LHC production of forward-center and forward-forward di-jets in the $k_t$-factorization $unintegrated$ parton distribution frameworks

The present work is devoted to study the high-energy $QCD$ events, such as the di-jet productions from proton-proton inelastic collisions at the $LHC$ in the forward-center and the forward-forward configurations, using the $unintegrated$ parton distribution functions ($UPDF$) in the $k_t$-factorization framework. The $UPDF$ of $Kimber$ et. al. ($KMR$) and $Martin$ et.al. ($MRW$) are generated in the leading order ($LO$) and next-to-leading order ($NLO$), using the $Harland-Lang$ et al. ($MMHT2014$) $PDF$ libraries. While working in the forward-center and the forward-forward rapidity sectors, one can probe the parton densities at very low longitudinal momentum fractions ($x$). Therefore, such a computation can provide a valuable test-field for these $UPDF$. We find very good agreement with the corresponding di-jet production data available from $LHC$ experiments. On the other hand, as we have also stated in our previous works, (i.e. the protons longitudinal and transverse structure function as well as hadron-hadron $LHC$ $W/Z$ production), the present calculations based on the $KMR$ prescriptions show a better agreement with the corresponding experimental data. This conclusion is achieved, due to the particular visualization of the angular ordering constraint ($AOC$), despite the fact that the $LO-MRW$ and the $NLO-MRW$ formalisms both employ better theoretical descriptions of the $Dokshitzer$-$Gribov$-$Lipatov$ -$Altarelli$-$Parisi$ ($DGLAP$) evolution equation, and hence are expected to produce better results. The form of the $AOC$ in the $KMR$ prescription automatically includes the re-summation of the higher-order $ln({1/x})$ type contributions, i.e. the $Balitski$-$Fadin$-$Kuraev$-$Lipatov$ ($BFKL$) logarithms, in the $LO$-$DGLAP$ evolution equation.