Phenomenological model of multiphoto-production of charged pion pairs on the proton

The production of charged pion pairs via multiphoton absorption from an intense X-ray laser wave colliding with an ultrarelativistic proton beam is studied. Our calculations include the contributions from both the electromagnetic and hadronic interactions where the latter are described approximately by a phenomenological Yukawa potential. Order-of-magnitude estimates for $\pi^+\pi^-$ production on the proton by two- and three-photon absorption from the high-frequency laser field are obtained and compared with the corresponding rates for $\mu^+\mu^-$ pair creation