Hooft ’ s representation of the β - function

It is demonstrated, that 't Hooft's renormalization scheme (in which the β-function has exactly the two-loop form) is generally in conflict with the natural physical requirements and specifies the type of the field theory in an arbitrary manner. It violates analytic properties in the coupling constant plane and provokes misleading conclusion on accumulation of singularities near the origin. It artificially creates renormalon singularities, even if they are absent in the physical scheme. The 't Hooft scheme can be used in the framework of perturbation theory but no global conclusions should be drawn from it. 1. It is well-known, that the renormalization procedure is ambiguous [1, 2]. Let for simplicity only the interaction constant g is renormalized. Any observable quantity A, defined by a perturbation expansion, is a function F (g 0 , Λ) of the bare value g 0 and the momentum cutoff Λ. According to the renormalization theory, A becomes independent on Λ, if it is expressed in terms of renormalized g: A = F (g 0 , Λ) = F R (g). (1) The renormalized coupling constant g is usually defined in terms of a certain vertex, e.g. the four-leg vertex Γ 4 (p i , m) in the gφ 4 theory, attributed to a certain length scale L through some choice of mass m and momenta p i. Two types of definition are conventionally used: (1) m is finite, p i = 0, and g = Γ 4 (0, m) corresponds to a length scale L ∼ m −1 ; (2) m = 0, p i ∼ µ, and g = Γ 4 (p i , 0) corresponds to a length scale L = µ −1 ; the condition p i ∼ µ is technically realized by the equality p i · p j = a ij µ 2 , where a ij are usually taken for the so called " symmetric point " , a ij = (4δ ij − 1)/3, though any other choice a ij ∼ 1 is possible. Already the choice either (1) or (2) with different constants a ij provides essential ambiguity of the renormalization scheme. In fact, the physical condition that g is determined by a vertex Γ 4 on the length scale L can be realized technically in many variants (e.g. using averaging over p i with some weight function localized on the scale L −1) 1. 1 The latter …