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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 [86Jump To The Next Citation Point130Jump To The Next Citation Point154Jump To The Next Citation Point200]. 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 violation.

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 Figure 1View Image.

View Image

Figure 1: Elements involved in threshold constraints.
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. pAa + pBa + ...= pCa + pDa + ..., where pa is the four momentum of the various particles A, B, C,D, .... 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. [201]).

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. n > 2 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 done.

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, A --> A + g, where A 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 A of the form

n E2 = p2 + m2 + f(n)--p-- (80) A EnP-l2
with (n) fA > 0. For simplicity, in this pedagogical example we shall not change the photon dispersion relation w = k. 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) [154Jump To The Next Citation Point]. The group velocity of A, v = dE/dp, is equal to one at a momentum
[ ] m2En - 2 1/n pth = ------Pl-(n) , (81) (n - 1)fA
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 n = 3 and (3) fe of O(1) at 10TeV, well below the maximum electron energies in astrophysical systems. We can rewrite Equation (81View Equation) for f (nA) and see that the expected constraint from the stability of A at some momentum p obs is
(n) m2En P-l2 fA < --------n--. (82) (n - 1)pobs
Therefore constraints can be much less than order one with particle energies much less than EPl. The orders of magnitude of constraints on f(n) A 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 [154Jump To The Next Citation Point].

- e
+ p

mass < 1 eV 0.511 MeV 938 GeV

p obs 100 TeV 50 TeV 1020 eV

n = 2 10- 28 10-15 10-22
n = 3 10- 14 10-2 10-14
n = 4 1 1012 10-6

Table 2: Orders of magnitude of vacuum Čerenkov constraint for various particles

For neutrinos, pobs comes from AMANDA data [123]. The pobs for electrons comes from the expected energy of the electrons responsible for the creation of ~ 50TeV gamma rays via inverse Compton scattering [188268] in the Crab nebula. For protons, the pobs is from AGASA data [267].

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 O(1) even for n = 4 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.

6.5.2 Assumptions

One must make a number of assumptions before one can analyze Lorentz violating thresholds in a rigorous manner.

Rotation Invariance
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 ---> p = |p |.

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 > 4 operators generate dispersion relations of the form
pn E2 = m2 + p2 + f(n)-n--2, (83) EPl
which do not satisfy this condition at momentum near the Planck scale if f (n) < 0. The turnover momentum pTO where the dispersion relation is no longer monotonically increasing is (n) 1/(n- 2) pTO = (- 2/(nf )) EPl. The highest energy particles known to propagate are the trans-GZK cosmic rays with energy 10-8EPl. Hence unless f (n) » 1, pTO is much higher than any relevant observational energy, and we can make the assumption of monotonicity without loss of generality.

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 [88Jump To The Next Citation Point] for single particle decays with n = 2 type dispersion and [215] for two in-two out particle interactions with general dispersion. Here we state the more general versions.

Theorem 1: The configuration at a threshold for a particle with momentum p1 is the minimum energy configuration of all other particles that conserves momentum.

Theorem 2: At a threshold all outgoing momenta are parallel to p1 and all other incoming momentum are anti-parallel.

6.5.4 New threshold phenomena

Asymmetric thresholds
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 pin and an electron/positron pair with momenta q1, q2. We will assume our Lorentz violating coefficients are such that the electron and positron have identical dispersion.20 Imagine that the dispersion coefficients f (n) for the electron and positron are negative and such that the electron/positron dispersion is given by the solid curve in Figure 2View Image. We define the energy Esymm to be the energy when both particles have the same momentum q1 = q2 = pin/2. This is not the minimum energy configuration, however, if the curvature of the dispersion relation (2 2 @ E/@p) at pin/2 is negative. If we add a momentum Dq to q2 and - Dq to q1, then we change the total energy by DE = DE2 - DE1. Since the curvature is negative, DE1 > DE2 and therefore DE < 0. The symmetric configuration is not the minimum energy configuration and is not the appropriate configuration to use for a threshold analysis for all pin.
View Image

Figure 2: Total outgoing particle energy in symmetric and asymmetric configurations.
Note that part of the dispersion curve in Figure 2View Image 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 pin/2 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 n = 2 type dispersion modifications (either positive or negative) or for strictly positive coefficients at any n. For further examples of the intricacies of asymmetric thresholds, see [154Jump To The Next Citation Point167Jump To The Next Citation Point].

Hard Čerenkov thresholds
Related to the existence of asymmetric thresholds is the hard Čerenkov threshold, which also occurs only when n > 2 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 @E/@p exceeds the phase velocity w/k 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

w(k) = E(p) - E(p - k). (84)
Dividing both sides by k and taking the soft photon limit k --> 0 we have
lim w(k)-= lim E(p)---E(p----k)-= @E--. (85) k-->0 k k-->0 k @p
Equation (85View Equation) 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 n > 2 dispersion. As an example, consider an unmodified electron and a photon dispersion of the form w2 = k2 - k3/EPl. The energy conservation equation in the threshold configuration is
m2 m2 k2 p + --- = p - k + ---------+ k - ----, (86) 2p 2(p - k) 2EPl
where p is the incoming electron momentum. Introducing the variable x = k/p and rearranging, we have
m2E ---3Pl-= x(1 - x). (87) p
Since all particles are parallel at threshold, x must be between 0 and 1. The maximum value of the right hand side is 1/4, and so we see that we can solve the conservation equation if 2 1/3 p > (4m EPl), which is approximately 23TeV. At threshold, x = 1/2 so this corresponds to emission of a hard photon with an energy of 11.5TeV.

Upper thresholds
Upper thresholds do not occur in Lorentz invariant physics. It is easy to see that they are possible with Lorentz violation, however. In figure 3View Image the region R in energy space spanned by Eout(Xk, p1) is bounded below, since each individual dispersion relation is bounded below. However, if one can adjust the dispersion E1(p1) 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 E1(p1) moves in and out of R.
View Image

Figure 3: An example of an upper and lower threshold. R is the region spanned by all Xk and E1(p1) is the energy of the incoming particle. Where E1(p1) enters and leaves R are lower and upper thresholds, respectively.
As a concrete example consider photon decay, + - g - --> e + e, with unmodified photon dispersion and an electron/positron dispersion relation of
2 2 2 7m2/3- 2 -p3- E = p + m - 2/3p + EPl, (88) 2E Pl
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 k is then
( ) k m2 7m2/3 k3 k = 2 --+ ---- ---2/3-k + ----- , (89) 2 k 8E Pl 8EPl
which reduces to
2 7m2/3- 2 -k3- 8m - E2/3 k + EPl = 0. (90) Pl
Equation (90View Equation) has two positive real roots, at V~ -- k = (4 ± 2 2)m2/3E1/P3l, corresponding to a lower and upper threshold at 14TeV and 82TeV, respectively. Such a threshold structure would produce a deficit in the observed photon spectrum in this energy band.21 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 5× 1019 GeV and the upper entering below the highest events at 3 × 1020 GeV [154Jump To The Next Citation Point]. The region of parameter space where such a scenario might happen is extremely small, however.

Helicity decay
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 eL ---> eR + g, with an unmodified photon dispersion and the electron dispersion relation
E2 = m2 + p2, (91) R (4) E2L = m2 + p2 + feL p2 (92)
for right and left-handed electrons. Furthermore, assume that f(4) > 0 eL. 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
2 (4) 2 p + m-- + f-eL-p = p - k + ---m-----+ |k|, (93) 2p 2 2(p - k)
where p is the incoming momentum and k is the outgoing photon momentum. We have assumed that the transverse momentum is zero, which gives us the minimum and maximum values of k. k is assumed to be less than p; 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 (93View Equation) for kmin, and kmax yields to lowest order in m and f(4) eL:
f(eL4)p- kmin = - 4 , (94) f(e4L) p2 kmax = p--2----(4)-2. m + feL p
From Equation (94View Equation) it is clear that when p2« m2/f (4) eL the phase space is highly suppressed, while for 2 2 (4) p « m /feL the phase space in k becomes of order p. The momentum 2 (4) 1/2 pth = (m /feL ) acts as an effective threshold, where the reaction is strongly suppressed below this energy. Constraints from helicity decay in the current literature [155Jump To The Next Citation Point] 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 [155Jump To The Next Citation Point].

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 [154Jump To The Next Citation Point156155Jump To The Next Citation Point]. 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 [224Jump To The Next Citation Point]. More generally, a simple calculation shows that the energy loss rate above threshold from the vacuum Čerenkov effect rapidly begins to scale as e2AEn/En P-l2, where A is a coefficient that depends on the coefficients of the Lorentz violating terms in the EFT. Similarly, the photon decay rate is e2AEn - 1/EnP-l 2. In both cases the reaction times for high energy particles are roughly (e2A) -1En- 2/En -1 Pl, which is far shorter than the required lifetimes for electrons and photons in astrophysical systems for n = 2,3.22 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.

Photon decay
Lorentz violating terms can be chosen such that photons become unstable to decay into electron-positron pairs [152Jump To The Next Citation Point]. We observe 50 TeV photons from the Crab nebula. There must exist then at least one stable photon polarization. The thresholds for n = 2,3 dispersion have been calculated in [154Jump To The Next Citation Point]. Demanding that these thresholds are above 50 TeV yields the following best constraints.

For n = 2 with CPT preserved we have (2) (2) 2 2 -16 fg - fe < 4m /pth = 4× 10 [154Jump To The Next Citation Point]. If we set d = 0 in Equation (39View Equation) so that there is no helicity dependence, this translates to the constraint kF /2 + c < 4 × 10-16. If d /= 0 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 -16 kF /2 + (c± d) < 4 .10.

For n = 3 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 50TeV 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 q in Equation (40View Equation) is |q|< 10-4 [127]. The level of constraints from threshold reactions at 50 TeV is around 10-2 [152167Jump To The Next Citation Point]. Since the birefringence constraint is so much stronger than threshold constraints, we can effectively set q = 0 and derive the photon decay constraint in the region allowed by birefringence. With this assumption, we can derive a strong constraint on both jR and jL by considering the individual decay channels g --> e-R + e+L and g --> e- + e+ L R, where L and R stand for the helicity. For brevity, we shall concentrate on g --> e- + e+ R L, 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 (3) feR and hence jR (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

m2 f(3) m2 f(3) k = p + ---+ -eR-p2 + k - p + ---------- -eR-(k - p)2, (95) 2p 2EPl 2(k - p) 2EPl
where k is the incoming photon momentum and p is the outgoing electron momentum. Cancelling the lowest order terms and introducing the variable z = 2p/k - 1, this can be rewritten as
2 3 ---4m--EPl--- k = f (3)(- z + z3) . (96) eR
To find the minimum energy configuration we must minimize the right hand side of Equation (96View Equation) with respect to z (keeping the right hand side positive). We note that since the range of z is between - 1 and +1, the right hand side of Equation (96View Equation) can be positive for both positive and negative (3) feR, 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 coefficient.

Returning to the calculation of the threshold, minimizing Equation 96View Equation), we find that the threshold momentum is

|| 2 V~ -||1/3 kth = ||m--EPl-6 3||. (97) | f (3) | eR
The absolute value here appears because we find the minimum positive value of Equation (96View Equation). Placing kth at 50 TeV yields the constraint |f(3)|< 0.25 eR and hence |jR| < 0.125. The same procedure applies in the opposite choice of outgoing particle helicity, so j L obeys this bound as well.
Vacuum Čerenkov
The 50 TeV 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.23 If one further assumes that the charged particles are electrons, it can then be inferred that 50 TeVelectrons must propagate. However, only one of the electron helicities may be propagating, so we can only constrain one of the helicities. For n = 2 the constraint is f(e2)- f(g2)< m2/p2th = 10 -16 [154Jump To The Next Citation Point], where fe(2) is the coefficient for one of the electron helicities. In terms of the mSME parameters this condition can be written as -16 c- |d| - kF/2 < 10. For n = 3 an added complication arises. If we consider just electrons as the source of the 50 TeV photons, then we have that either (3) feL or (3) f eR must satisfy
(3) (3) (3) (3) fe < 0.012 V~ ------------------ for fe > 0 and fg > - 3fe , (3) (3) 2 (3) (3) (3) (3) (3) (98) fg > f e - 0.048 - 2 0.024 - 0.048fe for fg < - 3fe < 0 or f g < fe < 0,
and the translation to q and jR,L is as before. Note that for the range of q allowed by birefringence, the relevant constraint is jR < 0.012 or jL < 0.012.

A major difficulty with the above constraint is that positrons may also be producing some of the 50 TeV photons from the Crab nebula. Since positrons have opposite dispersion coefficients in the n = 3 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 bound.

Photon annihilation
The high energy photon spectrum (above 10TeV) 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 n = 3 dispersion relation, based upon how far the threshold can move in the IR background. The constraints vary from O(1) to O(10). However, none of the analyses take into account the EFT requirement for n = 3 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 [263Jump To The Next Citation Point]. 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 [154Jump To The Next Citation Point17].

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, p + g - --> p + p0, 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 (5× 1019 eV), 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 19 5× 10 eV [135281]. A number of experiments have looked for the GZK cutoff, with conflicting results. AGASA found trans-GZK events inconsistent with the GZK cutoff at 2.5s [96], while Hi-Res has found evidence for the GZK cutoff (although at a lower confidence level; for a discussion see [263]). New experiments such as AUGER [113] 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 n = 4 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

2 2 (p + k) = (mp + mp) , (99)
as the outgoing particles are at rest at threshold. Here p is the UHECR proton 4-momentum and k is the soft photon 4-momentum. At threshold the incoming particles are anti-parallel, which gives a threshold energy for the GZK reaction of
( ) E -~ 3 × 1020 GeV . -w0--- , (100) GZK 2.7 K
where w0 is the energy of the CMBR photon. The actual GZK cutoff occurs at 5 × 1019 eV due to the tail of the CMBR spectrum and the particular shape of the cross section (the D 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 n = 2 dispersion, a constraint of (2) (2) -23 fp - fp < O(10 ) was derived in [88Jump To The Next Citation Point8710]. The case of n = 3 dispersion with f(p3)= f(p3) was studied in [13013213148474926162516612Jump To The Next Citation Point1672649], while the possibility of (3,4) (3,4) fp /= fp was studied in [154Jump To The Next Citation Point]. A simple constraint [154Jump To The Next Citation Point] can be summarized as follows. If we demand that the GZK cutoff is between 2× 1019 eV and 7 × 1019 eV then for (3) (3) fp = f p we have |f(p3)|< O(10 -14). If fp(3) /= f(p3) then there is a wedge shaped region in the parameter space that is allowed [154Jump To The Next Citation Point].

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 D-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 important.24 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 f(n) 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 cutoff.

UHECR Čerenkov
A complimentary constraint to the GZK analysis can be derived by recognizing that 1019- 1020 eV protons reach us - a vacuum Čerenkov effect must be forbidden up to the highest observed UHECR energy [88Jump To The Next Citation Point154Jump To The Next Citation Point119Jump To The Next Citation Point]. The direct limits from photon emission, treating a 19 5 × 10 eV proton as a single constituent are (2) (2) fp - fg < 4× 10-22 [154Jump To The Next Citation Point86Jump To The Next Citation Point119Jump To The Next Citation Point] for n = 225, f(p3)- f(g3)< O(10 -14)for n = 3 [154Jump To The Next Citation Point], and f(p4)- f (4g) < O(10 -5) for n = 4 [154Jump To The Next Citation Point]. Equivalent bounds on Lorentz violation in a conjectured low energy limit of loop quantum gravity have also been derived using UHECR Čerenkov [190]. Čerenkov emission for UHECR has been used most extensively in [119Jump To The Next Citation Point], 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 energies26, 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. n = 2 dispersion) and assume that all bosons propagate with speed 1 while all fermions have a maximum speed of 1- e. Let us take the case e < 0. 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 20 10 eV proton not radiate yields the bound e > - 10-23, similar to the standard Čerenkov bound above. If instead e > 0, then + - e e 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 e < 10-23. Combined with the above bound we have |e| < 10-23, which is a strong two sided bound. The parton approach yields two-sided bounds on dimension six operators of order (4) -2 |f |< O(10 ) 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 -15 10.27 For the exact constraints and assumptions, see [119Jump To The Next Citation Point]. Note that if one treated electrons, positrons, and protons as the fundamental constituents with only n = 2 dispersion and assigned each a common speed 1 - e, 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 (n) f does yield comparable constraints. Finally, we comment that [119] 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 [224] have analyzed the emission of gravitons from a high energy particle with n = 2 type dispersion and find the rate to be

dE Gp4 2 --- = ----(cp - 1) , (101) dt 3
where cp is the speed of the particle and G 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 cp - 1 < 2 × 10- 15. This bound assumes that the cosmic rays are protons, uses the highest record energy 3 × 1020 eV, 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 d /= 0 in the mSME.

The corresponding bounds for n = 3,4 type dispersion are not known, but one can easily estimate their size. The particle speed is approximately 1 + f (n)(E/EPl)n -2. For a proton at an energy of 20 10 eV (-8 10 EPl) the constraint on the coefficient (3) f is then of -7 O(10 ). Note though, that in this case only one of the UHECR protons must satisfy this bound due to helicity dependence. Similarly, the n = 4 bound is of O(10).

Equation (101View Equation) 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 [105Jump To The Next Citation Point]. Demanding that high energy cosmic rays not emit these extra modes and assuming no significant Lorentz violation for cosmic rays yields the bounds

-15 - c1- c3 < 1 × 10 , (c1 + c3)2(c2+ 3c1c3 - 2c4) -----------1-2-------------< 1.4× 10-31, c1 (c3--c4)2- -30 (102) |c + c | < 1 × 10 , 1 4 c4- c2 - c3 -19 ------------< 3 × 10 c1
on the coefficents in Equation (48View Equation). The next to last bound requires that (c4 - c2- c3)/c1 > 10- 22. If, as the authors of [105Jump To The Next Citation Point] argue, no gravity-aether mode can be superluminal, then these bounds imply that every coefficient is generically bounded by |c| < 10- 15 i. There is, however, a special case given by c3 = - c1,c4 = 0,c2 = c1/(1- 2c1) where all the modes propagate at exactly the speed of light and hence avoid this bound.
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