Generalized dual symmetry of nonabelian theories and the freezing of α_s

C. R. Das, L. V. Laperashvili, H. B. Nielsen

Published 2005-11-23, updated 2006-08-23Version 6

The quantum Yang-Mills theory, describing a system of fields with non-dual (chromo-electric g) and dual (chromo-magnetic \tilde g) charges and revealing the generalized dual symmetry, is developed by analogy with the Zwanziger formalism in QED. The renormalization group equations (RGEs) for pure nonabelian theories are analysed for both constants, \alpha = g^2 / 4\pi and \tilde\alpha = {\tilde g}^2 / 4\pi. The pure SU(3) \times \widetilde{SU(3)} gauge theory is investigated as an example. We consider not only monopoles, but also dyons. The behaviour of the total SU(3) \beta-function is investigated in the whole region of \alpha \equiv \alpha_s: 0 \le \alpha < \infty. It is shown that this \beta-function is antisymmetric under the interchange \alpha \longleftrightarrow 1/\alpha and is given by the well-known perturbative expansion not only for \alpha \ll 1, but also for \alpha \gg 1. Using an idea of the Maximal Abelian Projection by t' Hooft, we considered the formation of strings - the ANO flux tubes - in the Higgs model of scalar monopole (or dyon) fields. In this model we have constructed the behaviour of the \beta-function in the vicinity of the point \alpha = 1, where it acquires a zero value. Considering the phase transition points at \alpha \approx 0.4 and \alpha \approx 2.5, we give the explanation of the freezing of \alpha_s. The evolution of \alpha_s^{-1}(\mu) and the behaviour of V_{eff}(\mu) are investigated for both, perturbative and non-perturbative regions of QCD. It was shown that the effective potential has a minimum, ensured by the dual sector of QCD. The gluon condensate F^2_0, corresponding to this minimum, is predicted: F^2_0 \approx 0.15 GeV^4, in agreement with the well-known results.