the weak coupling constant and W^\mu~ is the field which creates the vector boson W^-~. 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~.
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>
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
Line: 1 to 1
Changed:
<
<
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 misalignment in flavour space of the up and down components of the SU(2)_L~ 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 u_i=(u,c,t)~ and d_i=(d,s,b), while g~ is the weak coupling constant and W^\mu~ is the field which creates the W^-~ 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}\
>
>

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 SU(2)_L~ 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 u_i=(u,c,t)~ and d_i=(d,s,b), while g~ is the weak coupling constant and W^\mu~ is the field which creates the W^-~ 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}\
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
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).
R_u, R_t~ 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}},\
>
>
\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).

R_u, R_t~ 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}}.
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 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~.
>
>

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~.
Revision 10
29 Jun 2010 - Main.MarcoCiuchini
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 misalignment in flavour space of the up and down components of the SU(2)_L~ 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 u_i=(u,c,t)~ and d_i=(d,s,b), while g~ is the weak coupling constant and W^\mu~ is the field which creates the W^-~ 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 SU(2)_L~ 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 u_i=(u,c,t)~ and d_i=(d,s,b), while g~ is the weak coupling constant and W^\mu~ is the field which creates the W^-~ 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).
R_u, R_t~ 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}}.
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 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~.
>
>
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~.
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 SU(2)_L~ 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 u_i=(u,c,t)~ and d_i=(d,s,b), while g~ is the weak coupling constant and W^\mu~ is the field which creates the W^-~ 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 SU(2)_L~ 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 u_i=(u,c,t)~ and d_i=(d,s,b), while g~ is the weak coupling constant and W^\mu~ is the field which creates the W^-~ 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).
R_u, R_t~ 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 SU(2)_L~ 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 u_i=(u,c,t)~ and d_i=(d,s,b), while g~ is the weak coupling constant and W^\mu~ is the field which creates the W^-~ 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
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 SU(2)_L 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 u_i=(u,c,t) and d_i=(d,s,b), while g is the weak coupling constant and W^\mu is the field which creates the W^- 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 disalignment in flavour space of the up and down components of the SU(2)_L~ 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 u_i=(u,c,t)~ and d_i=(d,s,b), while g~ is the weak coupling constant and W^\mu~ is the field which creates the W^-~ 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}\
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 \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}},\
>
>
\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).
R_u, R_t~ 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}}.
Changed:
<
<
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).
>
>
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>
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 SU(2)_L 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 u_i=(u,c,t) and d_i=(d,s,b), while g is the weak coupling constant and W^\mu is the field which creates the W^- 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 SU(2)_L 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 u_i=(u,c,t) and d_i=(d,s,b), while g is the weak coupling constant and W^\mu is the field which creates the W^- 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
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 SU(2)_L 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}},\
>
>
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 SU(2)_L 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 u_i=(u,c,t) and d_i=(d,s,b), while g is the weak coupling constant and W^\mu is the field which creates the W^- 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}}.
Changed:
<
<
The sign + (-) in the formula for \delta corresponds to \cos\gamma<0~(\cos\gamma>0).
>
>
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).
Revision 3
11 Jun 2010 - Main.MarcoCiuchini
Line: 1 to 1
  Under construction
Changed:
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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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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 SU(2)_L 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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