A Mathematica based Version of the CKMfitter Package

6 Chapter 2. Theory
VCKM is a complex 3×3 matrix and, hence, has 18 independent parameters: nine real
parts and nine imaginary parts of its nine complex matrix elements. An important
property is its unitarity, given by the relation:
VCKMV † †
CKM = V CKMVCKM = 1 , (2.8)
which ensures the conservation of probability. Due to unitarity and the freedom of
phase redefinition, the number of independent parameters is reduced to four and
the CKM matrix can be parameterized, e. g. by three Euler angles and one global
phase.
2.2.2 The Standard Parameterization of VCKM
The Standard Parameterization of the CKM matrix was proposed by Chau and
Keung [11] and is advocated by the Particle Data Group (PDG) [9]. It is obtained
by the product of three unitary complex rotation matrices, where the rotations are
characterized by Euler angles θ12, θ13 and θ23, which are the mixing angles between
the generations, and one overall CP-violating phase δ. The result is:
VCKM =
⎛
⎜
⎝
c12c13 s12c13 s13e −iδ
−s12c23 − c12s23s13e iδ c12c23 − s12s23s13e iδ s23c13
s12s23 − c12c23s13e iδ −c12s23 − s12c23s13e iδ c23c13
⎞
⎟
⎠ , (2.9)
where cij = cosθij and sij = sinθij for i < j = 1, 2, 3. The cij and sij are positive
for θij > 0. The unitarity relation (2.8) is strictly satisfied.
2.2.3 The Wolfenstein Parameterization of VCKM
As a result of the observed hierarchy between the different CKM matrix elements,
Wolfenstein [12] proposed a parameterization in terms of the four parameters A,
λ, ρ and η. It is an expansion of VCKM in λ � |Vus| and defined to all orders in λ
by [13]:
s12 ≡ λ
s23 ≡ Aλ 2
s13e −iδ ≡ Aλ 3 (ρ − iη) .
(2.10)
Thus, up to order of λ4 , the CKM matrix can be written as:
⎛
⎜
VCKM = ⎜
⎝
1 − λ2
2
λ Aλ3 −λ 1 −
(ρ − iη)
λ2
2
Aλ2 ⎞
⎟ + O
⎟
⎠
� λ 4� . (2.11)
Aλ 3 (1 − ρ − iη) −Aλ 2 1