Phase Transition Couplings in U(1) and SU(N) Regularized Gauge Theories

Using a 2-loop approximation for $\beta$-functions, we have considered the corresponding renormalization group improved effective potential in the Dual Abelian Higgs Model (DAHM) of scalar monopoles and calculated the phase transition (critical) couplings in U(1) and SU(N) regularized gauge theories. In contrast to our previous result $\alpha_{crit} \approx 0.17$, obtained in the one-loop approximation with the DAHM effective potential (see Ref.[20]), the critical value of the electric fine structure constant in the 2-loop approximation, calculated in the present paper, is equal to $\alpha_{crit}\approx 0.208$ and coincides with the lattice result for compact QED [10]: $\alpha_{crit}^{lat} \approx 0.20\pm 0.015$. Following the 't Hooft's idea of the "abelization" of monopole vacuum in the Yang--Mills theories, we have obtained an estimation of the SU(N) triple point coupling constants, which is $\alpha_{N,crit}^{-1}=\frac{N}{2}\sqrt{\frac{N+1}{N-1}} \alpha_{U(1),crit}^{-1}$. This relation was used for the description of the Planck scale values of the inverse running constants $\alpha_i^{-1}(\mu)$ (i=1,2,3 correspond to U(1), SU(2) and SU(3) groups), according to the ideas of the Multiple Point Model [16].