The parton distribution functions of the photon and the structure functionF 2 γ (x,Q 2)

We present new parametrisations of the parton distribution functions of the photon including the first parametrisation in next-to-leading order QCD. We take into account some recent theoretical considerations pertaining to the gluon content of the photon,g ^γ. We argue that if an evolution is started at very lowQ ² and a fit to allF ₂ ^γ data performed with no constraints on the gluon distribution, then physically unreasonable gluon distributions may result. Our results support recent indications thatQ ₀ ² ≤1 GeV² is too low a value from which to start a perturbative evolution. Starting our evolution atQ ₀ ² =5.3 GeV², we evolve up inQ ² using a modified version of Rossi's Ansatz. The limited lever arm inQ ² leads to limited sensitivity to the QCD scale parameter Λ, though there is a preference for low values in the 0.1–0.2 GeV range. We also present new parametrisations of the singular asymptotic quark and gluon distribution functions of the photon which we believe are more accurate than those in current use.     

Constraints on gluon evolution at smallx

The BFKL and the unified angular-ordered equations are solved to determine the gluon distribution at smallx. The impact of kinematic constraints is investigated. Predictions are made for observables sensitive to the gluon at smallx. In particular comparison is made with measurements at the HERA electron-proton collider of the proton structure functionF ₂(x, Q ²) as a function of lnQ ², the charm componentF _c ² (X, Q ²) and diffractiveJ/Ψ photoproduction.     

Constraints on gluon evolution at smallx

The BFKL and the unified angular-ordered equations are solved to determine the gluon distribution at smallx. The impact of kinematic constraints is investigated. Predictions are made for observables sensitive to the gluon at smallx. In particular comparison is made with measurements at the HERA electron-proton collider of the proton structure functionF ₂(x, Q ²) as a function of lnQ ², the charm componentF _c ² (X, Q ²) and diffractiveJ/?? photoproduction.     

Analysis of the logarithmic slope of <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript> from the Regge gluon density behavior at small <Emphasis Type="Italic">x</Emphasis>

We study the accuracy of the Regge behavior of the gluon distribution function for an approximate relation that is frequently used to extract the logarithmic slopes of the structure function from the gluon distribution at small x. We show that the Regge behavior analysis results are comparable with HERA data and are also better than other methods that expand the gluon density at distinct points of expansion. We also show that for Q ² = 22.4 GeV², the x dependence of the data is well described by gluon shadowing corrections to the GLR-MQ equation. The resulting analytic expression allows us to predict the logarithmic derivative ∂F ₂(x, Q ²)/∂lnQ ² and to compare the results with the H1 data and a QCD analysis fit with the MRST parameterization input.     

A new approach to calculate the gluon polarization

We derive the Leading-Order (LO) master equation to extract the polarized gluon distribution G(x,Q ²)=xδg(x,Q ²) from polarized proton structure function, g^p₁(x,Q²)g^{p}_{1}(x,Q^{2}). By using a Laplace-transform technique, we solve the master equation and derive the polarized gluon distribution inside the proton. The test of accuracy which is based in our calculations on two different methods, confirms that we achieve to the correct solution for the polarized gluon distribution. To determine the polarized gluon distribution xδg(x,Q ²) more accurately, we only need to have more experimental data on the polarized structure functions, g₁^p(x,Q²)g_{1}^{p}(x,Q^{2}). Our result for polarized gluon distribution is in good agreement with some phenomenological models.     

Deep-inelastic scattering in κ factorization and the anatomy of the differential gluon structure function of the proton

The differential gluon structure function of the proton, ?(x, Q ²), introduced by Fadin, Kuraev, and Lipatov in 1975 is extensively used in small-x QCD. We report here the first determination of ?(x, Q ²) from experimental data on the small-x proton structure function F _2p (x, Q ²). We give convenient parametrizations for ?(x, Q ²) based partly on the available DGLAP evolution fits (GRV, CTEQ, and MRS) to parton distribution functions and on realistic extrapolations into the soft region. We discuss the impact of soft gluons on various observables. The x dependence of the so-determined ?(x, Q ²) varies strongly with Q ² and does not exhibit simple Regge properties. Nonetheless, the hard-to-soft diffusion is found to give rise to a viable approximation of the proton structure function F _2p (x, Q ²) by the soft and hard Regge components with intercepts Δ_soft=0 and Δ_hard ～ 0.4.     

The role of screening corrections in smallx-behaviour of structure functions

We consider the influence of shadowing effects on the proton structure function in the small-x interval. The gluon-gluon shadowing are noticeable in the HERA kinematical region while the screening of the quark component of the structure function effects negligibly the gluon distribution. The only noticeable effect is the decreasing of sea quark densities at small-x. The explicit form of the gluon distribution in the proton depends significantly on the form of used boundary condition atQ ²=Q ₀ ² . We consider also difference of results obtained with the help of Kuraev-Lipatov-Fadin and Altarelli-Parisi evolution equations.     

The role of gluon-gluon shadowing in smallx-behaviour of structure functions

We consider influence of gluon shadowing effects on the proton structure function in the smallx interval. These effects are noticeable in the HERA kinematical region. The explicit form of the gluon distribution in the proton depend significantly on the form of used boundary condition atQ ²=Q ₀ ² . We consider also difference of results obtained with the help of Kuraev-Lipatov-Fadin and Altarelli-Parisi evolution equations.     

A solution of the DGLAP equation for gluon at low x

We obtain a solution of the DGLAP equation for the gluon at low x first by expanding the gluon in a Taylor series and then using the method of characteristics. We test its validity by comparing it with that of Glück, Reya and Vogt. The convergence criteria of the approximation used are also discussed. We also calculate εF ₂(x,Q)²/ε In Q ² using its approximate relations with the gluon distribution at low x. The predictions are then compared with the HERA data.     

Unpolarized coupled DGLAP evolution equation at small-x

In this paper, we have obtained the solution of the unpolarized coupled Dokshitzer–Gribove–Lipatov–Alterelli–Parisi (DGLAP) evolution equation in leading order at the small-x limit. Here, we have used a Taylor series expansion, separation of functions and then the method of characteristics to solve the evolution equations. We have also calculated t-evolution of singlet and gluon distribution functions and the results are compared with E665 and NNPDF data for singlet structure function and GRV1998 and MRST2004 gluon parametrizations. It is shown that our results are in good agreement with the parametrizations especially at small-x and high-Q ² region. From global parametrizations and our results, we have seen that the singlet and gluon distribution functions increase when Q ² increases for fixed values of x.     

Estimate of the screening corrections to gluon distributions in the smallx region

The screening corrections to gluon distributions in a proton corresponding to the triple gluonic ladder diagram are estimated. They are found to be relatively small: their values does not exceed 10% of the leading order QCD gluon distribution forx?10^?4 andQ ² ?100 GeV².     

Measurement of the longitudinal structure function and the smallx gluon density of the proton

At HERA energies the smallx region (x?10^?2) can be explored atQ ² values large enough that leading twist QCD calculations are valid. We show how measurement of the longitudinal structure function,F _L (x,Q ²), can lead to accurate measurement of the gluon structure function at such smallx values. Experimental systematic errors are discussed fully and requirements for the measurement outlined. We conclude that it should be possible to distinguish between the widely varying gluon distributions which are currently allowed.     

Shadowing correction to the gluon distribution behavior at small x

We determined the saturation exponent of the gluon distribution using the solution of the QCD nonlinear Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (NLDGLAP) evolution equation at small x . The very small-x behavior of the gluon distribution is obtained by solving the Gribov, Levin, Ryskin, Mueller and Qiu (GLR-MQ) evolution equation with the nonlinear shadowing term incorporated. The form of the initial condition for the equation is determined. We find, with decreasing x , the emergence of a singular behavior and the eventual taming (at R = 5 GeV^-1) and the essential taming (at R = 2 GeV^-1) of this singular behavior by the shadowing term. The nonlinear gluon density functions are calculated and compared with the results for the integrated gluon density from the Balitsky-Kovchegov (BK) equation for the different values of Q². It is shown that the results for the gluon density function are comparable with the results obtained from the BK equation solution. Also we show that for each x , the Q²-dependence of the data is well described by gluon shadowing corrections to the GLR-MQ equation. The resulting analytic expression allows us to predict the logarithmic derivative \( {\frac{{\partial F^{s}_{2} (x,Q^{2})}}{{\partial \ln Q^{2}}}}\) and to compare the results with H1 data and a QCD analysis fit.     

Observation of QCD effects in transverse momenta of e+e− jets

Probability distributions of the total transverse momenta (K⊥) of charged particles produced in hadronic jets in e⁺e^? annihilations have been measured for center of mass energies in the range from 9.2 to 31.6 GeV. A linear increase of the average K_⊥² with Q² is observed. The data are successfully compared with high order QCD predictions (according to a simple q$\text{q}$ picture supplemented by multiple emission of soft gluons). Deviations from this picture at the highest energies and large K_⊥ are then analyzed in terms of hard gluon bremsstrahlung and qualitative agreement is found with first order QCD predictions. Scaling “in the mean” is found to be valid both for jet and single particle transverse momenta.     

Neutrino and antineutrino charged-current inclusive scattering in iron in the energy range 20<E v <300 GeV

Inclusive charged-current interactions of high-energy neutrinos and antineutrinos have been studied with high statistics in a counter experiment at the CERN Super Proton Synchrotron. The energy dependence of the total cross-sections, the longitudinal structure function, and the nucleon structure functionsF ₂,xF ₃, and \(\bar q^{\bar v} \) are determined from these data. The analysis of theQ ²-dependence of the structure functions is used to test quantum chromodynamics, to determine the scale parameter Λ and the gluon distribution in the nucleon.     

A study of QCD scaling violations by direct resolution of Altarelli-Parisi type equations

We present a detailed study of scaling violations for non-singlet, singlet and glue distribution functions in the framework of several approximation schemes of QCD. Our formalism consists of direct resolution of the Altarelli-Parisi type equations and leads to a simple exponential form for the function q(x, Q²). This form is very suitable for the analysis of experimental data and for the exploration of different evolution schemes. In particular, we examine the implications on the QCD scaling parameter Λ and the gluon parameter n_G.     

Pionic parton distributions

The gluon and sea distributions of the pion are uniquely determined by the requirement of avalence-like structure of the input parton distributions at some low resolution scale. These (dynamical) results are obtained with practically no free parameters, just using the experimentally determined pionic valence distribution combined with the constraints for the pionic gluon distribution provided by direct-γ data. Simple parametrizations of the resulting parton distributions are presented in the range 10^?5?x<1 and 0.3?Q ²?10⁸ GeV² as obtained from the leading-and higher-order evolution equations.     

The Q<Superscript>2</Superscript> dependence of polarized and unpolarized proton structure functions in the relativistic quark exchange framework

In this article we intend to discuss the evolution of polarized and unpolarized structure functions in the (x,Q²) plane. We analyze the proton data on the spin dependence asymmetry A₁(x,Q²), by making the dynamical assumption that at low resolution energies, the hadrons consist only of valence quarks and the scaling violation of F₂(x,Q²) at low x comes only from the gluons density. While the sea quark and the gluon distributions are calculated using the inverse Mellin technique and the various moments of the valence quarks, the valence quark distribution itself is obtained from the relativistic quark-exchange model. A comparison is made with the corresponding available experimental data. Finally in agreement with the data, it is demonstrated that there is no significant Q²-dependence of asymmetry A₁(x,Q²) for x ranging 0.014 ≤ x ≤ 0.25. Received: 11 September 1999 / Revised version: 8 December 1999     

The gluon distributions in the smallx region beyond the leading order

The gluon distributions in a proton are calculated in the region of smallx (10^?4?x?10^?2) and largeQ ² taking into account the corrections which correspond to the sum of the leading powers of log 1/x beyond the leading logQ ² approximation. It is shown that they become significant for very small values ofx only provided however that the non-leading terms in the limit of smallx are also consistently included. The leading log 1/x approximation gives the gluon distributions which can differ by a factor of 2 from their leading logQ ² counterpart.