  Cooper pairs featuring a nonzero center-of-mass crystal momentum
${\boldsymbol Q}=(\pi,\pi, \dots)$ and an off-diagonal long-range order
($\eta$-pairing states) constitute exact eigenstates of a Hubbard model [C. N.
Yang, Phys. Rev. Lett. 63, 2144 (1989)]. Here we show that the $\eta$-pairing
states are rendered dynamically unstable via coupling to dynamical
electromagnetic fields. The instability is caused by "tachyonic"
electromagnetic fields and unstable plasma modes for attractive and repulsive
interactions, respectively. The typical time scale of the growth of the
instability is of the order of femtoseconds for electron systems in solids,
which places a strict bound on the lifetime of the $\eta$-pairing states. The
decay of the $\eta$-pairing states leads to enhanced light emission with
frequencies shifted from the Hubbard interaction strength for the repulsive
case and unattenuated electromagnetic penetration for the attractive case.
