ABSTRACT
At high energy the running coupling constant of QCD approaches to zero. Theories possessing such a property are called asymptotically free. First, this property of non-Abelian theories was ascertained in works by Gross and Wilczek [24, 25] and Politzer [26, 27] as well. It should be stressed, besides non-Abelian theories of gauge fields, none of the renormalizable field theory can be asymptotically free [2, 28]. Thus, the distinguishing feature of the Yang-Mills theories from all other field theories consists in the fact that the coupling constant falls with a decrease of distance. In the well-known QED the effective coupling constant decrease may be intuitively understood reasoning from the fact that, when increasing a distance between charges, a dielectric shielding of charges is strengthened by a cloud of virtual electron-positron pairs. Therefore, our task is to establish the reason of the origin of the shielding effect in non-Abelian theories. At first we investigate Green function properties at large transferred momenta in a space-
like region. These properties prove to be useful both in calculating radiative corrections to processes at high energies and in studying the effective coupling constant evolution. It is evident that in the large momenta region the mass terms in the Lagrangian may be omitted so as to take them into account as corrections on the following stages of calculations. This is why we are constrained by the case of the massless QCD. In the previous section we found the renormalized Green functions in the one-loop approx-
imation. It is worth noting that these functions must be defined by way of the renormalized coupling constant gs rather than the initial one gs0. Thus, the renormalized Green functions become dependent both on gs and on the normalization point κ. In QED the normalization point is fixed by the condition that a photon has zero momentum and an electron is on the mass shell (pˆ = m). The advantage of such a scheme follows from Thirring’s theorem [29] which reads: at zero photon energy the Compton scattering amplitude in all orders of constant α is exactly given by the classical formula. Hence, for the fundamental theory parameters α and me to be determined, classical expressions may be used. In QCD such a chosen scheme based on physical reasons is absent. Moreover, in QCD, as in any non-Abelian gauge theory, there are badly controllable infrared singularities and so the normalization point is taken under space-like momenta where vertices have no singularities. At that the numerical value of the color charge must be determined by comparing the theoretical and experimental values of the cross sections for the process, in which the vertex with external momenta at the normalization point enters as the subprocess. However, according
Particle and
to the contemporary conceptions free quarks and gluons must not exist. Therefore, we should discuss the problem concerning changes to appear in observables under variation of the normalization point. Since the initial Lagrangian involves only one parameter, the initial coupling constant
gs0, there must exist a dependence between gs and κ, which reveals oneself in the existence of the invariant charge
gs = gs(p 2/κ2, gs(κ)), (4.1)
to remain constant at a variation of κ and gs(κ). Derive a differential equation defining connection between κ and gs(κ). In accordance
with the multiplicative renormalization program considered, we introduce renormalization constants ZG(ν) and Zq(ν) in such a way that the finite limit exists
lim ν→0
(nG,nq)(ki; pj , ν).
