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We study quantum mechanically the influence of an external laser field on the antihydrogen (H-) formation cross sections in ground and excited states (2s,2p) via the three-body recombination process inside a dense plasma of antiprotons and positrons, supposed to be the most efficient mechanism for cold and trapped antihydrogen. The plasma screening is considered in the framework of a Debye- Hückel potential. The laser polarization is chosen to be parallel to the incident momentum of the passive positron. The modifications due to the presence of the laser field are found to be quite significant both quantitatively and qualitatively for all the states. In the presence of a laser, the H formation cross sections are found to be suppressed significantly with respect to the field-free situation particularly in the ground state, while the excited slate cross sections are in general enhanced at very low incident energies but are again suppressed at higher energies except for some special cases. The sensitivity of the formation cross sections with respect to the Debye length are also studied both in the laser-assisted and in the field-free situations. At very low incident energies, the ground state multiphoton formation is found to enhance with the increase in Debye screening, while the reverse is true for the excited states. These findings should have important implications for future H- studies. [ABSTRACT FROM AUTHOR]