Dokshitzer-Gribov-Lipatov-Altarelli-Parisi Research Articles

Evolution of structure functions in momentum space

We formulate the momentum-space Dokshitzer–Gribov–Lipatov-Altarelli–Parisi (DGLAP) evolution equations for structure functions measurable in deeply inelastic scattering. We construct a six-dimensional basis of structure functions that allows for a full three flavor structure and thereby provides a way to calculate perturbative predictions for physical cross sections directly without unobservable parton distribution functions (PDFs) and without the associated scheme dependence. We derive the DGLAP equations to first non-zero order in strong coupling αs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _\ extrm{s}$$\\end{document}, but the approach can be pursued to arbitrary order in perturbation theory. We also numerically check our equations against the conventional PDF formulation.

Asymptotic gluon density within the color dipole picture in light of HERA high-precision data

We present an analysis of the most precise set of HERA data within the color dipole formalism, by using an analytical gluon density, based on the double-logarithm approximation of the DGLAP equations in the asymptotic limit of the scaling variable, $\sigma=\log{(1/x)}\log{(\log{(Q^2/Q_ 0^2)})}\rightarrow \infty$. Fits to data, including charm and bottom quarks are performed and demonstrate the efficiency of the model in describing the reduced cross section, $\sigma_{r}$, in the wide range $Q^2:(1.5,500)$ GeV$^2$ for two dipole models including parton saturation effects. We also give predictions to $F_{2}^{c\bar{c}}$ , $F_{2}^{b\bar{b}}$ and $F_{L}$, all describing the data reasonably well in the range $Q^2:(2.5,120)$ GeV$^2$. Total cross sections of exclusive photoproduction of $J/\psi$ and $\rho$ are also calculated and successfully compared to HERA data and recent measurements at LHCb.

Unintegrated gluon distributions from the color dipole cross section in the BGK saturation model

We evaluate the unintegrated gluon distribution of the proton starting from a parametrization of the color dipole cross section including Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution and saturation effects. To this end, we perform the Fourier-Bessel transform of σ(x,r)/α(r). At large transverse momentum of gluons we match the so-obtained distribution to the logarithmic derivative of the collinear gluon distribution. We check our approach by calculating the proton structure function FL(x,Q2) finding good agreement with HERA data.

Analytical solution of the DGLAP equations using the generating function method

Recently, we generated the evolution equations of the parton distribution functions (PDF) usually used in the hadrons phenomenology by the stochastic modeling of the non-equilibrium statistical mechanics in the momentum space. The evolution equations obtained from the stochastic modeling are the same as the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations, but they are acquired by a mathematical procedure based on the non-equilibrium statistical mechanics and Markov processes. In this paper, we present a new analytical solution of the parton evolution equation (DGLAP equation) for the non-singlet quark distribution function using the generating function method. This method is a general technique for solving the master equation of Markov processes.

Analytical solution of the evolution equations of the parton distribution functions in the small-x region through the Kramers-Moyal expansion of the master equation of Markov processes

Recently, we generated the evolution equations of the parton distribution functions (PDF) usually used in the hadrons phenomenology using the stochastic modeling of the non-equilibrium statistical mechanics in the momentum space. The evolution equations obtained from stochastic modeling are the same as the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations, but are obtained by a more simplistic mathematical procedure based on the non-equilibrium statistical mechanics and the theory of Markov processes. In this paper, we analytically solve the parton evolution equation for the non-singlet quark distribution function in the small-x (the longitudinal momentum fraction) region through the Kramers-Moyal expansion of the master equation. Finally, we compare the cutoff dependent non-singlet quark distribution function obtained from the analytical solution by considering the strong ordering and the angular ordering constraints with the ordinary non-singlet quark distribution function produced by the MMHT2014 group. In general, we show that our results at the small x and moderate Q2 (the energy scale) are in good agreement with the results of the MMHT2014 group.

Signatures of gluon saturation from structure-function measurements

We study experimentally observable signals for nonlinear QCD dynamics in deep inelastic scattering (DIS) at small Bjorken variable $x$ and moderate virtuality $Q^2$, by quantifying differences between the linear Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution and nonlinear evolution with the Balitsky-Kovchegov (BK) equation. To remove the effect of the parametrization freedom in the initial conditions of both equations, we first match the predictions for the DIS structure functions $F_2$ and $F_{\rm L}$ from both frameworks in a region in $x,Q^2$ where both frameworks should provide an accurate description of the relevant physics. The differences in the dynamics are then quantified by the deviations when one moves away from this matching region. For free protons we find that the differences in $F_2$ remain at a few-percent level, while in $F_{\rm L}$ the deviations are larger, up to $10\,\%$ at the EIC and $40\,\%$ at the LHeC kinematics. With a heavy nucleus the differences are up to $10\,\%$ in $F_2$, and can reach $20\,\%$ and $60\,\%$ in $F_{\rm L}$ for the EIC and the LHeC, respectively.

Analytic derivation of the nonlinear gluon distribution function

In the present article, two analytical solutions based on the Laplace transforms method for the linear and non-linear gluon distribution functions have been presented at low values of $x$. These linear and non-linear methods are presented based on the solutions of the Dokshitzer-Gribov- Lipatov-Altarelli-Parisi (DGLAP) evolution equation and the Gribov-Levin-Ryskin Mueller-Qiu (GLR-MQ) equation at the leading-order accuracy in perturbative QCD respectively. The gluon distributions are obtained directly in terms of the parametrization of structure function $F_{2}(x,Q^{2})$ and its derivative and compared with the results from the parametrization models. The $n_{f}$ changes at the threshold are considered in the numerical results. The effects of the non-linear corrections are visible as $Q^{2}$ decreases and vanish as $Q^{2}$ increases. The nonlinear corrections tame the behavior of the gluon distribution function at low $x$ and $Q^{2}$ in comparison with the parametrization models.

Applying different angular ordering constraints and kt -factorization approaches to the single inclusive hadron production in the e+e− annihilation processes

We study the differential cross section of the single inclusive ${e}^{+}{e}^{\ensuremath{-}}$ annihilation to the hadrons via $\ensuremath{\gamma}$-production, in the different ${k}_{t}$-factorization frameworks. In order to take into account the transverse momenta of the incoming partons, for the first time, we apply the Kimber et al. (KMR) method to calculate the unintegrated parton fragmentation functions (UFFs) from the ordinary integrated one, i.e., the parton fragmentation functions (FFs), which satisfy the similar Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations, such as those of parton distribution functions (PDFs). Also, by utilizing the different angular ordering constraints the results corresponding to the Martin et al. (MRW) in the leading order (LO) and the next-to-leading-order (NLO) are obtained. The LO sets of DSS library for the input FFs is used. The numerical results are compared with the experimental data in the different energies which are reported by the different collaborations, such as TASSO, AMY, MARK II, CELLO, DELPHI, SLD, ALEPH and Belle with the other $\mathrm{QCD}+\mathrm{fragmentation}$ models such as pythia6.4 and 8.2 parton showers. The behaviors of the normalized differential cross sections and the multiplicity versus the ``transverse momentum'' (${p}_{\ensuremath{\perp}}$) are discussed. The final results demonstrate that the KMR and MRW UFFs give a good description of data and there is not much significant difference between the above three schemes. On the other hand, our results become closer to the data for the lower values of ${p}_{\ensuremath{\perp}}$ and the higher values of center of mass energies.

Numerical solution of DGLAP equations by the Tau spectral method

In this paper, a special kind of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations is introduced. Since it is not possible to solve DGLAP integral equations analytically, the numerical solutions of these equations can be of interest. Here, the Tau spectral method is used for solving this integral equation and offer an approximate solution. Finally, this solution is compared with solution obtained experimentally for $Q_0^2=0.35 GeV^2$.

Physical limits in the color-dipole model bounds

The ratio of the cross-sections for the transverse and longitudinal virtual photon polarizations, $$\sigma ^{\gamma ^{*}\mathrm{p}}_\mathrm{L/T}$$ , at high photon–hadron energy scattering is studied. The relationship between the gluon distributions obtained using the color-dipole model and standard gluons obtained from the Dokshitzer–Gribov–Lipatov–Altarelli-Parisi (DGLAP) evolution and the Altarelli–Martinelli equations is investigated. It is shown that the color-dipole bounds are dependent on the gluon distribution behavior. This behavior is considered by expansion and Laplace-transform methods. Numerical calculations at leading order (LO) and next-to-leading order (NLO) approximations and a comparison with the color-dipole model (CDM) bounds indicate the range of validity of this method at small dipole sizes, $$r\sim 1/Q{\ll }1/Q_\mathrm{s}$$ .

An evaluation of the proton structure functions F2 and FL at small x

We describe the determination of the DIS structure functions F2 and FL by using the singlet Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) and Altarelli-Martinelli equations at small values of x. The determination of the longitudinal structure function is presented as a parameterization of F2(x,Q2) and its derivative. Analytical expressions for σr(x,Q2) in terms of the effective parameters of the parameterization of F2(x,Q2) and FL(x,Q2) are presented. This analysis is enriched by including the higher-twist effects in calculation of the reduced cross sections which is important at low-x and low-Q2 regions. Numerical calculations and comparison with H1 data demonstrate that the suggested method provides reliable FL(x,Q2) and σr(x,Q2) at low x in a wide range of the low absolute four-momentum transfers squared (1.5 GeV2<Q2<120 GeV2) at moderate and high inelasticity. Expanding the method to low and ultra low values of x can be considered in the process analysis of new colliders. We compare the obtained longitudinal structure function with respect to the LHeC simulated uncertainties (Agostini et al. (2020) [4]) with the results from CT18 (Hou et al. (2021) [54]) parametrization model.

Extracting the parton distribution functions evolution equations using the stochastic modeling in the non-equilibrium statistical mechanics

In this paper, using the stochastic modeling of the non-equilibrium statistical mechanics in the momentum space, the evolution equations of the parton distribution functions (PDF) usually used in the hadrons phenomenology are generated. These stochastic modeling PDF evolution equations are the same as those of the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) ones, but they can be obtained by a more simplistic mathematical procedure based on the non-equilibrium statistical mechanics and the theory of Markov processes.

Application of the constituent quark exchange model to the parton distributions and the EMC ratios of 12C and 14N nuclei

The quark exchange model (QEM) is reformulated for the A=12 and 14 systems to obtain the constituent quark distributions of C12 and N14 nuclei, respectively. Afterwards, the different types of the point-like parton distribution functions (PDFs), i.e., the valence quarks, the sea quarks and the gluons, are extracted from the constituent quark model (CQM), at the hadronic scale Q02=0.34 GeV2 to generate the constituent quark exchange model (CQEM = QEM ⊕ CQM) PDFs. From the resulted PDFs, the structure functions and the European Muon Collaboration (EMC) ratios of the above-mentioned nuclei at the appropriate hadronic scale are calculated. To make our results more comparable with the available experimental data, we evolve the PDFs, by using the standard Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGALP) evolution equations, to the higher hard scales. Subsequently, the EMC ratios of C12 and N14 nuclei at some higher energy scales, corresponding to the scales of available data, are calculated, at the leading (LO) and the next-to-leading (NLO) orders of pQCD. By doing so, it is observed that the EMC ratios do not significantly depend on the hard scale Q2, and the outcomes are consistent with the various experimental data such as HERMES, BCDMS, JLab, SLAC, NMC, and EMC. Especially, in the mid-range of Bjorken x values, the results are so desirable. Like our previous works for the H4e and L6i nuclei, we again observe that at a fixed scale Q2, the LO and the NLO EMC ratios with high precision are approximately the same. Therefore, one can conclude that at a given hard scale, the LO approximation is good enough for calculating the EMC ratios of light nuclei.

SUSY-Like Relation of the Splitting Functions in Evolution of Gluon and Quark Multiplicities

We show the new relationship [1] between the anomalous dimensions, resummed through next-to-next-to-leading-logarithmic order (NNNLO), in the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations for the first Mellin moments $${{D}_{{q,g}}}({{\mu }^{2}})$$ of the quark and gluon fragmentation functions, which correspond to the average hadron multiplicities in jets initiated by quarks and gluons, respectively. This relationship strongly improves previous treatments by allowing for an exact solution of the evolution equations. The fit of the world data of $${{D}_{{q,g}}}({{\mu }^{2}})$$ for charged hadrons measured in $${{e}^{ + }}{{e}^{ - }}$$ annihilation leads to $$\alpha _{s}^{{(5)}}({{M}_{Z}}) = 0.1205\begin{array}{*{20}{c}} { + 0.0016} \\ {0.0020} \end{array}.$$

Dijets with Large Rapidity Separation at CMS

Search for asymptotic Balitsky–Fadin–Kuraev–Lipatov (BFKL) effects at CMS in dijet events with large rapidity separation between jets in proton-proton collisions at 7 TeV is presented. CMS data on cross-section ratios and azimuthal decorrelations for dijets with large rapidity separation between jets are compared with predictions by various Monte Carlo (MC) event generators based on Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) and BFKL evolutions. None of DGLAP-based generators can describe all measured observable by CMS at 7 TeV. NLL BFKL predictions are consistent with CMS data on dijet azimuthal decorrelations at 7 TeV. More data at higher LHC energies are needed to reveal possible effects.

The non-singlet structure function of light and heavy nuclei up to next-to-leading order analysis at low x region

We compute the non-singlet structure function of some various light and heavy nuclei using the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation at leading and next-to-leading order analysis at low x and low and moderate Q2 in x≤0.05 and 1≤Q2≤12 GeV2 region. Using the bounded nuclear parton distribution functions from the HKN parametrization group, we obtain the nuclear non-singlet structure function. Its dynamical behavior as well as nuclear factor are investigated. We compare our results with the new nCTEQ15 set of nuclear parton distribution functions. Our results are in reasonably good agreement with this new global fit.

Ultra-high energy neutrino-nucleon scattering and gluon saturation

Ultra-high energy neutrinos are an enigma; among their many poorly understood aspects are their origins and how they interact with nucleons when they reach the Earth. Due to the hard scale ([Formula: see text]) involved in neutrino-nucleon scattering and for a large range of neutrino energies, it is appropriate to describe the target nucleon in terms of its partons — quarks and gluons — and their evolution with [Formula: see text] as governed by the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations of perturbative Quantum ChromoDynamics (pQCD). Nevertheless, at the highest neutrino energies, the scattering cross-section is dominated by the contribution of small [Formula: see text] gluons of the target where one expects DGLAP evolution equations to break down due to high gluon density effects (gluon saturation). Here, we give a brief overview of gluon saturation physics in QCD and its effects on ultra-high energy neutrino-nucleon (nucleus) scattering cross-section.

SUSY-Like Relation in Evolution of Gluon and Quark Jet Multiplicities

We show the new relationship [1] between the anomalous dimensions, resummed through next-to-next-to-leading-logarithmic order, in the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations for the first Mellin moments D{q,g}(\mu2) of the quark and gluon fragmentation functions, which correspond to the average hadron multiplicities in jets initiated by quarks and gluons, respectively. This relationship, which is independent of the number of quark flavors, strongly improves previous treatments by allowing for an exact solution of the evolution equations. So far, such relationships have only been known from supersymmetric QCD.

Phenomenological Behavior of the Hard Pomeron Intercept

The hard pomeron intercept value is determined by the decoupling solutions of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations at low-x. The unpolarized proton structure functions at next-to-next-to-leading order (NNLO) analysis are calculated and compared with new Hadron-Electron Ring Accelerator (HERA) data.

Evidence of quasi-partonic higher-twist effects in deep inelastic scattering at HERA at moderate $$Q^2$$ Q 2

The combined HERA data for the inclusive deep inelastic scattering (DIS) cross sections for the momentum transfer Q^2 > 1,mathrm {GeV}^2 are fitted within the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) framework at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) accuracy, complemented by a QCD-inspired parameterisation of twist 4 corrections. A modified form of the input parton density functions is also included, motivated by parton saturation mechanism at small Bjorken x and at a low scale. These modifications lead to a significant improvement of the data description in the region of low Q^2. For the whole data sample, the new benchmark NNLO DGLAP fit yields chi ^2/mathrm{d.o.f. } simeq 1.19 to be compared to 1.46 resulting from the standard NNLO DGLAP fit. We discuss the results in the context of the parton saturation picture and describe the impact of the higher-twist corrections on the derived parton density functions. The resulting description of the longitudinal proton structure function F_{mathrm {L}} is consistent with the HERA data. Our estimates of higher-twist contributions to the proton structure functions are comparable to the leading-twist contributions at low Q^2 simeq 2 GeV^2 and x simeq 10^{-5}. The x-dependence of the twist 4 corrections obtained from the best fit is consistent with the leading twist 4 quasi-partonic operators, corresponding to an exchange of four interacting gluons in the t-channel.