1. A Pedagogical Discussion on Neutrino Wave Packet Evolution
CHENG-HSIEN LI & YONG-ZHONG QIAN
School of Physics and Astronomy, University of Minnesota at Twin Cities
Introduction
After OPERA Collaboration reported their false superluminal neutrino results in 2011, the
authors of [1] claimed that the off-axis neutrinos may be detected earlier than those traveling
along the axis due to the spreading of neutrino wave packet in the transverse direction. When
the edge of the wave packet reaches the detector, the distance traveled by the center of the
wave packet is shorter than the source-detector distance, and therefore a superluminal effect
is observed as depicted below.
An off-axis neutrino may be detected due to spreading in the transverse direction.
The argument was soon refuted by [2] which pointed out that the neutrino wave packet will
asymptotically evolve into a spherical wave front with Gaussian distribution. The authors
of [2], however, treated neutrinos as a continuous flux, instead of discrete wave packets, and
obtained a classical electromagnetism-like solution for the Gaussian beam (no localization in
the propagation direction). They inferred the spherical wave front indirectly from the phase
term in the solution.
For pedagogical purpose, we revisit the time evolution of a Gaussian wave packet with
a sharply peaked momentum distribution. We extend the solution commonly found in
literature by including higher order terms in the energy expansion. The spherical shape of
the wave packet will be recovered after higher order terms are considered.
Methods
Assumptions
• The mass and spin of the neutrino are neglected. Natural units are adopted.
• The initial state (at t = 0) of the neutrino is an isotropic Gaussian wave packet with spatial
width σ0 (⇒momentum width 1
2σ0
) in each direction and average momentum k0 = k0 ˆz.
• The momentum distribution is sharply peaked, i.e. k0σ0 ≫ 1.
Procedures
1. Shift the integration variable k → k + k0 so that the dispersion relation can be expanded
around k = 0:
ωk+k0
= k0





1 +
kz
k0
+
k2
⊥
2k2
0
−
kzk2
⊥
2k3
0
+
4k2
z k2
⊥ − k4
⊥
8k4
0
+ · · ·
commonly neglected in literature





, k2
⊥ ≡ k2
x + k2
y. (1)
2. Keep terms up to quadratic order in the exponent and expand those beyond by eiǫ
=
1 + iǫ + O(ǫ2
):
Ψ(r, t) =
d3
k
(2π)3
Momentum distribution
(8πσ2
0)
3
4 e−σ2
0k2
Plane wave
eik·r+ik0z
Quadratic phase
e−ik0t(1+kz/k0+k2
⊥/2k2
0)
× 1 +
it
2k2
0
kzk2
⊥ −
it
8k3
0
4k2
z k2
⊥ − k4
⊥ + higher order
Polynomial of ki
. (2)
3. Correction terms can be efficiently computed by the following identity:
∞
−∞
dk kn
e−a(k−b)2+c
∞
−∞
dk e−a(k−b)2+c
=
n
i=0
n
i
bn−i
×
0 , if i is odd.
(2a)−i/2
(i − 1)!! , if i is even.
(3)
The complex coefficients a, b, and c are obtained from completing the square of the expo-
nent.
Results
While in principle the wave function can be calculated to any desired order, we present here
only the very first order correction from (2). The zeroth order solution [3] is also presented for
reference:
Ψ(0)
(r, t) =
1
(2πσ2
0)
3
4 (1 + it
2k0σ2
0
)
exp −
x2
+ y2
4σ2
0(1 + it
2k0σ2
0
)
−
(z − t)2
4σ2
0
exp {ik0(z − t)}
Ψ(r, t) ≈ Ψ(0)
(r, t)



1 −
t
4k2
0σ3
0
×
(z − t)
σ0 1 + it
2k0σ2
0

1 −
x2
+ y2
4σ2
0 1 + it
2k0σ2
0





. (4)
The leading correction term is proportional to (z − t), which gives rise to the asymmetry
between the space-like and time-like regions and ”bends” the flat Gaussian distribution. A
graphical comparison of the probability contours of Ψ(0)
(r, t) and Ψ(r, t) at different time is
made in the following.
3985 3990 3995 4000 4005 4010 4015
40
20
0
20
40
z Σ0
xΣ0
t 4000 Σ0
7985 7990 7995 8000 8005 8010 8015
100
50
0
50
100
z Σ0
xΣ0
t 8000 Σ0
11 985 11 990 11 995 12 000 12 005 12 010 12 015
150
100
50
0
50
100
150
z Σ0
xΣ0
t 12000 Σ0
15 985 15 990 15 995 16 000 16 005 16 010 16 015
200
100
0
100
200
z Σ0
xΣ0
t 16000 Σ0
The blue and red are respectively the 2σ contour plot of the probability density calculated from
Ψ(0)
(r, t) and Ψ(r, t). The gray dashed curve depicts the circumference of radius t. These plots are
made by assuming k0σ0 = 100 and y = 0. The horizontal scale is uniform for all figures.
Discussions
• The wave packet is generated with some position uncertainty. Therefore, the apparent
overlap of the wave packet (either before or after correction) with the space-like region
must not be interpreted as a superluminal effect. A wave packet with the correct geome-
try always obeys special relativity within its intrinsic position uncertainties.
• Significant deviation of Ψ(r, t) from Ψ(0)
(r, t) in the probability contour appears at larger
time. The wave packet conforms to a spherical wave front after correction.
• The fact that the convergence radius of the complex exponential function is infinitely
large ensures the existence of the solution of (2). An additional criteria must be satisfied
to guarantee that (4) is a good approximation. That is, the first correction term in (2) gives
the largest contribution among all correction terms when
k0t ×
kzk2
⊥
k3
0
>
1
n!
k0t ×
kzk2
⊥
k3
0
n
for all n ≥ 2 =⇒
t/σ0
(k0σ0)2
2. (5)
• This perturbative method is limited by the above criteria. Higher order corrections must
be included when the propagation time exceeds ∼ 2k2
0σ3
0. However, this work demon-
strates that higher order terms change the shape of the wave packet.
References
1. D. V. Naumov, V. A. Naumov, arXiv:1110.0989 (2011).
2. D. Bernard, arXiv:1110.2321 (2011).
3. See for example, C. Giunti, C. W. Kim, U. W. Lee, PRD 44 3635 (1991).
Acknowledgement
This work was supported in part by the U.S. DOE under DE-FG02-87ER40328.