ABSTRACT
In this chapter, the concept of running coupling constants is analysed, and put in the broader context of the hypothesis of grand unification of all gauge forces except gravity. From the general expression, it follows that the coupling constant can be either an increasing or a decreasing function of https://www.w3.org/1998/Math/MathML" display="inline"> Q 2 . The discussion of Chapter 13 and 16 shows that the first and second case apply to QED and QCD, respectively. The https://www.w3.org/1998/Math/MathML" display="inline"> Q 2 -dependence of the QED coupling constant https://www.w3.org/1998/Math/MathML" display="inline"> α leads to the appearance of a singularity, known as Landau pole, which can be interpreted as a cutoff corresponding to the limit of the validity of the theory. In the case of QCD, on the other hand, asymptotic freedom has been exploited to determine the behaviour of https://www.w3.org/1998/Math/MathML" display="inline"> α s by measuring the deviations from the scaling limit in high-energy collisions. The different https://www.w3.org/1998/Math/MathML" display="inline"> Q 2 -dependence also has implications relevant to the formulation of the theory on a space-time lattice with finite spacing a. In the continuum limit, that is, for https://www.w3.org/1998/Math/MathML" display="inline"> Λ = ( 1 / a ) → ∞ , QED only exists as a free theory, while QCD does not involve any pathology. The theory of grand unification proposed by Georgi and Glashow, based on the minimal symmetry group containing the Standard Theory, is outlined, and its predictions at a mas scale https://www.w3.org/1998/Math/MathML" display="inline"> M G U T ∼ 10 15 G e V are compared to the high energy behaviour of the coupling constants of the Standard Theory. The https://www.w3.org/1998/Math/MathML" display="inline"> β function describing the evolution of the coupling of self interactions of the only physical scalar field occurring the in the unitary gauge, referred to as Brout-Englert-Higgs boson, is discussed.
