We are reporting on the ongoing effort of the Monte Carlo (MC) modelling of NLO DGLAP QCD evolution in the fully unintegrated form. The resulting parton shower MC is performing on its own the NLO QCD evolution, contrary to all known programs of this kind which are limited to LO level only. We overview this new MC scheme, for the non-singlet subset of the gluonstrahlung diagrams. Precision numerical tests of this new scheme are also shown.
Presently available perturbative QCD calculations combining hard process matrix element with the Parton Shower Monte Carlo programs feature hard process matrix element calculated often beyond the leading order (LO), that is including complete next-to-leading-order (NLO), or even next-to-next-to-leading-order (NNLO) corrections, while Parton Shower is only at the leading order (LO). We report here on a work in progress which demonstrate feasibility of constructing Parton Shower Monte Carlo (PSMC) featuring complete NLO corrections to QCD evolution with respect to the logarithm of the factorizat…
We investigate the infrared singularity structure of Feynman diagrams entering the next-to-leading-order (NLO) DGLAP kernel (non-singlet). We examine cancellations between diagrams for two gluon emission contributing to NLO kernels. We observe the crucial role of color coherence effects in cancellations of infra-red singularities. Numerical calculations are explained using analytical formulas for the singular contributions.
We show how parton distributions unintegrated over the parton transverse momentum, k_t, may be generated, at NLO accuracy, from the known integrated (DGLAP-evolved) parton densities determined from global data analyses. A few numerical examples are given, which demonstrate that sufficient accuracy is obtained by keeping only the LO splitting functions together with the NLO integrated parton densities. However, it is important to keep the precise kinematics of the process, by taking the scale to be the virtuality rather than the transverse momentum, in order to be consistent with the calculatio…
Feynman diagrams with two real partons contributing to the next-to-leading-order singlet gluon-quark DGLAP kernel are analysed. The infra-red singularities of unintegrated distributions are examined numerically. The analytical formulae are also given in some cases. The role of the colour coherence effects is found to be crucial for cancellations of the double- and single-logarithmic infra-red singularities.
We present the interplay between the new IR-improved DGLAP-CS theory and the precision of NLO parton shower/ME matched MC`s as it is realized by the new MC Herwiri1.031 in interface to MC@NLO. We discuss phenomenological implications using comparisons with recent LHC data on single heavy gauge boson production.
We show for a first time ever a prototype of a fully exclusive QCD NLO parton shower for the initial state (albeit for a limited set of diagrams). It is based on the rigorous theorems of the collinear factorisation, however the standard DGLAP evolution kernels are replaced by their exclusive versions. Contrary to the standard DGLAP approach, the constructed parton shower provides fully exclusive events, i.e. four-momenta of all emitted partons. At the inclusive level it is identical to the MS-bar scheme of the DGLAP evolution.
The next-to-leading order (NLO) evolution of the parton distribution functions (PDFs) in QCD is a common tool in the lepton-hadron and hadron-hadron collider data analysis. The standard NLO DGLAP evolution is formulated for inclusive (integrated) PDFs and done using inclusive NLO kernels. We report here on the ongoing project, called KRKMC, in which NLO DGLAP evolution is performed for the exclusive multiparton (fully unintegrated) distributions (ePDFs) with the help of the exclusive kernels. These kernels are calculated within the two-parton phase space for the non-singlet evolution, using Cu…
The next-to-leading order (NLO) evolution of the parton distribution functions (PDF's) in QCD is the "industry standard" in the lepton-hadron and hadron-hadron collider data analysis. The standard NLO DGLAP evolution is formulated for inclusive (integrated) PDFs and is done using inclusive NLO kernels. We report here on the ongoing project, called KRKMC, in which NLO DGLAP evolution is performed for the exclusive multiparton (fully unintegrated) distributions (ePDF's) with the help of the exclusive kernels. These kernels are calculated within the two-parton phase space for bremsstrahlung subse…
In this paper the singlet and non-singlet structure functions have been obtained by solving Dokshitzer, Gribove, Lipatov, Alterelli, Parisi (DGLAP) evolution equations in leading order (LO) and next to leading order (NLO) at the small x limit. Here we have used a Taylor Series expansion and then the method of characteristics to solve the evolution equations. We have also calculated t and x-evolutions of deuteron structure function and the results are compared with the New Muon Collaboration (NMC) data.
Theoretical predictions show that at low values of Bjorken $x$ the spin structure function, $g_1$ is influenced by large logarithmic corrections, $ln^2(1/x)$, which may be predominant in this region. These corrections are also partially contained in the NLO part of the standard DGLAP evolution. Here we calculate the non-singlet component of the nucleon structure function, $g_1^{NS}=g_1^p-g_1^n$, and its first moment, using a unified evolution equation. This equation incorporates the terms describing the NLO DGLAP evolution and the terms contributing to the $ln^2(1/x)$ resummation. In order to …
We present a parton shower which implements the DGLAP evolution of parton densities and fragmentation functions at next-to-leading order precision up to effects stemming from local four-momentum conservation. The Monte-Carlo simulation is based on including next-to-leading order collinear splitting functions in an existing parton shower and combining their soft enhanced contributions with the corresponding terms at leading order. Soft double counting is avoided by matching to the soft eikonal. Example results from two independent realizations of the algorithm, implemented in the two event gene…
We compute the longitudinal proton structure function F_L from the k_T factorization scheme, using the unified DGLAP/BFKL resummation approach at small x for the unintegrated gluon density. The differences between the k_T factorization, collinear factorization and dipole approaches are analyzed and discussed. The comparisons with the HERA data are made and predictions for the proposed Large Hadron-Electron Collider are also provided.
We perform a detailed NLO analysis of the combined CCFR xF_3 and F_2 structure functions data and extract the value of alpha_s, parameters of distributions and higher-twist (HT) terms using a direct solution of the DGLAP equation. The value of alpha_s(M_Z)=0.1222\pm 0.0048 (exp) \pm 0.0026 (theor) is obtained. Our results has a larger central value and errors than the original one of the CCFR collaboration due to model independent parameterization of the HT contributions. The dependence of HT contributions on the QCD renormalization scale is studied.
In this work, we present an analytical solution for QCD$\otimes$QED coupled Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations at the leading order (LO) accuracy in QED and next-to-leading order (NLO) accuracy in perturbative QCD using double Laplace transform. This technique is applied to obtain the singlet, gluon and photon distribution functions and also the proton structure function. We also obtain contribution of photon in proton at LO and NLO at high energy and successfully compare the proton structure function with HERA data \cite{12} and APFEL results \cite{7}. Some…
In this paper we present a new and efficient analytical solutions for evolving the QED$\otimes$QCD DGLAP evolution equations in mellin space and obtain the parton distribution functions (PDFs) in perturbative QCD including the QED corrections. The validity of our analytical solutions, which have done in the next to leading order QCD and the leading order QED approximations, are checked with the initial parton distributions from newly released CT14QED global analysis code (Phys. Rev.D93,114015(2016)). The evolved parton distributions are in good agreement with CT14QED code and also with those f…
We analytically solved the QED $\otimes$ QCD coupled DGLAP evolution equations at leading order (LO) quantum electrodynamics (QED) and next to leading order (NLO) quantum chromodynamics (QCD) approximations, using the Laplace transform method and then computed the proton structure function in terms of the unpolarized parton distributions functions. Our analyitical solutions for parton densities are in good agreement with those from APFEL (A PDF Evolution Library) (Computer Physics Communications 185, 1647-1668 (2014)) and CT14QED (Phys. Rev. D 93, 114015 (2016)) global parameterizations. We al…
We present the current status of the application of our approach of {\it exact} amplitude-based resummation in quantum field theory to precision QCD calculations, by realistic MC event generator methods, as needed for precision LHC physics.In this ongoing program of research, we discuss recent results as they relate to the interplay of the attendant IR-Improved DGLAP-CS theory of one of us and the precision of exact NLO matrix element matched parton shower MC's in the Herwig6.5 environment in relation to recent LHC experimental observations. There continues to be reason for optimism in the att…
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 …
The evidence from HERA for parton saturation, and other low-$x$ effects beyond the conventional DGLAP formalism, is recalled and critically reviewed in the light of new data and analyses presented at the conference.
Using repeated Laplace transform techniques, along with newly-developed accurate numerical inverse Laplace transform algorithms, we transform the coupled, integral-differential NLO singlet DGLAP equations first into coupled differential equations, then into coupled algebraic equations, which we can solve iteratively. After Laplace inverting the algebraic solution analytically, we numerically invert the solutions of the decoupled differential equations. Finally, we arrive at the decoupled NLO evolved solutions F_s(x,Q^2)=calF_s(F_{s0}(x),G_0(x)) and G(x,Q^2)=calG(F_{s0}(x),G_0(x)), where calF_s…
Kwiecinski, Martin, Stasto [13] argue for inclusion of DGLAP terms into BFKL evolution of unintegrated gluon density. The equation was reformulated by Oliveira, Martin, Ryskin [6] employing the opening angle θ = k/xp as the evolution variable. It leads to a description of a θ-integrated gluon density. This paper is a numerical study of these two similar combined BFKL+DGLAP formulations. It is a demonstration of feasibility of the new approach. The different ways of subtracting the contribution common for BFKL and DGLAP proposed in [13] and [6] are compared. The numerical tests confirm that the…
We explain particular, unique, approximate solutions of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations and also solutions of DGLAP evolution equations by using regge behaviour of structure functions and method of characteristic for t and x-evolutions of singlet and non-singlet structure functions in leading order (LO) and next-to-leading order (NLO). Hence t-evolution of deuteron, proton, neutron and difference of proton and neutron and x-evolution of deuteron, proton and neutron structure functions in LO and NLO at low-x from DGLAP evolution equations. The results …
We consider the photoproduction of D* mesons associated with two hadron jets at HERA collider in the framework of the kt-factorization approach. The unintegrated gluon densities in a proton are obtained from the full CCFM, from unified BFKL-DGLAP evolution equations as well as from the Kimber-Martin-Ryskin prescription. Resolved photon contributions are reproduced by the initial-state gluon radiation. We investigate different production rates and make comparison with the recent experimental data taken by the ZEUS collaboration. Special attention is put on the specific dijet correlations which …
We introduce the new IR-improved Dokshitzer-Gribov-Lipatov-Altarelli-Parisi-Callan-Symanzik (DGLAP-CS) kernels recently developed by one of us into the HERWIG6.5 to generate a new MC, HERWIRI1.0(31), for hadron-hadron scattering at high energies. We use MC data to compare the parton shower generated by the standard DGLAP-CS kernels and that generated by the new IR-improved DGLAP-CS kernels. The seamless interface to MC@NLO, MC@NLO/HERWIRI, is illustrated. We show comparisons with FNAL data and we discuss some possible LHC phenomenology implications.
Results for the two real parton differential distributions needed for implementing a next-to-leading order (NLO) parton shower Monte Carlo are presented. They are also integrated over the phase space in order to provide solid numerical control of the MC codes and for the discussion of the differences between the standard $\bar{MS}$ factorization and Monte Carlo implementation at the level of inclusive NLO evolution kernels. Presented results cover the class of non-singlet diagrams entering into NLO kernels. The classic work of Curci-Furmanski-Pertonzio was used as a guide in the calculations.
We present a phenomenological study of the current status of the application of our approach of {\it exact} amplitude-based resummation in quantum field theory to precision QCD calculations, by realistic MC event generator methods, as needed for precision LHC physics. We discuss recent results as they relate to the interplay of the attendant IR-Improved DGLAP-CS theory of one of us and the precision of exact NLO matrix-element matched parton shower MC's in the Herwig6.5 environment as determined by comparison to recent LHC experimental observations on single heavy gauge boson production and de…
We performed the detailed NLO analysis of the combined CCFR $xF_3$ and $F_2$ structure functions data and extracted the value of $α_s$, parameters of distributions and higher-twist (HT) terms using the direct solution of the DGLAP equation. The value of $α_s(M_Z)=0.1222\pm 0.0048 (exp)\pm 0.0040(theor)$ was obtained. The result has larger central value and errors, than the original result of the CCFR collaboration, in view of the incorporation into the fits of the HT terms as the free model independent parameters. The $x$-shapes of the HT contributions to $xF_3$ and $F_2$ are in agreement with…
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 comparis…
A new method to test the valence quark distribution of nucleon obtained from the maximum entropy method using Gottfried sum rule by performing DGLAP equations with the GLR-MQ-ZRS corrections and original LO~/~NLO DGLAP equations are outlined. The test relies on a knowledge of the unpolarized electron-proton structure function $F_2^{ep}$ and electron-neutron structure function $F_2^{en}$ and the assumption that Bjorken scaling is satisfied. In this work, the original Gottfried summation value obtained by the integrals of the structure function at different Q$^{2}$ is in accordance with the theo…
We use comparison with recent LHCb data on single Z/γ^* production and decay to lepton pairs as a vehicle to study the current status of the application of our approach of {\it exact} amplitude-based resummation in quantum field theory to precision QCD calculations, by realistic MC event generator methods, as needed for precision LHC physics. This represents an extension of the phase space of our previous studies based on comparison with CMS and ATLAS data, as the pseudo-rapidity range measured by the LHCb for leptons in the data we study is 2.0<η< 4.5 to be compared with |η|<4.6(2.4) in our p…
In a previous paper, hereafter referred to as I, we have analyzed the 7 TeV LHC data on W + jets events from the standpoint of IR-improved DGLAP parton shower effects, using the IR-improved Herwiri1.031 parton shower MC in comparison with the Herwig6.5 parton shower MC in the context of the exact $O(α_s)$ matrix element matched parton shower framework provided by MG5\_aMC@NLO. In the current paper, we extend this analysis to the LHC 8 and 13 TeV data to investigate the energy dependence of the results obtained in I. Specifically, W~+ jet events are generated in the MADGRAPH5\_aMC@NLO framework…
The non-singlet structure functions have been obtained by solving Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations in leading order (LO) and next-to-leading order (NLO) at the small-x limit. Here a Taylor series expansion has been used and then the method of characteristics has been applied to solve the evolution equations. Results are comopared with the Fermilab experiment E665 data and New Muon Collaboration (NMC) data.
By implementing the new IR-improved Dokshitzer-Gribov-Lipatov-Altarelli-Parisi-Callan-Symanzik (DGLAP-CS) kernels recently developed by one of us in the HERWIG6.5 environment we generate a new MC, HERWIRI1.0(31), for hadron-hadron scattering at high energies. We use MC data to illustrate the comparison between the parton shower generated by the standard DGLAP-CS kernels and that generated by the new IR-improved DGLAP-CS kernels. The interface to MC@NLO, MC@NLO/HERWIRI, is illustrated. Comparisons with FNAL data and some discussion of possible implications for LHC phenomenology are also present…
NLO QCD fits using the DGLAP formalism have been made to the high precision ZEUS $e^+p$ reduced cross-section data and to fixed target structure function data, in order to detrmine parton distribution functions and the value of $α_s(M^2_Z)$, taking full account of correlated systematic errors. Preliminary results are reported.
We study the sea quark contribution to the BFKL kernel in the framework of Mueller's dipole model using the results of our earlier calculation. We first obtain the BFKL equation with the running coupling constant. We observe that the ``triumvirate'' structure of the running coupling found previously for non-linear evolution equations is preserved for the BFKL equation. In fact, we rederive the equation conjectured by Levin and by Braun, albeit for the unintegrated gluon distribution with a slightly unconventional normalization. We obtain the leading-N_f contribution to the NLO BFKL kernel in t…
We introduce the physical factorisation scheme, which is necessary to describe observables which are `not completely inclusive'. We derive the formulae for NLO DGLAP evolution in this scheme, and also for the `rotation' of the conventional MSbar PDFs into the physical representation. Unlike, the MSbar prescription, where, for example, the gluon PDF at NLO obtains an admixture of the quark-singlet PDF, and vice-versa, the physical approach does not mix parton PDFs of different types. That is, the physical approach retains the precise quantum numbers of each PDF. The NLO corrections to DGLAP evo…
We show how it is possible to rewrite the BFKL equation for the unintegrated gluon distribution, in terms of integrated gluons, similar to that used in DGLAP. We add to our equation the next-to-leading log terms which provide exact energy-momentum conservation and account for the kinematic constraint in real gluon emission. In this way the equation includes the major part of the higher-order corrections to BFKL evolution. We discuss the possibility to obtain a unified BFKL-DGLAP evolution equation relevant to processes at the LHC where both log(1/x) and logQ^2 are large simultaneously.
This is the introductory part of my PhD thesis which consists of two parts, the separate introduction and four published articles. The introduction begins by a technically detailed description of the DGLAP evolution and the fast numerical solving method for the DGLAP equations, which has been used in the numerical works of the published articles of this thesis. A write-up of the next-to-leading order (NLO) calculations for the deeply inelastic scattering (DIS) and the Drell-Yan (DY) dilepton production cross-sections is also included. The formalism of the inclusive single hadron production at …
A method to construct Monte-Carlo event generators at arbitrarily non-leading order is explained for the case of a non-gauge theory. A precise and correct treatment of parton kinematics is provided. Modifications of the conventional formalism are required: parton showering is not exactly the same as DGLAP evolution, and the external line prescription for the hard scattering differs from the LSZ prescription. The prospects for extending the results to QCD are discussed.
We show that the conventional prescription used for DGLAP parton evolution at NLO has an inconsistent treatment of the contribution from the infrared (IR) region. We illustrate the problem by studying the simple example of QED evolution, treating the electron and photon as partons. The deficiency is not present in a physical approach which removes the IR divergency and allows calculation in the normal 4-dimensional space.
We compute the leading order (LO) $qg\to q γ$ and next-to-leading order (NLO) $gg\to q{\bar q} γ$ contributions to inclusive photon production in proton-proton (p+p) collisions at the LHC. These channels provide the dominant contribution at LO and NLO for photon transverse momenta $k_{γ\perp}$ corresponding to momentum fractions of $x\leq 0.01$ in the colliding protons. Our computations, performed in the dilute-dense framework of the Color Glass Condensate effective field theory (CGC EFT), show that the NLO contribution dominates at small-$x$ because it is sensitive to $k_\perp$-dependent unin…
A semi-numerical solution to Dokshitzer- Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations at leading order (LO), next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) in the small-x limit is presented. Here we have used Taylor series expansion method to solve the evolution equations and, t- and x-evolutions of the singlet structure functions have been obtained with such solution. We have also calculated t- and x-evolutions of deuteron structure functions F_2^d, and the results are compared with the E665 data and NMC data. The results are also compared to those obtaine…
We report on the progress in calculating NLO DGLAP splitting functions for $x<1$ using the New Principal Value prescription, which is a modification of the standard Principal Value approach proposed by Curci, Furmanski and Petronzio in 1980. The new prescription reproduces the standard results on the inclusive (integrated) level, but simplifies individual contributions and restricts the cancellations between real and virtual diagrams which makes it useful for Monte Carlo simulations.
We studied the effects of NLO $Q^2$ evolution of generalized parton distributions (GPDs) using the aligned-jet model for the singlet quark and gluon GPDs at an initial evolution scale. We found that the skewness ratio for quarks is a slow logarithmic function of $Q^2$ reaching $r^S=1.5-2$ at $Q^2=100$ GeV$^2$ and $r^g \approx 1$ for gluons in a wide range of $Q^2$. Using the resulting GPDs, we calculated the DVCS cross section on the proton in NLO pQCD and found that this model in conjunction with modern parameterizations of proton PDFs (CJ15 and CT14) provides a good description of the availa…
We propose a modified version of the Balitsky-Kovchegov (B-K) evolution equation, which includes the main NLO corrections. We use the result that the main NLO corrections to the BFKL kernel are the LO DGLAP corrections. We present a numerical solution of the modified nonlinear equation, and compare with the solution of the unmodified B-K equation. We show that the saturation momentum has a sharp increase in the LHC energy range. Our numerical solution indicates that the influence of the pre-asymptotic corrections, related to the full anomolous dimensions of the DGLAP equation, are rather large…
We investigate the role of the choice of the upper phase space limit Q in the Curci-Furmanski-Petronzio (CFP) factorization scheme, which exploits dimensional regularization MS scheme. We examine how the choice of Q influences the evaluation of the standard DGLAP (inclusive) evolution kernels, gaining experience needed in the construction of the exclusive Monte Carlo modelling of the NLO DGLAP evolution. In particular, we uncover three types of mechanisms which assure the independence on Q of the inclusive DGLAP kernels calculated in the CFP scheme. We use the examples of three types of the Fe…
We present two Monte Carlo algorithms of the Markovian type which solve the modified QCD evolution equations at the NLO level. The modifications with respect to the standard DGLAP evolution concern the argument of the strong coupling constant alpha_S. We analyze the z - dependent argument and then the k_T - dependent one. The evolution time variable is identified with the rapidity. The two algorithms are tested to the 0.05% precision level. We find that the NLO corrections in the evolution of parton momentum distributions with k_T - dependent coupling constant are of the order of 10 to 20%, an…
A new NLO DGLAP QCD fit to recent inclusive diffractive DIS data from the H1 collaboration is presented. Diffractive parton distributions are extracted, including their experimental and theoretical uncertainties. The parton distributions are used for comparisons with recent diffractive final state data from HERA and the TEVATRON.
The new IR-improved Dokshitzer-Gribov-Lipatov-Altarelli-Parisi-Callan-Symanzik (DGLAP-CS) kernels recently developed by one of us is implemented in the HERWIG6.5 environment to generate a new MC, HERWIRI1.0(31), for hadron-hadron scattering at high energies. The comparison between the parton shower generated by the standard DGLAP-CS kernels and that generated by the new IR-improved DGLAP-CS kernels is illustrated using MC data. This is done for some of the respective exact {\cal O}(α_s) corrected spectra using the seamless interfaces to MC@NLO while making comparisons with FNAL data. Some disc…
A framework to include triple collinear splitting functions into parton showers is presented, and the implementation of flavor-changing NLO splitting kernels is discussed as a first application. The correspondence between the Monte-Carlo integration and the analytic computation of NLO DGLAP evolution kernels is made explicit for both timelike and spacelike parton evolution. Numerical simulation results are obtained with two independent implementations of the new algorithm, using the two independent event generation frameworks Pythia and Sherpa.
The ultrahigh energy neutrino-nucleon cross sections are computed taking into account different phenomenological implementations of the non-linear QCD dynamic s. Based on the color dipole framework, the results for the saturation model supplemented by DGLAP evolution as well as for the BFKL formalism in the geometric scaling regime are presented. They are contrasted with recent calculations using NLO DGLAP and unified BFKL-DGLAP formalisms.
The QCD evolution of both unpolarized and polarized generalized parton distributions (GPDs) to next-to-leading order (NLO) accuracy is presented, in both the DGLAP and ERBL regions, for two appropriately symmetrized input distributions based on conventional parton density functions. To illustrate the relative size of the NLO corrections a comparison is made with leading order evolution of the same distributions. For the first time, NLO results are given for both small and large values of the skewedness parameter, $ζ= x_{bj}$, i.e. for all of the kinematic range relevant to HERA and HERMES.
We present particular and unique solutions of singlet and non-singlet Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations in leading order (LO) and next-to-leading order (NLO) and gluon, sea and valence quark Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations in leading order (LO) by applying Taylor expansion method at small-x. We obtain t-evolutions of deuteron, proton, neutron, difference and ratio of proton and neutron, gluon, light sea and valence quark structure functions and x-evolution of deuteron,gluon and light sea quark structure functions at sm…
We present a quantitative study of the $νN$ cross section in the neutrino energy range $10^4<E_ν<10^{14}$ GeV within two transversal QCD approaches: NLO DGLAP evolution using different sets of PDFs and BK small-$x$ evolution with running coupling and kinematical corrections. We show that the non-linear effects embodied in the BK equation yield a slower raise in the cross section for $E_ν\gtrsim 10^{8}$ GeV than the usual DGLAP based calculation. Finally, we translate this theoretical uncertainty into upper bounds for the ultra-high-energy neutrino flux for different experiments.
We show that already at the NLO level the DGLAP evolution kernel Pqq starts to depend on the choice of the evolution variable. We give an explicit example of such a variable, namely the maximum of transverse momenta of emitted partons and we identify a class of evolution variables that leave the NLO Pqq kernel unchanged with respect to the known standard MS-bar results. The kernels are calculated using a modified Curci-Furmanski-Petronzio method which is based on a direct Feynman-graphs calculation.
We propose a matrix evolution equation in (x,kt)-space for flavour singlet, unintegrated quark and gluon densities, which generalizes DGLAP and BFKL equations in the relevant limits. The matrix evolution kernel is constructed so as to satisfy renormalization group constraints in both the ordered and antiordered regions of exchanged momenta kt, and incorporates the known NLO anomalous dimensions in the MSbar scheme as well as the NLx BFKL kernel. We provide a hard Pomeron exponent and effective eigenvalue functions that include the n_f-dependence, and give also the matrix of resummed DGLAP spli…
