Cosmic Rays and the Search for
a Lorentz Invariance Violation
Wolfgang Bietenholz^{1}^{1}1Present address: Institut
für Theoretische Physik, Universität Regensburg,
D93040 Regensburg, Germany. EMail:
John von Neumann Institut (NIC)
Deutsches ElektronenSynchrotron (DESY)
Platanenallee 6, D15738 Zeuthen, Germany
DESY08072
This is an introductory review about the ongoing search for a signal of Lorentz Invariance Violation (LIV) in cosmic rays. We first summarise basic aspects of cosmic rays, focusing on rays of ultra high energy (UHECRs). We discuss the GreisenZatsepinKuz’min (GZK) energy cutoff for cosmic protons, which is predicted due to photopion production in the Cosmic Microwave Background (CMB). This is a process of modest energy in the proton rest frame. It can be investigated to a high precision in the laboratory, if Lorentz transformations apply even at factors . For heavier nuclei the energy attenuation is even faster due to photodisintegration, again if this process is Lorentz invariant. Hence the viability of Lorentz symmetry up to tremendous factors — far beyond accelerator tests — is a central issue.
Next we comment on conceptual aspects of Lorentz Invariance and the possibility of its spontaneous breaking. This could lead to slightly particle dependent “Maximal Attainable Velocities”. We discuss their effect in decays, Čerenkov radiation, the GZK cutoff and neutrino oscillation in cosmic rays.
We also review the search for LIV in cosmic rays. For multi TeV rays we possibly encounter another puzzle related to the transparency of the CMB, similar to the GZK cutoff, due to electron/positron creation and subsequent inverse Compton scattering. The photons emitted in a Gamma Ray Burst occur at lower energies, but their very long path provides access to information not far from the Planck scale. We discuss conceivable nonlinear photon dispersions based on noncommutative geometry or effective approaches.
No LIV has been observed so far. However, even extremely tiny LIV effects could change the predictions for cosmic ray physics drastically.
An Appendix is devoted to the recent hypothesis by the Pierre Auger Collaboration, which identifies nearby Active Galactic Nuclei — or objects next to them — as probable UHECR sources.
Preface : This overview is designed for nonexperts who are interested in a cursory look at this exciting and lively field of research. We intend to sketch phenomenological highlights and theoretical concepts in an entertaining and selfcontained form (as far as possible), avoiding technical details, and without claiming to be complete or rigorous.
Contents
 1 Cosmic rays
 2 Lorentz Invariance and its possible violation
 3 Cosmic rays
 4 Conclusions
 A New development since Nov. 2007: The AGN Hypothesis
 B Table of shorthand notations
1 Cosmic rays
1.1 Discovery and basic properties of cosmic rays
Cosmic rays consist of particles and nuclei of cosmic origin with high energy. Part of the literature restricts this term to electrically charged objects, but in this review we adapt the broader definition, which also embraces cosmic neutrinos and photons.
The discovery of cosmic rays dates back to the beginning of the century. At that time electroscopes and electrometers were developed to a point which enabled the reliable measurement of ionising radiation. Around 1909 the natural radioactivity on the surface of the Earth was already wellexplored, and scientists turned their attention to the radiation above ground. If its origin was solely radioactivity in the Earth, a rapid reduction would have been predicted with increasing hight.
In 1910 first observations on balloons led to contradictory results: K. Bergwitz did report such a rapid decrease [1], whereas A. Gockel could not confirm it [2]. Still in 1910 Th. Wulf — who had constructed an improved electrometer — performed measurements on the Eiffel tower. He found some reduction of the radiation compared to the ground, but this effect was clearly weaker than he had expected [3]. Although he admitted that the iron masses of the tower could affect the outcome, he concluded that either the absorption in the air is weaker than expected, or that there are significant sources above ground.
In 1912 V.F. Hess evaluated his measurements during seven balloon journeys [209]: he found only a very mild reduction of the radiation (hardly 10 %) for heights around 1000 m. Rising further he observed a slight increase, so that the radiation intensity around 2000 was very similar that that on ground. As he went even higher (up to 5350 m) he found a significant increase, which was confirmed in independent balloonborne experiments by W. Kolhörster in 1913 up to 6300 m [5].
In Ref. [6] V.F. Hess presented a detailed analysis of his measurements in heights of , which were most reliable. These results agreed essentially with those by A. Gockel, who had also anticipated the observation of an intensified radiation when he rose even higher [2]. Hess ruled out the hypothesis of sources mainly inside the Earth. He added that sources in the air would require e.g. a RaC density in high sectors of the atmosphere, which exceeds the density measured near ground by about a factor of . He concluded that a large fraction of the “penetrating radiation” does apparently originate outside the Earth and its atmosphere. In a specific test during an eclipse in 1912, and in the comparison of data that he obtained at day and at night, he did not observe relevant changes. From this he inferred further that the sun is unlikely to be a significant source of this radiation [209],^{2}^{2}2Radiation from the sun occurs at low energy, up to . It won’t be addressed here, and we adapt a notion of cosmic rays starting at higher energy. which we now denote as cosmic rays (the earlier German term was “Höhenstrahlung”).
Another seminal observation was made by P. Auger in the Alps (1937) [7]. Several Geiger counters, which were separated by tens or hundreds of meters, detected an event at practically the same time. Such a correlation, in excess of accidental coincidence, was also reported in the same year by a collaboration in Berlin [8]. Later B. Rossi commented that he had made similar observations already in 1934. P. Auger gave the correct interpretation of this phenomenon, namely the occurrence of extended air showers caused by cosmic rays: a high energy primary particle of cosmic origin triggers a large cascade in the terrestrial atmosphere [7]. Based on the comparison of air shower measurements at sea level and on Jungfraujoch (in Switzerland, 3500 m above sea level), and some simple assumptions about the shower propagation, P. Auger et al. conjectured that primary particles occur at energies of at least [9].
Today we know that the primary particles carry energies of about . The emerging cascades typically involve (in their maxima) secondary particle for each GeV of the primary energy, so an air shower can include up to particles. The energy dependence of the flux is illustrated in Figure 1. Its intensity falls off with a power law (new fits shift the exponent to ), up to minor deviations (‘‘knee’’^{3}^{3}3In addition, a small “second knee” has been observed around . It is not visible in Figure 1, but we will comment on it in Subsection 1.3., “ankle”). We will come back extensively to the behaviour in the highest energy sector.
The composition of the rays depends on the energy. Here we are particularly interested in ultra high energy cosmic rays (UHECRs), with energies , which are mostly of extragalactic origin. There one traditionally assumed about of the primary particles to be protons, about helium nuclei (), plus small contributions of heavier nuclei. At lower energies these fractions change. In that regime, the magnetic field of in our galaxy confines charged particles as diffuse radiation, so that the galactic component dominates [12]. In the range the rays mainly consist of heavier nuclei [13], and for there are protons, and C, N and O nuclei, according to Ref. [10]. As we consider still lower energies, also leptons and photons contribute significantly, see Sections 2 and 3. We quote here typical numbers from the literature, but many aspects of the energy dependent composition are controversial [12].
The arrival directions of charged rays are essentially isotropic. This is explained by their deviation in interstellar magnetic fields: such fields occur at magnitudes of inside the galaxies, and of nG extragalacticly, which is  over a large distance  sufficient for a sizable deflection. As a consequence the origin of charged rays from far distances can hardly be located.^{4}^{4}4Recent developments could change this picture for the UHECRs, see Appendix A. Of course, the approximately straight path length increases with the energy, so that UHECR directions may be conclusive if the source is nearby. As an example, a proton of has in our galaxy a Larmor radius, , which exceeds the galaxy radius (, cf. Appendix B).
The origin of cosmic rays is still mysterious. The first proposal was the Fermi mechanism (“second order”), which is based on particle collisions in an interstellar magnetic cloud [14]. Collisions which are (almost) headon are statistically favoured and lead to acceleration. A later version of the Fermi mechanism (“first order”) refers to shock waves in the remnant after a supernova [15]. However, these mechanisms can explain the cosmic rays at best in part; they cannot provide sufficient energies for UHECRs. In particular the first order mechanism could only attain about , and it predicts a flux — but the observed flux is close to a behaviour , as Figure 1 shows. A variety of scenarios has been suggested later, for comprehensive reviews we refer to Refs. [16]. They can roughly be divided into two classes:

Bottomup scenarios : Certain celestial objects are equipped with some mechanism to accelerate particles to these tremendous energies (which can exceed the energies reached in terrestrial accelerators by at least 7 orders of magnitude). The question what these objects could be is puzzling. Pulsars^{5}^{5}5Pulsars are rotating magnetised neutron stars. [17] and quasars^{6}^{6}6A quasar, or quasistellar radio source, is an extremely bright centre of a young galaxy. [18] are among the candidates that were considered, and recently Active Galactic Nuclei (AGN)^{7}^{7}7An Active Galactic Nucleus is the environment of a supermassive black hole in the centre of a galaxy (its mass is estimated around , where is the solar mass). It absorbs large quantities of matter and emits highly energetic particles, in particular photons, electrons and positrons (we repeat that the acceleration mechanism is not known). Only a small subset of the galaxies have active nuclei. Our galaxy does not belong to them; the nearest AGN are located in Centaurus A and Virgo; this will be of importance in Appendix A. attracted attention as possible UHECR sources, see Appendix A. However, a convincing explanation for the acceleration mechanism is outstanding (see Ref. [19] for a recent discussion, and Ref. [20] for a modern theory).

Topdown scenarios : The UHECRs originate from the decay or annihilation of superheavy particles spread over the Universe. These particles were generated in the very early Universe, so here one does not need to explain where the energy comes from (see e.g. Refs. [21, 22]). One has to explain, however, what kind of particles this could be: magnetic monopoles have been advocated [23], while another community refers to exotic candidates called “wimpzillas” [24] — but there are no experimental signals for any of these hypothetical objects. Moreover, this approach predicts a significant UHECR flux with and as primary particles, which is disfavoured by recent observations: in particular Ref. [25] established above an upper bound of 2 % for the photon flux. The negative impact on topdown scenarios is further discussed in Refs. [26, 27]. Ref. [28] had reported earlier measurements with similar conclusions.
1.2 The Cosmic Microwave Background
We proceed to another highlight of the historic development: in 1965 A.A. Penzias and R.W. Wilson discovered (rather accidentally) the Cosmic Microwave Background (CMB) [29], which gave rise to a breakthrough for the evidence in favour of the Big Bang scenario. It is estimated that the CMB emerged after about years. At that time the cooling of the plasma reached a point where hydrogen atoms were formed. Now photons could scatter off these electrically neutral objects, so that the Universe became transparent (for reviews, see Refs. [30]). The CMB that we observe today — after a period of of further cooling — obeys to a high accuracy Planck’s formula for the energy dependent photon density in black body radiation,
(1.1) 
where is the photon energy (for )
and is Boltzmann’s constant. This shape, plotted in Figure
2, is in excellent agreement with the most precise observation,
which was performed by the Cosmic Background Explorer (COBE) satellite
[31].
Any deviation from the black body formula (1.1)
over the wave length range
is
below of the CMB peak.^{8}^{8}8Sizable deviations seem to
occur, however, at much larger wave length, ,
due to an additional radio
background, the details of which are little known.
We add that there is also a Cosmic Neutrino Background, which decoupled
already about after the Big Bang, and its present temperature
is estimated as . We do not consider it here because
it has no significant impact on cosmic rays.
The temperature is identified as
[32]. This implies that the mean photon
energy and wave length are given by and .^{9}^{9}9The term “microwave” usually
refers to wave lengths in the range of about
, so it does apply to the CMB.
The resulting photon density in the Universe
amounts to . Interesting further aspects of the
CMB — which are, however, not directly relevant for our discussion —
are reviewed for instance in Ref. [33].
1.3 The GreisenZatsepinKuz’min cutoff
In the subsequent year the knowledge about cosmic rays and about the CMB led to an epochmaking theoretical prediction, which was worked out independently by K. Greisen at Cornell University [34], and by G.T. Zatsepin and V.A. Kuz’min at the Lebedev Institute [35]. They expected the flux of cosmic rays to drop abruptly when the energy exceeds the ‘‘GZK cutoff’’ ^{10}^{10}10Since this “cutoff” is not sharp, it is a bit arbitrary where exactly to put it. Eq. (1.2) gives a value which is roughly averaged over the literature, and which we are going to embed into phenomenological and theoretical considerations.
(1.2) 
The reason is that protons above this energy interact with background photons to generate pions. The dominant channel for this photopion production follows the scheme
(1.3) 
These two channels cover of the decays [32]. If even more energy than the threshold for this transition is available, excited proton states (, ) and higher resonances (, , ) can contribute to the photopion production as well. In these cases one may end up with or also with ; at even higher energy the production of three pions is possible too.
It is obvious that a proton with energy looses energy under photopion production, until it drops below the threshold for this process.^{11}^{11}11In the original studies [34, 35] terms like “ resonance” did not occur, but the authors knew effectively about the photopion production. They inferred this spectacular conclusion, although Ref. [34] only consists of 2 pages without any formula, and Ref. [35] is just a little more extensive. Its original Russian version has been translated but only few people have read (the page number is often quoted incorrectly). Nevertheless both works are of course topcited on a renowned level. We are now going to take a somewhat more quantitative look at transition (1.3) (in natural units, ).
We denote the proton energy at the threshold for the resonance as . In these considerations one refers to the FriedmannRobertsonWalker metrics (see Appendix B)  which is comoving with the expanding Universe  as the ‘‘laboratory frame’’ (we will call it the ‘‘FRW laboratory frame’’).^{12}^{12}12In terms of a CMB decomposition in spherical harmonics, the monopole contribution sets the temperature of . The dipole term is frame dependent; the requirement to make it vanish singles out the maximally isotropic frame [32], which we could also refer to in this context. We first consider the relativistic invariant
(1.4)  
and are the 3momenta of the proton and CMB photon. We arrived at the term by assuming a headon collision (and ). The last term refers to a baryon at rest, which sets the energy threshold. Thus we obtain an expression for . As an exceptionally high photon energy we insert — photons with even higher energy are very rare due to the exponential decay of the Planck distribution (1.1). This leads to^{13}^{13}13In eq. (1.2) we increased this energy threshold slightly, which is consistent with the unlikelihood of exact headon collisions.
(1.5) 
Further kinematic transformations yield a simple expression for the inelasticity factor , which represents the relative energy loss of the proton under photopion production [36],
(1.6) 
This relative loss still refers to the FRW laboratory frame, but formula (1.6) holds for general protonphoton scattering angles.
Let us now change the perspective and consider the Mandelstam variable in the rest frame of the proton, where we denote the photon 4momentum as
(1.7) 
The photon energy is related to by the Doppler effect,
(1.8) 
and are the proton velocity and the scattering angle in the FRW laboratory frame. The Lorentz factor can take a remarkable magnitude; for instance at the proton threshold energy it amounts to
(1.9) 
By averaging over the angle one arrives at [36]
(1.10) 
Hence a proton with perceives the CMB photons as quite energetic radiation.^{14}^{14}14The velocity of the proton in the centreofmass frame is given by . At it is small on the relativistic scale, hence the proton rest frame referred to above is not that far from the centreofmass frame.
Combining eqs. (1.6), (1.7) and (1.10) we can determine the inelasticity factor at a given proton energy (and averaged scattering angle),
(1.11)  
We see that the inelasticity is significant already at the energy threshold, and beyond it (gradually) rises further.
We are now prepared to tackle the question which ultimately matters
for the GZK cutoff: if a proton with travels
through the Universe, how long is the decay time of its energy ?
This question was analysed by F.W. Stecker [36],
who derived the following formula for the energy decay time ,
(1.12) 
is the photon threshold energy for photopion production in the rest frame of a proton (which has FRW laboratory energy ). The product of with the logarithm emerges from the Planck distribution (1.1) after integration. is the total crosssection for the photopion production,^{15}^{15}15Strictly speaking it is cleaner to set the lower bound in this integral to zero and rely on the strong suppression below due to , but the form (1.12) is more intuitive. Moreover eq. (1.12) simplifies the energy attenuation to a continuous process — its discrete nature gives rise to minor corrections. which was explored experimentally already in the 1950’s, so Refs. [34, 35, 36] could refer to it. The corresponding experiment with protons at rest, exposed to a ray beam of about , had been reported for instance in Ref. [37]. The crosssection varies mildly with the energy, in the magnitude of . Its profile was reproduced in Ref. [36], along with the measured inelasticity factor . It is remarkable that this simple experiment at modest energy provides relevant information about the fate of UHECRs. Based on the knowledge about , Figure 2 in Ref. [36] displays the mean free path length for the proton, for instance
(1.13) 
A later fit to the experimental data, which refers directly to the product , is worked out in Ref. [38]. It includes corrections like the higher photopion production channels, which still reduce the free path length a little.^{16}^{16}16The same is true for the somewhat higher CMB temperature in scattering processes long ago (although this effect is marginal). Various CMB temperatures have been considered in Ref. [35].
Refs. [34, 35] pointed out already that the corresponding energy attenuation for heavier nuclei is even stronger. The main reason here is the photodisintegration into lighter nuclei due to interactions with CMB photons (see Ref. [39] for a detailed analysis). The fragments carry lower energies; the energy per nucleon remains approximately constant. Hence we are in fact dealing with an absolute limitation for the energy of cosmic rays consisting of nucleons.^{17}^{17}17Regarding nonnuclear rays, we stress that neutrinos are not subject to any theoretical energy limit in the CMB, but no UHECR neutrinos have been observed yet. Numerous works have later reconsidered the attenuation of protons and heavier nuclei in the CMB in detail, see e.g. Refs. [40, 41] and references therein.
At last one might object that protons may still travel very long distances with energies above if they start at a much higher energy. However, at the attenuation is strongly intensified. (For instance the value of the lower integral bound in eq. (1.12) decreases; once the integral captures the peak in Figure 2, many more photons can contribute to the photopion production). As a result, extremely high energies decrease rapidly, and the final superGZK path does not exceed the corresponding path length at drastically. This feature is shown in a plot by the Pierre Auger Collaboration, which we reproduce in Figure 3.
It seems natural to assume a homogeneous distribution of UHECR sources in the Universe, both for bottomup and for topdown scenarios (cf. Subsection 1.1). Then one could expect a pileup of primary particles just below . As a correction one should take into account another process, which dominates somewhat below , and which enables the CMB to reduce the proton energy further: in the energy range the electron/positron pair production
(1.14) 
causes this effect, though the inelasticity is much smaller than in the case of photopion production (consider the ratio in eq. (1.11)).
Ref. [41] presents an updated overview, which incorporates all relevant effects. This leads to Figure 4 for the attenuation lengths for various primary nuclei. For the proton it is composed as , which refers to photopion production and pair creation (at even lower energy adding also the inverse attenuation length due to the redshift becomes significant).
All in all, one concludes the following: if protons — or heavier nuclei — hit our atmosphere with energies , they are expected to come from a distance of maximally
(1.15) 
If we rise the “cutoff” to (cf. footnote 10) the range decreases to . For instance this distance reaches out to the Virgo cluster of galaxies (its centre is about from here). is a large distance compared to our galaxy (or Milky Way) — our galactic plane has a diameter of about . But is short compared to the (comoving) radius of the visible Universe. On that scale the sources of superGZK radiation should be nearby,
(1.16) 
Despite the pair production, some pileup seems to occur at energies (cf. footnote 23). The status of the subsequent dip is discussed in Refs. [43]. As we mentioned in footnote 3, a “second knee” in the cosmic flux was observed by various collaborations around . As a possible interpretation it could be the pileup at the lower end of the pair creation threshold [12].
However, in this review we want to focus on the issue of superGZK energies in cosmic rays. Not even in our vicinity — given by the range (1.15) — we know of any acceleration mechanism, which could come near such tremendous energies. Therefore the detection of superGZK events could represent a mystery — and perhaps a hint for new physics. So let us now summarise the actual observations in this respect.
1.4 Observations of superGZK events
Even before the GZK cutoff was established from the theoretical side, one spectacular event had been reported by the Volcano Ranch Observatory in the desert of New Mexico [44]. The energy of the cosmic particle that triggered this event was estimated at , which exceeds already the GZK cutoff (1.2). Refs. [34, 35] both quoted this report and commented that they do not expect any events at even higher energies (Greisen added that he found even that event “surprising”). In fact, identifying the energy of the primary particle is a delicate issue — which we will address below — and the corresponding techniques were at an early stage.
In any case this initiated a longstanding and controversial challenge to verify the validity of the GZK cutoff. In 1971 yet another superGZK event was reported from an air shower detected near Tokyo [45].^{18}^{18}18For that event the energy was estimated even as , but it is not often quoted in the literature. This inspired a largescale installation in Akeno (170 km west of Tokyo), which is known as AGASA (Akeno Giant Air Shower Array); a historic account is given in Ref. [46].
Meanwhile also the Fly’s Eye project in Utah went into operation, and in 1991 it reported an event with primary particle energy [47]. Up to Ref. [45], this is the highest cosmic ray energy ever reported. In macroscopic units it corresponds to .^{19}^{19}19It is popular to compare this to the kinetic energy of a tennis balls. Its mass is typically , so that these are the kinetic energy for a speed of , which is somewhat below the speed attained in a professional game.
Up to the 21 century numerous new superGZK events were detected, in particular by AGASA. According to the data of this collaboration, the spectrum — i.e. the flux of UHECR rays — continues to follow the powerlike decrease in the energy that we saw in Figure 1, with hardly any extra suppression as is exceeded. This contradiction to the theoretical expectation triggered an avalanche of speculations. The AGASA results have been considered essentially consistent with the data of further air shower detectors in Yakutsk (Russia, at Lake Baikal) [48] and in Haverah Park (England, near Leeds) [49]. An overview of their superGZK statistics is given in Figure 5.
This observation disagrees, however, with the data of the HiRes (High Resolution Fly’s Eye) observatory, which concludes that the cosmic rays do respect the GZK cutoff [51]. HiRes is the successor experiment of Fly’s Eye since 1994, with a refined technology.^{20}^{20}20Initially the HiRes results seemed to agree as well, but here and in the following we refer to the results presented later by this Collaboration.
A comparison of the cosmic ray spectra as determined by AGASA (1990  2004) and by HiRes (1997  2006) is shown in Figure 6. Commented overviews over the results of these collaborations are included in Ref. [40, 52].
Figure 5 also shows that the overall statistics of these UHECR observations is modest. Note that the flux above is around primary particles per minute and , but above it drops to (as indicated in Figure 1), so the search for UHECR takes patience. One may question if there is really a significant contradiction between HiRes and the other groups mentioned above. In fact, an analysis in Ref. [54] concludes that this discrepancy between AGASA and HiRes might also be explained by statistical fluctuations, if the energy scale is corrected in each case by (which is well below the systematic uncertainty).
Nevertheless a large community assumed that a contradiction is likely and wondered about possible reasons. An obvious suspicion is that this may actually be a discrepancy between different methods.

To sketch these two methods, we first address the air shower, which is illustrated in Figure 7. At an early stage the huge energy transfer of an UHECR primary particle onto the molecules in the atmosphere generates — among other effects — a large number of light mesons (pions, kaons ), which rapidly undergo leptonic (or photonic) decays. Of specific interest for the observer are the muons emerging in this way: thanks to the strong time dilation they often survive the path of about 25 km to the surface of the Earth (although their mean life time is only ). Another consequence of the extremely high speed is that the air shower is confined to a narrow cone — typically the directions of motion of the secondary particles deviate by less than from the primary particle direction in the shower maximum (a few hundred meters from the core). The method applied by AGASA, Yakutsk and Haverah Park is an array of numerous Čerenkov detectors — spread over a large area on ground — which identify the trajectories and energies of highly energetic secondary particles. They use tanks with pure water, where Čerenkov radiation is amplified by photomultipliers. Detecting in particular a set of leptons (and photons) belonging to the same shower  with precise time of arrival, speed and direction  provides valuable information about the shower. Its evolution in the atmosphere, and ultimately the primary particle energy, are reconstructed by means of sophisticated numerical methods. (Of course, the information recorded on Earth is still insufficient for a unique reconstruction, so that maximal likelihood methods^{21}^{21}21A synopsis of this method is given in Ref. [32]. have to be applied to trace back the probable scenario.)

The method used by HiRes — and previously by Fly’s Eye — observes the showers in the atmosphere, before they arrive at Earth. The showers excite molecules of the air, which subsequently emit UV or bluish light when decaying to their ground state (with wave lengths . Although this light is weak, it is visible from Earth to powerful telescopes, along with light collecting mirrors, at least in nights with hardly any moonshine or clouds. (These telescopes are structured in a form similar to the compound eye of an insect.) In 1976 Volcano Ranch achieved the first successful observation, which was confirmed by the detection of the same air shower on ground. This air fluorescence light ^{22}^{22}22We adapt this term from the literature, although it actually suggests that the primary particle must be a photon. The appropriate term for this effect would be “scintillation”, or more generally “luminescence”. is emitted isotropically. It is also illustrated in Figure 7, along with the Čerenkov radiation in the air as a further effect. The direct nature of this observation is a valuable virtue, but an obvious disadvantage is the limitation of the detection time to . In this method, the observable height grows with the energy of the primary particle, which complicates the interpretation of the data [52].
We add a couple of qualitative remarks, without going into details. An obvious question is how protons as primary particles can be distinguished from heavier nuclei. For the latter the air showers arrive at the maximal number of particles at a higher point. The depth in the atmosphere to that point can be parametrised as [55, 27]
(1.17) 
where is the atomic mass number. This point can approximately be identified by either of these two methods: for the fluorescence method this observation is quite direct, and on ground it is characterised by the electron/muon ratio. Further criteria — such as the time profile of the signal and the curvature of the shower front — are reviewed in detail in Refs. [56], and summarised in Ref. [12].
In particular the Fly’s Eye record of seemed to originate from a quite heavy primary nucleus, such as oxygen. However, the identification is not easy at all, and in practice the criteria are not always consistent. Only cosmic rays can be distinguished quite clearly from other primary particles (they penetrate much deeper into the atmosphere, hence the air shower maximum is closer to Earth).
In the course of a long flight through the CMB,
heavy nuclei are expected to
break apart (photodisintegration), as we mentioned
before.
Therefore a dominance of protons in the UHECRs suggests that the
primary particles come from relatively far distances, and vice versa.
The status of UHECR observations to this point is reviewed in detail in Refs. [16, 40]. In light of the dilemma between the results obtained with these two techniques, the Pierre Auger Collaboration designed a new project in Argentina (near Malagüe, province of Mendoza) with the goal to clarify the situation [42]. Its planning started in 1992 and it is in stable operation since January 2004, while part of the equipment has still been installed. The concept of the Pierre Auger project is the combination of both techniques described above:

On Earth it involves 1600 detectors with 12 tonnes of water, where three photomultipliers monitor the Čerenkov light caused by air showers. The installation of these tanks has been terminated in 2008. They are distributed over an area of 3000 km on a triangular grid of spacing (for comparison: air showers due to UHECRs have diameters of about at the terrestrial surface).

In addition 24 telescopes are searching for fluorescence light from 4 wellseparated sites — all of them have already been operating since 2004.
The Čerenkov array is suitable for collecting large statistics, while the telescopes are important for a reliable calibration of the energy measurement. This can be achieved best based on the data from air showers, which are observed in both ways, hybrid events, by evaluating a variety of correlations [57]. With a single detection technique the energy calibration has been notoriously problematic.
In Refs. [57, 58, 11] the Pierre Auger Collaboration presented data collected until the middle of 2007, which we reproduce in Figure 8. The exposure to this point already exceeds the total exposure which was accumulated by HiRes and AGASA by about a factor of 2 resp. 4. The systematic uncertainty in the energy measurement is estimated around and the statistical error around (which is rather harmless in this context).
The vertical arrows in this plot mark the upper limits for the energy bins with C.L. [11], based on the FeldmanCousins method for Poisson distributions [59]. The small labels indicate the number of events detected in the corresponding bin, and the error bars represent the statistical uncertainty of the flux. Below (as given in eq. (1.2)) we recognise the powerlike behaviour of the energy spectrum. In the intervals , and beyond its slope was measured resp. , which rules out a single power law by 6 standard deviations. Hence for the flux drops clearly below the extrapolated power law. Nevertheless a considerable number of new superGZK events have been observed.^{23}^{23}23If one rescales the flux with the factor — as in Figure 6 — the shape turns into a small peak below (which could be interpreted as a pileup, as we mentioned in Subsection 1.3), and the suppression beyond the threshold appears weaker. Confronted with the fluxes measured earlier by AGASA and HiRes, these new data are closer to the latter. They are certainly consistent with an extra damping once the energy threshold for the resonance is exceeded. On the other hand, the sizable number of superGZK cosmic rays asks for an explanation and keeps the door open for speculations. An interesting interpretation of the UHECR arrival directions was published by the Pierre Auger Collaboration in November 2007, see Appendix A.
In this review we focus on Lorentz Invariance Violation (LIV) as one attempt to explain a possible excess of UHECRs compared to the theoretical prediction. We should add, however, that the “boring” outcome of full consistency with the GZK cutoff is by no means disproved — recently that scenario has actually been boosted, see again Appendix A.
2 Lorentz Invariance and its possible violation
So far we have assumed Lorentz Invariance (LI) to hold. If fact, it played a key rôle in the derivation of the GZK cutoff: the transformation of the scattering process of an UHECR proton and a CMB photon to the rest frame of the proton established the link to the experimentally known crosssection. Also the determination of the inelasticity factor for the proton under photopion production depends on LI, since it refers again to the proton rest frame (regardless whether one relies on theory, as in eq. (1.11), or on laboratory experiments). The same holds for the photodisintegration of heavier nuclei, and for electron/positron pair creation.
LI was therefore crucial for the analysis of the energy attenuation of superGZK cosmic rays. Here we relied on Lorentz transformations with extreme Lorentz factors of the magnitude
(2.1) 
see eq. (1.9) (we still set ).
There are no direct experimental tests if LI
still holds without any modifications for such extreme boosts;
accelerator experiments are limited to
Lorentz factors .
If the validity of the GZK cutoff will ultimately be confirmed,
then this observation may be considered an indirect piece of
evidence for LI under tremendous boosts.
If, on the other hand, the final analysis
reveals a mysterious excess of superGZK events,
it might indicate new physics. In that case, one point
to question about the standard picture is LI — so let us
take a closer look at it.
LI is a central characteristic of relativity:

In Special Relativity Theory (SRT) LI holds as a global symmetry.
The transformation rules were derived by H.A. Lorentz (1899 and 1904), H. Poincaré declared them a Law of Nature, A. Einstein provided a consistent physical picture (1905) and H. Minkowski embedded it into the geometry of a 4d space (1907). 
In General Relativity Theory (GRT) LI holds as a local symmetry (A. Einstein, around 1915). In each spacetime point the frame can be chosen so that locally the metrics takes the Minkowski form , which holds globally in SRT.
It is therefore tempting — and possible — to write GRT in the terminology of a gauge theory (see for instance Ref. [60], Section 6.4). However, as a quantum field theory this formulation is not renormalisable: the renormalisation group flow does not lead to a conformal field theory (i.e. to scale invariance) at high energy, as reviewed recently in Ref. [61]. Therefore our established description of particle physics (local quantum field theory) is incompatible with GRT, which applies to very low energy and describes longrange gravity successfully.
Hence LI plays an essential rôle in both, SRT and GRT, although this rôle is not the same. In this review we mostly address high particle energies up to the maximum that has been observed, hence SRT is in general the appropriate framework. At specific points, however, we are also going to comment on conceivable connections to GRT  this becomes relevant in view of speculations up to the Planck scale.^{24}^{24}24For approaches to quantum gravity, which discuss in particular the rôle of LI and its possible violation, see e.g. Refs. [62, 63, 64, 65].
In general it is untypical for global symmetries to hold exactly, so in SRT one could puzzle why this should be the case for LI (and the related CPT invariance, see Subsection 2.1).^{25}^{25}25In this spirit, the Standard Model of particle physics appears more natural in the now established form, which incorporates neutrino masses, and therefore only approximate chiral symmetry. On the other hand, gauge symmetries are exact, hence in this respect the GRT picture appears helpful (as long as no violation of LI is observed).
Nevertheless for our further discussion the framework of SRT is essential, since it allows us to apply particle physics as described by quantum field theory. In field theory, the manifestation of LI as a global symmetry is well established. Some field , which may represent a scalar, a spinor, a 4vector or some tensor, transforms in a (finite dimensional) representation of the Lorentz group ,
(2.2)  
see for instance Ref. [60]. In particular scalar quantities — like the Lagrangian of a relativistic field theory — are Lorentz invariant ().
In contrast to the rotation group, the Lorentz group is noncompact. The extrapolation of LI to arbitrarily high energy leads to the generic UV divergences in quantum field theories. The formulation on noncommutative spaces — to be addressed in Subsection 3.3 — was originally motivated as an attempt to weaken these UV divergences. Further early suggestions to deviate from LI in quantum field theories include Refs. [66].
2.1 Link to the CPT Theorem
According to the CPT Theorem, particle physics is invariant
if three discrete transformations are performed simultaneously:
C : flips particles into the corresponding antiparticles and vice versa.
This changes the signs of all couplings to gauge fields,
hence C is
denoted as charge conjugation.
P : parity, space reflection, .
T : inversion of the time direction, .
The CPT Theorem states that, assuming locality and LI,
CPT invariance is inevitable. This was recognised first
by W. Pauli and G. Lüders [67] and put on
a rigorous basis by R. Jost [68].
The basic idea is an analytic continuation of the Lorentz
group to imaginary time (Wick rotation), so one arrives at a reflection
at the origin of 4d Euclidean space.
This reflection captures P and T, and the Wick rotation implies that C is included as well. The combination amounts to an
antiunitary symmetry transformation.
Applying the CPT transformation twice, one also
obtains a derivation of the SpinStatistics Theorem.
(A historically and mathematically precise account is
given in Ref. [69]).
Only recently O.W. Greenberg turned the consideration around and proved the following Theorem [70]: Any breaking of CPT invariance necessarily entails a Lorentz Invariance Violation (LIV).
This Theorem is very general, it applies whenever CPT invariance breaks in any Wightman function.
On the other hand, if we only assume LIV, then CPT symmetry may be broken or not, i.e. there are many ways to break LI while leaving CPT symmetry intact. The leading LIV terms are linear in the energy resp. momenta, and these terms break CPT too; they are CPT odd. If we only admit LIV terms starting quadratically in the energy, then CPT can be preserved [71], so it is natural to assume
(2.3) 
2.2 Status of experimental LI tests
According to Greenberg’s Theorem, LI tests can be performed indirectly by verifying CPT invariance. Experiments probe in particular the CPT prediction of identical masses for particles and their antiparticles. This holds for instance for , up to , and for , up to (in both cases this is the bound for the relative deviation with C.L.) [32]. Ongoing tests investigate the strangeness driven oscillation and conclude [73]
(2.4) 
which corresponds to a remarkable relative precision of .
Similar high precision LI tests via CPT measure the magnetic moments of elementary particles and antiparticles [74]. A further indirect method to test LI in part deals with the equivalence principle [75]. However, direct tests are even more powerful, and we will explain below why we are more interested in CPTconserving LIV parameters.
The status of numerous experimental LI tests by means of a variety of methods has been reviewed extensively in Refs. [72, 76, 77]. Atomic physics gives access to excellent precisions for specific LIV parameters. Such parameters describe, for instance, a particle spin coupling to a conceivable “tensor background field”. We will come back to these objects in the following Subsection. Here we anticipate that a large number of such parameters can be introduced as additional couplings in the Standard Model. In light of the discussion in Subsection 2.1, we stress that these LIV terms are local, but they can be even or odd under CPT transformation. According to Ref. [78] they are all bounded by , in a form which factors out the Planck scale (2.6).
These precisions are impressive, and all results are in agreement with LI — no LIV has ever been observed. One may wonder if even higher precision is still of interest, or if it is already sufficient for all phenomenological purposes. In the following we would like to argue that this is not necessarily the case. The precision achieved so far may be insufficient with respect to scenarios, which could drastically affect physics on the Planck sale, or even the propagation of cosmic rays.

We first address the CPT conserving class of LIVs, which exclude terms of [71], so we can assume the violation effect to be , as in relation (2.3). Let us consider the case where LI is violated dramatically on the Planck scale, where one expects both, gravity effects and quantum effects to be strong. For instance the magnitude of (dimensionless) LIV effects at such energies could be
(2.5) where is the Planck mass, i.e. the energy scale set by the gravitational constant ,
(2.6) Accelerator experiments reach energies , so they could only be confronted with such LIV effects . It is therefore conceivable that a CPT conserving LIV has been overlooked up to now, although it plays an essential rôle on the Planck scale.

We move on to the case where LI and CPT are violated. Then we expect LIV effects . Let us assume that even a surprisingly strong LIV parameter of this kind, say with a magnitude of , has been overlooked. It would be amplified to on the Planck scale, so even there it would be of minor importance.
Therefore the CPT conserving LIV terms are more interesting. In addition to this estimate of magnitudes one might also argue that breaking one fundamental law of standard quantum field theory — which has been a principal pillar of physics since the century — is a very daring step already. Hence it appears reasonable to study this step in detail, rather than damaging yet another highly established principle on top of it.
Cosmic rays represent indeed a unique opportunity to
observe effects not too far below the Planck scale. The point
is not only that they carry the highest particle energies
in the Universe, but — even more importantly
— their extremely long flight, which could accumulate
tiny effects of new physics also at low or moderate energy (we will
sketch examples in Subsection 3.4). If there is any hope to find
experimental hints related to theoretical approaches like string
theory^{26}^{26}26String theories start from a higher
dimension and first need a spontaneous LIV in that space to explain
why we experience only 4 extended dimensions. This requires a
nonperturbative mechanism, which is not well understood.
One attempt to formulate string theory beyond perturbation theory
is the 10d IIB matrix model (or IKKT model) [79];
a possible spontaneous splitting into extended and compact
directions is discussed in Refs. [80], and for its
4d counterpart in Refs. [81].
As a different aspect,
LIV effects on the Casimir force in a 5d brane world are
discussed in Ref. [82].
or (loop) quantum gravity,^{27}^{27}27Conceivable loop quantum gravity
effects on cosmic rays are discussed in Refs. [83].
So far we can restrict parameters in specific quantum gravity approaches
based on the bounds on LIV parameters [84].
then it is likely to involve cosmic rays (although also
LHC might have a chance [85]).^{28}^{28}28We mention
at the sideline that the scattering
of UHECR primary particles on atmospheric molecules takes place
at higher energies in the centreofmass frame than any scattering
in accelerator experiments. In particular the protonproton
collisions at LHC (with 7 TeV per beam) will be equivalent
in the centreofmass energy to the scattering of a
proton on a fixed target. This happens about times
per second in the terrestrial atmosphere, in addition to the cosmic
rays hitting other astronomical bodies, which obviously invalidates
popular concerns about the safety of LHC [86]. Here, however, we are interested
in interactions of UHECRs with CMB photons, which
involve modest centreofmass energies, see Subsection 1.3.
These hypothetical theories tend to install new fields in the
vacuum, in addition to the Higgs field of the Standard Model.
We are now going to glimpse at the mechanism how such
new background fields could lead to spontaneous LIV.
2.3 Standard Model Extension (SME)
A systematic approach to add local LIV terms to the Standard Model has been worked out by A. Kostelecký and collaborators starting 10 years ago [87]. This collaboration provides a kind of encyclopedia of such terms, which is reviewed — along with its experimental bounds — in Ref. [88]. The original motivation emerged from string theory [89], which is, however, not needed for the resulting Standard Model Extension (SME).
We take a look at a prototype of a LIV term in this extension. Consider (in a shorthand notation) the Lagrangian of some fermion field , which may represent a quark or a lepton,
(2.7) 
We start with the free kinetic term, followed by the Yukawa coupling to the Higgs field . The Standard Model assumes to undergo Spontaneous Symmetry Breaking (SSB), so that one of its components takes a nonvanishing expectation value, say . This leads to the fermion mass . A mass term cannot be written into the Lagrangian directly if we want to keep the freedom to couple the left and the righthanded chirality components of the fermion field independently to a gauge field; this is the situation in the electroweak sector of the Standard Model.^{29}^{29}29On the other hand, a fermion mass term is allowed in a vector theory like QCD, where both chirality components of a quark couple to the gluons in the same manner.
Let us now consider the third term in the Lagrangian (2.7), which is one of the possible extensions beyond the Standard Model. The Yukawatype coupling is another free parameter, which couples the fermion to a new background field of the Higgstype, albeit with a tensor structure. Again we assume SSB, which could, for instance, lead to
(2.8) 
This additional term will obviously distort the fermion dispersion relation compared to the Standard Model, which amounts to a spontaneous LIV.
In contrast to the Higgs field we are now confronted with the question if new background fields — which may have a vector or a tensor structure — are Lorentz transformed as well. For a simple change of the observer’s inertial frame, all fields — including the background fields — are transformed, and remains invariant. However, the question if particles, which are described by fields like , really perceive LI depends on a transformation of these fields only in a constant background [90, 72]. Kostelecký et al. denote this as an “active” (or “particle”) Lorentz transformation, and in this respect LI may break spontaneously, as in eq. (2.8).^{30}^{30}30Here one reintroduces a distinction, which one had hoped to overcome since the historic work by Einstein.
In this way we can construct a nonstandard dispersion relation for any particle, depending on its coupling to the tensor field or further new background fields. The inclusion of gauge interactions proceeds in the familiar way (one promotes some global symmetries to local ones by means of covariant derivatives).
Of course the LIV parameter is just one example — the mass term and kinetic term in the Lagrangian (2.7) could both be extended by including a background field for each element of the Clifford algebra (further terms are written down in Section 3.5). Another example, which was considered earlier [91], is the addition of an extra gauge term of the ChernSimons type to modify QED,^{31}^{31}31In nonAbelian gauge theory, LIV through ChernSimonstype terms is studied in Refs. [92].
(2.9) 
Here is a vector of dimension mass (and ). Each component has vanishing covariant derivatives in all frames. This vector introduces a “preferred direction”, where also all the ordinary derivatives vanish — Ref. [91] associates it with the preferred frame in a galaxy. It identifies an astrophysical bound of based on radio galaxies (the method will be sketched in Subsection 3.5). Recently there have been attempts to achieve even higher precision for this bound based on the rotation of CMB photon polarisation vectors [93].
Note that breaks also CPT invariance spontaneously. Examples for CPT conserving LIV terms in the fermionic and gauge part of the extended Lagrangian are the LIV term in eq. (2.7), or (where is another background tensor field). Kostelecký et al. have identified more than 100 LIV parameters in this way, including CPT breaking terms [94]. The resulting model preserves a number of properties of the Standard Model, like energy and momentum conservation, gauge invariance and locality. For massive fermions at low energy, and small LIV parameters, also energy positivity and causality are safe [95]. In the framework of SRT the photon is identified with the NambuGoldstone boson of the SSB of Lorentz symmetry. That collaboration also discussed various scenarios for an interpretation in the GRT framework [87, 64, 88].
In this parametrisation, the current status is summarised in Ref. [78] as follows: all (dimensionless) LIV coefficients are (as we mentioned before). For those which break CPT the upper bound is as tiny as , which further motivates our focus on CPT even terms. A detailed overview of the various bounds is given in Ref. [77].
An obvious question is why LIV should become manifest only at huge energies, although SSB is typically a low energy phenomenon. In principle even a LIV detected a low energy would not necessarily imply LIV at high energies. Of course we are interested in the opposite situation, where LIV is significant only at high energies, so we have to assume the SSB to persist over all energies that we consider. This setting requires tiny LIV parameters multiplying momenta of the fields (as in the examples above). Then the deviation from LI is only visible at huge momenta and a contradiction to known phenomenology can be avoided.
Although consistent, this scenario implies a severe fine tuning problem. On tree level one might hope for a somehow natural suppression of the LIV parameters by a factor (where is the electroweak mass scale) [96]. If one includes 1loop radiative corrections, however, this effect is lost and the magnitude of LIV effects is just given by ( being an electroweak Yukawa coupling) [97], which is far from the suppression needed. If we consider LIV nevertheless, we have to add this problem to the list of unsolved hierarchy problems, like the small vacuum angle in QCD, the small particle masses (on Planck scale) or the small cosmological constant. On the other hand, the Standard Models of particle physics and cosmology are well established, despite the presence of these unsolved problems.
2.4 Maximal Attainable Velocities (MAVs)
Within the same framework we now switch to the pragmatic perspective of Ref. [71]. We saw in the previous Subsection that LIV can be arranged for. Now we assume that this mechanism has been at work, and we proceed to an effective Lagrangian, which includes explicit LIV terms. Such an effective Lagrangian may also capture hadrons as composite particles (like Chiral Perturbation Theory).
Following Ref. [71] we leave, however, other Standard Model properties intact as far as possible. We therefore require CPT and gauge invariance to persist, and rotation symmetry to hold in a ‘‘preferred frame’’.^{32}^{32}32In contrast to Ref. [91], the following considerations on cosmic rays can hardly employ the preferred frame in a galaxy, because the primary particle paths extend over much larger distances. One might refer to the frame which renders the CMB maximally isotropic, cf. footnote 12. Moreover we do not break power counting renormalisability, i.e. we only add terms of mass dimension .
Let us outline which terms are admitted by these conditions. For a bosonic field we can add a purely spatial kinetic term
(2.10) 
(Equivalently we could add a term with only time derivatives instead). For a fermion field we insert a spatial kinetic term as well, where the LIV parameters can be distinct for positive or negative chirality,
(2.11) 
Gauge interactions require covariant derivatives also in these nonstandard terms. For the pure gauge part we first consider the Abelian gauge group , where the electric and the magnetic field are given by , . The above conditions allow for three independent terms,^{33}^{33}33Without insisting on CPT invariance, the ChernSimons term (2.9) would be added to this list.
(2.12) 
The first two among these terms are LI, so we work with the third one. Also in YangMills gauge theories Ref. [71] uses the corresponding term
(2.13) 
where the index runs over the generators of the Abelian factors in the gauge group.
In this way one obtains a quasiStandard Model with 46 LIV
parameters. In view of the above list the large number may
come as a surprise at first sight, but it can be understood
by the numerous new fermion generation mixings (for one case,
the phenomenological impact on neutrino oscillation will
be discussed in Subsection 2.5.4).
Coleman and Glashow verified that all these parameters
preserve the vanishing gauge anomaly,
so that the model remains gauge invariant on quantum level.
We proceed to another prototype illustration and consider a neutral scalar field [71]. In some background the renormalised propagator is written as
(2.14) 
Geometrically we assume the usual Minkowski space with , , and is the renormalised mass at . The functions and are not specified, but obeys the conventional normalisation, and is adapted to it, (both functions are smooth in this point).
Let us now turn on a tiny LIV parameter and consider its leading order. The poles in the propagator are shifted to
(2.15) 
The renormalised mass is modified, . However, we are more interested in the feature of the dispersion relation (2.15): the parameter takes the rôle, which is usually assigned to . From the group velocity
(2.16) 
it is obvious that this is the speed that the particle approaches asymptotically at large .
Following this pattern, each particle type P can pick up its own Maximal Attainable Velocity (MAV) ; it might slightly deviate from the speed , which establishes the Minkowski metrics. A tiny value of the dimensionless parameter justifies the above linear approximations, and it is compatible with the desired setting where LIVs only become noticeable at huge momenta. We mentioned before that this imposes a new hierarchy problem (which we have to live with) and that we are referring here to an effective approach, which captures composite particles, unlike Subsection 2.3.
2.5 Applications of distinct MAVs
Let us now address some applications that we are led to if we assume certain particles to posses MAVs distinct from . We still follow Ref. [71] and focus on applications of potential interest in cosmic ray propagation.^{34}^{34}34It has also been suggested to test MAVs in air showers [98].
2.5.1 Decay at ultra high energy
We consider the possibility that some particle with index decays into a set of particles with indices . At ultra high energy all masses are negligible, but we assume individual MAVs for the particles involved, . We further define as the minimal MAV among the decay products. Then the energetic decay condition reads
(2.17) 
The interesting observation is that tiny differences in the MAVs can be arranged such that a usual decay cannot take place anymore if the primary particle carries ultra high energy. This happens when the condition (2.17) is violated, i.e. if is smaller than any of the .
On the other hand, new decays could be allowed at high energy, such as the photon decay , see Subsections 2.5.2 and 3.5, or the radiative muon decay . A further example is the inverse decay , which tightens the bounds on specific LIV parameters [99] based on the new results for UHECR that we review in Appendix A.
2.5.2 Vacuum Čerenkov radiation
Next we consider a particle with a MAV that slightly exceeds the speed of light, i.e. it exceeds the MAV of the photon in vacuum,
(2.18) 
This particle can be accelerated to a speed . Then it will emit vacuum Čerenkov radiation, so it slows down intensively. The energy required for vacuum Čerenkov radiation has to fulfil
(2.19) 
where is the particle mass, and we consider the leading order
in .
As we discussed in Section 1, we know from cosmic rays that protons support an energy up to over a considerable path, hence this energy does not fulfil inequality (2.19). Coleman and Glashow infer
(2.20) 
where and are the mass and the parameter (see eq. (2.18)) of the proton.
Cosmic electrons and positrons are observed only up to , hence the Čerenkov constraint on their parameter is much less stringent, On the other hand, the absence of the photon decay constrains from below [100]. In total one arrives at
(2.21) 
2.5.3 Impact on the GZK cutoff
In this generalised framework we now reconsider the headon collision of relation (1.3). For the MAVs we could check all kinds of scenarios — they are all conceivable in the effective ansatz of Subsection 2.4. Here we pick out the particularly interesting option that only the proton has a slightly slower MAV,
(2.22) 
(in the notation of eq. (2.20): ). The condition for a resonance in the scattering on a CMB photon was written down before in eq. (1.4). The following requirement still holds,
(2.23) 
where are the proton energy and momentum, and is the photon energy, all in the FRW laboratory frame. The evaluation of the righthandside is now modified due to the nonstandard dispersion relation of the proton,
(2.24) 
where we assumed and . If we additionally make use of , we arrive at a new energy threshold condition, where one term is added compared to eq. (1.4),
(2.25) 
Once more, the extra term is suppressed by the factor , but it can become powerful at very large energies. The consequence of this modification is illustrated in Figure 9: the righthandside increases linearly with the proton energy and reaches the standard threshold (for ) at the value given in eq. (1.5). A small leads to an increased threshold energy, and above a critical value the resonance is avoided,
(2.26) 
Hence the critical parameter is tiny indeed, and this is all it takes to remove the GZK cutoff, at least in view of the dominant channel of photopion production.^{35}^{35}35The idea that LIV could invalidate the GZK cutoff was first expressed in Refs. [101]. Later it has also been discussed in the SME framework [102]. (This is of course compatible with the condition (2.20), which only requires .) The experimental bounds for the LIV parameters in the fundamental SME Lagrangian are below (cf. Subsection 2.3). However, it is not obvious how this translates into bounds on effective MAV differences (remember that here we started from a renormalised propagator).^{36}^{36}36A possible transition from SME parameters to hadrons by means of a parton model was discussed in Ref. [103].
Let us stress once more that this is not in contradiction to the fact that this resonance is observed at low energy — in that case the masses contribute significantly to the energies and the above approximations are not valid any longer.
For the dominant candidate for a photopion production channel is an excited proton state,
(2.27) 
But at ultra high energy can only decay if , according to condition (2.17).^{37}^{37}37Bounds on based on UHECR are discussed in Refs. [65, 104]. With a suitable choice of the MAVs we could close that channel too, and so on.
Just before Ref. [71] appeared, Ref. [18] attracted attention to the observation that the top 5 superGZK events known at that time all arrived from the direction of a quasar (cf. footnote 6). This inspired Coleman and Glashow to suggest an unconventional hypothesis: the superGZK primary particles could be neutrons. Assuming particle specific MAVs, the following arguments yield a consistent picture (see also Ref. [105]):

For (where refers to the decay products in the sense of relation (2.17)) the decay of the neutron can be avoided.

Since we already assumed a slightly reduced neutron MAV, we might add to protect the neutron from the GZK cutoff.

Unlike the proton, the neutron is hardly deflect by interstellar magnetic fields, hence the scenario of an approximately straight path — which is required to make sense of the observation in Ref. [18] for quasars at large distances — becomes realistic.
This suggestion is currently not discussed anymore, but its consistency is beautiful enough to be reviewed nevertheless. Nowadays superGZK events have been reported (see Section 1), and the quasar hypothesis is out of fashion as well. However, the hypothesis of clustered UHECR directions has recently been revitalised, see Appendix A. Also the idea of electrically neutral primary particles is still considered, see e.g. Refs. [106].^{38}^{38}38However, neutral primary particles are not required for the hypothesis discussed in Appendix A, since it assumes nearby UHECR sources. They include pions [65], photons [107] (though Ref. [25] puts a narrow bound on their UHECR flux), neutrinos [108] and hadrons of quarks, which could be stable in SUSY theories [109].
2.5.4 Impact on neutrino oscillation
If the MAVs can deviate from , there are three bases for the neutrino states:

the flavour basis with the states ,

the basis of the neutrino masses,

the basis of the MAVs.
In this framework neutrino oscillation could occur even at vanishing neutrino masses, due to a flavour mixing in the MAV basis.^{39}^{39}39A discussion in the SME terminology is given in Ref. [110]. However, in light of experimental data this scenario has soon been discarded [111, 112], so we start here from the usual point of view that the observed neutrino oscillation is evidence for nonvanishing neutrino masses.
We do consider, however, the possibility that the mass driven oscillation could still be modified as a subleading effect by additional MAV mixing. This scenario was investigated [113] based on the data by the MACRO Collaboration in Gran Sasso (Italy). The study can be simplified by focusing on the oscillation , which is most relevant for the observation of cosmic neutrinos. In this 2flavour picture the two parameters
(2.28) 
modify the life time of .
As an example, Figure 10 shows the survival probability of over a distance of (roughly a path across the Earth passing near the centre) at (still in natural units, ). The three curves refer to the MAV mixing angles with . We see that the effect of such a mixing is very significant for neutrino energies of . If we assume neutrino masses , this corresponds to a Lorentz factor , similar to the primary proton in a UHECR (see eq. (1.9)). Cosmic neutrinos in this energy range occur,^{40}^{40}40One suspects that there are cosmic neutrinos at much higher energies as well — they are not restricted by any GZKtype energy cutoff (cf. footnote 17) nor deviated by magnetic fields, so they could open new prospects to explore the Universe. Several projects are going to search for them systematically, see Section 4. and their behaviour is again suitable for a sensitive test of LI at ultra high factors.
The MACRO Collaboration detected upward directed muons generated by neutrinos in the state through the scattering on a nucleon,
(2.29) 
By monitoring multiCoulomb scattering they reconstructed the energy and finally . They observed 58 events with and compared this flux to the one of at low energies. It turned out that the oscillation curve obtained solely by mass mixing (corresponding to the curve for in Figure 10) works very well. Including the additional parameters (2.28) does not improve the fits to the data. Therefore the scenario of additional mixing due to different MAVs is not supported, and Ref. [113] concludes
(2.30) 
Of course the exact bound on depends on the mixing angle —
it becomes even more stringent when
approaches .
This result is in agreement with similar analyses based on the SuperKamiokande K2K data [112, 114]. Ref. [112] introduced a parameter to distinguish scenarios for the origin of neutrino mixing as follows:
(2.31) 
The fit to the K2K data led to , supporting the standard picture and strongly constraining the alternatives. Similarly Ref. [114] generalised the scenario corresponding to by admitting vector and tensor background fields — as in Subsection 2.3 — and arrived qualitatively at the same conclusion.
3 Cosmic rays
In this section we are going to consider the photons themselves as they appear in cosmic rays — so far they entered our discussion only in the background. Their identification as primary particles is on relatively safe grounds (cf. Section 1).
3.1 Another puzzle for high energy cosmic rays ?
The highest energies in cosmic rays were observed around (see e.g. Refs. [117]). They originate from the Crab Nebula, which is the remnant of a supernova — recorded by Chinese and Arab astronomers in the year 1054 — at a distance of .
The sources of the strongest rays reaching us from outside our galaxy are blazars. Blazars are a subset of the AGN, see footnote 7.^{41}^{41}41Appendix A addresses the hypothesis that AGN could emit UHECRs as well. A few hundred blazars are known. A prominent example is named Markarian 501; in 1997 the High Energy Gamma Ray Astronomy (HEGRA) satellite detected rays with from an outburst that this blazar had emitted [118]. It is located at a distance of — this can be determined from the redshift (). Here also the direction of the source is known, in contrast to fartravelling charged cosmic rays (gravitational deflections are small). Another example, which is noteworthy in this context, is Markarian 421, at a distance of 110 Mpc, which emitted photons that arrived with [119].
It was suggested that the observation of these highly energetic rays could pose a new puzzle, which is similar to the GZK cutoff [120]. Again the question is why the CMB can be transparent for rays of such high energies. In this case, one expects electron/positron pair creation of an UV photon in the ray and an IR photon in the background,
(3.1) 
if sufficient energy is involved. This pair creation is followed by inverse Compton scattering on further CMB photons, triggering a cascade. In the centreofmass frame both photons in the transition (3.1) have the energy
(3.2) 
where is the Lorentz factor between centreofmass and FRW laboratory frame (cf. Section 1). The condition for pair creation is obvious,
(3.3) 
If we insert once more the exceptionally large CMB photon energy that we referred to before in Sections 1 and 2, , the threshold for is almost . If we repeat the considerations about the CMB photon density and the crosssection for this pair creation, in analogy to Subsection 1.3, also the observed energies of photons coming from far distances (outside our galaxy) might be puzzling [121]. The mean free path length is given by a formula similar to eq. (1.12),
(3.4) 
As in eq. (1.4), is one of the (LI) Mandelstam variables. The differential photon density of the CMB follows Planck’s formula (1.1). The minimal CMB photon energy is , see eq. (3.3). The boundaries for the angular integral over are also obvious, and is the crosssection. It peaks just above the threshold , and at high energy () it decays as [21]
(3.5) 
Ref. [122] evaluated this formula and found for instance for a path length , so that the radiation from Markarian 421 and 501 appears amazing. Ref. [122] suggested a generalisation of formula (3.4) to a LIV form, inserting ad hoc a deformed photon dispersion relation, which extends . However, the question if this puzzle persists in the LI form, after taking all corrections into account, is controversial [123, 124, 76]. One issue is our poor knowledge about the radio background radiation, which dominates over the CMB at wave lengths (cf. footnote 8). Ref. [21] studied the effective penetration length of the photon cascade under different assumptions for the extragalactic magnetic field and the radio background. Over a broad interval for these two parameters, Figure 14 in that work implies
(3.6) 
In light of these numbers the observations of Markarian 421 and 501 appear less puzzling, but the issue is still under investigation. As in the case of the GZK cutoff we discuss possible solutions for such a puzzle — if it exists — based on LIV.
Also this multiTeV