  Prolonged experimental attempts to find magnetic monopoles (i.e., elementary
particles with an isolated magnetic charge in three dimensions) have not yet
been successful despite intensive efforts made since Dirac's proposal in 1931.
Particle physicists have predicted the possible collision and pair annihilation
of two magnetic charges with opposite signs. However, if such annihilation
exists, its experimental observation would be difficult because its energy
scale is predicted to be tremendously high ($\sim$10$^{16}$ GeV). In the
present work, we theoretically predict using the Floquet theory that a pair of
slightly gapped Dirac-cone bands in a weakly-charge-ordered organic conductor
$\alpha$-(BEDT-TTF)$_2$I$_3$, which behave as magnetic charges with opposite
signs in the momentum space, exhibit pair annihilation under irradiation with
linearly polarized light. This photoinduced pair annihilation is accompanied by
a non-topological phase transition to the Floquet normal insulator phase in
contrast to the well-known circularly-polarized-light-induced topological phase
transition to the Floquet Chern insulator phase. We discuss that
$\alpha$-(BEDT-TTF)$_2$I$_3$ has a peculiar band structure capable of realizing
a suitable experimental condition (i.e., off-resonant condition) and a charge
ordered state providing a required staggered site potential and thereby
provides a rare example of materials that can be used to observe the predicted
pair annihilation phenomenon. The feasibility of experimental observation is
also discussed.
