ABSTRACT
This chapter is devoted to the discussion of the limits on the mass of the scalar fields needed to enable the mechanism of spontaneous symmetry breaking through generation of the masses of quarks, leptons and vector bosons in the Standard Theory. The limits are dictated by the requirement of stability of the theory up to some energy scale, e.g. the energy of grand unification or the Plank mass, https://www.w3.org/1998/Math/MathML" display="inline"> M P l a n k = 1.2 × 10 19 G e V , obtained combining Newton's gravitational constant, the speed of light in vacuum and the quantum of action https://www.w3.org/1998/Math/MathML" display="inline"> ℏ . The Plank mass may be interpreted as a cutoff allowing to relieve the so-called triviality issue, originating form the occurrence of the Landau pole in QED. The https://www.w3.org/1998/Math/MathML" display="inline"> β function describing the evolution of the coupling constant associated with interactions of the only physical scalar field appearing in the unitary gauge, referred to as Brout-Englert-Higgs boson, is discussed. The results of this analysis show the presence of a Landau pole, analogue to the one encountered in QED. The demand that no such pole occur below a given energy https://www.w3.org/1998/Math/MathML" display="inline"> Λ determines an upper limit for the Higgs boson mass, https://www.w3.org/1998/Math/MathML" display="inline"> M H . Depending on the coupling between the Higgs boson and the top quark, a lower limit on https://www.w3.org/1998/Math/MathML" display="inline"> M H , corresponding to the region of https://www.w3.org/1998/Math/MathML" display="inline"> β ( λ ) < 0 and implying that the theory is unstable, also emerges. The upper and lower bounds on https://www.w3.org/1998/Math/MathML" display="inline"> M H resulting from theoretical studies—updated to take into account the measured mass of the top quark—are reported, and the behaviour of the effective coupling constant https://www.w3.org/1998/Math/MathML" display="inline"> λ obtained using the observed value of https://www.w3.org/1998/Math/MathML" display="inline"> M H is discussed.
