Natural TeV cutoff of the Higgs field from a multiplicative Lagrangian

The various types of the nonstandard Lagrangian can be added to the standard Lagrangian with the invariant of the equation of motion in the low energy limit. In this paper, we construct the multiplicative Lagrangian of a complex scalar field giving the approximated Klein-Gordon equation from the inverse problem of the calculus of variation. Then, this multiplicative Lagrangian with arbitrary high cutoff is applied to the toy model of the Higgs mechanism in U(1)-gauge symmetry in order to study the simple effects in the Higgs physics. We show that, after spontaneous symmetry breaking happens, the Higgs vacuum expectation value is free from the Fermi-coupling constant and the Higgs field gets the natural cutoff in TeV scale. The other relevant coupling constants, the UV sensitivity of Higgs mass due to the loop correction, some applications on the strong CP problem as well as anomalous small fermion mass, and the cosmological constant problem are also discussed.