  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].
