JHEP06(2016)100
Here θ is a CP and CPT violating complex parameter while δ violates CP and T. In terms
of physical parameters, except for the phase of q/p which is convention dependent,
2
the
effective Hamiltonian can be written as [17]
H =
µ −
∆µ
2
θ
p
q
∆µ
2
√
1 − θ
2
q
p
∆µ
2
√
1 − θ
2
µ +
∆µ
2
θ
!
. (2.9)
2.2 The entangled system
In a B factory operating at the Υ(4S) peak, our initial two-meson state is Einstein-
Podolsky-Rosen [18] entangled,
3
|Ψ
0
i =
1
√
2
|B
0
d
i|
¯
B
0
d
i − |
¯
B
0
d
i|B
0
d
i
=
1
√
2(p
L
q
H
+ p
H
q
L
)
|B
L
i|B
H
i − |B
H
i|B
L
i
, (2.10)
which maintains its antisymmetric entangled character in the H eigenstate basis. This
implies the antisymmetric character of the two meson state at all times and for any two
independent linear combinations of B
0
d
and
¯
B
0
d
. The corresponding evolution is therefore
given in a simple way. The transition amplitude for the decay of the first state into |fi at
time t
0
, and then the second state into |gi at time t + t
0
, is
hf, t
0
; g, t + t
0
|T |Ψ
0
i =
e
−i(µ
H
+µ
L
)t
0
√
2(p
L
q
H
+ p
H
q
L
)
e
−iµ
H
t
A
L
f
A
H
g
− e
−iµ
L
t
A
H
f
A
L
g
, (2.11)
where the decay amplitudes of the eigenstates into the final state f are A
H,L
f
≡ hf|T |B
H,L
i.
Squaring and integrating over t
0
, the double decay rate I(f, g; t) is obtained:
I(f, g; t) =
e
−Γ t
4Γ|p
L
q
H
+ p
H
q
L
|
2
e
i∆M t/2
e
∆Γ t/4
A
H
f
A
L
g
− e
−i∆M t/2
e
−∆Γ t/4
A
L
f
A
H
g
2
. (2.12)
This expression is very useful to realize the following expected symmetry property: up to
the global exponential decay factor e
−Γ t
, the combined transformations t → −t and f g
should be the identity. Expanding the t dependence and taking the approximation ∆Γ = 0,
valid for the neutral B
0
d
states, one can write:
I(f, g; t) = e
−Γ t
hΓ
f
ihΓ
g
i
Γ
n
C
h
[f, g] + C
c
[f, g] cos(∆M t) + S
c
[f, g] sin(∆M t)
o
, (2.13)
with hΓ
f
i defined below. Therefore, starting from an entangled state as in eq. (2.10), and
using the Quantum Mechanical evolution in eq. (2.11), the following symmetry properties
arise from the previous remark:
C
h
[f, g] = C
h
[g, f], C
c
[f, g] = C
c
[g, f], S
c
[f, g] = −S
c
[g, f]. (2.14)
2
Although q/p is phase convention dependent, in the CP or T invariant limits, its phase is fixed relative
to the convention adopted for the action of the CP operator on |B
0
d
i and |
¯
B
0
d
i.
3
See references [19–21] to consider corrections to this assumption.
– 4 –
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