Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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1
76 5 Phenomenology o
f
Neutrino
O
scillations
|
˜
V
μ
VV
1
|
2
=
ˆ
Δ
12
ˆ
Δ
13
˜
Δ
12
˜
Δ
13
|
V
μ
V
V
1
|
2
+
ˆ
Δ
22
ˆ
Δ
23
˜
Δ
1
2
˜
Δ
1
3
|
V
μ
VV
2
|
2
+
ˆ
Δ
32
ˆ
Δ
33
˜
Δ
1
2
˜
Δ
13
|
V
μ
VV
3
|
2
,
|
˜
V
μ
VV
2
|
2
=
ˆ
Δ
11
ˆ
Δ
13
˜
Δ
2
1
˜
Δ
2
3
|
V
μ
V
V
1
|
2
+
ˆ
Δ
21
ˆ
Δ
23
˜
Δ
21
˜
Δ
23
|
V
μ
VV
2
|
2
+
ˆ
Δ
31
ˆ
Δ
33
˜
Δ
21
˜
Δ
23
|
V
μ
VV
3
|
2
,
|
˜
V
μ
VV
3
|
2
=
ˆ
Δ
1
1
ˆ
Δ
1
2
˜
Δ
31
˜
Δ
32
|
V
μ
V
V
1
|
2
+
ˆ
Δ
21
ˆ
Δ
22
˜
Δ
31
˜
Δ
32
|
V
μ
VV
2
|
2
+
ˆ
Δ
31
ˆ
Δ
32
˜
Δ
31
˜
Δ
32
|
V
μ
VV
3
|
2
;
(
5.62
)
a
n
d
|
˜
V
τ
VV
1
|
2
=
ˆ
Δ
12
ˆ
Δ
1
3
˜
Δ
12
˜
Δ
13
|
V
τ
VV
1
|
2
+
ˆ
Δ
22
ˆ
Δ
23
˜
Δ
12
˜
Δ
13
|
V
τ
VV
2
|
2
+
ˆ
Δ
3
2
ˆ
Δ
33
˜
Δ
12
˜
Δ
13
|
V
τ
VV
3
|
2
,
|
˜
V
τ
VV
2
|
2
=
ˆ
Δ
11
ˆ
Δ
1
3
˜
Δ
21
˜
Δ
23
|
V
τ
VV
1
|
2
+
ˆ
Δ
21
ˆ
Δ
23
˜
Δ
21
˜
Δ
23
|
V
τ
VV
2
|
2
+
ˆ
Δ
3
1
ˆ
Δ
33
˜
Δ
21
˜
Δ
23
|
V
τ
VV
3
|
2
,
|
˜
V
τ
VV
3
|
2
=
ˆ
Δ
11
ˆ
Δ
12
˜
Δ
3
1
˜
Δ
32
|
V
τ
VV
1
|
2
+
ˆ
Δ
21
ˆ
Δ
22
˜
Δ
3
1
˜
Δ
32
|
V
τ
VV
2
|
2
+
ˆ
Δ
31
ˆ
Δ
32
˜
Δ
3
1
˜
Δ
3
2
|
V
τ
VV
3
|
2
,
(
5.63
)
whe
r
e
ˆ
Δ
ji
≡
m
2
j
−
˜
m
2
i
a
n
d
˜
Δ
ji
≡
Δ
˜
m
2
j
i
=˜
m
2
j
−
˜
m
2
i
h
ave bee
n
defi
n
ed
.N
ote
that
ˆ
Δ
ji
=
ˆ
Δ
jj
+
˜
Δ
j
i
h
o
ld
s, an
db
ot
h
ˆ
Δ
jj
a
n
d
˜
Δ
ji
c
an be directl
y
read o
ff
f
rom Eq.
(
5.43
)
. Of course, we hav
e
|
˜
V
αi
VV
|
2
=
|
V
αi
V
V
|
2
i
nth
e
A
→
0
limit.
W
e continue to ca
l
cu
l
at
e
˜
V
αi
VV
˜
V
∗
βi
VV
in
te
rm
sof
V
αi
VV
V
∗
βi
VV
(
for
α
=
β
)
. The latte
r
c
an form three unitarity trian
g
les, which are named a
s
e
f
or
(
α,
β
)
=
(
μ
,
τ
),
μ
f
or
(
α, β
)
=
(
τ,e
)
an
d
τ
for
(
α,
β
)
=
(
e,
μ
)
in Fig. 3.10, in the comple
x
plane in vacuum. Their e
ff
ective counterparts in matter are then re
f
erred t
o
as
˜
e
,
˜
μ
an
d
˜
τ
. Taking account of Eqs.
(
5.55
)
,
(
5.57
)
and
(
5.58
)
,onema
y
easil
y
write down a full set of equations o
f
˜
V
αi
V
V
˜
V
∗
βi
VV
(
fo
r
α
=
β
)
:
˜
X
⎛
⎜
⎛⎛
⎝
⎜⎜
˜
V
α
VV
1
˜
V
∗
β
V
V
1
˜
V
α
VV
2
˜
V
∗
β
V
V
2
˜
V
α
VV
3
˜
V
∗
β
V
V
3
⎞
⎟
⎞⎞
⎠
⎟⎟
=
X
+
A
δ
αe
+
δ
e
β
Y
⎛
⎝
⎛⎛
V
α
VV
1
V
∗
β
VV
1
V
α
VV
2
V
∗
β
VV
2
V
α
VV
3
V
∗
β
VV
3
⎞
⎠
⎞⎞
,
(
5.64
)
where
˜
X
,
X
a
n
d
Y
h
avebeengiveninEq.
(
5.60
)
. The solutions to Eq.
(
5.64
)
a
re
(
Xing and Zhang, 2005; Zhang and Xing, 2005
)
˜
V
μ
VV
1
˜
V
∗
τ
V
V
1
=
(
ˆ
Δ
21
+
A
)
Δ
31
˜
Δ
21
˜
Δ
31
V
μ
VV
1
V
∗
τ
V
V
1
+
(
ˆ
Δ
1
1
+
A
)
Δ
32
˜
Δ
21
˜
Δ
31
V
μ
VV
2
V
∗
τ
V
V
2
,
˜
V
μ
VV
2
˜
V
∗
τ
V
V
2
=
(
ˆ
Δ
32
+
A
)
Δ
21
˜
Δ
3
2
˜
Δ
21
V
μ
VV
2
V
∗
τ
V
V
2
+
(
ˆ
Δ
22
+
A
)
Δ
31
˜
Δ
3
2
˜
Δ
21
V
μ
VV
3
V
∗
τ
V
V
3
,
˜
V
μ
V
V
3
˜
V
∗
τ
V
V
3
=
(
ˆ
Δ
1
3
+
A
)
Δ
23
˜
Δ
13
˜
Δ
23
V
μ
V
V
3
V
∗
τ
V
V
3
+
(
ˆ
Δ
33
+
A
)
Δ
21
˜
Δ
13
˜
Δ
23
V
μ
V
V
1
V
∗
τ
V
V
1
,
(
5.65
)
f
or the effective triangl
e
˜
e
;
an
d
5.2 Neutrino
O
scillations and
Q
uantum
C
oherence 17
7
˜
V
τ
VV
1
˜
V
∗
e
VV
1
=
ˆ
Δ
21
Δ
31
˜
Δ
21
˜
Δ
31
V
τ
VV
1
V
∗
e
VV
1
+
ˆ
Δ
11
Δ
32
˜
Δ
21
˜
Δ
3
1
V
τ
VV
2
V
∗
e
VV
2
,
˜
V
τ
VV
2
˜
V
∗
e
VV
2
=
ˆ
Δ
32
Δ
21
˜
Δ
3
2
˜
Δ
21
V
τ
VV
2
V
∗
e
VV
2
+
ˆ
Δ
22
Δ
31
˜
Δ
3
2
˜
Δ
21
V
τ
VV
3
V
∗
e
VV
3
,
˜
V
τ
VV
3
˜
V
∗
e
VV
3
=
ˆ
Δ
1
3
Δ
23
˜
Δ
13
˜
Δ
23
V
τ
VV
3
V
∗
e
VV
3
+
ˆ
Δ
33
Δ
21
˜
Δ
13
˜
Δ
23
V
τ
VV
1
V
∗
e
VV
1
,
(
5.66
)
f
or the effective triangl
e
˜
μ
;
an
d
˜
V
e
VV
1
˜
V
∗
μ
VV
1
=
ˆ
Δ
21
Δ
31
˜
Δ
21
˜
Δ
31
V
e
VV
1
V
∗
μ
VV
1
+
ˆ
Δ
11
Δ
32
˜
Δ
21
˜
Δ
31
V
e
VV
2
V
∗
μ
VV
2
,
˜
V
e
VV
2
˜
V
∗
μ
VV
2
=
ˆ
Δ
3
2
Δ
2
1
˜
Δ
3
2
˜
Δ
2
1
V
e
VV
2
V
∗
μ
VV
2
+
ˆ
Δ
22
Δ
31
˜
Δ
3
2
˜
Δ
21
V
e
VV
3
V
∗
μ
VV
3
,
˜
V
e
VV
3
˜
V
∗
μ
VV
3
=
ˆ
Δ
13
Δ
2
3
˜
Δ
13
˜
Δ
23
V
e
VV
3
V
∗
μ
VV
3
+
ˆ
Δ
33
Δ
21
˜
Δ
13
˜
Δ
23
V
e
VV
1
V
∗
μ
VV
1
,
(
5.67
)
f
or the e
ff
ective trian
g
le
˜
τ
,
w
h
er
e
Δ
ji
≡
Δ
m
2
ji
=
m
2
j
−
m
2
i
has bee
n
defi
n
ed
.
O
ne ma
y
use the sides o
f
˜
α
(
for
α
=
e
,
μ
,τ
)
to calculate the effectiv
e
J
arlskog invariant
˜
J
a
nd then obtain the relationshi
p
between
˜
J
a
nd
J
as
given in Eq.
(
5.50
)
.Theareaso
f
˜
e
,
˜
μ
a
n
d
˜
τ
are a
ll
equa
l
t
o
˜
J
/
2.
W
e remark that the results obtained above are onl
y
valid
f
or neutrino
s
propagating in matter.
A
s for antineutrinos propagating in matter, the rele-
vant
f
ormulas can be directly written out throu
g
h the replacements
V
=
⇒
=
=
V
∗
(
o
r
J
=
⇒−J
==
)
and
A
=
⇒−
==
A
. For a neutrino oscillation ex
p
eriment wit
h
L
1
000 km, the de
p
endence of terrestrial matter effects on the neutrin
o
b
eam ener
g
y is approximate
ly
A
≈
2
.
28
×
10
−
4
eV
2
E/
[
GeV
](
Mocioiu and
Shrock, 2000
)
. Typically takin
g
Δ
21
≈
8
×
10
−
5
eV
2
,
Δ
32
≈
2
.
3
×
1
0
−
3
e
V
2
,
θ
12
≈
33
◦
,
θ
13
≈
3
◦
,
θ
23
≈
4
5
◦
a
n
d
δ
≈
90
◦
in vacuum
,
we show the numerical
d
ependence o
f
˜
J
/
J
o
n E in Fi
g
.5.1
f
or both neutrinos and antineutrinos
.
We see that
˜
J
i
s in most cases smaller than
J
. This unfortunate feature
m
akes it hard to directly measure leptonic CP or T violation in any realisti
c
l
on
g
-baseline neutrino oscillation experiments. I
f
the neutrino beam ener
gy
i
ssmall
(
e.g.,
E
∼
1 GeV or smaller
)
, the matter-induced suppression of
˜
J
i
s not significant. In this case a study of CP violation and unitarity triangle
s
i
n a medium-baseline neutrino oscillation experiment mi
g
ht be more
f
easibl
e
(
Minakata and Nunokawa, 2000; Xing, 2001a
).
5
.2 Neutrino
O
scillations and
Q
uantum
C
oherence
A
lthough the phenomenology of neutrino oscillations has been established,
a
number o
f
basic issues associated with the theory o
f
neutrino oscillations
1
78 5 Phenomenology o
f
Neutrino
O
scillations
Fig. 5.1 Terrestrial matter effects on the Jarlskog invariant of CP violation for
neutrinos
(
ν
wit
h
+
A
a
n
d
V
)
and antineutrinos
(
ν
w
i
th
−
A
a
n
d
V
∗
)
in a realisti
c
long-baseline oscillation experiment
(
Zhang and Xing, 2005. With permission from
Springer Science+Business Media
)
a
re sti
ll
un
d
er
d
e
b
ate. In t
h
is section we intro
d
uce t
h
ewave-pac
k
et approac
h
f
or describing neutrino oscillations and analyze the conditions of quantum
c
o
h
erence un
d
er w
h
ic
h
neutrino osci
ll
ations can ta
k
ep
l
ace
.
5.2.1 A Paradox o
f
Neutrino Oscillations
The treatment o
f
neutrino oscillations in
S
ection 5.1 is actuall
y
oversimple,
a
lthou
g
h it can lead to the correct results. Now let us clarify the same ener
gy
or same momentum assumption for neutrino mass eigenstates composing
a
g
iven neutrino
fl
avor state, because it has o
f
ten been used in the calculatio
n
of neutrino oscillation
p
robabilities.
I
f the same energ
y
E is assumed for three neutrino mass eigenstates, the
y
m
ust possess di
ff
erent momenta
p
k
=
E
2
−
m
2
k
(
fo
r
k
=1
,
2
,
3)
which can
app
rox
i
mate t
o
p
k
=
E
−
m
2
k
/
(
2
E
)
in the relativistic limit. Then a give
n
n
eutrino
fl
avor state propa
g
ates in the one-dimensional space
x
as fo
ll
ows:
|
ν
α
(
x
)
=
3
k
=
1
V
∗
αk
VV
e
i
p
k
x
|
ν
k
.
(
5.68
)
One ma
y
obtain the amplitude o
f
ν
α
→
ν
β
ν
o
scillations by projecting this
state o
n
to
|
ν
β
ν
at the detecto
r
w
i
th
x
=
L
.
T
h
e osci
ll
ation pro
b
a
b
i
l
ities are
P
(
ν
α
→
ν
β
ν
)
=
3
j
=
1
3
k
=1
V
αj
V
V
V
βk
V
V
V
∗
αk
V
V
V
∗
βj
V
V
e
i(
p
k
−
p
j
)
L
5.2 Neutrino
O
scillations and
Q
uantum
C
oherence 17
9
=
3
j
=
1
3
k
=1
V
αj
VV
V
βk
V
V
V
∗
αk
VV
V
∗
βj
V
V
e
xp
2
−
i
Δ
m
2
kj
2
E
L
6
.
(
5.69
)
This result is the same as the one given in Eq.
(
5.5
)
. If the same momentum
p
i
s assumed
f
or three neutrino mass ei
g
enstates, nevertheless, they must hav
e
d
ifferent energie
s
E
k
=
p
2
+
m
2
k
(
fo
r
k
=
1
,
2
,
3)
which can approximate t
o
E
k
=
p
+
m
2
k
/
(2
p
)
in the relativistic limit. In this case a given neutrino flavor
s
tate evo
l
ves on
ly
in time,
|
ν
α
(
t
)
=
3
k
=1
V
∗
αk
V
V
e
−
i
E
k
t
|
ν
k
,
(
5.70
)
a
nd the
p
robabilities of
ν
α
→
ν
β
ν
o
scillations turn out to b
e
P
(
ν
α
→
ν
β
ν
)
=
3
j
=
1
3
k
=1
V
αj
VV
V
βk
VV
V
∗
αk
VV
V
∗
βj
VV
e
−
i
(
E
k
−
E
j
)
t
=
3
j
=
1
3
k
=1
V
αj
VV
V
βk
VV
V
∗
αk
VV
V
∗
βj
VV
e
x
p
2
−
i
Δ
m
2
kj
2
p
t
6
.
(
5.71
)
Takin
g
account o
f
p
≈
E
m
k
a
n
d
t
≈
L
i
nt
h
e natura
l
unit s
y
stem
,
we find that the results obtain in Eqs.
(
5.69
)
and
(
5.71
)
are identical. An
i
mmediate question is why these two totally different assumptions lead to th
e
s
ame result. In
f
act, these two assumptions contradict each other and neithe
r
of them is correct. One ma
y
consider the deca
y
mod
e
π
+
→
μ
+
+
ν
μ
a
nd
a
nalyze its kinematics in the rest frame of the pion to demonstrate that th
e
s
ame ener
g
y or momentum assumption must be wron
g
.
S
uppose a neutrin
o
m
ass eigenstat
e
|
ν
k
w
ith the eigenvalu
e
m
k
is produced in this decay, then
i
ts momentum an
d
ener
g
yare
g
iven
by
p
2
k
=
m
2
π
−
m
2
μ
−
m
2
k
2
−
4
m
2
k
m
2
μ
4
m
2
π
,
E
2
k
=
m
2
π
−
m
2
μ
+
m
2
k
2
4
m
2
π
,
(
5.72
)
which satisfy
E
2
k
=
p
2
k
+
m
2
k
.
It is straightforward to verify that neither the
ener
g
ies nor the momenta are equal
f
or three neutrino mass ei
g
enstates with
n
on-degenerate masses.
T
he above paradox originates from the naive treatment of neutrino mass
ei
g
enstates as the plane waves. The reason is simply that particles are all
generated via the production processes occurring in a finite space volum
e
a
nd a finite time interval, which exclude the plane-wave description. As a
m
atter o
ff
act, particles in the production, propa
g
ation and detection are the
wave packets. In the wave-packet language the momenta of three neutrin
o
1
80 5 Phenomenology o
f
Neutrino
O
scillations
m
ass ei
g
enstates are continuous
l
y
d
istri
b
ute
d
an
dh
i
ghl
ypea
k
e
d
aroun
d
t
h
e
a
vera
g
ed values, so are the correspondin
g
ener
g
ies.
A
fter the elapsed time
t
a
n
d
at t
h
e
d
istance
L
from the source, the phase difference between the mas
s
e
ig
enstates
|
ν
k
a
n
d
|
ν
j
ν
r
eads
(
Akhmedov and Smirnov, 2009
)
Δ
φ
k
j
=
E
k
−
E
j
E
t
−
p
k
−
p
j
L
,
(
5.73
)
whe
r
e
E
j
E
=
p
2
j
+
m
2
j
a
n
d
E
k
=
p
2
k
+
m
2
k
h
o
ld
.Expan
d
ing t
h
eenerg
y
d
ifference in Eq.
(
5.73
)
in terms of the neutrino momentum and mass-square
d
d
ifferences, we hav
e
Δ
E
kj
≡
E
k
−
E
j
E
=
v
g
Δ
p
Δ
kj
+
Δ
m
2
kj
2
E
,
(
5.74
)
w
h
e
r
e
v
g
is the avera
g
e
g
roup velocity,
E
≡
(
E
k
+
E
j
E
)
/
2 denotes the avera
g
e
energy, an
d
Δ
p
Δ
kj
≡
p
k
−
p
j
represents the momentum difference. Substituting
E
q.
(
5.74
)
into Eq.
(
5.73
)
, we obtain the phase differenc
e
Δ
φ
k
j
=
Δ
m
2
k
j
2
E
t
−
(
L
−
v
g
t
)
Δ
p
Δ
k
j
.
(
5.75
)
A
lternatively, one may expand the momentum difference in Eq.
(
5.73
)
i
n
terms o
f
the ener
g
y and mass-squared di
ff
erences
:
Δ
p
Δ
k
j
=
1
v
g
Δ
E
kj
−
Δ
m
2
kj
2
p
,
(
5.76
)
whe
r
e
p
≡
(
p
k
+
p
j
)
/
2 is the average momentum. Inserting Eq.
(
5.76
)
into
E
q.
(
5.73
)
, one arrives at the phase difference
Δ
φ
k
j
=
Δ
m
2
k
j
2
p
L
−
1
v
g
(
L
−
v
g
t
)
Δ
E
k
j
.
(
5.77
)
It becomes evident that the phase difference in Eq.
(
5.75
)
can reduce to th
e
c
onventiona
l
resu
l
t
Δ
φ
kj
=
Δ
m
2
kj
t
/
(
2
E
)
in Eq.
(
5.71
)
in the same momentum
a
ssumption. Li
k
ewise
,
Δ
φ
k
j
i
nEq.
(
5.77
)
can reproduce the one in Eq.
(
5.69
)
i
n the same energy assumption. These two assumptions are unnecessary be-
c
ause the phase differences in Eq.
(
5.75
)
and
(
5.77
)
turn into the conventiona
l
o
n
es
i
ft
h
eco
n
d
i
t
i
on
L
=
v
g
t
i
s fulfilled
(
Akhmedov and Smirnov, 2009
)
.The
i
nter
p
retation of this condition is as follows: the center of the wave
p
acket
should be located a
r
ou
n
d the detecto
r
at the d
i
sta
n
ce
L
;
name
l
y
,
|
L
−
v
g
t
|
s
hould be smaller than the e
ff
ective width o
f
the wave
p
acket
σ
x
, which de-
pends on the production and detection of neutrinos. Beyond this range
(
i.e.,
|
L
−
v
g
t
|
σ
x
)
the neutrino wave functions will be highly suppressed, causing
the oscillator
y
terms i
n
P
(
ν
α
→
ν
β
ν
)
to disappear. One may therefore draw
two conclusions:
(
1
)
the same energy and same momentum assumptions are
5.2 Neutrino
O
scillations and
Q
uantum
C
oherence 181
u
nnecessary;
(
2
)
the conventional formulas of neutrino oscillation probabili-
ties are valid for relativistic neutrinos if the loss of
q
uantum coherence, du
e
to either the wave-packet separation or the localization of the source an
d
d
etector, is negligible
(
Cohe
n
et al
., 2009; Akhmedov and Smirnov, 2009
)
.
B
efore discussing the quantum coherence in neutrino oscillations, w
e
briefly review the interference phenomena and the concept of coherence i
n
optics
(
Bonn and Wolf, 1980
)
. In the famous Young’s double-slit experiment,
the light from a monochromatic source passes through two neighboring slits
a
nd finally arrives at the screen. It turns out that the light intensity on the
s
creen varies between the minimum and maximum
f
rom point to point. Th
e
i
nterference occurs due to the
p
hase difference which develo
p
s for the tw
o
l
i
g
ht beams
f
rom the slits with di
ff
erent optical paths. This is an ele
g
an
t
experiment to demonstrate the wave nature o
f
li
g
ht. However, any realis
-
tic light source is not exactly monochromatic. For instance, the excited gas
c
an emit li
g
ht whose spectrum consists o
f
bri
g
ht sharp lines. I
f
one o
f
thes
e
l
ines is input into the Michelson inter
f
erometer, one ma
y
observe the circular
f
ringes. It has been found that the visibility of the fringes decreases as th
e
optical path di
ff
erence between two inter
f
erin
g
beams increases.
S
uppose th
e
real li
g
ht is actually composed of wave trains, whose frequencies vary in th
e
range of
Δ
ν
a
roun
d
t
h
e average va
l
ue
ν
0
.
It can be shown that the duration of
the wave t
r
a
in
s
Δ
t
and the
f
requency ran
g
e
Δ
ν
f
ul
fi
ll the reciprocit
y
relation
Δ
t
Δ
ν
1
/
(4
π
)
, which is rather analogous to the Heisenberg uncertaint
y
relation in quantum mechanics. Let us define the coherence length as
(
Bon
n
a
nd Wolf, 1980
)
l
co
h
≡
c
Δ
t
∼
c
Δ
ν
=
λ
2
Δ
λ
,
(
5.78
)
w
h
e
r
e
c
is the speed o
f
li
g
ht in vacuum and
λ
i
stheavera
g
ewavelen
g
th
.
H
ence the circular fringes disappear if the difference of the optical paths
b
ecomes muc
hl
ar
g
er t
h
an t
h
eco
h
erence
l
en
g
t
h.
T
he
q
uantum coherence in neutrino oscillations seems more com
p
licated
t
h
an t
h
ec
l
assica
l
co
h
erence in optics. It is sti
ll b
eing
d
e
b
ate
d
t
h
at un
d
er
which conditions the coherence o
f
neutrino mass ei
g
enstates is lost,
f
or ex
-
a
mple, in the analysis of the GSI anomaly
(
Ivano
v
e
lal.
,
2008
;
Giunti
,
2008
;
L
ipkin, 2008
)
or the M¨ossbauer neutrino oscillations
(
Akhmedo
v
et a
l
.
, 2009;
Bi
l
en
ky
et al
.
,
2009
)
. But it becomes clear that the wave-packet approach
i
s helpful and probably unavoidable in solving such problems
(
Kayser, 1981
;
K
i
e
r
s
et al.
,
1996; Grimus and Stockinger, 1996; Campagne, 1997; Grimu
s
et
al.
,
1998; Shtanov, 1998; Giunti, 2004
)
and other paradoxes
(
Akhmedov an
d
Smirnov, 2009
)
. A formal treatment of neutrino oscillations has also bee
n
performed in the framework of quantum field theories
(
Cardall and Chung
,
1
999; Ioannisian and Pila
f
tsis, 1999;
C
ardall, 2000; Beuthe, 2002;
G
iunti,
2
002
;
Blason
e
e
tal., 2002
).
1
82 5 Phenomenology o
f
Neutrino
O
scillations
5.2.2 The Wave-packet Approac
h
N
ow we use the wave-packet language to describe neutrino oscillations. Le
t
u
ssupposet
h
at neutrinos are initia
ll
ypro
d
uce
d
via t
h
ewea
k
c
h
arge
d
-current
i
nteractions at the source.
A
fter propa
g
atin
g
adistance
L
d
urin
g
t
h
etim
e
T
, they arrive at the detector and are detected via the weak charged-current
i
nteractions. Analogous to the spectral line of the real light, the neutrino mass
ei
g
enstates s
h
ou
ld b
e represente
db
yt
h
ewavepac
k
ets. For simp
l
icity, we on
ly
c
onsider the one-dimensional s
p
ac
e
x
i
n the direction from the source to the
d
etector.
S
ince both the production and detection o
f
neutrinos are localized
i
na
fi
nite space-time volume, we expect that a neutrino
fl
avor ei
g
enstate is a
l
inear superposition of the wave packets of neutrino mass eigenstates. Mor
e
explicitly, a neutrino
fl
avor state at any tim
e
t
c
an be written as
(
Giunti an
d
K
im, 1998;
G
iunti
et al.
,
1993, 1998
)
|
ν
α
(
t
)
=
3
k
=
1
V
∗
αk
VV
d
pg
d
s
k
(
p
)
e
−
i
E
k
(
p
)
t
|
ν
k
(
p
)
,
(
5.79
)
whe
r
e
E
k
(
p
)=
p
2
+
m
2
k
is the neutrino ener
g
y
f
or each momentum mod
e
i
nthewave
p
acket. The distribution function in the momentum s
p
ace can b
e
taken to be
G
aussian:
g
s
k
(
p
)=
1
2
π
(
σ
s
p
)
2
1
/
4
e
x
p
−
(
p
−
p
k
)
2
4
(
σ
s
p
)
2
,
(
5.80
)
whe
r
e
p
k
i
s the average momentum for the mass eigenstat
e
|
ν
k
,an
d
σ
s
p
i
sthe
m
omentum width o
f
the wave packet. The superscript “s” reminds us o
f
th
e
wave packet at the source. Note that the neutrino mass eigenstates can hav
e
d
i
ff
erent avera
g
emomenta
p
k
a
n
d
ener
g
ies
E
k
≡
E
k
(
p
k
)
in this wave-packe
t
f
ormulation. In the coordinate s
p
ac
e
|
ν
k
(
p
)
=e
i
px
|
ν
k
holds, and thus th
e
n
eutrino wave function of the initial state read
s
|
ν
α
(
x, t
)
=
1
2
π
(
σ
s
x
)
2
1
/
4
3
k
=1
V
∗
αk
VV
e
x
p
i(
p
k
x
−
E
k
t
)
−
(
x
−
v
k
t
)
2
4
(
σ
s
x
)
2
|
ν
k
(
5.81
)
w
i
th
σ
s
x
b
eing t
h
ewave-pac
k
et wi
d
t
h
in t
h
ecoor
d
inate space.
σ
s
x
a
n
d
σ
s
p
a
re associate
d
wit
h
eac
h
ot
h
er via t
h
e uncertaint
y
re
l
atio
n
σ
s
x
σ
s
p
=
1
/
2. In
a
ddition, the group velocity of the wave packet is defined a
s
v
k
≡
d
E
k
(
p
)
d
p
d
p
=
p
k
=
p
k
E
k
.
(
5.82
)
So
E
k
(
p
)
≈
E
k
+
v
k
(
p
−
p
k
)
holds if the distribution function
g
s
k
(
p
)
is sharpl
y
pea
k
e
d
at
p
=
p
k
(
i.e.
,
σ
s
p
E
2
k
/
m
k
is satisfied
)
. The neutrino flavor stat
e
|
ν
β
ν
at the detector is also a su
p
er
p
osition o
f
the wave
p
ackets o
f
neutrin
o
m
ass eigenstates:
5.2 Neutrino
O
scillations and
Q
uantum
C
oherence 18
3
|
ν
β
ν
=
3
j
=1
V
∗
βj
V
V
d
pg
d
d
j
(
p
)
|
ν
j
ν
(
p
)
.
(
5.83
)
H
ere the distribution function
g
d
j
(
p
)
can also be taken to be Gaussian, and it
c
orrespon
d
stot
h
ewave-pac
k
et wi
d
t
h
σ
d
p
at t
h
e detecto
r. N
ote t
h
at t
h
eflavor
st
a
t
e
|
ν
β
ν
is to be experimentally identified at the detector, so we i
g
nore it
s
evolution in time. The corresponding wave function in the coordinate space
c
an be
fig
ured out in a similar way
:
|
ν
β
ν
(
x
−
L
)
=
1
2
π
(
σ
d
x
)
2
1
/
4
3
j
=1
V
∗
αj
V
V
e
x
p
i
p
i
j
(
x
−
L
)
−
(
x
−
L
)
2
4
(
σ
d
x
)
2
|
ν
j
ν
,
(
5.84
)
where
L
is the distance between the source and detector
,
and
σ
d
x
=
1
/
(2
σ
d
p
)
d
enotes the wave-packet width in the coordinate space. The amplitude o
f
ν
α
→
ν
β
ν
t
ransition is then
g
iven by the inte
g
ration o
f
ν
β
ν
(
x
−
L
)
|
ν
α
(
x
,
T
)
over the coordinat
e
x
.W
e obtain
A
α
β
(
L,
T
)
=
!
2
σ
s
x
σ
d
x
σ
2
x
3
j
=
1
V
∗
αj
VV
V
βj
VV
e
xp
i
p
j
L
−
E
j
E
T
−
(
L
−
v
j
T
)
2
4
σ
2
x
(
5.85
)
wit
h
σ
2
x
=(
σ
s
x
)
2
+(
σ
d
x
)
2
. The oscillation probabilit
y
P
αβ
P
(
L,
T
)=
|
A
α
β
(
L,
T
)
|
2
tu
rn
souttobe
P
αβ
PP
(
L
,
T
)
∝
3
j
=
1
3
k
=
1
V
αj
VV
V
βk
VV
V
∗
αk
VV
V
∗
βj
VV
e
x
p
i
(
p
k
−
p
j
)
L
−
(
E
k
−
E
j
E
)
T
−
1
4
σ
2
x
(
L
−
v
k
T
)
2
+(
L
−
v
j
T
)
2
)
.
(
5.86
)
H
owever
,
the arrival time of neutrinos at the detector has never been mea
-
s
ured in current neutrino oscillation experiments.
S
o one should avera
g
ethe
oscillation probabilit
y
over
T
.
The inte
g
ration over
T
is Gaussian too
,
and it
c
an be performed by finding the minimum of the argument of the exponential
f
unction. The minimum is achieved at the poin
t
T
=
(
v
k
+
v
j
)
L
v
2
k
+
v
2
j
−
2i
(
E
k
−
E
j
E
)
σ
2
x
v
2
k
+
v
2
j
.
(
5.87
)
Substituting Eq.
(
5.87
)
into Eq.
(
5.86
)
and imposing the normalization con
-
d
i
t
i
o
n
P
αe
PP
(
L
)+
P
αμ
PP
(
L
)
+
P
ατ
PP
(
L
)
= 1, we can get the oscillation probabilit
y
(
Giunti and Kim, 1998; Giunti, 2002; Giunti and Kim, 2007
)
P
αβ
PP
(
L
)
=
3
j
=
1
|
V
αj
VV
|
2
|
V
βj
VV
|
2
+
2R
e
j
<k
V
αj
VV
V
βk
VV
V
∗
αk
VV
V
∗
βj
VV
e
x
p
⎡
⎣
−
i
2
π
L
λ
kj
−
L
L
coh
k
j
2
1
84 5 Phenomenology o
f
Neutrino
O
scillations
−
2
π
2
(
1
−
ξ
)
σ
x
λ
k
j
2
⎤
⎦
,
(
5.88
)
whe
r
e
ξ
is defi
n
ed v
i
a
E
j
E
=
E
+
ξ
m
2
j
/
(
2
E
)
wit
h
E
b
ein
g
t
h
eavera
g
eener
gy
of neutrinos in the production process
(
whe
n
m
2
j
→
0
)
,
λ
k
j
≡
4
π
E
/
Δ
m
2
k
j
d
enote the neutrino oscillation lengths, an
d
L
c
oh
k
j
≡
4
√
2
σ
x
E
2
/
|
Δ
m
2
kj
|
s
tan
d
f
or the coherence lengths. Comparing between Eq.
(
5.88
)
and Eq.
(
5.5
)
,we
observe that there are two additional terms in the ex
p
onential in the wave-
p
acket a
pp
roach. When the condition
s
σ
x
λ
kj
an
d
L
L
co
h
kj
are satisfied
,
h
owever, Eq.
(
5.88
)
can reproduce Eq.
(
5.5
)
. Let us give some comments o
n
the physical implications of these two extra terms
(
Giunti and Kim, 2007
).
(
1
)
The coherence of neutrino mass eigenstates is determined by the size
s
of
L/
L
coh
k
j
a
n
d
σ
x
/λ
k
j
.
A
ll the current neutrino oscillation experiments satisf
y
t
h
econ
d
itions
σ
x
λ
kj
an
d
L
L
c
oh
kj
,
so the plane-wave treatment of
n
eutrino osci
ll
ations is accurate enou
gh
an
d
can
l
ea
d
to t
h
e correct resu
l
t.
(
2
)
I
f
L
L
c
oh
k
j
holds, the se
p
aration between different neutrino wave
packets is larger than the effective wave-packet widt
h
σ
x
.
In
th
i
scasethe
i
nter
f
erence amon
g
neutrino mass ei
g
enstates disappears and the oscillator
y
terms in Eq.
(
5.88
)
are exponentially suppressed, so the probabilities of neu-
trino oscillations are just given by the first term in Eq.
(
5.88
)
. A similar result
h
as been obtained in Eq.
(
5.6
)
, where the probabilities are averaged over the
n
eutrino energy spectrum and propagation distances
.
(
3
)
I
f
σ
x
λ
k
j
i
ssatis
fi
ed, we hav
e
σ
s
x
λ
k
j
o
r
σ
d
x
λ
kj
o
r
both.
In each case neutrino oscillations have alread
y
taken place within the sourc
e
or the detector. As a result, the oscillatory phenomena cannot be observe
d
because o
f
the exponential suppression.
(
4
)
It has been pointed out that the detection may restore the coherenc
e
of neutrino mass eigenstates
(
Kiers et a
l
., 1996
)
. This observation comes fro
m
the last term in the exponential in Eq.
(
5.88
)
. Suppose the energy resolution
becomes more accurate, then the momentum uncertaint
y
σ
d
p
b
ecomes smaller
.
In
th
i
s case
σ
x
g
ets
l
ar
g
er, so
d
oes t
h
eco
h
erence
l
en
g
t
hs
L
coh
k
j
∝
σ
x
.I
t
i
s
i
n
p
ractice difficult to observe such a restoration of coherence because of a
n
a
vera
g
eovert
h
eener
g
y reso
l
ution an
dd
etection position
.
5
.2.
3C
oherence of
C
osmic Neutrinos
If neutrino mass ei
g
enstates are incoherent at the detector, or if the detectio
n
of neutrinos has to average the oscillation probabilities over the neutrino en-
er
g
y spectra and propa
g
ation distances, the neutrino
fl
avor conversion is just
d
escribed by the classical probabilities as given in Eq.
(
5.6
)
. The detection o
f
n
eutrino oscillations depends both on the coherence of neutrinos and on the
experimental setup.
C
oncernin
g
the
fi
rst condition, one ou
g
ht to care
f
ully ex-
a
mine the
p
roduction
p
rocesses and
p
ro
p
erties of neutrino sources, and the
n
5.2 Neutrino
O
scillations and
Q
uantum
C
oherence 18
5
d
etermine the widths o
f
neutrino wave packets. The experimental setup
f
o
r
n
eutrino detection is also im
p
ortant.
A
swehavementioned,thearrivaltim
e
of neutrinos at the detector is not measured in current neutrino oscillatio
n
experiments. I
f
the ener
g
y resolution is so accurate that the neutrino mass
eigenstate arriving at the detector can be determined, one will be unable t
o
observe neutrino oscillations
(
Kayser, 1981
)
. In the following let us exam
-
i
ne the coherence o
f
neutrinos
f
rom various sources.
S
uch an anal
y
sisisver
y
c
rucial in the discussion of ultrahigh-energy cosmic neutrino oscillations
.
I
n the coordinate space the wave-packet width of neutrinos is
σ
x
,
w
h
ic
h
is
r
elated to the
m
o
m
e
n
tu
m
w
i
dth
σ
p
t
h
rou
gh
t
h
e uncertainty re
l
atio
n
σ
x
∼
σ
−
1
p
. On the other hand, the group velocity of the wave packet correspondin
g
to t
h
e neutrino mass ei
g
enstat
e
|
ν
k
i
s
g
iven
b
y
v
k
=
p
k
/E
k
.
Th
us a d
i
ffe
r
e
n
ce
between two
g
roup velocities o
f
neutrino mass ei
g
enstates read
s
Δ
v
k
j
=
p
k
E
k
−
p
j
E
j
E
=
Δ
m
2
j
k
2
E
2
,
(
5.89
)
where
Δ
m
j
k
≡
m
2
j
−
m
2
k
an
d
E
≡
(
E
k
+
E
j
E
)
/
2
. For simplicit
y
,weonl
y
con-
c
entrate on the order-of-magnitude estimation by considering two neutrino
flavors and omitting all the subscripts. Then Eq.
(
5.89
)
canbewrittena
s
Δ
v
=
Δ
m
2
/
(
2
E
2
)
. After a propagation distanc
e
L
,
the se
p
aration betwee
n
two wave pac
k
ets i
s
d
L
=
L
v
Δ
v
=
Δ
m
2
2
E
2
L,
(
5.90
)
w
h
e
r
e
v
is the avera
g
e
g
roup velocity. I
f
this separation si
g
ni
fi
cantly exceeds
the wave-packet width
(
i.e.,
d
L
σ
x
)
, there will be no overlap between the
wave packets o
f
two neutrino mass ei
g
enstates, implyin
g
that the coherenc
e
will be lost. With the help of Eq.
(
5.90
)
, one may convert the condition
d
L
σ
x
into the following form:
L
2
E
2
σ
x
Δ
m
2
∼
L
c
oh
,
(
5.91
)
where the coherence len
g
th is de
fi
ned a
s
L
c
o
h
≡
4
π
E
2
σ
x
/
Δ
m
2
.N
ote t
h
at t
h
e
c
oherence lengths defined in Eq.
(
5.88
)
are subject to the case of Gaussia
n
wave packets. The condition for the coherence loss in Eq.
(
5.91
)
is consisten
t
with the discussions below Eq.
(
5.88
)
,where
L
L
c
oh
leadstoahi
g
h sup-
pression of the oscillatory terms. For simplicity, we ignore the spread of th
e
wave packet caused b
y
the presence o
f
di
ff
erent momentum modes. Note also
that the wave-
p
acket width σ
x
i
n question is actuall
y
σ
s
x
,
the width associated
with the production mechanism at the source. Although the coherence ma
y
b
e restore
d
at t
h
e
d
etector wit
h
an exce
ll
ent ener
g
y reso
l
ution, an avera
ge
over the location of the source erases the oscillations. Hence we focus on the
production of neutrinos and the wave-packet width at the source.