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In a series of papers, we have investigated the compatibility of the$Kimber$-$Martin$-$Ryskin$ ($KMR$) and $Martin$-$Ryskin$-$Watt$ ($MRW$)$unintegrated$ parton distribution functions ($UPDF$) as well as thedescription of the experimental data on the proton structure functions. Thepresent work is a sequel to that survey, via calculation of the transversemomentum distribution of the electro-weak gauge vector bosons in the$k_t$-factorization scheme, by the means of the $KMR$, the $LO\;MRW$ and the$NLO\;MRW$ $UPDF$, in the next-to leading order ($NLO$). To this end, we havecalculated and aggregated the invariant amplitudes of the corresponding$involved$ diagrams in the $NLO$, and counted the individual contributions indifferent frameworks. The preparation process for the $UPDF$ utilizes the $PDF$of $Martin$ et al, $MSTW2008-LO$, $MSTW2008-NLO$, $MMHT2014-LO$ and$MMHT2014-NLO$ as the inputs. Afterwards, the results have been analyzedagainst each other, as well as the existing experimental data. Our calculationshow excellent agreement with the experiment data. It is however interesting topoint-out that, the calculation using the $KMR$ framework illustrates astronger agreement with the experimental data, despite the fact that the$LO\;MRW$ and the $NLO\;MRW$ formalisms employ a better theoretical descriptionof the $DGLAP$ evolution equation. This is of course due to the use of thedifferent implementation of the angular ordering constraint in the $KMR$approach, in which automatically includes the re-summation of $ln({1/x})$,$BFKL$ logarithms, in the $LO$-$DGLAP$ evolution equation.

physical review. d/physical review. d.

The $NLO$ production of the $W^{\pm}$ and $Z^0$ vector bosons via hadron  collisions in the frameworks of $KMR$ and $MRW$ unintegrated parton  distribution functions