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.     

Twist-4 gluon recombination corrections for deep inelastic structure functions

We calculate twist-4 coefficient functions for the deep inelastic structure function F₂(x,Q²) associated to 4-gluon operator matrix elements for general values of the Bjorken variable x and study the numerical effect on the slope ∂F₂(x,Q²)/∂logQ². It is shown that these contributions diminish the strongly rising twist-2 terms towards small values of x.     

A measurement of the proton structure function F2(x,Q2)

A measurement of the proton structure function F₂(x, Q²) is reported for momentum transfers squared Q² between 4.5 GeV² and 1600 GeV² and for Bjorken x between 1.8 × 10⁻¹⁴ and 0.13 using data collected by the HERA experiment H1 in 1993. It is observed that F₂ increases significantly with decreasing x, confirming our previous measurement made with one tenth of the data available in this analysis. The Q² dependence is approximately logarithmic over the full kinematic range covered. The subsample of deep inelastic events with a large pseudo-rapidity gap in the hadronic energy flow close to the proton remnant is used to measure the “diffractive” contribution to F₂.     

F 2 c at low x

We study the heavy-quark contributions to the proton structure function F ₂(x,Q ²) at next-to-leading order using compact formulas at small values of Bjorken??s x variable. The formulas provide a good agreement with the modern HERA data for F ₂ ^c (x,Q ²).     

The longitudinal structure function F L(x, Q 2) and the gluon distribution G(x, Q 2) at low x

We obtain a relation between the longitudinal structure function F _L(x, Q ²), F ₂(x, Q ²) and G(x, Q ²) at small x, using the formalism recently reported by one of the authors [2]. We also obtain a relation between F _L(x, Q ²), F ₂(x, Q ²) and its slope (dF ₂(x, Q ²))/(dlnQ ²). This provides us with the determination of the longitudinal structure function F _L(x, Q ²) from F ₂(x, Q ²) data and hence extract the gluon distribution G(x, Q ²).     

Measurement of the proton structure function F 2 at low x and low Q 2 at HERA

We report on a measurement of the proton structure function F ₂ in the range 3.5 × 10^?5 ≤ x ≤ 4 × 10^?3 and 1.5 GeV² ≤ Q ² ≤ 15 GeV² at the ep collider HERA operating at a centre-of-mass energy of ${sqrt s} = 300 {? GeV}$. The rise of F ₂ with decreasing x observed in the previous HERA measurements persists in this lower x and Q ² range. The Q ² evolution of F ₂, even at the lowest Q ² and x measured, is consistent with perturbative QCD.     

Discriminative deep inelastic tests of strong interaction field theories

It is demonstrated that recent measurements of ∫¹₀F₂(x, Q²)dx eliminate already all strong interaction field theories except QCD. A detailed study of scaling violations of F₂(x, Q²) in QCD shows their insensitivity to the gluon content of the hadron at presently measured values of Q².     

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.     

Structure functions of nuclei for all x and Q 2

We propose a modified version of the quark cluster model to describe the structure functions of nuclei over the complete kinematical region of the variables x = Q ²/2m _N ν and Q ². The model accounts for shadowing effects at very small x and anti-shadowing in the region 0.1 ≤ x ≤ 0.3. The experimental observation that the ratio of structure functions F ₂ ^A /F ₂ ^D is less than unity in the region x ? 0.5 is related to the influence of multiquark clusters. The model agrees with all available data on nuclear structure fuctions for a large variety of nuclei, and can be used to make predictions for kinematical regions not yet studied experimentally.     

How to measure the charm density in the proton

We study two experimental ways to measure the heavy-quark content of the proton: using the Callan-Gross ratio R(x, Q ²) = F _L /F _T and/or the azimuthal cos(2φ) asymmetry in DIS. Our approach is based on the following observations. First, the ratio R(x, Q ²) = F _L /F _T and azimuthal cos(2φ) asymmetry in heavy-quark leptoproduction are stable, both parametrically and perturbatively, within pQCD. Second, both these quantities are sensitive to resummation of the mass logarithms of the type α_sln(Q ²/m ²). We conclude that the heavy-quark densities in the nucleon can, in principle, be determined from high-Q ² data on the Callan-Gross ratio and/or the azimuthal asymmetry. In particular, the charm content of the proton can be measured in future studies at the proposed Large Hadron-Electron (LHeC) and Electron-Ion (EIC) Colliders.     

A model for FL and R = FL/FT at low x and low Q2

A model for the longitudinal structure function F_L at low x and low Q² is presented, which includes the kinematical constraint F_L ～ Q⁴ as Q² → 0. It is based on the photon-gluon fusion mechanism suitably extrapolated to the region of low Q². The contribution of quarks having limited transverse momentum is treated phenomenologically assuming that it is described by the soft pomeron exchange mechanism. The ratio R = F_L/(F₂ ? F_L), with the F₂ appropriately extrapolated to the region of low Q², is also discussed.     

Total cross sections and nucleon structure functions in the gargamelle SPS neutrino/antineutrino experiment

Total neutrino and antineutrino cross sections in the energy range 15 to 150 GeV, and the nucleon structure functions, F₂(x, Q²) and xF₃(x, Q²) in the Q² range 0.5 to 50 (GeV/c)² have been measured using a data sample of 3000 neutrino and 3800 antineutrino events. The structure functions show a weak Q² dependence at different x values.     

Scaling and non-scaling in a rest frame quark-parton model of the proton

Electron-proton deep inelastic scattering is treated as the incoherent scattering of electrons by bound Dirac partons in the proton rest frame. An approximate bound state wave function is used for the initial parton, while the final parton is considered free. A good fit is obtained to the structure function F₁(x,Q²) in the range x > 0.15, Q² > 2 GeV. The subsequent prediction for F₂(x,Q²) is not as good, indicating a small additional contribution by longitudinal photons for W < 2.5 GeV. The parton momentum distribution is found to contain transverse momentum of 400–600 MeV, increasing with x.     

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.     

Charm production and parton distributions

Recent accurate data on F ₂(x,Q) and on F ₂ ^c (x,Q) from HERA at small-x require a more precise treatment of charm production in the global analysis of parton distributions. We improve on existing global QCD analyses by implementing the leptoproduction formalism of Aivazis et al. which represents a natural generalization of the conventional zero mass QCD parton framework to include heavy quark mass effects. We also perform analyses based on the fixed-flavor-number scheme, which is widely used in the literature, and demonstrate their uses and limitations. We discuss the implications of the improved treatment of heavy quark mass effect in practical applications of PQCD and compare our results with recent related works.     

F L structure function and heavy quark contributions

In this paper we obtain the heavy-quark contribution to the longitudinal structure functions F _L (x, Q ²). Since F _L structure functions contains rather large heavy flavor contributions in the small x region, we need to use the massive operator matrix elements, which contribute to the heavy flavor Wilson coefficients in unpolarized deeply inelastic scattering in the region Q ²?>?>?m ². The method of QCD analysis, based on the Jacobi polynomials method, is also described. Our results for longitudinal structure function are in good agreement with the available experimental data.     

Anomalous cross section for photon-photon scattering in gauge theories

We consider the deep inelastic structure functions of the photon in an asymptotically free gauge theory. In contrast to the case of a hadronic target, we find that the shortdistance analysis determines the shape and magnitude and not merely the Q² dependence of the structure functions. The structure functions of the free quark theory are renormalized by finite, calculable factors. For example, at x = 0.1, we find that F₂ will, at large Q², exceed the free quark result by a factor 1.751, while for x = 0.5, F₂ is suppressed asymptotically, relative to the free quark theory, by a factor 0.964, and at x = 0.8, by a factor 0.611.     

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.