Multiple Lorentz groups  a toy model for superluminal muon neutrinos
Abstract
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 model particles.
The model is discussed with regard to the Supernova 1987 (SN 1987) velocity bound on electron antineutrinos 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.
pacs:
03.30.+p, 11.55.Fv, 13.15.+g, 14.60.StI Introduction
At the end of September 2011, the OPERA collaboration announced the remarkable detection of superluminal muon neutrinos at
the level OPERA:2011zb (). Their experimental result with statistical and systematical error was given by:
(1.1) 
Unfortunately, on February, 2012 two error sources had become evident, which were likely to ruin their result. First, a fiber connection to a computer card had not been attached properly. Second, there had been a problem with the clock at OPERA used between the synchronizations 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 timeofflight measurement was given:
(1.2) 
This number states the deviation of the muon neutrino timeofflight from the time that light needs to travel the distance from CERN to the Gran Sasso underground laboratory. Hence, the deviation is now consistent with zero.
The physics community had considered the result given by Eq. (1.1) 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 gravitational origin. The authors of Contaldi:2011 (); Besida:2011fi (); vanElburg:2011ze (); Alicki:2011qw (); Henri:2011wu (); Broda:2011du () tried to figure out how the OPERA result could be explained by possible systematical errors. Beyond that, in Davoudiasl:2011gb () a crosscheck for the result was proposed. It was demonstrated that muon neutrinos traveling with superluminal velocity can produce signatures for highlyboosted 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 energy quickly by the emission of electron positron pairs Cohen:2011hx (). If muon neutrinos moved faster than light, the process would be energetically possible above a certain neutrino energy threshold resulting in a copious production of electron positron pairs. On the other hand, the authors of Mohanty:2011 (); Lingli:2011kh () discussed that the Cohen–Glashow constraint can be avoided. This may be the case when, for example, Lorentzviolating effects depend quadratically on the neutrino energy or if Lorentzviolation is not fixed but covariant with the neutrino fourmomentum.
In arXiv:1111.4994 () 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 Cohen:2011hx () are not applicable.
In the article Bi:2011nd () further constraints on the deviation of the neutrino velocity from the speed of light were given by considering pion decay and TeVneutrinos detected by ICECUBE. These gave severe bounds on Lorentz 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 wellknown that dispersion relations of particles will be modified, if they propagate through a medium. In Cacciapaglia:2011ax (); AmelinoCamelia:2011dx () the superluminal motion of muon neutrinos is interpreted in the framework of deformed dispersion relations, which are a lowenergy manifestation of Lorentzviolating physics at the Planck scale. In these theories the vacuum behaves as an effective medium.
Such a medium can result from standard model physics, as well. For example, in Nakanishi:2011vv () superluminal neutrinos 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 Nojiri:2011ju () a model is proposed which describes a spontaneous breakdown of Lorentz symmetry by a scalar background field that is added to the action via a Lagrange multiplier. This framework leads to a modified neutrino dispersion relation depending on the momentum of the neutrino.

The neutrino velocity can be modified by Fermi point splitting (for a recent review see arXiv:1111.4627 ()), which removes the degeneracy of zeros of the fermionic energy spectrum Klinkhamer:2011mf ().

The neutrino dispersion relation can change because of environmental effects caused by fields that accumulate at the position of the Earth. These may lead to an effective metric, in which the neutrino propagates with superluminal velocity Dvali:2011mn (); Oda:2011kh ().
^{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 Alles:2011wq (). In the article 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 modified gravity. In the article Saridakis:2011eq () particle propagation in a HořavaLifshitz 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 larger than the speed of light for a special HořavaLifshitz scenario.

In Klinkhamer:2011pw () the existence of a sterile neutrino that travels with a superluminal velocity is proposed. Sterile neutrinos 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 Winter:2011zf ().
Furthermore, assuming superluminal neutrino propagation at a certain energy, neutrinos may propagate with a velocity at very high energies leading to a different neutrino horizon. In Lacki:2011wv () bounds from astrophysical observations are set on for very high neutrino energies.
Ii Extension of the Lorentz group — neutrinos and a hidden sector
Let us, for now, take Eq. (1.1) 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 Eq. (1.1) in the rest of the paper.
ii.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 example, 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 Gedankenexperiments will be performed, whose concept was initiated in Liberati:2001sd ().

We consider some hypothetical beings living in a fluid. They are assumed to consist of fluid atoms, which are held together solely by phononmediated 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 in the fluid Liberati:2001sd (). We expect the beings to build a rocket, which can be accelerated by phonon emission. From the relativistic addition of velocities it follows that the maximum attainable velocity of the rocket is given by .

Einstein found that the dynamics of particles in our universe is governed by the “standard” Lorentz group with the invariant speed of light . His theory has been heralded or substantiated by various experiments MichelsonMorley:1887 (); Kennedy:1932 (); Ives:1938 (). In the second Gedankenexperiment humans build a rocket which is accelerated by a light engine, namely the emission of photons. In this case, the relativistic addition of velocities leads to as the upper limit of the rocket velocity.

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. Fig. 1. This leads to a neutrino dynamics which is based on a “hidden sector Lorentz group” with an invariant velocity . The third Gedankenexperiment is to build a rocket consisting of neutrinos with an acceleration 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 different levels of the Lorentz group each with a distinct invariant velocity can be realized
in nature. The coupling constant is assumed to be small enough such that neutrino propagation is not affected too much to violate existing
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. 2. Each new coupling of particles to massless
gauge bosons
ii.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 () obey a kinematics resting upon special relativity, but with the speed of light replaced by the speed of the hidden sector gauge boson :
(2.3) 
Here is the relativistic neutrino energy of the th mass eigenstate, the neutrino momentum, and
its mass.
(2.4) 
The front velocity, which corresponds to the velocity of the highest frequency forerunners of a wave, is then given by Brillouin1960 ():
(2.5) 
It equals the signal velocity of a function shaped pulse in configuration space. Hence, any possible distortion of a signal does not play a role for the front velocity. The case is related to superluminal, to luminal, and to subluminal motion.
If the gauge boson is assumed to have spin 1 analogously to the photon, at the level of Lagrange densities the ordinary minimal coupling procedure can be performed with again replaced by :
(2.6a)  
(2.6b)  
(2.6c)  
(2.6d)  
(2.6e)  
(2.6f)  
(2.6g) 
where are the neutrino spinor fields describing a specific mass eigenstate and
are the standard Dirac matrices. The covariant derivative contains the vector field of the gauge boson
and the charge , to which couples. Both a Dirac mass term and two possible choices for Majorana
mass terms ChengLi:1984 () with Dirac mass and Majorana masses , are given.
Note that there is a in the zeroth component of the partial derivative . In the context of the Lorentzviolating
Standard Model Extension Colladay:1998fq () the nonzero Lorentzviolating coefficients can be found in the lefthanded neutrino sector of
in their Eq. (9).
(2.7) 
The coefficient matrix is both symmetric and traceless:
(2.8) 
This resembles the CPTeven nonbirefringent modified Maxwell theory coefficients in the photon sector BaileyKostelecky2004 (); Altschul:2006zz (), which is clear, since both sectors are related by a coordinate transformation — at least at first order in the Lorentzviolating coefficients Altschul:2006zz ().
ii.3 Extension of the toy model to three neutrino flavors
The neutrino masses are eigenvalues to the mass eigenstates . However, the weak interaction gauge bosons couple to flavor eigenstates with . The transformation from mass to flavor eigenstates and (vice versa) is governed by the unitary PNMS matrix :
(2.9a)  
When, for simplicity, the CPviolating phases are set to zero,  
(2.9b) 
where, for brevity, and have been used Nakamura:2010zzi (). Here,
for are the neutrino mixing angles. Kinematik measurements of neutrino masses (e.g. for pion decay
and beta decay
(2.10a)  
(2.10b)  
(2.10c) 
Since in Eq. (II.2) the neutrino mass eigenvalues are multiplied by , the maximum velocity of each neutrino flavor will be defined in the following way:
(2.11a)  
(2.11b)  
(2.11c) 
If we assume
(2.12a)  
(2.12b) 
it is sufficient to linearize the equations above:
(2.13) 
Here, . From the latter equation follows the simplified result
(2.14a)  
(2.14b) 
In the rest of the article the neutrino velocities will be approximated by 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 neutrino flavors, is to account for the Supernova 1987 (SN 1987) bound on electron antineutrinos
(2.15) 
An electron neutrino produced by a weak interaction process is a mixture of neutrino mass eigenstates according to the relation
(2.16) 
After the neutrino has traveled through space and reaches a distance from the origin of its production it holds
(2.17) 
where has been assumed Nakamura:2010zzi (). Hence, the initial electron neutrino state then corresponds to a mixture of all flavor eigenstates. However, the initial composition of mass eigenstates remains the same, because quantum mechanically the statement
(2.18) 
is valid. As a result of that, also the velocity of the neutrino does not change during its propagation, since it is determined by the initial composition of mass eigenstates. The antineutrinos coming from the supernova were detected as electron antineutrinos on Earth. For this reason the bound of Eq. (2.15) 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 only couples to the first mass eigenstate, to the second, and to the third, via the respective charge for , 2, and 3, respectively. If any sector obeys a different invariant velocity , the constraint of Eq. (2.15) 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 , , and are Nakamura:2010zzi ():
(2.19a)  
(2.19b)  
(2.19c) 
With the lower bound on and the upper bound on we obtain the PNMS matrix
(2.20) 
Current experimental data imply that neutrinos are almost massless. Concretely, this means
from neutrino oscillation data Nakamura:2010zzi () and
(2.21) 
which is obtained from WMAP observations Komatsu:2008hk (). Therefore, an approximate value of directly follows from Eq. (1.1):
(2.22) 
Assuming and we obtain:
(2.23) 
Hence, Eq. (1.1) 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 and if the second moves faster than . Since the first eigenstate propagates slower than , it need not necessarily couple to any hidden sector. In contrast, the second mass eigenstate has to couple to a traveling faster than light.
ii.4 Challenges of the model and introduction of sterile neutrinos
The argument of Cohen:2011hx () resulting in the rapid loss of the neutrino energy by electron positron emission relies on fundamental principles: fourmomentum 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.

We could assume the energy loss of muon neutrinos to be compensated by a Compton scattering type process, where gauge bosons 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 distribution of 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.

An alternative proposal is that a neutrino itself is part of the hidden sector making it to some kind of superluminal, 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 neutrino flavors. This idea has already been suggested in other publications, see e.g. Klinkhamer:2011pw (); Klinkhamer:2011js () and references therein. Reference Marfatia:2011bw () states that sterile neutrino models may be in conflict with the atmospheric neutrino data measured at SuperKamiokande. However, the models considered in the latter article only involve one sterile neutrino and one single mixing angle with this neutrino. Conclusions for models with more mixing angles have not been obtained.
According to the second item of the list above, we extend the toy model by sterile neutrino mass eigenstates. Then the transformation between the flavor and mass eigenstates is governed by a unitary matrix :
(2.24) 
Following Eq. (2.14) we can write:
(2.25) 
For our toy model we consider the simplest case with one single sterile neutrino, hence . In principle, this sterile neutrino mass eigenstate
mixes with the active neutrino mass eigenstates for . This mixing can be described by introducing three additional
mixing angles , , and . The corresponding mixing matrix can then be constructed from
as follows:
(2.26) 
where . In what follows, we examine a subspace of the free toy model parameters, which is sevendimensional. It is spanned by the three sterile neutrino mixing angles , , and by the invariant velocities of the neutrino mass eigenstates , , , . 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 . The single sterile neutrino is assumed to travel with superluminal speed: we therefore set .
As a result, only the sterile mixing angles remain as free parameters. In Fig. 3 three cases are considered, where in each one of these angles is fixed: , , and . We would like to explore, whether in each case the remaining two angles can be chosen such that the electron antineutrino velocity respects the SN 1987 bound of Eq. (2.15) and that the muon neutrino velocity lies in the error band of Eq. (1.1). 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 , , and that becomes evident from the third panel. With these values we obtain the following results for the velocities of the three active neutrino flavors:
(2.27a)  
(2.27b)  
(2.27c) 
Whether there exists a choice of angles that does not contradict existing atmospheric neutrino data — as was proposed in Marfatia:2011bw () — will not be examined here.
To summarize, within the toy model presented a superluminal sterile neutrino mass eigenstate can be introduced, such that the electron neutrino respects the SN 1987 bound and the muon neutrino travels with the superluminal velocity that is given by Eq. (1.1). For the parameters chosen above the tau neutrino is then slightly superluminal, as well.
Iii Conclusions
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 neutrino, 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 light.
First of all, every physical model describing superluminal 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 presentday physics, which are difficult to circumvent. Honestly, the latter is also a severe problem for the current model, unless the superluminal neutrino itself is part of a hidden sector, hence sterile.
Besides that, the toy model makes the following predictions that can be verified or falsified by experiment:

If the muon neutrino moves with a superluminal velocity, its velocity is isotropic and does not depend on the neutrino energy (besides any massrelated dependence).

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 model builders.
Acknowledgments
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 acknowledges support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of Karlsruhe Institute of Technology.
Footnotes
 These numbers can be found in the updated version 2.0 of OPERA:2011zb ().
 Reference Iorio:2011ay () gives new experimental bounds on the mass scale that is characteristic for the model presented in Dvali:2011mn ().
 Remark that the phonon is not a gauge boson, but a Goldstone boson (massless excitation) resulting from the spontaneously broken translation symmetry in a solid. This does not matter for the argument, though.
 We assume that all mass eigenstates propagate with the same momentum . Whenever we refer to kinematics we stick to the notation of with an index (denoting the mass or flavor eigenstate) for the neutrino mass.
 The question, whether the neutrino is a Dirac or a Majorana particle, has not been answered so far.
 Alternatively, the effective Hamiltonian given by Eq. (14) in Kostelecky:2003cr () can be considered.
 Furthermore, currently no experimental data concerning these phases are on hand Nakamura:2010zzi ().
 Neutrinoless double beta decay that occurs for Majorana neutrinos leads to a different definition of the “flavor eigenstate mass” arXiv:0909.2104 ().
 An antineutrino is assumed to travel with the same velocity as the corresponding neutrino. This makes sense, since the model presented corresponds to a CPTeven term of the Lorentzviolating Standard Model Extension. See the end of Sec. II.2 for a brief discussion concerning this issue.
 Note that under the assumptions taken, the unit should be replaced by , as well. But since the mass values given have been obtained in the context of special relativity, where the speed of light is the invariant velocity, we keep .
 The latter choice is reasonable, since according to Eqs. (1.1) and (2.15). We keep in mind that the constraints on and were obtained at different neutrino energies, but this does not play a role in our model, though.
 with all CPviolating phases set to zero
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