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 1.
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. [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. 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.
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 the form
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) [154]. The group velocity of , , is equal to one at a momentum 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 is 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 [154].For neutrinos, comes from AMANDA data [123]. 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 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 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 manner.
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 [88] for single particle decays with 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 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 are anti-parallel.
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, we have 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 .
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 [224]. 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 .^{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.
For with CPT preserved we have [154]. 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 [127]. 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 coefficient.
Returning to the calculation of the threshold, minimizing Equation 96), we find that the threshold momentum is
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 well.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 bound.
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 [154]. A simple constraint [154] 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 [154].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 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 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.
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 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 [105]. 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 [105] 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.http://www.livingreviews.org/lrr-2005-5 |
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