Superluminal propagation from IR physics

One may believe that front velocities of waves in a given theory coincide with the UV limit of phase velocities for any dispersion relations. This implies that IR physics is irrelevant to the discussion of propagation speed of waves. We first consider a theory that contains higher spatial derivatives in the wave equation and prove that front velocities coincide with the UV limit of phase velocities, at least, if parity is conserved. However, we also show that front velocities do not coincide with the UV limit of phase velocities in general dispersion relations. We explicitly give several examples in which front velocities are superluminal owing to an IR or intermediate energy scale property of dispersion relations even if the UV limit of phase velocities is luminal. Our finding conveys the important caution that not only UV physics but also IR physics can be significant to superluminality.