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 Diracs 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.
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