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

Abstract

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 $F_{2}^{s}(x,Q^{2})$ and G(x,Q ²) as a function of their initial parameterization at the starting scale $Q_{0}^{2}$. 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 $Q_{0}^{2}=1\ \mathrm{GeV}^{2}$.

Appendices

Appendix A

The explicit forms of the functions a(x), b(x), c(x) and d(x) are

(18)

Here the splitting functions are given by [20, 21, 58]

(19)

with C _A =N _c =3, $C_{F}=\frac{N_{c}^{2}-1}{2N_{c}}=\frac{4}{3}$ and $T_{f}=\frac{1}{2}N_{f}$. The convolution integrals in (18) which contains plus prescription, ()₊, can be easily calculated by [59]

(20)

Appendix B

The explicit forms of the functions M(x,t),N(x,t),P(x,t) and Q(x,t) are

(21)

where the strong coupling constant, α _s , up to NNLO is given by Eqs. (6)–(7). The explicit forms of the second- and third-order splitting functions are, respectively [31–33],

(22)

where

(23)

and

(24)

where L0=ln(z), L1=ln(1−z) and D0=1/(1−z).