Renormalization group and the deep structure of the proton

Abstract

The spirit of the renormalization group approach lies entirely in the observation that in a specific theory the renormalization constants such as the couplings, the masses, are arbitrary mathematical parameters which can be varied by changing arbitrarily the renormalization prescription. For example, given a scale of mass μ, prescriptions can be chosen by doing subtractions of the relevant amplitudes at the continously varying points μ e t , t being an arbitrary real parameter. A representation of such a renormalization group transformatio n μ→ μ=μ e t is the transformation g→ itg ( t) of the renormalized coupling into a continously varying coupling cons the so-called “running coupling constant”. If, for the theory under investigation there exists a domain of the t space where g ( t) i small, then because we are ignorant of how to handle field theory beyond the perturbative approach, one should focus attention on the experimental range in which g (t) “runs” with small values . Indeed, taking momenta q 0 e t = q, corresponds to working with momenta q 0 and mass scale μ e −1, which by a renormalization group transformation as above is the same as working with μ and g(t)(t = log q q 0 ) . In non-Abelian gauge theories, as t increases, g ( t) behaves like 1 t →0 and it is possible to find q and q 0 ranges such that perturbation theory can be applied (asymptotic freedom). The study of physical quantities at different values of their energy momenta is therefore reflected by a study of these quantities at moderate energies with couplings which are not constants but vary with the energies at which one is working. The introduction of couplings varying with energy momentum is the major outcome of the renormalization group formalism. One is therefore able to follow the evolution of these couplings and masses through large appropriate domains of energy. Applicability of naïve perturbation expansion to non-Abelian gauge theories of strong interactions in the domain of high energy momentum transfer is one result from the renormalization group way of looking at these problems and will probably remain its only use. Non-trivial fixed points are unlikely to occur. Emphasis is put in this paper exclusively on deep inelastic scattering of lepton off hadrons, using the light cone operator product expansion. All other processes like lepton-antilepton annihilation into hadrons, lepton pair production in hadronic collisions, large p T production in such collisions are deliberately not discussed, though we know that asymptotic freedom might apply there too; therefore a perturbation treatment in terms of the running coupling constant is sensible for parton induced reactions. Generalities on the factorization procedure are mentioned in the conclusion of this paper. Problems other than those raised in field theory must be treated on a more general basis. Noteworthy examples of critical behaviour show the existence of non-trivial fixed points and non-trivial become the applications of the renormalization group approach. Reference to these phenomena will be only shortly made in the conclusion of this paper, as excellent recent reviews exist already on the subject.