Unintegrated Parton Distribution Functions Research Articles

Investigation of the Z boson production via hadron–hadron collisions in the k_t and (z, k_t)-factorization frameworks

In this paper, we study the Z boson production via the proton–proton (p–p) collisions within the k_t and (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-factorization (k_t is the partonic transverse momentum), the KATIE parton level event generator is used, while for the (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)-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 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 and (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)-factorization, predicts the p–p DCS closer to the experimental data with respect to those of k_t-factorization framework.

Far-forward production of charm mesons and neutrinos at forward physics facilities at the LHC and the intrinsic charm in the proton

We discuss production of far-forward $D$ mesons/antimesons and neutrinos/antineutrinos from their semileptonic decays in pp-collisions at the LHC. We include the gluon-gluon fusion $gg \to c\bar{c}$, the intrinsic charm (IC) $gc \to gc$ as well as the recombination $gq \to Dc$ partonic mechanisms. The calculations are performed within the $k_T$-factorization approach and the hybrid model using different unintegrated parton distribution functions (uPDFs) for gluons from the literature, as well as within the collinear approach. We compare our results to the LHCb data for forward $D^{0}$-meson production at $\sqrt{s} = 13$ TeV for different rapidity bins in the interval $2 < y < 4.5$. The IC and recombination models are negligible at the LHCb kinematics. Both the mechanisms start to be crucial at larger rapidities and dominate over the standard charm production mechanisms. At high energies there are so far no experiments probing this region. We present also energy distributions for forward electron, muon and tau neutrinos to be measured at the LHC by the currently operating FASER$\nu$ experiment, as well as by future experiments like FASER$\nu2$ or FLArE, proposed very recently by the Forward Physics Facility project. Again components of different mechanisms are shown separately. For all kinds of neutrinos (electron, muon, tau) the subleading contributions, i.e. the IC and/or the recombination, dominate over light meson (pion, kaon) and the standard charm production contribution driven by fusion of gluons for neutrino energies $E_{\nu} \gtrsim 300$ GeV. For electron and muon neutrinos both the mechanisms lead to a similar production rates and their separation seems rather impossible. On the other hand, for $\nu_{\tau} + {\bar \nu}_{\tau}$ neutrino flux the recombination is further reduced making the measurement of the IC contribution very attractive.

Single and pair J/ψ production in the improved color evaporation model using the parton Reggeization approach

In the article, we study single and pair $J/\ensuremath{\psi}$ hadroproduction in the improved color evaporation model via the parton Reggeization approach. The last one is based on the ${k}_{T}$ factorization of hard processes in multi-Regge kinematics, the Kimber-Martin-Ryskin-Watt model for unintegrated parton distribution functions, and the effective field theory of Reggezied gluons and quarks, suggested by L.N. Lipatov. We compare contributions from the single and double parton scattering mechanisms in the pair $J/\ensuremath{\psi}$ production. The numerical calculations are realized using the Monte-Carlo event generator KaTie.

Production of three isolated photons in the parton Reggeization approach at high energies

We study a large-$p_T$ three-photon production in proton-proton collisions at the LHC. We use the leading order (LO) approximation of the parton Reggeization approach consistently merged with the next-to-leading order corrections originated from the emission of additional jet. For numerical calculations we use the parton-level generator KaTie and modified KMR-type unintegrated parton distribution functions which satisfy exact normalization conditions for arbitrary $x$. We compare our prediction with data from ATLAS collaboration at the center-of-mass energy $\sqrt{s}=8$ TeV. We find that the inclusion of the real next-to-leading-order corrections leads to a good agreement between our predictions and data with the same accuracy as for the next-to-next-to-leading calculations based on the collinear parton model of QCD. At higher energies ($\sqrt{s}=13$ and 27 TeV) parton Reggeization approach predicts larger cross sections, up to $\sim 15$ \% and $\sim 30$ \%, respectively.

Production of forward heavy-flavor dijets at the LHCb within the kT -factorization approach

We calculate differential cross sections for $c \bar c$- and $b \bar b$-dijet production in $pp$-scattering at $\sqrt{s} = 13$ TeV in the $k_T$-factorization and hybrid approaches with different unintegrated parton distribution functions (uPDFs). We present distributions in transverse momentum and pseudorapidity of the leading jet, rapidity difference between the jets and the dijet invariant mass. Our results are compared to recent LHCb data on forward production of heavy flavour dijets, measured recently for the first time individually for both, charm and bottom flavours. We found that an agreement between the predictions and the data within the full $k_T$-factorization is strongly related to the modelling of the large-$x$ behaviour of the gluon uPDFs which is usually not well constrained. The problem may be avoided following the hybrid factorization approach. Then a good description of the measured distributions is obtained with the Parton-Branching, the Kimber-Martin-Ryskin, the Kutak-Sapeta and the Jung setA0 CCFM gluon uPDFs. We calculate also differential distributions for the ratio of $c \bar c$ and $b \bar b$ cross sections. In all cases we obtain the ratio close to 1 which is caused by the condition on minimal jet transverse momentum ($p_{T}^{\mathrm{jet}} > 20$ GeV) introduced in the experiment, that makes the role of heavy quark mass almost negligible. The LHCb experimental ratio seems a bit larger. We discuss potentially important for the ratio effect of $c$- or $b$-quark gluon radiative corrections related to emission outside of the jet cone. The found effect seems rather small. More refine calculation requires full simulation of $c$- and $b$-jets which goes beyond of the scope of the present paper.

On validity of different PDFs sets using the proton k_t-factorization structure functions and the Gaussian k_t-dependence of KMR UPDFs

In this work, we discuss: (i) The ratios of different parton distribution functions (PDFs), i.e., MMHT2014, CJ15, CTEQ6l1, HERAPDF15, MSTW2008, HERAPDF20 and MSHT20, and the corresponding Kimber–Martin–Ryskin (KMR) unintegrated parton distribution functions (UPDFs) sets versus the hard scale Q^2, to find out the sensibility of the KMR UPDFs with respect to the input PDFs sets. It is shown that there is not much difference between the different input-PDFs or corresponding UPDFs sets ratios. (ii) Then, the dependence of proton k_t-factorization structure functions on the different UPDFs sets which can use the above PDFs sets as input, are presented. The results are compared with the experimental data of ZEUS, NMC and H1+ZEUS at the hard scale Q^2 =27 and 90~text {GeV}^2, and a reasonable agreement is found, considering different input PDFs sets. (iii) Furthermore, by fitting a Gaussian function, which depends on the transverse momentum k_t, to the KMR UPDFs and averaging over x (the fractional parton momentum), we obtain the average transverse momentum, langle k_t^2rangle , in the scale range Q^2=2.4{-}1.44times 10^4~text {GeV}^2, which is in agreement with the other groups predictions, 0.25{-}0.44~text {GeV}^2 at Q^2=2.4 text {GeV}^2. (iv) Finally we explore the average transverse momentum for which, the results of proton structure function with the KMR UPDFs and that of the Gaussian k_t-dependent, are consistent to each other. Through the above report, at each step the parton branching (PB) UPDFs, i.e., the transverse momentum dependent PDFs (PB TMDPDFs) are considered for comparisons with the corresponding KMR UPDFs output.

Validity check of the KATIE parton level event generator in the kt -factorization and collinear frameworks

In the present paper, we check and study the validity of the \textsc{KaTie} parton level event generator, by calculating the inclusive electron-proton (ep) dijet and the Proton-proton (p-p) Drell-Yan electron-pair productions differential cross sections in the $k_t$- and collinear factorization frameworks. The Martin-Ryskin-Watt (MRW) unintegrated parton distribution functions (UPDFs) are used as the input UPDFs. The results are compared with those of ZEUS ep inclusive dijet and ATLAS p-p Drell-Yan electron-pair productions, experimental data. The \textsc{KaTie} parton level event generator can directly calculate the cross sections in the $k_t$-factorization framework. It is noticed that the lab to the Breit transformation in this generator is not correctly implemented by its author, so the produced output does not cover the ep ZEUS experimental data in which the mentioned transformation is applied. By fixing the above transformation in the \textsc{KaTie} generator code, we could appropriately produce the ep inclusive dijet differential cross section, in comparison with those of ZEUS data. It is also shown that the MRW at the NLO level, with the angular ordering constraint can successfully predict the ATLAS p-p Drell-Yan data. Finally, as it is expected, we conclude that the $k_t$-factorization is an appropriate tool for the small longitudinal parton momenta and high center of mass energies, with respect to the collinear one.

Analysis of Drell-Yan lepton pair production in the p−p(p¯) colliders using different angular ordering constraints and the kt -factorization approach

In this work, the $p\ensuremath{-}p(\overline{p})$ Drell-Yan lepton pair production (DY) differential cross sections at hadron colliders, such as LHC and TEVATRON, are studied in the ${k}_{t}$-factorization framework. In order to take into account the transverse momenta of incoming partons, we use the unintegrated parton distribution functions of Kimber et al. (KMR) and Martin et al. (MRW) in the leading order (LO), and next-to-leading-order (NLO) levels with the input MMHT2014 PDF libraries. Based on the different off shell partonic matrix elements, we analyze the behaviors of DY differential cross sections with respect to the invariant mass, the transverse momentum and the rapidity as well as the specific angular correlation between the produced leptons. The numerical results are compared with the experimental data, in different energies, which are reported by various collaborations, such as CDF, CMS, ATLAS, and LHCb. It is shown that the NLO-MRW and KMR schemes predict closer results to the data compared to the LO-MRW, since we do not have fragmentation. It is demonstrated that while the ${q}^{*}+{\overline{q}}^{*}\ensuremath{\rightarrow}{\ensuremath{\gamma}}^{*}/Z+g\ensuremath{\rightarrow}{l}^{+}+{l}^{\ensuremath{-}}+g$ subprocess has a negligible contribution, it has a sizable effect in the low dilepton transverse momentum. In agreement with the NNLO, perturbative quantum chromodynamics (pQCD), report (pythia, sherpa, etc.) by including the higher-order perturbative contributions the better results are archived. On the other hand as the scale of energy increases, for the LHC energies, the Compton subprocess, i.e., ${q}^{*}+{g}^{*}\ensuremath{\rightarrow}{\ensuremath{\gamma}}^{*}/Z\ensuremath{\rightarrow}{l}^{+}+{l}^{\ensuremath{-}}+q$, has the largest contribution to the differential cross section in the most intervals of some observables, as is expected. The variation of the differential cross section with respect to the various variables such as the invariant mass, the center of mass energy, etc. are discussed. In order to validate our results, we also consider the strong ordering constraint and the KaTie parton-level event generator.

P−p(p¯) cross section of isolated single photon production in the kt factorization and different angular ordering unintegrated parton distributions frameworks

In the present work, it is intended to calculate and study the single and double differential cross-sections of the prompt single-photon production as a function of produced single-photon transverse momentum and rapidity, in the high-energy P-P(\={P}) colliders, such as LHC and TEVATRON. The differential cross-sections of prompt single-photon production are calculated in the $k_t$-factorization frameworks using various angular ordering unintegrated parton distribution functions (UPDF), namely the Kimber et al. and Martin et al. procedures. These scheme-dependent UPDF are generated in the leading and next-to-leading order levels to predict and analyze the different partonic contributions to the above cross-sections. The above two procedures utilize the phenomenological parton distribution functions (PDF) libraries of Martin et al., i.e., MMHT2014. It is shown that the calculated prompt single-photon production differential cross-sections in the above frameworks are relatively successful in generating satisfactory results compared to the experimental data of different collaborations, i.e., CDF (2017), ATLAS (2017), CMS (2011) and D0 (2006), as well as the other theoretical predictions such as collinear factorization Monte Carlo calculations (the JETPHOX, SHERPA, PYTHIA, and MCFM methods). Also, for a closer precision, the differential cross-section for the NLO gluon-gluon , quark-gluon and quark-(anti)quark processes are calculated. An extensive discussion and comparison are made regarding (i) the behavior of the contributing partonic sub-processes, (ii) the possible double-counting between the $2\to 2$ and $2\to 3$ sub-processes, i.e., gluons-fusion, in our calculated prompt single-photon production differential cross-sections and (iii) the sensibility check of our results to the different angular ordering constraints.

High-energy factorization for the Drell-Yan process in pp and pp¯ collisions with new unintegrated PDFs

The formalism for uniform description of Drell-Yan transverse-momentum spectrum is presented in a framework of High-Energy Factorization, which smoothly interpolates between Collins-Soper-Sterman formalism at $|{\bf q}_T|\ll Q$ and usual Collinear Parton Model at $|{\bf q}_T|\sim Q\ll \sqrt{S}$. The new formula for deriving Unintegrated Parton Distribution Functions(UPDFs) from collinear ones is introduced, which leads to excellent description of the shape of $Z$-boson $|{\bf q}_T|$-spectrum at high energies up to $|{\bf q}_T|/\sqrt{S}\simeq 0.02$. Description of normalized $|{\bf q}_T|$-distributions at low energies is achieved via the fit of non-perturbative parameters of quark UPDFs. Reasonable description of angular distributions of leptons in the dilepton center-of-mass frame is also obtained with new UPDFs.

Pathologies of the Kimber-Martin-Ryskin prescriptions for unintegrated PDFs: Which prescription should be preferred?

We discuss the different Kimber-Martin-Ryskin (KMR) prescriptions for unintegrated parton distribution functions (uPDFs). We show that the strong-ordering (SO) and the angular-ordering (AO) cutoffs lead to strong discrepancies between the obtained cross sections. While the result obtained with the AO cutoff overestimates the heavy-flavor cross section by about a factor of 3, the SO cutoff gives the correct answer. We also solve the issue of the KMR uPDF definitions mentioned by Golec-Biernat and Sta\ifmmode \acute{s}\else \'{s}\fi{}to [Phys. Lett. B 781, 633 (2018)] and show that, in the case of the AO cutoff, the KMR uPDFs are ill defined.

A detailed study of charm content of a proton in the frameworks of the Kimber-Martin-Ryskin and Martin-Ryskin-Watt approaches

The charm structure function, (F2cc‾(x,Q2)), is calculated in the framework of the kt-factorization formalism by using the unintegrated parton distribution functions (UPDF), which are generated through the Kimber et al. (KMR) and Martin et al. (MRW) procedures. The Martin group (MMHT2014) parton distribution functions (PDF) is used as the input PDF for the corresponding UPDF. The resulted F2cc‾(x,Q2) is compared with the predicted data and the calculations given by the ZEUS and H1 collaborations, the parton pQCD theory, i.e. the general-mass variable-flavor-number scheme (GMVFNS), the LO collinear procedure and the saturation model introduced by Golec-Biernat and Wu¨sthoff (GBW), respectively. In general, it is shown that the calculated charm structure functions based on the stated above two UPDF schemes are consistent with the experimental data and other theoretical predictions. Also, a short discussion is presented regarding the KMR and MRWUPDF behaviors.

Consistent treatment of charm production in higher-orders at tree-level within kT -factorization approach

We discuss production of $c \bar c$-pairs within $k_T$-factorization approach (off-shell initial partons) with unintegrated parton distribution functions (uPDFs). We present a consistent prescription which merges the standard leading-order (LO) $k_T$-factorization calculations for this process with tree-level next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) matrix elements. For the first time we include in this framework 2 $\to$ 3 and 2 $\to$ 4 processes with extra partonic emissions for single particle distributions as well as for correlation observables. The use of the KMR uPDF leads to a good description of the existing charm ($D$-meson) data already at the leading-order. On the other hand, a new Parton-Branching (PB) uPDF strongly underestimates the same experimental data. A direct inclusion of the higher-orders at tree-level leads to an overestimation of the data, especially for the KMR uPDF. This suggests a significant double-counting. We propose a simple method how to avoid the double-counting. Our procedure leads to a much better description of the experimental data when including the higher-order contributions. Then with the KMR uPDF we get similar results (both for single particle and correlation observables) as for the standard calculations of the 2 $\to$ 2 processes. For the PB uPDF inclusion of the higher-orders considerably improves description of the experimental data. We conclude that the LO calculation with the KMR uPDF effectively includes the higher-orders which is not the case for the PB uPDF.

A detailed study of the LHC and TEVATRON hadron–hadron prompt-photon pair production experiments in the angular ordering constraint kt-factorization approaches

Based on the kt-factorization framework, it is intended to report a detailed study of the differential cross section of the isolated prompt-photon pairs (IPPP) production in high-energy inelastic hadron–hadron collisions. The two scheme-dependent unintegrated parton distribution functions (UPDF) in which the angular ordering constraints (AOC) are imposed, namely the Kimber–Martin–Ryskin (KMR) and the Martin–Ryskin–Watt (MRW) approaches, in the leading and the next-to-leading orders (LO and NLO) are considered, respectively. These two prescriptions (KMR and MRW) utilize the phenomenological parton distribution function (PDF) libraries of Martin et al, i.e. the MMHT2014. The computations are performed in accordance with the different experimental constraints of the latest existing experiment of the D0, CDF, CMS and ATLAS collaborations. It is shown that the above frameworks are capable of producing acceptable results, compared to the experimental data, the pQCD and some Monte Carlo calculations (i.e. 2γNNLO, SHERPA, DIPHOX and RESBOS). It is also concluded that the KMR framework produces better results in the higher center-of-mass energies, while the same thing can be argued about the LO-MRW prescription in lower energies. Additionally, these two schemes show different behavior in the regions where the fragmentation and higher pQCD effects become important. A clear prediction for the various shoulders and tails, which are detected experimentally, are observed and discussed. The possible double countings between and processes are studied. Finally, in agreement to the work of Golec-Biernat and Stasto, it is shown that there is not any dispute about the application of the AOC and the cut-off, in the above prescriptions, at least in the calculation of the various IPPP differential cross sections. However, mostly the LO and NLO-MRW frameworks are the upper and lower bands of the KMR prescriptions, otherwise they predicate the data.

The EMC ratios of 4He, 3He and 3H nuclei in the kt factorization framework using the Kimber–Martin–Ryskin unintegrated parton distribution functions

The unintegrated parton distribution functions (UPDFs) of 3H, 3He and 4He nuclei are generated to calculate their structure functions (SFs) in the kt-factorization approach. The Kimber–Martin–Ryskin (KMR) formalism is applied to evaluate the double-scale UPDFs of these nuclei from their single-scale parton distribution functions (PDFs), which can be obtained from the constituent quark exchange model (CQEM). Afterwards, these SFs are used to calculate the European Muon Collaboration (EMC) ratios of these nuclei. The resulting EMC ratios are then compared with the available experimental data and good agreement with data is achieved. In comparison with our previous EMC ratios, in which the conventional PDFs were used in the calculations, the accord of the present outcomes with experiment at the small x region becomes impressive. Therefore, it can be concluded that the kt dependence of partons can reproduce the general form of the shadowing effect at the small x values in above nuclei.

Application of the Kimber-Martin-Ryskin and Martin-Ryskin-Watt unintegrated parton distributions to the EMC ratio of a Li6 nucleus in the kt -factorization framework

In the present work, it is intended to calculate the unintegrated parton distribution functions (UPDFs) of the $^6{Li}$ nucleus, which depend not only on the longitudinal momentum fraction $x$ (the Bjorken variable) and the factorization scale $\mu$ of partons, but also on their transverse momentum ($k_t$). Therefore, the KMR and MRW procedures are applied to generate the $k_t$-dependent parton distributions from the familiar integrated parton densities (PDFs), which were determined in our recent related work, using the constituent quark exchange model (CQEM) for the $^6Li$ nucleus. Then, the resulting UPDFs of $^6{Li}$ are compared with the UPDFs of free proton from our previous work. Afterwards, from the $k_t$-factorization formalism, the structure function (SF) of $^6{Li}$ nucleus is computed to extract its European Muon Collaboration (EMC) ratio. The results are compered with those generated from the $^6Li$ nucleus PDFs and the available NMC experimental data. It is observed that, especially in the small $x$ region, where the $k_t$ dependence of partons are important in the calculations of the hadron structure functions, the present EMC ratios are extremely improved compared with those of our previous work, and show an excellent agreement with the NMC experimental data.

Study of inclusive single-jet production in the framework of kt -factorization unintegrated parton distributions

The present work is intended to study the double-differential cross section of the inclusive single-jet production as the functions of the transverse momentum and the rapidity of the jet in the high-energy hadron-hadron collisions. The angular-ordering-constraint ${k}_{t}$-factorization framework is used to calculate the above cross section that is available experimentally. The conditions are taken in accordance with the LHC experiments. The results are compared and analyzed using the existing CMS LHC data. The scheme-dependent unintegrated parton distribution functions (UPDF) of Kimber-Martin-Ryskin (KMR) and Martin-Ryskin-Watt (MRW) in the leading-order and the next-to-leading order (NLO) are used to predict the input partonic UPDF. The utilized phenomenological frameworks prove to be relatively successful in generating satisfactory results compared to the different experiment data, such as CMS (8 and 13 TeV). Extensive discussions and comparisons are made regarding the behavior of the contributing partonic subprocesses. Finally, it is shown that the application of the KMR UPDF to the single-jet differential cross sections have better agreement with the CMS data; on the other hand, they are very similar to those of NLO-MRW.

On the use of the KMR unintegrated parton distribution functions

We discuss the unintegrated parton distribution functions (UPDFs) introduced by Kimber, Martin and Ryskin (KMR), which are frequently used in phenomenological analyses of hard processes with transverse momenta of partons taken into account. We demonstrate numerically that the commonly used differential definition of the UPDFs leads to erroneous results for large transverse momenta. We identify the reason for that, being the use of the ordinary PDFs instead of the cutoff dependent distribution functions. We show that in phenomenological applications, the integral definition of the UPDFs with the ordinary PDFs can be used.

Semi-NLO production of Higgs bosons in the framework of kt-factorization using KMR unintegrated parton distributions

The cross-section for the production of the Standard Model Higgs boson has been calculated using a mixture of LO and NLO partonic diagrams and the unintegrated parton distribution functions (UPDF) of the Kimber–Martin–Ryskin (KMR) from the kt-factorization framework. The UPDF are prepared using the phenomenological libraries of Martin–Motylinski–Harland Lang–Thorne (MMHT2014). The results are compared against the existing experimental data from the CMS and the ATLAS collaborations and available pQCD calculation. It is shown that, while the present calculation is in agreement with the experimental data, it is comparable with the pQCD results. It is also concluded that the K-factor approximation is comparable with the semi-NLOkt-factorization predictions.

A phenomenological study on the production of Higgs bosons in the cSMCS model at the LHC

In the present work, we intend to predict the production rates of the Higgs bosons in the simplest extension of the Standard Model (SM) by a neutral complex singlet (cSMCS). This model has an additional source of CP violation and provides strong enough first-order electroweak phase transition to generate the baryon asymmetry of universe (BAU). The scalar spectrum of the cSMCS includes three neutral Higgs particles with the lightest one considered to be the 125 GeV Higgs boson found at LHC. The SM-like Higgs boson comes mostly from the SM-like SU(2) doublet, with a small correction from the singlet. To predict the production rates of the Higgs bosons, we use a conventional effective LO QCD framework and the unintegrated parton distribution functions (UPDF) of Kimber–Martin–Ryskin (KMR). We first compute the SM Higgs production cross-section and compare the results to the existing theoretical calculations from different frameworks as well as the experimental data from the CMS and ATLAS collaborations. It is shown that our framework is capable of producing sound predictions for these high-energy QCD events in the SM. Afterwards we present our predictions for the Higgs boson production in the cSMCS.