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

If the final state is a CP -eigenstate CP |f〉 = ηf |f〉 with an eigenvalue ηf = ±1, then the time
dependent asymmetry
becomes
af (t) = Γ(B0 (t) → f) − Γ(B 0 (t) → f)
Γ(B 0 (t) → f) + Γ(B 0 (t) → f)
(2.40)
af (t) = Af cos(∆Mt) + Sf sin(∆Mt), (2.41)
neglecting the small width difference of the B meson. Here, the direct and indirect (or mixinginduced)
CP asymmetries are written as
Af = |λf | 2 − 1
|λf | 2 , (2.42)
+ 1
Sf =
2 Imλf
. (2.43)
1 + |λf | 2
Since the absolute value of q/p is approximately 1, direct CP -violation |Af | = 0 requires
|Af | = |Af |, which could happen if Af is a sum of (more than one) decay amplitudes having
different CP -phases. Indirect CP -violation, on the other hand, proves the quantum mechanical
interference between the mixing and decay amplitudes.
2.4.1 Measurement of sin 2φ1
The mixing-induced asymmetry provides a variety of methods to measure the angles of the
Unitarity Triangle (Fig. 2.1). It was first proposed by Bigi, Carter, and Sanda in 1980–1981
[13–15], and gave strong motivation to construct the present KEK B Factory. The best known
example is the case where the final state is J/ψK0 S , whose quark level process is b → ccs followed
by K0 − K 0 mixing. Provided that the decay is dominated by a single amplitude, the ratio of
decay amplitudes is given by
A J/ψK 0 S
A J/ψK 0 S
=
VcbV ∗
cs
V ∗
cb Vcs
VcsV ∗
cd
V ∗
csVcd
, (2.44)
The overall sign is (−1) 2 = 1, where one minus sign appears from a phase convention for the B
and B meson states, and the other is due to the CP odd final state J/ψK0 S . Together with the
phase in the mixing
q
p −e−iφM
V ∗
tbVtd = −
VtbV ∗
, (2.45)
cd
the entire ratio λ J/ψK 0 S becomes
V ∗
tbVtd λJ/ψK0 = −
S VtbV ∗
VcbV
td
∗
cs
V ∗
cbVcs VcsV ∗
cd
V ∗
csVcd
= −e 2iφ1 . (2.46)
Thus, one can precisely measure the angle φ1 from the time-dependent asymmetry
a J/ψK 0 S (t) = − sin(2φ1) sin(∆Mt). (2.47)
There exists an additional decay amplitude through the penguin diagram b → scc, which involves
. Using the unitarity of the CKM matrix VtsV
∗
, the
the CKM factor VtsV ∗
tb
35
tb
= −VcsV ∗
cb − VusV ∗ ub