The LO and the NLO unintegrated parton distributions in the modified DGLAP formalism

The leading order (LO) and the next-to-leading order (NLO) unintegrated parton distribution functions (UPDF) are calculated by using the latest version of integrated parton distribution functions (PDF) of Martin et al. (MSTW2008) as the inputs. Similar to our previous works, rather than the Ciafaloni–Catani–Fiorani–Marchesini (CCFM) evolution equations, the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) collinear approximation is used to consider the dependence of parton distributions on the second scale, kt2, the partons transverse momenta, beside the first scale, μ2, which is included in the last step of DGLAP evolution equation (Kimber et al. procedure). The three-dimensional UPDF are presented in terms of different [x,μ2]-planes and in the range of CERN LHC energies and the parametrization procedure for the various values of kt2. It is shown that the two-scale UPDF behave similar to their corresponding PDF at large kt2≃106GeV2. In both LO and NLO levels at each kt2 a peak is observed around μ2=kt2 especially at x≃10−4 (x⩽10−4) for the gluons (quarks). In contrast to the complication which exists in the parameterized PDF i.e. the negative gluon distribution at small x and μ2, the UPDF are always positive except at large x (≃1) which is mainly due to the angular ordering which makes numerical instability in this region (the values of UPDF are very small). We hope present results could help a better understanding of the LHC data at CERN.