Letter of Intent for KEK Super B Factory Part I: Physics

= − g2
√2
u c t
L γµ ⎛
VCKM
ˆ ⎜
⎝
d
s
b
⎞
⎟
⎠
L
W † µ + h.c., (2.5)
where Wµ denotes the W boson, and g2 is the gauge coupling corresponding to the gauge group
SU(2)L. In the low energy effective Hamiltonian it appears as the Fermi constant GF / √ 2 =
g2 2 /8M 2 W . Due to the misalignment between the up-type and down-type quark fields, the charged
current induces transitions among different generations.
In contrast, the neutral current is flavor-conserving, which is ensured by the unitarity of the
CKM matrix, and thus Flavor Changing Neutral Currents (FCNC) are absent at the tree level
in the Standard Model. This is the Glashow-Iliopoulos-Maiani (GIM) mechanism [3]. Even
including loop corrections, the FCNC interaction vanishes in the limit of degenerate (up-type)
quark masses, due to the unitarity of the CKM matrix.
The CKM matrix is a unitary N × N matrix with N(= 3) number of generations, and thus
contains N 2 parameters in general. However, 2N − 1 phases may be absorbed by rephasing
the 2N quark fields (one overall phase is related to the total baryon number conservation and
is irrelevant for the quark mixing), and (N − 1) 2 independent parameters remain. Of these,
1
2N(N +1) are real parameters, which correspond to rotation angles among different generations,
while 1
2N(N − 1) are imaginary parameters, which are sources of CP -violation. In the threegeneration
Standard Model, there are 3 mixing angles and 1 CP -phase.
The standard parametrization of the CKM matrix is the following: [4]
VCKM =
⎛
⎜
⎝
c12c13 s12c13 s13e −iδ
−s12c23 − c12s23s13e iδ c12c23 − s12s23s13e iδ s23c13
s12s23 − c12c23s13e iδ s23c12 − s12c23s13e iδ c23c13
⎞
⎟
⎠ , (2.6)
where cij = cos θij and sij = sin θij with θij (ij = 12, 13 and 23) the mixing angles, and δ is the
complex phase. It is known experimentally that the angles are small and exhibit the hierarchy
1 ≫ s12 ≫ s23 ≫ s13. To make this structure manifest, the Wolfenstein parametrization [5] is
often used, in which one sets λ = |Vus| 0.22 and
VCKM =
⎛
⎜
⎝
1 − λ 2 /2 λ Aλ 3 (ρ − iη)
−λ 1 − λ 2 /2 Aλ 2
Aλ 3 (1 − ρ − iη) −Aλ 2 1
⎞
⎟
⎠ + O(λ 4 ), (2.7)
with A, ρ and η being real parameters of order unity. In this parametrization the source of
CP -violation is carried by the most off-diagonal elements Vub and Vtd.
Among these four parameters, λ and A are relatively well known from corresponding semileptonic
decays: |Vus| = 0.2196 ± 0.0026 from Kl3 decays and |Vcb| = (41.2 ± 2.0) × 10 −3 from
inclusive and exclusive b → clνl decays [4]. The determination of the other two parameters ρ
and η is conveniently depicted as a contour in the plane of (ρ, η). It corresponds to the unitarity
relation of the CKM matrix applied to the first and third columns
VudV ∗ ub + VcdV ∗
cb + VtdV ∗
tb = 0. (2.8)
This relation may be presented in the complex place as in Fig. 2.1 (a), which is called the
“unitarity triangle” Since VcdV ∗
cb is real to a good approximation (up to O(λ7 )), it is convenient
to normalize the triangle by |VcdV ∗
cb | = Aλ3 so that the apex has the coordinate (ρ, η) where
ρ = ρ(1 − λ 2 /2), η = η(1 − λ 2 /2), (2.9)
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