Fermion mass mixing and vacuum triviality in the renormalization group procedure for effective particles

Abstract: Renormalization group procedure for effective particles is applied to the model quantum theory of free fermions to which one adds an interaction in the form of a mass mixing term. If one used a standard approach based on the instant form of dynamics, the theory would suffer from a generic vacuum problem caused by a divergent production of virtual quanta out of a bare vacuum and it would require an adjustment of its degrees of freedom to the added interaction term before quantization, considered a means of avoiding the quantum vacuum problem. In the effective particle approach, the quantum vacuum problem is dealt with instead by using the front form of dynamics, where the pair production is excluded by momentum conservation. The corresponding Hamiltonian includes mass parameters through constraint equations while the required quantum field operators are constructed independently of all mass parameters, including the parameters that appear in the added mass mixing interaction term. Then the masses and states of physical fermions emerge at an end of the non-perturbative calculation that is carried out entirely in one and the same interacting quantum theory with a trivial vacuum and no quantization adjustment. An a priori infinite set of renormalization group equations for all momentum modes of fermion fields is reduced to just one equation for a two-by-two mass squared matrix, thanks to 7 kinematical symmetries of the front form (the instant form has only 6). For strong mass mixing interactions, the fermion model solutions qualitatively differ from the analogous, earlier found boson model solutions by the absence of tachyons.