Journal of Modern Physics, 2012, 3, 1398-1407
http://dx.doi.org/10.4236/jmp.2012.310177 Published Online October 2012 (http://www.SciRP.org/journal/jmp)
Multiple Lorentz Groups—A Toy Model for
Superluminal Muon Neutrinos
Institute for Theoretical Physics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
Received July 23, 2012; revised August 31, 2012; accepted September 9, 2012
In this article an idea is presented, which allows for the explanation of superluminal muon neutrinos. It is based on the
introduction of a new superluminal, massless gauge boson coupling to the neutrino only, but not to other standard mod-
el particles. The model is discussed with regard to the Supernova 1987 (SN 1987) velocity bound on electron antineu-
trinos and the Cohen-Glashow constraint on superluminal neutrino propagation. The latter can be circumvented if—
within the framework of the model—a sterile neutrino mixing with the active neutrino mass eigenstates is introduced.
The suggestion of a sterile neutrino accounting for superluminal neutrinos has already been proposed in several papers.
It is possible to choose mixing angles with the sterile neutrino sector such that the model respects both the SN 1987
bound and the muon neutrino travels superluminally.
Keywords: Special Relativity; Dispersion Relations; Neutrino Interactions; Non-Standard-Model Neutrinos
At the end of September 2011, the OPERA collaboration
announced the remarkable detection of superluminal
muon neutrinos at the 6.2
-level . Their experimen-
tal result with statistical and systematical error was given
0.24 10 .
Unfortunately, on February, 2012 two error sources
had become evident, which were likely to ruin their re-
sult. First, a fiber connection to a computer card had not
been attached properly. Second, there had been a prob-
lem with the clock at OPERA used between the synchro-
nizations with the Global Positioning System.
At the 25th International Conference on Neutrino
Physics and Astrophysics in Kyoto on June 8th, 2012 a
final update on the OPERA time-of-flight measurement
This number states the deviation of the muon neutrino
time-of-flight from the time that light needs to travel the
distance from CERN to the Gran Sasso underground la-
boratory. Hence, the deviation is now consistent with
The physics community had considered the result giv-
en by Equation (1) with care, since the deviation from the
speed of light lay several orders of magnitude above
what would be expected, if it was from quantum gravita-
tional origin. The authors of [2-7] tried to figure out how
the OPERA result could be explained by possible sys-
tematical errors. Beyond that, in  a cross-check for the
result was proposed. It was demonstrated that muon neu-
trinos traveling with superluminal velocity can produce
signatures for highly-boosted tt-quark pairs at the LHC,
where one or both quarks decay semileptonically.
Furthermore, on the one hand, Cohen and Glashow
showed that a superluminal neutrino would lose its en-
ergy quickly by the emission of electron positron pairs
. If muon neutrinos moved faster than light, the proc-
would be energetically possible above
a certain neutrino energy threshold resulting in a copi-
ous production of electron positron pairs. On the other
hand, the authors of [10,11] discussed that the Cohen-
Glashow constraint can be avoided. This may be the
case when, for example, Lorentz-violating effects de-
pend quadratically on the neutrino energy or if Lor-
entz-violation is not fixed but covariant with the neu-
In  two models were investigated, where the first
gave rise to deformed energy conservation laws and the
second resulted in deformed momentum conservation
laws. For these models the bounds of  are not applica-
1These numbers can be found in the updated version 2.0 of .
opyright © 2012 SciRes. JMP
M. SCHRECK 1399
In the article  further constraints on the deviation
of the neutrino velocity from the speed of light were
given by considering pion decay and TeV-neutrinos de-
tected by ICECUBE. These gave severe bounds on Lor-
entz symmetry violation in the neutrino sector clashing
with the experimental OPERA result.
Although the OPERA result has now proven to be
wrong, it stimulated theoretical ideas in this field and led
to many new models, which can perhaps be applied to
other realms of physics. It may also be the case that a
certain neutrino species indeed travels superluminally.
However, the deviation from the speed of light is then
expected to be much smaller than the value measured by
OPERA. For this reason some representative examples
for models that try to describe superluminal neutrinos
will be listed:
It is well-known that dispersion relations of particles
will be modified, if they propagate through a medium.
In [14,15] the superluminal motion of muon neutrinos
is interpreted in the framework of deformed disper-
sion relations, which are a low-energy manifestation
of Lorentz-violating physics at the Planck scale. In
these theories the vacuum behaves as an effective
Such a medium can result from standard model phys-
ics, as well. For example, in  superluminal neu-
trinos are explained by the assumption that Earth is
surrounded by a special kind of matter consisting of
separated quarks. If the wave functions of quarks are
entangled, they can form colorless objects and, hence,
are confined, even when they are spatially separated
by a large distance.
In  a model is proposed which describes a spon-
taneous breakdown of Lorentz symmetry by a scalar
background field that is added to the action via a La-
grange multiplier. This framework leads to a modified
neutrino dispersion relation depending on the mo-
mentum of the neutrino.
The neutrino velocity can be modified by Fermi point
splitting (for a recent review see ), which re-
moves the degeneracy of zeros of the fermionic en-
ergy spectrum .
The neutrino dispersion relation can change because
of environmental effects caused by fields that accu-
mulate at the position of the Earth. These may lead to
an effective metric, in which the neutrino propagates
with superluminal velocity [20,21]2. Furthermore, in
the context of general relativity, a particle traveling
along a geodesic path in a metric different from the
Minkowski metric can be investigated . In the ar-
ticle previously mentioned the mean velocity of such
a particle is calculated with the assumption that the
observer stays at rest. The average velocity can be
larger than the speed of light, even if the velocity as a
local property defined in a spacetime point is smaller.
This is investigated for a Schwarzschild metric.
A further alternative is to consider models of modi-
fied gravity. In the article  particle propagation in
a Hořava-Lifshitz modified gravitational background
is considered. The authors derive the Dirac equation
for a fermion traveling through such a background.
The condition for the existence of nontrivial solutions
of the Dirac equation leads to a modified neutrino
dispersion relation. The neutrino velocity can be lar-
ger than the speed of light for a special Hořava-Lif-
In  the existence of a sterile neutrino that travels
with a superluminal velocity is proposed. Sterile neu-
trinos cleverly get around the Cohen-Glashow bound,
since they do not couple to the Z boson. An analysis
involving sterile superluminal neutrinos is presented
e.g. in .
Furthermore, assuming superluminal neutrino propa-
gation at a certain energy, neutrinos may propagate with
a velocity at very high energies leading to a dif-
ferent neutrino horizon. In  bounds from astrophysi-
cal observations are set on
vc for very high neutrino
2. Extension of the Lorentz Group—
Neutrinos and a Hidden Sector
Let us, for now, take Equation (1) as it stands, since the
idea of superluminal muon neutrinos proposed in the
current article is purely theoretical and does not rely on
the OPERA result. The goal is to describe a superluminal
neutrino species without quantum gravity effects, but by
the introduction of new particles coupling to neutrinos.
For the analysis presented as follows the OPERA value
can be chosen just as an example (independently of its
correctness) in order to demonstrate the model proposed.
That is why we will often refer to Equation (1) in the rest
of the paper.
2.1. Invariant and Maximum Velocity
The foundations of special relativity are the relativity
principle and the constancy (invariance) of the speed of
light. As a result, the Galilei group of classical mechanics
is replaced by the Lorentz group, which leads, for exam-
ple, to the relativistic law of addition of velocities. The
fact that the speed of light is the maximum attainable
velocity of all particles does not directly follow from the
Lorentz group, since it only delivers an invariant velocity
at first. In order to understand this, three Gedankenex-
periments will be performed, whose concept was initi-
2Reference  gives new experimental bounds on the mass scale M*
that is characteristic for the model presented in .
Copyright © 2012 SciRes. JMP
ated in .
1) We consider some hypothetical beings living in a
fluid. They are assumed to consist of fluid atoms, which
are held together solely by phonon-mediated forces. The
beings do not feel other forces such as electromagnetism
or gravitation. Their dynamics is expected to be governed
by an acoustic Lorentz group with the invariant velocity
being the velocity of sound
c in the fluid . We
expect the beings to build a rocket, which can be acceler-
ated by phonon emission. From the relativistic addition
of velocities it follows that the maximum attainable ve-
locity of the rocket is given by
2) Einstein found that the dynamics of particles in our
universe is governed by the “standard” Lorentz group
with the invariant speed of light c. His theory has been
heralded or substantiated by various experiments [29-31].
In the second Gedankenexperiment humans build a rock-
et which is accelerated by a light engine, namely the
emission of photons. In this case, the relativistic addition
of velocities leads to c as the upper limit of the rocket
3) Now we are ready to discuss the central idea of this
article. The basic assumption is that the photon is not the
gauge boson which moves with the highest velocity. We
adopt neutrinos carrying a new charge differing from
all charges of the standard model. This charge is to be
mediated by a postulated massless gauge boson
moving with a speed . Neutrinos couple to these
gauge bosons, which form—possibly together with other
unknown particles—a hidden sector. The latter does not
interact with any other particle of the standard model, cf.
Figure 1. This leads to a neutrino dynamics which is
based on a “hidden sector Lorentz group” with an in-
variant velocity . The third Gedankenexperiment is to
build a rocket consisting of neutrinos with an accelera-
tion process working by the emission of gauge bosons
The limiting velocity of the rocket is then given by ,
which is larger than the speed of light.
The consequences from this deliberation is that differ-
ent levels of the Lorentz group each with a distinct in-
variant velocity can be realized in nature. The coupling
constant is assumed to be small enough such that neu-
trino propagation is not affected too much to violate ex-
isting bounds on the interaction of neutrinos with matter.
The size of the coupling is not important for now—only
its existence. The situation described is depicted in Fig-
ure 2. Each new coupling of particles to massless gauge
bosons3 opens a new sector with a maximum attainable
velocity for these particles from left to right.
2.2. Modified Neutrino Kinematics and
Lagrangian of the Hidden Sector
The gauge boson
is assumed to couple to neutrino
mass eigenstates. This seems a more natural choice than
the coupling to flavor eigenstates, since the hidden sector
does not know anything about neutrino flavors. Hence,
the neutrino mass eigenstates i obey a
kinematics resting upon special relativity, but with the
speed of light replaced by the speed of the hidden
sector gauge boson
is the relativistic neutrino energy of the i-th
mass eigenstate, p
the neutrino momentum, and i
its mass4. The modified neutrino dispersion relation in
Equation (3) is isotropic and gives the deformation.
If we suppress the mass eigenstate index i for a moment
and interpret neutrinos as matter waves with frequency
and three-momentum k, we have to carry out the re-
, which leads to:
The front velocity, which corresponds to the velocity
of the highest frequency forerunners of a wave, is then
given by :
fr, lim .
It equals the signal velocity of a
pulse in configuration space. Hence, any possible distor-
tion of a signal does not play a role for the front velocity.
The case is related to superluminal, >1 1
luminal, and to subluminal motion.
If the gauge boson
is assumed to have spin 1 ana-
logously to the photon, at the level of Lagrange densities
the ordinary minimal coupling procedure can be per-
formed with c again replaced by :
4We assume that all mass eigenstates propagate with the same momen-
tum p. Whenever we refer to kinematics we stick to the notation of m
with an index (denoting the mass or flavor eigenstate) for the neutrino
3Remark that the phonon is not a gauge boson, but a Goldstone boson
(massless excitation) resulting from the spontaneously broken transla-
tion symmetry in a solid. This does not matter for the argument, though.
Copyright © 2012 SciRes. JMP
12 SciRes. JMP
Figure 1. Hidden sector that decouples from all the standard model particles except the neutrino. The neutrino is assumed to
carry a charge, which massless hidden sector gauge bosons ˆ
couple to. These move with a velocity that is larger than
the speed of light c.
Figure 2. Illustration of the situation presented in the previous three Gedankenexperiments. The horizonal axis shows differ-
ent sectors, each containing a massless boson which transforms under the Lorentz group with a special invariant velocity.
The velocity of this boson sets the maximum attainable velocity for all particles coupling to the corresponding sector. In the
first sector the maximum velocity is given by the speed of sound cs of phonons. Note that the phonon takes a special role here,
since it is not a gauge boson. However, for the very general argument this is not of importance. In the second sector the pho-
ton sets the upper limit, which manifests itself as the speed of light c, whereas for the hidden sector it is the velocity of
. There is the possibility of further sectors whose invariant velocity may also be smaller than c. If in the latter
case a standard model particle couples to such a sector, its kinematics will still be governed by a Lorentz group with the in-
variant velocity c.
Copyright © 20
,, ,,Dirac Li RiRi Li
are the neutrino spinor fields de-
scribing a specific mass eigenstate and
standard Dirac matrices. The covariant derivative ˆ
contains the vector field ˆ
of the gauge boson ˆ
the charge , to which
couples. Both a Dirac mass
term and two possible choices for Majorana mass terms
 with Dirac mass
and Majorana masses 1
are given5. Here,
is a left-handed,
handed neutrino spinor, and C in Equation (12) denotes
the charge conjugation operator.
Note that there is a in the zeroth component of the
. In the context of the Lorentz-vio-
lating Standard Model Extension  the nonzero Lor-
entz-violating coefficients can be found in the left-
handed neutrino sector of lept on
in their Equation
(9).6 If we write this term in the mass eigenstate basis
, the and denote the corresponding coefficients by
-matrix is both diagonal in the eigenstate coeffi-
cients i, j and diagonal in the Lorentz indices. The latter
holds, since the model is isotropic. This leads to
The coefficient matrix
is both symmetric and
This resembles the CPT-even nonbirefringent modi-
fied Maxwell theory coefficients
in the photon
6Alternatively, the effective Hamiltonian given by Equation (14) in 
can be considered.
5The question, whether the neutrino is a Dirac or a Majorana particle,
has not been answered so far.
sector [36,37], which is clear, since both sectors are re-
lated by a coordinate transformation—at least at first
order in the Lorentz-violating coefficients .
2.3. Extension of the Toy Model to Three
The neutrino masses i are eigenvalues to the mass ei-
. However, the weak interaction gauge bos-
ons couple to flavor eigenstates
The transformation from mass to flavor eigenstates and
(vice versa) is governed by the unitary
When, for simplicity, the CP-violating phases are set
to zero7, the matrix U reads
23 23 1313
12 1312 13
12 2312 231312 2312 23
12 2312231312 231223
00sc s c
sc csscc ss
ss ccscs sc
where, for brevity,
and cos ij
used . Here, ij
the neutrino mixing angles. Kinematik measurements
of neutrino masses (e.g. for pion decay and beta decay8)
lead to “masses of neutrino flavors,” which are the
weighted average of the neutrino mass eigenvalues
Since in Equation (3) the neutrino mass eigenvalues
i are multiplied by , the maximum velocity of each
neutrino flavor will be defined in the following way:
=1 ,2, 3
If we assume
1, ,cccc c
it is sufficient to linearize the equations above:
. From the latter equation follows
the simplified result
1,2 , 3
In the rest of the article the neutrino velocities will be
approximated by i for the mass eigenstates and by
for the flavor eigenstates, since neutrino masses are
assumed to be much smaller than neutrino energies.
The Supernova 1987 Bound
The point of extending the toy model to all active neu-
trino flavors, is to account for the Supernova 1987 (SN
1987) bound on electron antineutrinos9 [40,41]:
An electron neutrino produced by a weak interaction
process is a mixture of neutrino mass eigenstates accord-
ing to the relation
After the neutrino has traveled through space and
reaches a distance L from the origin of its production it
1,2 , 3
where i has been assumed . Hence, the ini-
tial electron neutrino state then corresponds to a mixture
of all flavor eigenstates. However, the initial composition
of mass eigenstates remains the same, because quantum
9An antineutrino is assumed to travel with the same velocity as the
corresponding neutrino. This makes sense, since the model presented
corresponds to a CPT-even term of the Lorentz-violating Standard
Model Extension. See the end of Secition 2.2 for a brief discussion
concerning this issue.
7Furthermore, currently no experimental data concerning these phases
are on hand .
8Neutrinoless double beta decay that occurs for Majorana neutrinos
leads to a different definition of the “flavor eigenstate mass” .
opyright © 2012 SciRes. JMP
M. SCHRECK 1403
mechanically the statement
1, 2 ,3
is valid. As a result of that, also the velocity of the neu-
trino does not change during its propagation, since it is
determined by the initial composition of mass eigenstates.
The antineutrinos coming from the supernova were de-
tected as electron antineutrinos on Earth. For this reason
the bound of Equation (28) will be considered as a bound
on the velocity of electron neutrinos—regardless of
whether their flavor was different on their way to Earth.
We assume three distinct hidden sectors each with its
own gauge boson
, where 1
only couples to the first
mass eigenstate, 2
to the second, and 3
to the third,
via the respective charge for , 2, and 3, respec-
tively. If any sector obeys a different invariant velocity
, the constraint of Equation (28) does not
necessarily contradict a deviation from the speed of light
of the order of for one single neutrino flavor. This
will be shown as follows.
The current experimental values or bounds for the
three neutrino mixing angles 12
, and 13
With the lower bound on 23
and the upper bound on
we obtain the PNMS matrix
Current experimental data imply that neutrinos are al-
most massless. Concretely, this means ,,
from neutrino oscillation data  and
(95% CL),10 (36)
which is obtained from WMAP observations . There-
fore, an approximate value of ˆ
directly follows from
and we obtain11:
5.30 10,1.13 10
Hence, Equation (1) for muon neutrinos and the SN
1987 bound for electron neutrinos do not clash, if the
velocity of the first mass eigenstate is a little bit lower
than c and if the second moves faster than c. Since the
first eigenstate propagates slower than c, it need not nec-
essarily couple to any hidden sector. In contrast, the sec-
ond mass eigenstate has to couple to a
faster than light.
2.4. Challenges of the Model and Introduction of
The argument of  resulting in the rapid loss of the
neutrino energy by electron positron emission relies on
fundamental principles: four-momentum conservation
and the coupling of the neutrino sector to the Z boson.
Models for superluminal neutrino propagation have to
compete with the very general result mentioned, and this
is also the case for the toy model presented here.
1) We could assume the energy loss of muon neutrinos
to be compensated by a Compton scattering type process,
where gauge bosons ˆ
scatter with muon neutrinos.
However, this argument leads to additional problems.
First of all, the free parameters of the model (e.g. the
charge or the initial energy of a
boson) have to
be chosen such that this compensation is possible, which
requires extreme finetuning. If the momentum distribu-
tion of ˆ
is homogeneous and isotropic, the average
energy transfer to the neutrino will be zero. In principle,
the distribution may be anisotropic, but then neutrinos
might be deflected on their way from CERN to the Gran
Sasso underground laboratory.
2) An alternative proposal is that a neutrino itself is
part of the hidden sector making it to some kind of su-
perluminal, sterile neutrino
. Then the neutrino does
not couple to the Z boson, rendering the process
forbidden. The sterile neutrino may mix
with the active neutrino species leading to superluminal
propagation of at least some of the standard model neu-
trino flavors. This idea has already been suggested in
other publications, see e.g. [25,43] and references therein.
Reference  states that sterile neutrino models may be
in conflict with the atmospheric neutrino data measured
at Super-Kamiokande. However, the models considered
in the latter article only involve one sterile neutrino and
one single mixing angle with this neutrino. Conclusions
10Note that under the assumptions taken, the unit 2
eV cshould be
replaced by 2
eV c, as well. But since the mass values given have
een obtained in the context of special relativity, where the speed o
light c is the invariant velocity, we keep c.
11The latter choice is reasonable, since 5
according to Equations (1) and (28). We keep in mind that the con-
straints on e
were obtained at different neutrino energies,
but this does not play a role in our model, though.
Copyright © 2012 SciRes. JMP
for models with more mixing angles have not been ob-
According to the second item of the list above, we ex-
tend the toy model by
sterile neutrino mass eigen-
states. Then the transformation between the
vor and mass eigenstates is governed by a unitary
Following Equation (2.3) we can write:
For our toy model we consider the simplest case with
one single sterile neutrino, hence . In principle,
this sterile neutrino mass eigenstate 4
mixes with the
active neutrino mass eigenstates i
This mixing can be described by introducing three addi-
tional mixing angles 14
, and 34
. The corre-
sponding -mixing matrix
U can then be con-
structed from U as follows12:
0 1 0 0
10 0 1 0
1 0 0 0
0 0 1 0
0, 0, 00
where . In what follows, we examine a
subspace of the free toy model parameters, which is
seven-dimensional. It is spanned by the three sterile
neutrino mixing angles 14
and by the in-
variant velocities of the neutrino mass eigenstates 1,
2, 3, 4. Since this phase space is that large, it will
be reduced by the special choice below. We assume that
the invariant velocities of the three standard neutrino
mass eigenstates correspond to the speed of light, which
means 123 . The single sterile neutrino is
assumed to travel with superluminal speed: we therefore
As a result, only the sterile mixing angles remain as
free parameters. In Figure 3 three cases are considered,
where in each one of these angles is fixed: 14 π3
, and 34 π5
. We would like to explore,
whether in each case the remaining two angles can be
chosen such that the electron anti-neutrino velocity re-
spects the SN 1987 bound of Equation (28) and that the
muon neutrino velocity lies in the error band of Equa-
tion (1). In all plots overlapping regions are small, but
they exist. At least one of the three mixing angles has to
be rather large. A special possibility is 14 5π4
, and 34 π5
that becomes evident from the
third panel. With these values we obtain the following
results for the velocities of the three active neutrino fla-
Whether there exists a choice of angles that does not
contradict existing atmospheric neutrino data—as was
proposed in —will not be examined here.
To summarize, within the toy model presented a su-
perluminal sterile neutrino mass eigenstate can be intro-
duced, such that the electron neutrino respects the SN
1987 bound and the muon neutrino travels with the su-
perluminal velocity that is given by Equation (1). For the
parameters chosen above the tau neutrino is then slightly
superluminal, as well.
In this article a concept accounting for superluminal
muon neutrinos was presented. It is based on a multiple
Lorentz group structure. The dynamics of the neutrino is
assumed to obey the Lorentz group with an invariant
velocity that is larger than the speed of light. This will be
possible, if the neutrino couples to a hidden sector of
massless gauge bosons that move faster than photons.
Then the neutrino field transforms under the Lorentz
group with an invariant velocity which corresponds to
the velocity of these gauge bosons.
If an experiment measures a deviation of the neutrino
velocity that is much larger than the speed of light, this
will be very difficult to understand in the context of
physics at the Planck scale. The idea presented here leads,
in principle, to a modified dispersion relation of the neu-
trino, as well. However, the framework is not quantum
gravity, but special relativity and field theory with an
invariant velocity imposed that differs from the speed of
First of all, every physical model describing superlu-
minal muon neutrinos has to compete with the SN 1987
bound. This is a minor difficulty, since the toy model
presented here can be altered such that electron neutrinos
behave differently compared to muon neutrinos. More
severe is the Cohen-Glashow constraint that is based on
fundamental principles of present-day physics, which are
12With all CP-violating phases set to zero.
opyright © 2012 SciRes. JMP
Copyright © 2012 SciRes. JMP
(a) (b) (c)
Figure 3. Each panel shows the plane of a different pair of sterile neutrino mixing angles, where the remaining angle is set
to the special value given below the corresponding panel. Regions of the electron and the muon neutrino velocity are
shown, where , and
is the common choice. The blue areas depict the region
and the green areas show the region
12.370 320 24101 8110cc
. The condition
..0 320 3410
1 2.37cc .
is fulfilled for all possible mixing angles in each panel, so is
. (a) 14π3
; (b) 24 π5
; (c) 34 π5
difficult to circumvent. Honestly, the latter is also a se-
vere problem for the current model, unless the superlu-
minal neutrino itself is part of a hidden sector, hence ste-
Besides that, the toy model makes the following pre-
dictions that can be verified or falsified by experiment:
If the muon neutrino moves with a superluminal ve-
locity, its velocity is isotropic and does not depend on
the neutrino energy (besides any mass-related de-
It is not a local effect, i.e. muon neutrinos move with
a superluminal velocity in interstellar space, as well.
To find a physical theory both explaining superluminal
neutrinos without bothering already established data and
facts about the neutrino sector is a great challenge for
It is a pleasure to thank S. Thambyahpillai (KIT) and A.
Crivellin (University of Bern) for reading an early draft
of the paper and for useful suggestions. Furthermore, the
author thanks V. A. Kostelecký (Indiana University,
Bloomington) for helpful discussions. The author ac-
knowledges support by Deutsche Forschungsgemein-
schaft and Open Access Publishing Fund of Karlsruhe
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