We are dealing again with the limitations that pure-kinematics particle-reaction analyses suffer when the properties of the
incoming particles are not fully under control. The pure kinematics of the PKV0 test theory definitely forbids (for negative
of order 1 and
) pion production resulting from collisions between a
proton and a
CMBR photon. But it allows pion production resulting from collisions between a
proton and
more energetic photons, and in order to exclude that possibility one ends up formulating assumptions about
dynamics (the low density of relevant photons may be compensated for by an unexpected increase in cross
section).