This document analyzes the potential for overlap between neutrino wave packets emerging from different production processes in a source. It defines an "overlap indicator" to quantify the average number of wave packets a chosen one overlaps with. This indicator is calculated to be negligible for man-made sources like accelerators and reactors, assuming a sharp neutrino momentum distribution. Neutrino wave packet overlap could be significant for supernovae, but detecting any interference effects would be extremely challenging due to the weak nature of neutrino interactions.

1. Do Neutrino Wave Packets Overlap?
Cheng-Hsien Li & Yong-Zhong Qian
School of Physics and Astronomy, University of Minnesota
Objective
� To quantify the overlap among neutrino wave packets (WPs) emerging from
different production processes in the source.
Introduction
� The influence of one neutrino on another is generally considered negligible
unless in extreme environments with high neutrino number density such as
the interior of an exploding supernova or the early universe.
� In addition to actual physical interaction, the mutual interference among
neutrinos could possibly arise from the exchange symmetry of fermions,
which takes effect when the WPs overlap.
� Quantum particles are described as WPs, which encode their position and
momentum information. In the massless limit, neutrino WPs spread only in
the transverse direction as they evolve in time.
Figure 1: Schematic illustration of the spreading of a relativistic WP.
Methods
� Assumptions:
� Neglect neutrino mass and thus neutrino oscillations.
� Neutrino WPs are of simple Gaussian form with initial longitudinal width
al and transverse width at:
Ψ(�r, 0) =
1
(2π)3/4
ata
1/2
l
exp
�
−
ρ2
4a2
t
−
z2
4a2
l
+ ip0z
�
.
� The momentum distribution of the neutrino is sharp such that p0al � 1
and p0at � 1.
� Assume point-like source (see Discussions).
� Approach:
� Derive a 3D solution by making paraxial approximation to the wave
equation. The derived probability density in the far-field limit can be
expressed as |Ψ(�r, t)|2
= R(r, t)Θ(θ), where
R(r, t) ∝
1
r2
exp
�
−
(r − t)2
2a2
l
�
& Θ(θ) ∝ exp
�
−
θ2
2 · (2p0at)−2
�
.
� Define the 90%-probability volume as the neutrino ”size”. The volume
has a radial width τ = 4al and spans an angular size
θf ≈ 1.22 × (Eνat)−2
as shown in Fig. 2.
� Given a reference WP, consider its spatial overlap with other WPs of
similar energies; the energy difference is less than the intrinsic energy
uncertainty ΔEν ∼ 2/al in a WP.
� The spatial overlap requires WPs to be emitted within certain time
interval and in similar directions as shown in Fig. 3.
0 20 40 60 80 100 120
z/a
-60
-40
-20
0
20
40
60
ρ/a
0.000
0.001
0.002
0.003
0.004
t = 120 a
t = 80 a
2θf
4a
4a
Figure 2: Probability density at selected times
assuming at = al = a and Eνat = 10.
❒
Source
Figure 3: Illustration of overlap criteria.
Results:
� The overlap indicator η is defined to be the average number of WPs that a
chosen WP overlaps with:
η =
dΦ
dEν
× ΔEν × 2τ ×
ΔΩoverlap
ΔΩsource
≈
dΦ
dEν
×
96π
(Eνat)2ΔΩsource
.
� dΦ/dEν is the energy-differential production rate of the neutrino source.
� The dependence on al cancels and that on at remains. The initial widths
are unknown and theoretically difficult to compute.
Figure 4: 2D illustration of different degree of overlap (by varying unknown at).
� By naively assuming point-like sources, we use the typical energy-differential
rates in the following estimate:
Source ηpoint Distance Energy HBT
×(Eνat)2
Scale Scale Geometry
1. Accelerator 10−1
106
m GeV No
2. Reactor 100
103
− 105
m MeV No
3. The Sun 109
− 1019
1 AU 0.1 − 10 MeV No
4. Supernova 1032
− 1036
10 kpc 10 MeV Yes
Table 1: Estimate of different sources.
� Accelerator: NuMI LE configuration / Reactor: 3 GWth core.
Discussions
� Though subject to the unknown parameter at, the overlap indicator is
negligible for man-made sources provided that the neutrino momentum
distribution is sharp.
� One consequence of WP overlap is the Hanbury Brown and Twiss (HBT)
effect, which concerns (anti-)correlation in detector counting rate. The
effect requires confined production region (of size rp) and detection region
(of size rd) via rprdE/R � 1 [1]. Restricting the production region to
comply with HBT-geometry renders η negligible for sources 1-3 in Table 1.
Production Detection
a
b
c
d
� It is extremely difficult to observe HBT effect of neutrinos due to their
weakly-interacting nature. The overlap of WPs discussed in this work
should be of no concern to current experiments, unless there exists
non-trivial interference effect in detecting one of the overlapping neutrinos.
Reference: [1] U. Fano, Am. J. Phys. 29, 539 (1961).
Acknowledgments
This work was supported in part by the U.S. DOE
under DE-FG02-87ER40328. The authors gratefully
acknowledge the travel support from the Council of
Graduate Students, University of Minnesota and
the hospitality received from NuPhys2015.
Contact Information
� Email: cli@physics.umn.edu