6.5 Particle threshold interactions in EFT
When Lorentz invariance is broken there are a number of changes that can occur with threshold
reactions. These changes include shifting existing reaction thresholds in energy, adding additional thresholds
to existing reactions, introducing new reactions entirely, and changing the kinematic configuration at
threshold [86, 130, 154, 200]. By demanding that the energy of these thresholds is inside or outside a
certain range (so as to be compatible with observation) one can derive stringent constraints on Lorentz
In this section we will describe various threshold phenomena introduced by Lorentz violation in EFT
and the constraints that result from high energy astrophysics. Thresholds in other models are discussed in
Section 6.6. We will use rotationally invariant QED as the prime example when analyzing new threshold
behavior. The same methodology can easily be transferred to other particles and interactions. A diagram of
the necessary elements for threshold constraints and the appropriate sections of this review is shown in
Thresholds are determined by energy-momentum conservation. Since we are working in straight EFT
in Minkowski space, translational invariance implies that the usual conservation laws hold,
i.e. , where is the four momentum of the various particles
. Since this just involves particle dispersion, we can neglect the underlying EFT for the
general derivations of thresholds and threshold theorems. EFT comes back into the picture when we need to
determine (i) the actual dispersion relations that occur in a physical system to establish constraints and (ii)
matrix elements for actual reaction rates (cf. ).
||Elements involved in threshold constraints.
Threshold constraints have been looked at for reactions which have the same interaction vertices as in
Lorentz invariant physics. The reaction rate is therefore suppressed only by gauge couplings and
phase space. dispersion requires higher mass dimension operators, and these operators
will generically give rise to new interactions when the derivatives are made gauge covariant.
However, the effective coupling for such interactions is the same size as the Lorentz violation
and hence is presumably very small. These reactions are therefore suppressed relative to the
Lorentz invariant coupling and can most likely be ignored, although no detailed study has been
6.5.1 Required particle energy for “Planck scale” constraints
We now give another simple example of constraints from a threshold reaction to illustrate the required
energy scales for constraints on Planck scale Lorentz violation. The key concept for understanding how
threshold reactions are useful is that, as we briefly saw for the photon decay reaction in Section 6.4, particle
thresholds are determined by particle mass, which is a small number that can offset the large Planck energy.
To see this in more detail, let us consider the vacuum Čerenkov effect, , where is some
massive charged particle. In usual Lorentz invariant physics, this reaction does not happen due to
energy-momentum conservation. However, consider now a Lorentz violating dispersion relation for of
with . For simplicity, in this pedagogical example we shall not change the photon dispersion
relation . Čerenkov radiation usually occurs when the speed of the source particle exceeds the
speed of light in a medium. The same analysis can be applied in this case, although for more general
Lorentz violation there are other scenarios where Čerenkov radiation occurs even though the speed
condition is not met (see below) . The group velocity of , , is equal to one at a
and so we see that the threshold momenta can actually be far below the Planck energy, as it is controlled by
the particle mass as well. For example, electrons would be unstable with and of at
, well below the maximum electron energies in astrophysical systems. We can rewrite
Equation (81) for and see that the expected constraint from the stability of at some momentum
Therefore constraints can be much less than order one with particle energies much less than . The
orders of magnitude of constraints on estimated from the threshold equation alone (i.e. we have
neglected the possibility that the matrix elements are small) for various particles are given in
Table 2 .
||Orders of magnitude of vacuum Čerenkov constraint for various particles
For neutrinos, comes from AMANDA data . The for electrons comes from the
expected energy of the electrons responsible for the creation of gamma rays via inverse
Compton scattering [188, 268] in the Crab nebula. For protons, the is from AGASA
We include the neutrino, even though it is neutral, since neutrinos still have a non-vanishing interaction
amplitude with photons. We shall talk more about neutrinos in Section 6.8. The neutrino energies in this
table are those currently observed; if future neutrino observatories see PeV neutrinos (as expected) then the
constraints will increase dramatically.
This example is overly simplified, as we have ignored Lorentz violation for the photon. However, the
main point remains valid with more complicated forms of Lorentz violation: Constraints can be derived with
current data that are much less than even for Lorentz violation. We now turn to a
discussion of the necessary steps for deriving threshold constraints, as well as the constraints themselves for
more general models.
One must make a number of assumptions before one can analyze Lorentz violating thresholds in a rigorous
Almost all work on thresholds to date has made the assumption that rotational invariance holds.
If this invariance is broken, then our threshold theorems and results do not necessarily hold.
For threshold discussions, we will assume that the underlying EFT is rotationally invariant and
use the notation .
We will assume that the dispersion relation for all particles is monotonically increasing.
This is the case for the mSME with small Lorentz violating coefficients if we work in a
concordant frame. Mass dimension operators generate dispersion relations of the form
which do not satisfy this condition at momentum near the Planck scale if . The turnover
momentum where the dispersion relation is no longer monotonically increasing is
. The highest energy particles known to propagate are the trans-GZK
cosmic rays with energy . Hence unless , is much higher than any relevant
observational energy, and we can make the assumption of monotonicity without loss of
High energy incoming particle
If there is a multi-particle in state, we will assume that one of the particles is much
more energetic than all the others. This is the observational situation in reactions such as
photon-photon scattering or pion production by cosmic rays scattering off the cosmic microwave
background (the GZK reaction; see Section 6.5.6).
6.5.3 Threshold theorems
Eventually, any threshold analysis must solve for the threshold energy of a particular reaction. To do this,
we must first know the appropriate kinematic configuration that applies at a threshold. Of use will be a set
of threshold theorems that hold in the presence of Lorentz violation, which we state below. Variations on
these theorems were derived in  for single particle decays with type dispersion and  for
two in-two out particle interactions with general dispersion. Here we state the more general
Theorem 1: The configuration at a threshold for a particle with momentum is the minimum energy
configuration of all other particles that conserves momentum.
Theorem 2: At a threshold all outgoing momenta are parallel to and all other incoming momentum
6.5.4 New threshold phenomena
Asymmetric thresholds are thresholds where two outgoing particles with equal masses have
unequal momenta. This cannot occur in Lorentz invariant reactions. Asymmetric thresholds
occur because the minimum energy configuration is not necessarily the symmetric configuration.
To see this, let us analyze photon decay, where we have one incoming photon with momentum
and an electron/positron pair with momenta , . We will assume our Lorentz
violating coefficients are such that the electron and positron have identical dispersion.
Imagine that the dispersion coefficients for the electron and positron are negative
and such that the electron/positron dispersion is given by the solid curve in Figure 2. We
define the energy to be the energy when both particles have the same momentum
. This is not the minimum energy configuration, however, if the curvature of
the dispersion relation () at is negative. If we add a momentum to
and to , then we change the total energy by . Since the
curvature is negative, and therefore . The symmetric configuration is
not the minimum energy configuration and is not the appropriate configuration to use for a
threshold analysis for all .
Note that part of the dispersion curve in Figure 2 has positive curvature, as must be the case if at low
energies we have the usual Lorentz invariant massive particle dispersion. If we were considering the
constraints derivable when is small and in the positive curvature region, then the symmetric
configuration would be the applicable one. In general when it is appropriate to use asymmetric
thresholds or symmetric ones depends heavily on the algebraic form of the outgoing particle Lorentz
violation and the energy that the threshold must be above. The only general statement that can be
made is that asymmetric thresholds are not relevant when the outgoing particles have
type dispersion modifications (either positive or negative) or for strictly positive
coefficients at any . For further examples of the intricacies of asymmetric thresholds,
see [154, 167].
||Total outgoing particle energy in symmetric and asymmetric configurations.
Hard Čerenkov thresholds
Related to the existence of asymmetric thresholds is the hard Čerenkov threshold, which also
occurs only when with negative coefficients. However, in this case both the outgoing and
incoming particles must have negative coefficients. To illustrate the hard Čerenkov threshold,
we consider photon emission from a high energy electron, which is the rotated diagram of the
photon decay reaction. In Lorentz invariant physics, electrons emit soft Čerenkov radiation
when their group velocity exceeds the phase velocity of the electromagnetic
vacuum modes in a medium. This type of Čerenkov emission also occurs in Lorentz violating
physics when the group velocity of the electrons exceeds the low energy speed of light in vacuum.
The velocity condition does not apply to hard Čerenkov emission, however, so to understand
the difference we need to describe both types in terms of energy-momentum conservation.
Let us quickly remind ourselves where the velocity condition comes from. The energy conservation
equation (imposing momentum conservation) can be written as
Dividing both sides by and taking the soft photon limit we have
Equation (85) makes clear that the velocity condition is only applicable for soft photon emission.
Hard photon emission can occur even when the velocity condition is never satisfied, if the photon
energy-momentum vector is spacelike with dispersion. As an example, consider an unmodified
electron and a photon dispersion of the form . The energy conservation equation in
the threshold configuration is
where is the incoming electron momentum. Introducing the variable and rearranging,
Since all particles are parallel at threshold, must be between 0 and 1. The maximum value of the
right hand side is , and so we see that we can solve the conservation equation if
, which is approximately . At threshold, so this corresponds to
emission of a hard photon with an energy of .
Upper thresholds do not occur in Lorentz invariant physics. It is easy to see that they are
possible with Lorentz violation, however. In figure 3 the region R in energy space spanned by
is bounded below, since each individual dispersion relation is bounded below.
However, if one can adjust the dispersion freely, as would be the case if the incoming
particle was a unique species in the reaction, then one can choose Lorentz violating coefficients
such that moves in and out of R.
As a concrete example consider photon decay, , with unmodified photon dispersion
and an electron/positron dispersion relation of
chosen strictly for algebraic convenience. This dispersion relation has positive curvature everywhere,
implying that the electron and positron have equal momenta at threshold. The energy conservation
equation, where the photon has momentum is then
which reduces to
Equation (90) has two positive real roots, at , corresponding to
a lower and upper threshold at and , respectively. Such a threshold
structure would produce a deficit in the observed photon spectrum in this energy
Very little currently exists in the literature on the observational possibilities of upper thresholds. A
complicated lower/upper threshold structure has been applied to the trans-GZK cosmic ray events,
with the lower threshold mimicking the GZK-cutoff at and the upper entering below
the highest events at . The region of parameter space where such a scenario
might happen is extremely small, however.
||An example of an upper and lower threshold. R is the region spanned by all
and is the energy of the incoming particle. Where enters and leaves R are
lower and upper thresholds, respectively.
In previous work on the Čerenkov effect based on EFT it has been assumed that left and right
handed fermions have the same dispersion. As we have seen, however, this need not be the case. When
the fermion dispersion is helicity dependent, the phenomenon of helicity decay occurs. One of the
helicities is unstable and will decay into the other as a particle propagates, emitting some sort of
radiation depending on the exact process considered. Helicity decay has no threshold in the
traditional sense; the reaction happens at all energies. However, below a certain energy
the phase space is highly suppressed, so we have an effective threshold that practically
speaking is indistinguishable from a real threshold. As an example, consider the reaction
, with an unmodified photon dispersion and the electron dispersion relation
for right and left-handed electrons. Furthermore, assume that . The opening up of the phase
space can be seen by looking at the minimum and maximum values of the longitudinal photon
momentum. The energy conservation equation is
where is the incoming momentum and is the outgoing photon momentum. We have assumed
that the transverse momentum is zero, which gives us the minimum and maximum values of .
is assumed to be less than ; one can check a posteriori that this assumption is valid. It can be
negative, however, which is different from a threshold calculation where all momenta are necessarily
parallel. Solving Equation (93) for and yields to lowest order in and :
From Equation (94) it is clear that when the phase space is highly suppressed, while
for the phase space in becomes of order . The momentum
acts as an effective threshold, where the reaction is strongly suppressed below this energy. Constraints
from helicity decay in the current literature  are complicated and not particularly useful. Hence
we shall not describe them here, instead focussing our attention on the strict Čerenkov effect when
the incoming and outgoing particle have the same helicity. For an in-depth discussion of helicity decay
constraints see .
6.5.5 Threshold constraints in QED
With the general phenomenology of thresholds in hand, we now turn to the actual observational constraints
from threshold reactions in Lorentz violating QED. We will continue to work in a rotationally invariant
setting. Only the briefest listing of the constraints is provided here; for a more detailed analysis
see [154, 156, 155]. Most constraints in the literature have been placed by demanding that the threshold
for an unwanted reaction is above some observed particle energy. As mentioned previously, a necessary step
in this analysis is to show that the travel times of the observed particles are much longer than the reaction
time above threshold. A calculation of this for the vacuum Čerenkov has been done for QED with
dimension four Lorentz violating operators in . More generally, a simple calculation shows
that the energy loss rate above threshold from the vacuum Čerenkov effect rapidly begins
to scale as , where is a coefficient that depends on the coefficients of the
Lorentz violating terms in the EFT. Similarly, the photon decay rate is . In both
cases the reaction times for high energy particles are roughly , which is
far shorter than the required lifetimes for electrons and photons in astrophysical systems for
The lifetime of a high energy particle in QED above threshold is therefore short enough that we can
establish constraints simply by looking at threshold conditions.
Lorentz violating terms can be chosen such that photons become unstable to decay into
electron-positron pairs . We observe photons from the Crab nebula. There must
exist then at least one stable photon polarization. The thresholds for dispersion
have been calculated in . Demanding that these thresholds are above yields the
following best constraints.
For with CPT preserved we have . If we
set in Equation (39) so that there is no helicity dependence, this translates to the
constraint . If then both helicities of electrons/positrons must
satisfy this bound since the photon has a decay channel into every possible combination of
electron/positron helicity. The corresponding limit is .
For the situation is a little more complicated, as we must deal with photon and electron
helicity dependence, positron dispersion, and the possibility of asymmetric thresholds. The
Crab photon polarizations are unknown, so only the region of parameter space in which both
polarizations decay can be excluded. We can simplify the problem dramatically by noting that the
birefringence constraint on in Equation (40) is . The level of constraints from
threshold reactions at is around [152, 167]. Since the birefringence constraint is so
much stronger than threshold constraints, we can effectively set and derive the photon decay
constraint in the region allowed by birefringence. With this assumption, we can derive a strong
constraint on both and by considering the individual decay channels and
, where and stand for the helicity. For brevity, we shall concentrate on
, the other choice is similar. The choice of a right-handed electron and left-handed
positron implies that both particle’s dispersion relations are functions of only and
hence (see Section 4.1.4). The matrix element can be shown to be large enough
for this combination of helicities that constraints can be derived by simply looking at
the threshold. Imposing the threshold configuration and momentum conservation, and
substituting in the appropriate dispersion relations, the energy conservation equation becomes
where is the incoming photon momentum and is the outgoing electron momentum. Cancelling
the lowest order terms and introducing the variable , this can be rewritten as
To find the minimum energy configuration we must minimize the right hand side of Equation (96)
with respect to (keeping the right hand side positive). We note that since the range of is
between and , the right hand side of Equation (96) can be positive for both positive and
negative , which implies that the bound will be two sided.
As an aside, it may seem odd that photon decay happens at all when the outgoing particles have
opposite dispersion modifications, since the net effect on the total outgoing energy might seem to
cancel. However, this is only the case if both particles have the same momenta. We can always choose
to place more of the incoming momentum into the outgoing particle with a negative coefficient,
thereby allowing the process to occur. This reasoning also explains why the bound is two sided, as
the threshold configuration gives more momentum to whichever particle has a negative
Returning to the calculation of the threshold, minimizing Equation 96), we find that the threshold
The absolute value here appears because we find the minimum positive value of Equation (96).
Placing at yields the constraint and hence . The same
procedure applies in the opposite choice of outgoing particle helicity, so obeys this bound as
The photons observed from the Crab nebula are believed to be produced via inverse
Compton (IC) scattering of charged particles off the ambient soft photon background.
If one further assumes that the charged particles are electrons, it can then be inferred that
electrons must propagate. However, only one of the electron helicities may be propagating,
so we can only constrain one of the helicities.
For the constraint is , where is the coefficient for
one of the electron helicities. In terms of the mSME parameters this condition can be written as
. For an added complication arises. If we consider just electrons as
the source of the photons, then we have that either or must satisfy
and the translation to and is as before. Note that for the range of allowed by
birefringence, the relevant constraint is or .
A major difficulty with the above constraint is that positrons may also be producing some of
the photons from the Crab nebula. Since positrons have opposite dispersion
coefficients in the case, there is always a charged particle able to satisfy the Čerenkov
constraint. Hence by itself, this IC Čerenkov constraint can always be satisfied in the Crab
and gives no limits at all. However, as we shall see in Section 6.7 the vacuum Čerenkov
constraint can be combined with the synchrotron constraint to give an actual two-sided
The high energy photon spectrum (above ) from astrophysical sources such as
Markarian 501 and 421 has been observed to show signs of absorption due to scattering off
the IR background. While this process occurs in Lorentz invariant physics, the amount of
absorption is affected by Lorentz violation. The resulting constraint is not nearly as clear cut as
in the photon decay and Čerenkov cases, as the spectrum of the background IR photons and
the source spectrum are both important, neither of which is entirely known. Various authors
have argued for different constraints on the dispersion relation, based upon how far
the threshold can move in the IR background. The constraints vary from to .
However, none of the analyses take into account the EFT requirement for that opposite
photon polarization have opposite Lorentz violating terms. Such an effect would cause one
polarization to be absorbed more strongly than in the Lorentz invariant case and the other
polarization to be absorbed less strongly. The net result of such a situation is currently unknown,
although current data from blazars suggest that both polarizations must be absorbed to some
degree . Since even at best the constraint is not competitive with other constraints, and
since there is so much uncertainty about the situation, we will not treat this constraint in any
more detail. For discussions see [154, 17].
6.5.6 The GZK cutoff and ultra-high energy cosmic rays
The GZK cutoff
Ultra-high energy cosmic rays (UHECR), if they are protons, will interact strongly with
the cosmic microwave background and produce pions, , losing energy in
the process. As the energy of a proton increases, the GZK reaction can happen with lower
and lower energy CMBR photons. At very high energies (), the interaction
length (a function of the power spectrum of interacting background photons coupled with
the reaction cross section) becomes of order 50 Mpc. Since cosmic ray sources are probably
at further distances than this, the spectrum of high energy protons should show a cutoff
around [135, 281]. A number of experiments have looked for the GZK cutoff,
with conflicting results. AGASA found trans-GZK events inconsistent with the GZK cutoff at
, while Hi-Res has found evidence for the GZK cutoff (although at a lower confidence
level; for a discussion see ). New experiments such as AUGER  may resolve this issue
in the next few years. Since Lorentz violation shifts the location of the GZK cutoff, significant
information about Lorentz violation (even for type dispersion) can be gleaned from
the UHECR spectrum. If the cutoff is seen then Lorentz violation will be severely constrained,
while if no cutoff or a shifted cutoff is seen then this might be a positive signal.
For the purposes of this review, we will assume that the GZK cutoff has been observed and describe
the constraints that follow. We can estimate their size by noting that in the Lorentz invariant case the
conservation equation can be written as
as the outgoing particles are at rest at threshold. Here is the UHECR proton 4-momentum and
is the soft photon 4-momentum. At threshold the incoming particles are anti-parallel, which gives
a threshold energy for the GZK reaction of
where is the energy of the CMBR photon. The actual GZK cutoff occurs at due to
the tail of the CMBR spectrum and the particular shape of the cross section (the resonance).
From this heuristic threshold analysis, however, it is clear that Lorentz violation can become
important when the modification to the dispersion relation is of the same order of magnitude as
the proton mass. For dispersion, a constraint of was
derived in [88, 87, 10]. The case of dispersion with was studied
in [130, 132, 131, 48, 47, 49, 26, 16, 25, 166, 12, 167, 264, 9], while the possibility of
was studied in . A simple constraint  can be summarized as follows. If we
demand that the GZK cutoff is between and then for we
have . If then there is a wedge shaped region in the parameter space
that is allowed .
The numerical values of these constraints should not be taken too literally. While the order of
magnitude is correct, simply moving the value of the threshold for the proton that interacts with a
CMBR photon at some energy does not give accurate numbers. GZK protons can interact with any
photon in the CMBR distribution above a certain energy. Modifying the threshold modifies the
phase space for a reaction with all these photons in the region to varying degrees, which
must be folded in to the overall reaction rate. Before truly accurate constraints can be
calculated from the GZK cutoff, a more detailed analysis to recompute the rate in a Lorentz
violating EFT considering the particulars of the background photon distribution and
-resonance must be done. However, the order of magnitude of the constraints above is
roughly correct. Since they are so strong, the actual numeric coefficient is not particularly
Another difficulty with constraints using the GZK cutoff is the assumption that the source spectrum
follows the same power law distribution as at lower energies. It may seem that proposing a deviation
from the power law source spectrum at that energy would be a conspiracy and considered unlikely.
However, this is not quite correct. A constraint on will, by the arguments above, be such that
the Lorentz violating terms are important only near the GZK energy - below this energy we have
the usual Lorentz invariant physics. However, such new terms could then also strongly
affect the source spectrum only near the GZK energy. Hence the GZK cutoff could vanish
or be shifted due to source effects as well. Unfortunately, we have little idea as to the
mechanism that generates the highest energy cosmic rays, so we cannot say how Lorentz
violation might affect their generation. In summary, while constraints from the position of
the GZK cutoff are impressive and useful, their actual values should be taken with a
grain of salt, since a number of unaccounted for effects may be tangled up in the GZK
A complimentary constraint to the GZK analysis can be derived by recognizing that
protons reach us - a vacuum Čerenkov effect must be forbidden up to the highest observed
UHECR energy [88, 154, 119]. The direct limits from photon emission, treating a
proton as a single constituent are [154, 86, 119] for ,
for , and for . Equivalent
bounds on Lorentz violation in a conjectured low energy limit of loop quantum gravity have
also been derived using UHECR Čerenkov .
Čerenkov emission for UHECR has been used most extensively in , where two-sided limits
on Lorentz violating dimension 4, 5, and 6 operators for a number of particles are derived.
The argument is as follows. If we view a UHECR proton as actually a collection of constituent
partons (i.e. quarks, gauge fields, etc.) then the dispersion correction should be a function of
the corrections for the component partons. By evaluating the parton distribution function for
protons and other particles at high energies,
one can get two sided bounds by considering multiple reactions, in the same way one obtains
two sided bounds in QED. As a simple example, consider only dimension four rotationally
invariant operators (i.e. dispersion) and assume that all bosons propagate with speed
while all fermions have a maximum speed of . Let us take the case . A proton
is about half fermion and half gauge boson, while a photon is 80 percent gauge boson and 20
percent fermion. The net effect, therefore, is that a proton travels faster than a photon and
hence Čerenkov radiates. Demanding that a proton not radiate yields the bound
, similar to the standard Čerenkov bound above.
If instead , then pair emission becomes possible as electrons and positrons are
85 percent fermion and 15 percent gauge boson. Pair emission would also reduce the UHECR
energy, so one can demand that this reaction is forbidden as well. This yields the bound
. Combined with the above bound we have , which is a strong two
sided bound. The parton approach yields two-sided bounds on dimension six operators of order
for all constituent particles, depending on the assumptions made about
equal parton dispersion corrections. Bounds on the coefficients of CPT violating dimension five
operators are of the order .
For the exact constraints and assumptions, see . Note that if one treated electrons, positrons,
and protons as the fundamental constituents with only dispersion and assigned each a
common speed , one would obtain no constraints. Therefore the parton model is more
powerful. However, for higher dimension operators that yield energy dependent dispersion,
simply assigning electrons and protons equal coefficients does yield comparable constraints.
Finally, we comment that  does not explicitly include possible effects such as SUSY that
would change the parton distribution functions at high energy.
6.5.7 Gravitational Čerenkov
High energy particles travelling faster than the speed of graviton modes will emit graviton Čerenkov
radiation. The authors of  have analyzed the emission of gravitons from a high energy particle with
type dispersion and find the rate to be
where is the speed of the particle and is Newton’s constant. We have normalized the speed of
gravity to be one. The corresponding constraint from the observation of high energy cosmic rays is
. This bound assumes that the cosmic rays are protons, uses the highest record energy
, and assumes that the protons have travelled over at least 10 kpc. Furthermore, the bound
assumes that all the cosmic ray protons travel at the same velocity, which is not the case if CPT is violated
or in the mSME.
The corresponding bounds for type dispersion are not known, but one can easily estimate
their size. The particle speed is approximately . For a proton at an energy of
() the constraint on the coefficient is then of . Note though, that in
this case only one of the UHECR protons must satisfy this bound due to helicity dependence. Similarly, the
bound is of .
Equation (101) only considers the effects of Lorentz violation in the matter sector which give rise to a
difference in speeds, neglecting the effect of Lorentz violation in the gravitational sector. Specifically, the
analysis couples matter only to the two standard graviton polarizations. However, as we shall
see in Section 7.1, consistent Lorentz violation with gravity can introduce new gravitational
polarizations with different speeds. In the aether theory (see Section 4.4) there are three new
modes, corresponding to the three new degrees of freedom introduced by the constrained aether
vector. The corresponding Čerenkov constraint from possible emission of these new modes
has recently been analyzed in . Demanding that high energy cosmic rays not emit these
extra modes and assuming no significant Lorentz violation for cosmic rays yields the bounds
on the coefficents in Equation (48). The next to last bound requires that . If, as
the authors of  argue, no gravity-aether mode can be superluminal, then these bounds imply that
every coefficient is generically bounded by . There is, however, a special case given by
where all the modes propagate at exactly the speed of light and hence
avoid this bound.