Decoupling of the DGLAP evolution equations at next-to-next-to-leading order (NNLO) at low-x

We present a set of formulas to extract two second-order independent differential equations for the gluon and singlet distribution functions. Our results extend from the LO up to NNLO DGLAP evolution equations with respect to the hard-Pomeron behavior at low-x. In this approach, both singlet quarks and gluons have the same high-energy behavior at low-x. We solve the independent DGLAP evolution equations for the functions $F^{s}_{2}(x,Q^{2})$ and $G(x,Q^{2})$ as a function of their initial parameterization at the starting scale $Q^{2}_{0}$ . The results not only give striking support to the hard-Pomeron description of the low-x behavior, but give a rather clean test of perturbative QCD showing an increase of the gluon distribution and singlet structure functions as x decreases. We compared our numerical results with the published BDM (Block et al. Phys. Rev. D 77:094003 (2008)) gluon and singlet distributions, starting from their initial values at $Q^{2}_{0} =1 GeV^{2}$.