Unintegrated Parton Distribution Functions Research Articles

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.

KMR kt-factorization procedure for the description of the LHCb forward hadron–hadron Z0 production at [formula omitted

Quite recently, two sets of new experimental data from the LHCb and the CMS Collaborations have been published, concerning the production of the Z0 vector boson in hadron–hadron collisions with the center-of-mass energy ECM=s=13TeV. On the other hand, in our recent work, we have conducted a set of semi-NLO calculations for the production of the electro-weak gauge vector bosons, utilizing the unintegrated parton distribution functions (UPDF) in the frameworks of Kimber–Martin–Ryskin (KMR) or Martin–Ryskin–Watt (MRW) and the kt-factorization formalism, concluding that the results of the KMR scheme are arguably better in describing the existing experimental data, coming from D0, CDF, CMS and ATLAS Collaborations. In the present work, we intend to follow the same semi-NLO formalism and calculate the rate of the production of the Z0 vector boson, utilizing the UPDF of KMR within the dynamics of the recent data. It will be shown that our results are in good agreement with the new measurements of the LHCb and the CMS Collaborations.

Transverse momentum dependent (TMD) parton distribution functions generated in the modified DGLAP formalism based on the valence-like distributions

Transverse momentum dependent (TMD) parton distributions, also referred to as unintegrated parton distribution functions (UPDFs), are produced via the Kimber–Martin–Ryskin (KMR) prescription. The GJR08 set of parton distribution functions (PDFs) which are based on the valence-like distributions is used, at the leading order (LO) and the next-to-leading order (NLO) approximations, as inputs of the KMR formalism. The general and the relative behaviors of the generated TMD PDFs at LO and NLO and their ratios in a wide range of the transverse momentum values, i.e. [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are investigated. It is shown that the properties of the parent valence-like PDFs are imprinted on the daughter TMD PDFs. Imposing the angular ordering constraint (AOC) leads to the dynamical variable limits on the integrals which in turn increase the contributions from the lower scales at lower [Formula: see text]. The results are compared with our previous studies based on the MSTW2008 input PDFs and it is shown that the present calculation gives flatter TMD PDFs. Finally, a comparison of longitudinal structure function [Formula: see text] is made by using the produced TMD PDFs and those that were generated through the MSTW2008-LO PDF from our previous work and the corresponding data from H1 and ZEUS collaborations and a reasonable agreement is found.

LHC production of forward-center and forward-forward di-jets in the kt-factorization and transverse dependent 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. This provides us with much valuable case study, since such phenomena can provide a direct glimpse into the partonic behavior of a hadron in a dominant gluonic region. We use the unintegrated parton distribution functions (UPDF) in the kt-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). Such a model computation can provide simpler analytic description of data with respect to existing formalisms such as perturbative QCD. The differential cross-section calculations are performed at the center of mass energy of 7 TeV corresponding to CMS collaboration measurement. It is shown that the gluonic jet productions are dominant and a good description of data as well as other theoretical attempts (i.e. KS-linear, KS-nonlinear and rcBK) is obtained. The uncertainty of the calculations is derived by manipulating the hard scale of the processes by a factor of two. This conclusion is achieved, due to the particular visualization of the angular ordering constraint (AOC), that is incorporated in the definition of these UPDF.

Charm quark and meson production in association with single-jet at the LHC

We discuss charm quark/antiquark and charmed meson production in association with one extra jet (gluon, quark, antiquark) at the LHC. The calculations are performed both in collinear and $k_T$-factorization approaches. Different unintegrated gluon distribution functions are used in the $k_T$-factorization approach. Several predictions for the LHC are presented. We show distributions in rapidity and transverse momenta of $c$/$\bar c$ (or charmed mesons) and the associated jet as well as some two-dimensional observables. Interesting correlation effects are predicted, \textit{e.g.} in azimuthal angles $\varphi_{c\bar{c}}$ and $\varphi_{c\mathrm{\textit{-jet}}}$. We have also discussed a relation of the $2 \to 2$ and $2 \to 3$ partonic calculations in the region of large transverse momenta of charm quarks/antiquarks as well as the similarity of the next-to-leading order collinear approach and the $k_T$-factorization approach with the KMR unintegrated parton distribution functions. Integrated cross sections for $D^{0}$ + jet production for ATLAS detector acceptance and for different cuts on jet transverse momenta are also presented.

NLO production of W± and Z0 vector bosons via hadron collisions in the frameworks of Kimber-Martin-Ryskin and Martin-Ryskin-Watt unintegrated parton distribution functions

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 the description of the experimental data on the proton structure functions. The present work is a sequel to that survey, via calculation of the transverse momentum 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 have calculated and aggregated the invariant amplitudes of the corresponding $involved$ diagrams in the $NLO$, and counted the individual contributions in different 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 analyzed against each other, as well as the existing experimental data. Our calculation show excellent agreement with the experiment data. It is however interesting to point-out that, the calculation using the $KMR$ framework illustrates a stronger agreement with the experimental data, despite the fact that the $LO\;MRW$ and the $NLO\;MRW$ formalisms employ a better theoretical description of the $DGLAP$ evolution equation. This is of course due to the use of the different 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.

A new phenomenological investigation of KMR and MRW unintegrated parton distribution functions

The longitudinal proton structure function, $F_L(x,Q^2)$, from the $k_t$ factorization formalism by using the unintegrated parton distribution functions (UPDF) which are generated through the KMR and MRW procedures. The LO UPDF of the KMR prescription is extracted, by taking into account the PDF of Martin et al, i.e. MSTW2008-LO and MRST99-NLO and next, the NLO UPDF of the MRW scheme is generated through the set of MSTW2008-NLO PDF as the inputs. The different aspects of $F_L(x,Q^2)$ in the two approaches, as well as its perturbative and non-perturbative parts are calculated. Then the comparison of $F_L(x,Q^2)$ is made with the data given by the ZEUS and H1 collaborations. It is demonstrated that the extracted $F_L(x,Q^2)$ based on the UPDF of two schemes, are consistent to the experimental data, and by a good approximation, they are independent to the input PDF. But the one developed from the KMR prescription, have better agreement to the data with respect to that of MRW. As it has been suggested, by lowering the factorization scale or the Bjorken variable in the related experiments, it may be possible to analyze the present theoretical approaches more accurately.

The proton FL dipole approximation in the KMR and the MRW unintegrated parton distribution functions frameworks

In the spirit of performing a complete phenomenological investigation of the merits of Kimber–Martin–Ryskin (KMR) and Martin–Ryskin–Watt (MRW) unintegrated parton distribution functions (UPDF), we have computed the longitudinal structure function of the proton, FL(x,Q2), from the so-called dipole approximation, using the LO and the NLO-UPDF, prepared in the respective frameworks. The preparation process utilizes the PDF of Martin et al., MSTW2008-LO and MSTW2008-NLO, as the inputs. Afterwards, the numerical results are undergone a series of comparisons against the exact kt-factorization and the kt-approximate results, derived from the work of Golec-Biernat and Stasto, against each other and the experimental data from ZEUS and H1 Collaborations at HERA. Interestingly, our results show a much better agreement with the exact kt-factorization, compared to the kt-approximate outcome. In addition, our results are completely consistent with those prepared from embedding the KMR and MRW UPDF directly into the kt-factorization framework. One may point out that the FL, prepared from the KMR UPDF shows a better agreement with the exact kt-factorization. This is despite the fact that the MRW formalism employs a better theoretical description of the DGLAP evolution equation and has an NLO expansion. Such unexpected consequence appears, due to the different implementation of the angular ordering constraint in the KMR approach, which automatically includes the resummation of ln⁡(1/x), BFKL logarithms, in the LO-DGLAP evolution equation.

Dijet azimuthal decorrelations at the LHC in the parton Reggeization approach

We study inclusive dijet azimuthal decorrelations in proton-proton collisions at the CERN LHC invoking the hypothesis of parton Reggeization in t-channel exchanges at high energies. In the parton Reggeization approach, the main contribution to the azimuthal angle difference between the two most energetic jets is due to the Reggeon-Reggeon-Particle-Particle scattering, when the fusion of two Reggeized gluons into a pair of Yang-Mills gluons dominates. Using a high-energy factorization scheme with the Kimber-Martin-Ryskin unintegrated parton distribution functions and the Fadin-Lipatov effective vertices we obtain good agreement of our calculations with recent measurements by the ATLAS and CMS Collaborations at the CERN LHC.

The NLO unintegrated parton distribution functions (PDF) in the KMR and the MRW frameworks using the MSTW2008 PDF

Abstract The next-to-leading order (NLO) unintegrated parton distribution functions (UPDF) are calculated in the two schemes by using the latest version of the integrated parton distribution functions (PDF) of Martin et al. (MSTW2008) as the inputs and the final results are compared among each others. In the first method, so-called the Kimber–Martin–Ryskin (KMR) prescription, the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) collinear approximation is used, instead of the Ciafaloni–Catani–Fiorani–Marchesini (CCFM) evolution equations and the dependence of PDF on the second scale, k t (the partons transverse momenta), beside the first scale, μ (μ is the probe scale), is included in the last step of DGLAP evolution equation. In the second approach, which has been proposed by Martin, Ryskin and Watt (MRW), the current scale, the splitting functions, etc are modified in the KMR procedure. The three-dimensional UPDF are presented in terms of different [ x , k t 2 ] -planes for the large values of x (the longitudinal fraction of parton momentum) at μ 2 = 10 2 and 10 4 GeV 2 . It is shown that as in MRW calculations, at each μ 2 a kink is observed around k t 2 ≃ μ 2 and k t 2 ≃ 0.38 μ 2 for the KMR and the MRW formalisms, respectively. Finally, it is demonstrated that the MRW gluon UPDF decrease with respect to the KMR ones, except of the small k t 2 regions, i.e. the Higgs production domain, so it can be concluded that the application of the MRW procedure may improve the exclusive reaction cross sections.

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≃106 GeV2. 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.

TRANSVERSE MOMENTUM DISTRIBUTIONS FROM EFFECTIVE FIELD THEORY

We present a factorization theorem for the low transverse momentum (pT) and rapidity (Y) distribution of the Higgs and electroweak gauge bosons using the Soft-Collinear Effective Theory. In the region M ≫ pT ≫ ΛQCD, where M denotes the mass of the electroweak object, the factorization formula is given in terms of perturbatively calculable functions and the standard PDFs. For pT ~ ΛQCD, the factorization theorem is given in terms of non-perturbative Impact-parameter Beam Functions (iBFs) and an Inverse Soft Function (iSF). The iBFs correspond to completely unintegrated PDFs and can be interesting probes of momentum distributions in the nucleon. The iBFs and the iSF are grouped together and written as a product of a gauge invariant and non-perturbative Transverse Momentum Function (TMF) with the standard PDFs. We present results of NLL resummation for the Higgs and Z-boson distributions and give a comparison with Tevatron data.

The general behavior of NLO unintegrated parton distributions based on the single-scale evolution and the angular ordering constraint

To overcome the complexity of generalized two hard scale (kt,μ) evolution equation, well known as the Ciafaloni, Catani, Fiorani and Marchesini (CCFM) evolution equations, and calculate the unintegrated parton distribution functions (UPDF), Kimber, Martin and Ryskin (KMR) proposed a procedure based on (i) the inclusion of single-scale (μ) only at the last step of evolution and (ii) the angular ordering constraint (AOC) on the DGLAP terms (the DGLAP collinear approximation), to bring the second scale, kt into the UPDF evolution equations. In this work we intend to use the MSTW2008 (Martin et al.) parton distribution functions (PDF) and try to calculate UPDF for various values of x (the longitudinal fraction of parton momentum), μ (the probe scale) and kt (the parton transverse momentum) to see the general behavior of three-dimensional UPDF at the NLO level up to the LHC working energy scales (μ2). It is shown that there exits some pronounced peaks for the three-dimensional UPDF(fa(x,kt)) with respect to the two variables x and kt at various energies (μ). These peaks get larger and move to larger values of kt, as the energy (μ) is increased. We hope these peaks could be detected in the LHC experiments at CERN and other laboratories in the less exclusive processes.

Wilson lines in transverse-momentum dependent parton distribution functions with spin degrees of freedom

We propose a new framework for transverse-momentum dependent parton distribution functions, based on a generalized conception of gauge invariance which includes into the Wilson lines the Pauli term ∼ F μ ν [ γ μ , γ ν ] . We discuss the relevance of this nonminimal term for unintegrated parton distribution functions, pertaining to spinning particles, and analyze its influence on their renormalization-group properties. It is shown that while the Pauli term preserves the probabilistic interpretation of twist-two distributions—unpolarized and polarized—it gives rise to additional pole contributions to those of twist-three. The anomalous dimension induced this way is a matrix, calling for a careful analysis of evolution effects. Moreover, it turns out that the crosstalk between the Pauli term and the longitudinal and the transverse parts of the gauge fields, accompanying the fermions, induces a constant, but process-dependent, phase which is the same for leading and subleading distribution functions. We include Feynman rules for the calculation with gauge links containing the Pauli term and comment on the phenomenological implications of our approach.

The New Investigation of Kimber–Martin–Ryskin Unintegrated Partons

The three sets of conventional integrated parton distribution functions (PDFs) such as MSTW 2008 (Martin et al), GJR08 (Gluck et al) and CTEQ6.6 (Nadolsky et al) are used as the inputs to generate the unintegrated parton distribution functions (UPDFs) via the Kimber–Martin–Ryskin (KMR) procedure and the results are compared with each other. The UPDFs are obtained for several values of the squared transverse momentum component of parton \({(k_t^2)}\) and the longitudinal momentum fraction (x). The various aspects of the PDFs and the UPDFs ratios of GJR08 and CTEQ6.6 to MSTW 2008 are analyzed and discussed. In addition to our previous justification (Modarres and Hosseinkani in Nucl Phys A 815:40, 2009) about the stability and the reliability of KMR UPDFs, it is observed here that they are more convergent in the larger values of the scale \({k_t^2}\) (i.e. the suppression of discrepancies of different PDFs schemes) such that the UPDFs produced via the KMR approach are stable with respect to the variation of the input sets of PDFs and they also tend to a unique value by increasing the scale \({k_t^2}\).

Looking at saturation effects with [formula omitted] production in pA collisions

We compute the inclusive differential cross section production of the pseudo-scalar meson η ′ in high-energy proton-nucleus ( pA) collisions. We use an effective coupling between gluons and η ′ meson to derive a reduction formula that relates the η ′ production to a field-strength tensor correlator. We take into account saturation effects on the nucleus side by using the Color Glass Condensate formalism to evaluate this correlator. We derive new results for Wilson line - color charges correlators in the McLerran-Venugopalan model needed in the computation of η ′ production. The unintegrated parton distribution functions are used to characterize the gluon distribution inside the proton. We show that the cross section is sensitive to saturation effects so it can be utilized to estimate the value of the saturation scale.

Wilson-line and color charge density correlators and the production ofη'in the color glass condensate forppandpAcollisions

We compute the inclusive differential cross section production of the pseudoscalar meson {eta}{sup '} in high-energy proton-proton (pp) and proton-nucleus (pA) collisions. We use an effective coupling between gluons and {eta}{sup '} meson to derive a reduction formula that relates the {eta}{sup '} production to a field-strength tensor correlator. For pA collisions, we take into account saturation effects on the nucleus side by using the color glass condensate formalism to evaluate this correlator. We derive new results for Wilson-line color charge correlators in the McLerran-Venugopalan model needed in the computation of {eta}{sup '} production. The unintegrated parton distribution functions are used to characterize the gluon distribution inside protons. We show that in pp collisions, the cross section depends on the parametrization of unintegrated parton distribution functions, and thus it can be used to put constraints on these distributions. We also demonstrate that in pA collisions, the cross section is sensitive to saturation effects, so it can be utilized to estimate the value of the saturation scale.

The Kimber–Martin–Ryskin unintegrated partons via the MRST and GRV parametrizations

We consider the GRV98 (Glück et al.) and MRST99 (Martin et al.) sets of parton distribution functions (PDFs) as the inputs to obtain the unintegrated PDFs (UPDFs) via the Kimber–Martin–Ryskin (KMR) procedure. The ratios of MRST99 to GRV98 for both PDFs and UPDFs are compared. Since UPDFs are two scales dependent functions (namely k t 2 and μ 2 ), by fixing the values of k t 2 or μ 2 , their relative behaviors for the wide range of μ 2 or k t 2 are investigated. It is shown that by increasing k t 2 the UPDFs are shifted toward the larger values of the scale μ 2 . Finally, it is observed that the KMR procedure is not only stable and insensitive to the variation of input sets of partons such as those of GRV or MRST, but also it has this ability to suppress these discrepancies to make the unique and reliable outputs for UPDFs.

Next-to-leading order hard scattering using fully unintegrated parton distribution functions

We calculate the next-to-leading order fully unintegrated hard scattering coefficient for unpolarized gluon-induced deep inelastic scattering using the logical framework of parton correlation functions developed in previous work. In our approach, exact four-momentum conservation is maintained throughout the calculation. Hence, all nonperturbative functions, like parton distribution functions, depend on all components of parton four-momentum. In contrast to the usual collinear factorization approach where the hard scattering coefficient involves generalized functions (such as Dirac {delta} functions), the fully unintegrated hard scattering coefficient is an ordinary function. Gluon-induced deep inelastic scattering provides a simple illustration of the application of the fully unintegrated factorization formalism with a nontrivial hard scattering coefficient, applied to a phenomenologically interesting case. Furthermore, the gluon-induced process allows for a parametrization of the fully unintegrated gluon distribution function.

PION AND KAON PRODUCTION IN NUCLEON–NUCLEON COLLISIONS

Inclusive cross section for pion production in proton - proton collisions are calculated based on unintegrated parton distribution functions (uPDFs). In addition to purely gluonic terms the present approach includes also quark degrees of freedom. Phenomenological fragmentation functions from the literature are used. The new mechanisms are responsible for π+ - π- asymmetry. In contrast to standard collinear approach, application of 2 → 1 kt - factorization approach can be extended towards much lower transverse momenta, both at mid and forward rapidity region. The results of the calculation are compared with SPS and RHIC data.