Recently, the ATLAS data of isolated three-photon production showed that the next-to-leading order (NLO) collinear factorization is not enough to describe experimental data. Therefore, one needs to calculate the cross section beyond the NLO, and as showed later, these data can be well described by the NNLO calculation within the collinear factorization framework. However, it is shown that the $$k_t$$ k t -factorization can be quite successful in describing exclusive and high energy collision processes, henceforth we decided to calculate isolated three-photon production within this framework. In this work we use the Martin, Ryskin, and Watt unintegrated parton distribution functions (MRW UPDFs) at LO and NLO levels, in addition to parton branching (PB) UPDFs in order to calculate cross section which we utilize the KATIE parton level event generator. It will be shown that in contrast to collinear factorization, the $$k_t$$ k t -factorization can describe quiet well the three-photon production ATLAS data. Interestingly our results using the NLO-MRW and PB UPDFs can cover the data within their uncertainty bands, similar to the NNLO collinear results.

In this paper, we investigate the differential cross-sections of the inclusive jet and dijet productions of the ZEUS collaboration data at the center-of-mass energies of ∼300 GeV and 319 GeV using the k t and (z, k t ) − factorizations with the different unintegrated and double-unintegrated parton-distribution functions (UPDFs and DUPDFs), respectively. The KaTie event generator is used to calculate the differential cross-sections of the UPDFs, while for the input DUPDFs, the calculations are directly performed by evaluating the corresponding matrix elements. We check the effects of choosing the different implementations of angular or strong ordering constraints using the UPDFs and the corresponding DUPDFs of the Kimber-Martin-Ryskin (KMR), the leading-order (LO), and next-to-leading-order (NLO) Martin-Ryskin-Watt (MRW) approaches. The impacts of choosing the virtualities(1−z) in the differential cross-section predictions for the ZEUS experimental data are also investigated. It is observed that, as one would expect, the application of (z, k t ) − factorization is better than the k t − factorization framework for predictions of high-virtuality Q 2 with respect to the ZEUS collaboration data, and also that the results of the KMR and LO-MRW UPDFs and DUPDFs are reasonably close to each other and, in general, can describe the data. It is also observed that only in the case of k t -factorization does the inclusion of the Born level make our results overshoot the inclusive jet experimental data.

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In this paper, we study the Z boson production via the proton–proton (p–p) collisions within the $$k_t$$ k t and $$(z, k_t)$$ ( z , k t ) -factorization frameworks, using the Martin–Ryskin–Watt (MRW) unintegrated parton distribution functions (UPDFs) and the double unintegrated parton distribution functions (DUPDFs), respectively. For calculation of the differential cross section (DCS) within the $$k_t$$ k t -factorization ($$k_t$$ k t is the partonic transverse momentum), the KATIE parton level event generator is used, while for the $$(z, k_t)$$ ( z , k t ) -factorization, the DCS is directly computed. Up to the tree level partonic next-to-leading order (NLO) are included, beside the inclusion of branching ratios, in our calculation. It should be noted that Martin, Ryskin and Watt are originally calculated the same process, i.e., the Z boson production, within the $$(z, k_t)$$ ( z , k t ) -factorization framework, while including only the lowest order tree level partonic sub-process. However, the present report extends their work from two perspectives. First, the additional sub-processes are included, second, for the processes up to the next-to-leading order (NLO), a direct calculation by considering the final state leptons is imposed. Finally, we compare our results with the $$13\; TeV$$ 13 T e V data of the ATLAS, LHCb, CMS collaborations, the corresponding collinear factorization predictions and the Modarres, et al. reports. Our p–p DCS calculations show that the $$k_t$$ k t and $$(z, k_t)$$ ( z , k t ) -factorizations frameworks give relatively the same behavior in the central rapidity regions. While at the large rapidity regions, the $$(z, k_t)$$ ( z , k t ) -factorization, predicts the p–p DCS closer to the experimental data with respect to those of $$k_t$$ k t -factorization framework.