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.     

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.     

Does leading ln x resummation predict the rise of g 1 at small x?

We numerically analyse the evolution of the flavor non-singlet g ₁ structure function taking into account the all-order resummation of α_s ln² × terms which is expected to have much stronger effects than the DGLAP evolution in the small x region. We include a part of the next-to-leading logarithmic corrections coming from the resummed “coefficient function” which are not considered in the calculation of Blümlein and Vogt to respect the factorization scheme independence. It is pointed out that the resummed coefficient function gives unexpectedly large suppression factor over the experimentally accessible range of x and Q ². This fact implies that the next-to-leading logarithmic contributions are very important for the g ₁ structure function.     

HERA data and DGLAP evolution: Theory and phenomenology

We examine critically the evidence for deviations from next-to-leading order perturbative DGLAP evolution in HERA data. We briefly review the status of perturbative small-x resummation and of global determinations of parton distributions. We show that the geometric scaling properties of HERA data are consistent with DGLAP evolution, which is also strongly supported by the double asymptotic scaling properties of the data. However, backward-evolution of parton distributions into the low x  , low Q²$Q^2$ region does show evidence of deviations between the observed behaviour and the next-to-leading order predictions. These deviations cannot be explained by missing next-to-next-to-leading order perturbative terms, and are consistent with perturbative small-x resummation.     

Small x behaviour of the chirally-odd parton distribution h 1(x,Q 2)

The small x behaviour of the structure function h ₁ (x, Q ²) is studied within the leading logarithmic approximation of perturbative QCD. There are two contributions relevant at small x. The leading one behaves like (1/x)⁰ i.e. it is just a constant in this limit. The second contribution, suppressed by one power of x, includes the terms summed by the GLAP equation. Thus for h ₁ (x, Q ²) the GLAP asymptotics and Regge asymptotics are completely different, making h ₁(x, Q ²) quite an interesting quantity for the study of small x physics.     

Model for nucleon valence structure functions at allx,allp ⊥ and allQ 2 from the correspondence between QCD and DTU

We propose a simple parametrization of the nucleon valence structure functions at allx, allp _⊥ and allQ ². We use the DTU parton model to fix the parametrization at a reference point (Q ₀ ² =3 GeV²) and we mimic the QCD evolution by replacing the dimensioned parameters of the DTU parton model by functions depending onQ ². Excellent agreement is obtained with existing data.     

Analytical solution of the double-logarithmic integral equation with QCD running coupling

The analytical solution of the double-logarithmic integral equation with QCD running coupling describing small-x behaviour of the non-singlet structure function ? _NS(x,Q ²) has been found for any cut-off parameter μ. Analytical properties of the solution and a position of the right-most singularity in the complex ρ-plane which determines the asymptotics of ? _NS(x,Q ²) at small x have been studied. The asymptotical formula ? _NS(x,Q ²) = C ₁ x ^-λ1{ln^κ1(Q ²/Λ²) —ln^κ1 (μ ²/Λ²) + κ ₁ ln^κ1-1(Q ²/Λ²)[ψ(1) - ψ(λ₁)]} valid if x ? 1 and ln(Q ²/Λ²) ? 1 has been obtained where C ₁, λ₁ are constants, κ ₁ = g/λ₁, λ₁ < g = 8/(33 - 2gh _f), gh _f is a number of active flavours and ψ(ξ) denotes the digamma function.     

Model for nucleon gluon and sea quark structure functions at allx and allQ 2 from the correspondence between QCD and DTU

We propose a simple parametrization of the nucleon gluon and sea quark structure functions at allx, and allQ ². We use the DTU parton model to fix the parametrization at a reference point (Q ₀ ² =1.25 GeV² andA ²=0.01 GeV²) and we mimic the QCD evolution by replacing the dimensioned parameters of the DTU parton model by functions depending onQ ².     

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 EMC effect at low x in perturbative QCD

The EMC effect is studied in the perturbative QCD hard pomeron approach. In this framework the nuclear structure function can be expressed directly in terms of the proton one. In the limit x → 0 and for a nucleus with a constant density the effect is found to be a function of a single variable which combines its x-, Q- and A-dependence. The EMC ratio of the nuclear to nucleon structure functions goes to zero for x → 0 or/and A ? 1. At large Q ² the effect dies out as ln Q/Q. To describe the present data at 0. 005 < x < 0. 05 an attempt is made to take into account subasymptotic effects by introducing an anomalous dimension falling with x. This leads to a weaker x- and Q-dependence, which almost disappears as x grows and the anomalous dimension becomes small.     

The NLO Parton Distribution in the (x, Q 2)-Plane: A Relativistic Quark-Exchange Approach to A = 3 Mirror Nuclei

A next-to-leading order QCD analysis of polarized and unpolarized structure functions of the proton in the (x, Q²)-plane is discussed within the scheme of the radiation parton formalism. The valence quark distribution is obtained from the application of the relativistic quark-exchange model to A = 3 mirror nuclei, i.e., ³He and ³H. The sea quark and gluon distributions are calculated using the inverse Mellin technique in the NLO approximation. A comparison is made with the corresponding available experimental data. We find a good fit for F₂^p(x, Q²) to the data. It is shown that our new NLO calculation improves our previous works. We argue that the valence quark scenario at some μ₀² ≪ Q² is a reasonable assumption in the framework of the DGLAP evolution equation. In agreement with the data, it is demonstrated that the asymmetry A₁^p(x, Q²) has no significant Q²-dependence as we go to the small x, even at NLO limit. Finally we argue that for small x ≤ 0.2 it is a good approximation to consider ³He and ³H structure functions as those of neutrons and protons, respectively.     

The effect of screening on the x and Q2 behaviour of F2 slopes

A systematic study of ϖF₂(x, Q²)/ϖ ln Q² and ϖ ln F₂(x, Q²)/ϖ ln(1/x) is carried out in pQCD taking screening corrections into account. The result of calculations, which are different from the non-screened DGLAP prediction, are compared and shown to agree with the available experimental data as well as a pseudo data base generated from the ALLM'97 parameterization. This pseudo data base allows us to study in detail our predictions over a wider kinematic region than is available experimentally, and allows us to make suggestions for future experiments. Our results are compared with the GRV'94 parameterization (which is used as an input for our calculations) as well as the recently proposed MRST structure functions.     

The BFKL-Regge factorization and F2b , F2c , FL at HERA: physics implications of nodal properties of the BFKL eigenfunctions

The asymptotic freedom is known to split the leading-log BFKL pomeron into a series of isolated poles in the complex angular momentum plane. One of our earlier findings was that the subleading hard BFKL exchanges decouple from such experimentally important observables as small-x charm F ₂ ^c , beauty F ₂ ^b and the longitudinal structure functions of the proton at moderately large Q ². For instance, we predicted precocious BFKL asymptotics of F ₂ ^c (x, Q ²) with intercept of the rightmost BFKL pole a _IP(0) − 1 = Δ_IP ≈ 0.4. On the other hand, the small-x open beauty photo- and electro-production probes the vacuum exchange for much smaller color dipoles which entails significant subleading vacuum pole corrections to the small-x behavior. In view of the accumulation of the experimental data on small-x F ₂ ^c and F ₂ ^b we extend our 1999 predictions to the kinematical domain covered by new HERA measurements. Our parameter-free results agree well with the determination of F ₂ ^c , F _L and published H1 results F ₂ ^b on but slightly overshoot the very recent (2008, preliminary) H1 results on F ₂ ^b .     

An estimate of higher twist at small x_{\mathrmB} and low Q^2 based upon a Saturation Model

We investigate the influence of higher twist corrections to deep inelastic structure functions in the low- and small-x HERA region. We review the general features of the lowest-order QCD diagrams which contribute to twist-4 at small-x, in particular the sign structure of longitudinal and transverse structure functions which offers the possibility of strong cancellations in . For a numerical analysis we perform a twist analysis of the saturation model which has been very successful both in describing the structure function and the DIS diffractive cross section at HERA. As the main conclusion, twist 4 corrections are not small in or but in they almost cancel. This indicates the limitation on the use of the DGLAP formalism at small x and . We point out that analysis needs a large twist-4 correction. We also indicate the region of validity of the twist expansion. Received: 20 March 2000 / Published online: 9 August 2000     

The linked dipole chain model for DIS

The Linked Dipole Chain Model provides an interpolation between the regions of high Q² (DGLAP) and low x moderate Q² (BFKL) in DIS. It is a reformulation and a generalization of the results obtained by Ciafaloni, Catani, Fiorani and Marchesini, and it gives a unified treatment of “normal DIS”, boson-gluon fusion events and hard subcollisions in resolved photon-proton scattering. Thus the formalism provides a complete picture which incorporates all hard or semihard hadronic interactions in a simple way, which is suitable for a Monte Carlo treatment of both structure functions and final state properties. We also discuss non-leading effects which significantly reduce the increase of the structure function for very small x-values.     

Structure functions of bound nucleons: From the EMC effect to nuclear shadowing

We describe the nuclear structure functions in the whole range of the Bjorken variablex, by combining various effects in a many-step procedure. First, we present a QCD motivated model of nucleons, treated, in the limit of vanishingQ ², as bound states of three relativistic constituent quarks. Gluons and sea quarks are generated radiatively from the input valence quarks. All parton distributions are described in terms of the confinement (or nucleon's) radius. The results for free nucleons are in agreement with the experimental determinations. The structure functions of bound nucleons are calculated by assuming that the main effect of nucleon binding is stretching of nucleons. The larger size of bound nucleons lowers the valence momentum and enhances the radiatively generated glue and sea densities. In the small-x region the competitive mechanism of nuclear shadowing takes place. It also depends on the size of the nucleons. By combining stretching, shadowing and Fermi motion effects (the latter confined to very largex), the structure function ratio is well reproduced. Results are also presented for theA-dependence of the momentum integral of charged partons, the nuclear gluon distribution and the hadron-nuclei cross sections.     

Scaling violation in the pion structure function

TheQ ²-dependence of the pion structure function extracted from high transverse momentum π⁰ production cross sections is examined. We find, in thex→1 region, that this dependence can be parametrized by a power of (1?x) increasing as ln lnQ ². This is in agreement with the expectations of asymptotic freedom and also seen in the structure function extracted from dilepton production. It is not possible to fit all of theQ ²-dependence with a simple power law term.     

Testing QCD scaling violations in the HERA energy range

High energy electron-proton colliders open new kinematical domains for the investigation of scaling violations in nucleon structure functions. Focussing on HERA as a definite study case we explore the precision which can be expected in determinations of the QCD scale parameter Λ and the running coupling constant α _s (Q ²). Our results provide a quantitative basis for discussions of the possibilities and requirements of QCD tests in inclusive deep inelastic scattering at HERA energies.