In the framework of the first Born approximation, we investigate the scenario where in addition to a laser field, a nuclear Coulomb field is also present to affect a proton. We work in the approximation in which the proton is considered to be a structureless spin 1/2 Dirac particle with a mass m p . Furthermore, in the laboratory system, the fixed nucleus is treated as a point-like Coulomb potential. In the presence of a laser field, and taking into account the Coulomb effect, the proton will be described by distorted Dirac-Volkov wave functions. The introduction of the Coulomb effect to both the incident and scattered proton will enhance the relativistic differential cross sections (RDCSs). Regarding the physical picture, it is found that for the various kinetic energies of the incident proton, the Coulomb effect can be neglected at high kinetic energies in this particular geometry. Therefore, Dirac-Volkov states are largely sufficient to describe the laser-dressed protons. The behavior of the various RDCSs versus the atomic number Z is also presented.

In the framework of the first Born approximation and using the Dirac–Volkov formalism, we investigate the relativistic model describing the laser assisted elastic differential cross sections (DCSs) for positron scattering by the hydrogen atom. Both the DCSs for electron and positron are notably modified by the circularly polarized laser field, particularly in the regime of high energies and medium intensities. Comparing the two numerical DCSs, we recover the well-known result that the elastic scattering differential cross sections with and without laser field for a hydrogen atom by electron and positron impact are the same (Greiner and Reinhardt. Quantum Electrodynamics, 2nd ed., Springer Verlag, New York, Berlin, Heidelberg. 1994). The numerical results reveal that the differential cross section for positron – hydrogen atom scattering is significantly reduced with the increase of the electric field strength.

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The theory of triple differential cross-sections, developed previously within the framework of nonrelativistic Schrödinger formalism, is extended to the relativistic treatment by using the Dirac-Volkov formalism. At high energies, the colliding particles lose their Coulomb character and the channel of the symmetric coplanar geometry is opened. In the first Born approximation, we have studied theoretically the laser-assisted relativistic ionization of hydrogen atoms by positron impact in this geometry. Triple differential cross-sections are calculated by using two approaches: (i) RPWSG to describe relativistic plane wave in the symmetric geometry, (ii) DVPWSG (Dirac-Volkov plane wave in the symmetric geometry) in which we take full account of the relativistic dressing effects, has been proposed to study the influence of laser field on the colliding particles. To check the consistency of our model, we have used, for comparison, the nonrelativistic approach: nonrelativistic plane wave in the symmetric geometry. Numerical results for the variation of relativistic triple differential cross-sections and their dependences on laser field parameters (intensity, frequency) and incident positron energy are also presented. As for the laser modifications, by summing over a very large number of exchanged photons, the laser-assisted triple differential cross-sections usually tends to approach the laser-free triple differential cross-sections, obeying the famous Kroll-Watson sum rule for positron projectile. In the absence, of any experimental data at high energies for this particular ionization process, we are not in a position to compare the present results with the experimental findings.

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