Mass and coupling magnetic field dependence in a scalar theory with charged bosons from an environmentally friendly renormalization group analysis

We compute the running of the mass of a neutral boson and of its self-coupling in a simple model describing the self-interaction of three scalars, one of them neutral and the other two electrically charged, subject to the effects of a magnetic field, as functions of the field strength, at one-loop order. We resort to the environmentally friendly renormalization group approach, where the flow variable is taken as that describing the environmental conditions, in this case, the strength of the magnetic field. We find the magnetic field dependent mass and coupling beta functions and use them to set up the differential equations satisfied by the neutral scalar mass and coupling. We solve the resulting system of coupled equations both numerically and also analytically in the small-mass approximation. We find that the neutral scalar mass increases, while the coupling decreases with increasing field strength. The study is intended to set up the ideas to later use them in more sophisticated theories such as QED and QCD.