Dokshitzer–Gribov–Lipatov–Altarelli–Parisi Equations 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.

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.

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 $Q_0^2=0.35 GeV^2$.

Solution of QCD⊗QED coupled DGLAP equations at NLO

In this work, we present an analytical solution for QCD⊗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 [1] and APFEL results [2]. Some comparisons also have been done for the singlet and gluon distribution functions with the MSTW results [3]. In addition, the contribution of photon distribution function inside the proton has been compared with results of MRST [4] and with the contribution of sea quark distribution functions which obtained by MSTW [3] and CTEQ6M [5].

Q 2 evolution of xF 3(x,Q 2) structure function and Gross–Llewellyn Smith sum rule up to next-next-to-leading order at low x and low Q 2 using a Q 2-dependent Regge ansatz

A Q 2-dependent Regge ansatz for $$xF_3 (x,Q^2 )$$ structure function as the initial input is utilized to solve analytically the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi evolution equation in leading order, next-to-leading order and next-next-to-leading order. The solutions of the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi equations are also used to predict the Q 2 behavior of Gross–Llewellyn Smith sum rule. Dokshitzer–Gribov–Lipatov–Altarelli–Parisi evolved results for $$xF_3 (x,Q^2)$$ structure functions as well as obtained sum rule results are analyzed phenomenologically in comparison with other results taken from Chicago–Columbia–Fermilab–Rochester, Neutrino experiment at the Fermlab Tevatron, CERN Hybrid Oscillation Research Apparatus and CERN–Dortmund–Heidelberg–Saclay–Warsaw collaborations, and a very good agreement is observed in this regard.

Regge-like initial input and evolution of non-singlet structure functions from DGLAP equation up to next-next-to-leading order at low x and low Q 2

This is an attempt to study how the features of Regge theory, along with QCD predictions, lead towards the understanding of unpolarized non-singlet structure functions \(F_{2}^{\text {NS}} (x,Q^{2} )\) and xF3(x,Q2) at low x and low Q2. Combining the features of perturbative quantum chromodynamics (pQCD) and Regge theory, an ansatz for \(F_{2}^{\text {NS}} (x,Q^{2})\) and xF3(x,Q2) structure functions at small x was obtained, which when used as the initial input to Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equation, gives the Q2 evolution of the non-singlet structure functions. The non-singlet structure functions, evolved in accordance with DGLAP evolution equations up to next-next-to-leading order are studied phenomenologically in comparison with the available experimental and parametrization results taken from NMC, CCFR, NuTeV, CORUS, CDHSW, NNPDF and MSTW Collaborations and a very good agreement is observed in this regard.

Cross sections of the neutrino-nucleon interaction at ultra high energy

We determine the ultra-high-energy (UHE) neutrino-nucleon cross sections analytically both for charged and neutral currents to leading order by using the solutions of the non-singlet Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation and the singlet DGLAP equation, respectively. Next, we determine the cross sections for both charged and neutral currents at the one-loop level by carrying out a numerical integration of the double differential cross-section. Our analytical, as well as our numerically-determined, results are compared with the next-to-leading order (NLO) results of various authors. We find our numerically-determined result to be in sufficient agreement with the results obtained by those authors. On the other hand, our analytical result matches better with other results at the lower end of the energy spectrum of the UHE neutrino rather than with results at the higher end.

An improved analysis of Chicago–Columbia–Fermilab–Rochester neutrino data with Singlet Dokshitzer–Gribov–Lipatov–Altarelli–Parisi equation at low x

The present work reports t-evolution of singlet structure function using analytical solution of Taylor-approximated Dokshitzer–Gribov–Lipatov–Altarelli–Parisi equation. Our predictions are compared with Chicago–Columbia–Fermilab–Rochester neutrino data and are found to be quite compatible.

Evolution of singlet structure functions from DGLAP equation at next-to-next-to-leading order at small-x

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-evolution 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 obtained by the fit to $F_{2}^{d}$ produced by the NNPDF collaboration based on the NMC and BCDMS data.

Numerical solution of [formula omitted] evolution equations for fragmentation functions

Semi-inclusive hadron-production processes are becoming important in high-energy hadron reactions. They are used for investigating properties of quark–hadron matters in heavy-ion collisions, for finding the origin of nucleon spin in polarized lepton–nucleon and nucleon–nucleon reactions, and possibly for finding exotic hadrons. In describing the hadron-production cross sections in high-energy reactions, fragmentation functions are essential quantities. A fragmentation function indicates the probability of producing a hadron from a parton in the leading order of the running coupling constant αs. Its Q2 dependence is described by the standard DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi) evolution equations, which are often used in theoretical and experimental analyses of the fragmentation functions and in calculating semi-inclusive cross sections. The DGLAP equations are complicated integro-differential equations, which cannot be solved in an analytical method. In this work, a simple method is employed for solving the evolution equations by using Gauss–Legendre quadrature for evaluating integrals, and a useful code is provided for calculating the Q2 evolution of the fragmentation functions in the leading order (LO) and next-to-leading order (NLO) of αs. The renormalization scheme is MS¯ in the NLO evolution. Our evolution code is explained for using it in oneʼs studies on the fragmentation functions. Program summaryProgram title: ffevol1.0Catalogue identifier: AELJ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AELJ_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1535No. of bytes in distributed program, including test data, etc.: 10 191Distribution format: tar.gzProgramming language: Fortran77Computer: Tested on an HP DL360G5-DC-X5160Operating system: Linux 2.6.9-42.ELsmpRAM: 130 M bytesClassification: 11.5Nature of problem: This program solves time-like DGLAP Q2 evolution equations with or without next-to-leading order αs effects for fragmentation functions. The evolved functions can be calculated for Dgh, Duh, Du¯h, Ddh, Dd¯h, Dsh, Ds¯h, Dch, Dc¯h, Dbh and Db¯h of a hadron h.Solution method: The DGLAP integro-differential equations are solved by the Euler method for the differentiation of lnQ2 and the Gauss–Legendre method for the x integral as explained in Section 4 of the manuscript.Restrictions: This program is used for calculating Q2 evolution of fragmentation functions in the leading order or in the next-to-leading order of αs. Q2 evolution equations are the time-like DGLAP equations. The double precision arithmetic is used. The renormalization scheme is the modified minimal subtraction scheme (MS¯). A user provides initial fragmentation functions as the subroutines FF_INI and HQFF in the end of the distributed code FF_DGLAP.f. In FF_DGLAP.f, the subroutines are given by taking the HKNS07 (2) functions as an example of the initial functions. Then, the user inputs kinematical parameters in the file setup.ini as explained in Section 5.2 of the manuscript.Running time: A few seconds on HP DL360G5-DC-X5160.

Some remarks on dipole showers and the DGLAP equation

It has been argued recently that parton showers based on color dipoles conflict with collinear factorization and do not lead to the correct Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation. We show that this conclusion is based on an inappropriate assumption, namely, the choice of the gluon energy as evolution variable. We further show numerically that Monte Carlo programs based on dipole showers with ``infrared-sensible'' evolution variables reproduce the DGLAP equation both in asymptotic form as well as in comparison to the leading behavior of second-order QCD matrix elements.

Regge residues from Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution

We show that combining forward and backward evolution allows us to extract the residues of the triple-pole Pomeron and of the other singularities for $10<~{Q}^{2}<~1000{\mathrm{GeV}}^{2}.$ In this approach, the essential singularity generated by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution is considered as a numerical approximation to a triple-pole Pomeron. The ${Q}^{2}$-dependent form factors, unknown in Regge theory, are predicted by the DGLAP equation, and lead to a fit to the experimental data with a ${\ensuremath{\chi}}^{2}/dof$ of 1.02. In our case, Regge theory applies at all values of ${Q}^{2}.$ The method used here gives two main results: first, the parton content of the Pomeron is given at large ${Q}^{2}$ by DGLAP evolution, and second, we can evaluate the uncertainties on the gluon distribution which prove to be large at small x and small ${Q}^{2}.$

Fragmentation functions in supersymmetric QCD and ultrahigh energy cosmic ray spectra produced in top-down models

We present results from two different methods for the calculation of hadron spectra in QCD and supersymmetric QCD with large primary energies $\sqrt{s}$ up to ${10}^{16}\mathrm{GeV}.$ The two methods considered are a Monte Carlo (MC) simulation and the evolution of fragmentation functions described by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations. We find that the pion, nucleon, and all-hadron spectra calculated with the two methods agree well. The MC simulation is performed with new hadronization functions (in comparison with our previous work), motivated by low energy $(\sqrt{s}<{M}_{Z})$ data and the DGLAP equation. The hadron spectra calculated with both sets of hadronization functions agree well, which indicates that our method for calculating the hadronization function works successfully. The small difference in the calculated hadron spectra characterizes the uncertainties of this method. We calculate also the spectra of photons, neutrinos, and nucleons and compare them with other published results. The agreement is good for all x from $\ensuremath{\sim}{10}^{\ensuremath{-}5}$ up to $x<~0.3.$ The consistency of the spectra calculated by different methods allows us to consider the spectral shape as a signature of models with decays or annihilations of superheavy particles, such as topological defects or superheavy dark matter. The ultrahigh energy cosmic ray spectra from these sources are calculated.

A C++ code to solve the DGLAP equations applied to Ultra High Energy Cosmic Rays

We solve numerically the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equations for the evolution of fragmentation functions using the Laguerre method. We extend this method to include supersymmetric evolution. The solution to the DGLAP equations is particularly interesting to calculate the expected spectra of Ultra High Energy Cosmic Rays in models where they are produced by the decay of a massive particle X, M X >10 12 GeV.

MASS EFFECTS ON THE NUCLEON SEA STRUCTURE FUNCTIONS

Nucleon sea structure functions are studied using Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equations with the massive gluon–quark splitting kernels for strange and charm quarks, the massless gluon–quark splitting kernels for up and down quarks, and the massless kernels for all other splitting parts. The SU (2)f flavor symmetry for two light quarks, "up" and "down", is assumed. Glück–Reya–Vogt (GRV) and Martin–Roberts–Stirling (MRS) sets are chosen to be the base structure functions at [Formula: see text]. We evolve the sea structure functions from [Formula: see text] to Q2= 50 GeV 2 using the base structure function sets and DGLAP equations. Some (about 10%) enhancement is found in the strange quark distribution functions at low x (<0.1) in leading order of the DGLAP equations compared to results directly from those structure function sets at the value of Q2=50 GeV 2. We provide the value of κ and also show the behavior of [Formula: see text] after the evolution of structure functions.

DOKSHITZER–GRIBOV–LIPATOV–ALTARELLI–PARISI EVOLUTION AND THE RENORMALIZATION GROUP IMPROVED YENNIE–FRAUTSCHI–SUURA THEORY IN QCD

We show that the recently derived renormalization group improved Yennie–Frautschi–Suura (YFS) exponentiation of soft (k0→0) gluons in QCD is fully compatible with the usual Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution of the structure functions of hadrons in the respective hadron–hadron hard interactions. We show how to implement the YFS exponentiation without double or over counting effects already implied by the DGLAP equation. In this way, we arrive at a theory which allows for the development of realistic, multiple gluon Monte–Carlo event generators for hard hadron–hadron scattering processes in which the DGLAP evolved structure functions are correctly synthesized with the respective YFS exponentiated soft gluon effects in a rigorous way.