Mass shift of a charged scalar particle in an intense standing electromagnetic wave

A method for the approximate solution to the Klein-Gordon equation for a charged particle in the presence of two electromagnetic, noncollinear, transverse plane waves has been developed. The solution is a product of functions which are either solutions to linear, ordinary, differential equations of the Hill type or solutions to a Volkov equation. The mass-energy relation for the scalar particle in this approximation has a complicated dependence on the field strengths and on the particle momentum. In particular, we find that a slowly moving particle in an intense standing electromagnetic wave has a mass shift of order $\frac{v^2}{c^2}$ times that given by second-order perturbation theory. Here v is the component of particle velocity perpendicular to the direction of "propagation" of the waves.