### 6.4 Threshold constraints

We now turn our attention from astrophysical tests involving a single particle species to threshold reactions, which often involve many particle types. Before delving into the calculational details of energy thresholds with Lorentz violation, we give a pedagogical example that shows why particle decay processes (which involve rates) give constraints that are only functions of reaction threshold energies. Consider photon decay, (see Section 6.5.5 for details). In ordinary Lorentz invariant physics the photon is stable to this decay process. What forbids this reaction is solely energy/momentum conservation - two timelike four-momenta (the outgoing pair) cannot add up to the null four momentum of the photon. If, however, we break Lorentz invariance and assume a photon obeys a dispersion relation of the form
while electrons/positrons have their usual Lorentz invariant dispersion, then it is possible to satisfy energy conservation equation if (to see this intuitively, note that the extra term at high energies acts as a large effective mass for a photon). Therefore a photon can decay to an electron positron pair.

This type of reaction is called a threshold reaction as it can happen only above some threshold energy where is the electron mass. The threshold energy is translated into a constraint on in the following manner. We see photons from the Crab nebula [268], hence this reaction must not occur for photons up to this energy as they travel to us from the Crab. If the decay rate is high enough, one could demand that is above , constraining and limiting this type of Lorentz violation. For , , and so we can get a slightly better than constraint on from photons [152]. If, however, the rate is very small then even though a photon is above threshold it could still reach us from the Crab. Using the Lorentz invariant expression for the matrix element (i.e. just looking at the kinematical aspect of Lorentz violation) one finds that as increases above the rate very rapidly becomes proportional to . If a photon is above threshold, the decay time is then approximately . The travel time of a photon from the Crab is seconds. Hence if a photon is at all above threshold it will decay almost instantly relative to the observationally required lifetime. Therefore we can neglect the actual rate and derive constraints simply by requiring that the threshold itself is above .

It has been argued that technically, threshold constraints can’t truly be applicable to a kinematic model where just modified dispersion is postulated and the dynamics/matrix elements are not known. This isn’t actually a concern for most threshold constraints. For example, if we wish to constrain at by photon decay, then we can do so as long as is within 11 orders of magnitude of its Lorentz invariant value (since the decay rate goes as ). Hence for rapid reactions, even an enormous change in the dynamics is irrelevant for deriving a kinematic constraint. Since kinematic estimates of reaction rates are usually fairly accurate (for an example see [202201]) one can derive constraints using only kinematic models. In general, under the assumption that the dynamics is not drastically different from that of Lorentz invariant effective field theory, one can effectively apply particle reaction constraints to kinematic theories since the decay times are extremely short above threshold.

There are a few exceptions where the rate is important, as the decay time is closer to the travel time of the observed particle. Any type of reaction involving a weakly interacting particle such as a neutrino or graviton will be far more sensitive to changes in the rate. For these particles, the decay time of observed particles can be comparable to their travel time. As well, any process involving scattering, such as the GZK reaction () or photon annihilation () is more susceptible to changes in as the interaction time is again closer to the particle travel time. Even for scattering reactions, however, would need to change significantly to have any effect. Finally, is important in reactions like (), which are not observed in nature but do not have thresholds [15418332124]. In these situations, the small reaction rate is what may prevent the reaction from happening on the relevant timescales. For all of these cases, kinematics only models should be applied with extreme care. We now turn to the calculation of threshold constraints assuming EFT.