We propose a multiplicative formulation of the Higgs Lagrangian, derived from the inverse problem in the calculus of variations, as an alternative framework to investigate the fermion mass hierarchy. In this setup, fermion masses emerge as discrete quantities determined by a finite set of scaling factors, thereby allowing the observed charged-lepton and heavy-quark masses to be accommodated without introducing arbitrarily small parameters. In addition, this framework admits specific solutions where the Yukawa couplings of the charged leptons and heavy quarks converge to a universal value, approximately coinciding with the Higgs self-coupling. This numerical coincidence provides a potential hint of an underlying dynamical structure that correlates the Higgs sector with the fermion masses. Furthermore, the background-dependent Higgs self-interactions are found to decrease asymptotically, ensuring perturbative consistency in the large-field regime and suggesting possible extensions toward ultraviolet completion.
Supanyo et al. (Wed,) studied this question.