00 2 On the Fourier transformation of Renormalization Invariant Coupling

Integral transformations of the QCD invariant (running) coupling and of some related objects are discussed. Special attention is paid to the Fourier transformation, that is to transition from the space-time to the energy–momentum representation. The conclusion is that the condition of possibility of such a transition provides us with one more argument against the real existence of unphysical singularities observed in the perturba-tive QCD. The second one results in a technical comment on the way of " translation " of some singular long–range asymptotic behaviors to the infrared momentum region. It relates to the recent ALPHA collaboration results on the asymptotic behavior of the QCD effective coupling obtained by lattice simulation. Current practice of the QFT calculations, as a rule, employs expression of observables and of some other intermediate renormalization–group (RG) invariant, or covariant, objects in terms of the invariant coupling function (also referred to as the " invariant, or running coupling constant " 1). This invariant coupling ¯ α(Q 2) is a real function of a real positive argument Q 2 ≡ Q 2 − Q 2 0 , momentum transfer squared. The notion of invariant coupling (IC) has initially been introduced 2 — see, e.g., refs.[1, 2] on renormalization group — in terms of a product of real constants z i entering into finite Dyson renormalization transformations taken in the energy–momentum representation. Here, the IC is expressed in terms of scalar (i.e., , Lorentz–invariant) QFT amplitudes taken also in the momentum representation. For instance, in QED one has ¯ α(Q 2) ≡ αd Q 2 µ 2 , m 2 µ 2 ; α with d , the transverse photon propagator amplitude which is real in the Euclidean domain. In QCD, IC ¯ α s (Q 2) is usually defined in a similar way as a product of the expansion 1 In view of semantic absurdity of the last term, we use the expression " invariant coupling function " or " invariant coupling ". 2 Under the name of " invariant charge " (of electron), the natural one in the QED context.