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 [202, 201
]) 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 [154
, 183, 3, 2, 124]. 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.
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