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     

A new numerical method for inverse Laplace transforms used to obtain gluon distributions from the proton structure function

We recently derived a very accurate and fast new algorithm for numerically inverting the Laplace transforms needed to obtain gluon distributions from the proton structure function F₂^gp(x,Q²)F_{2}^{\gamma p}(x,Q^{2}). We numerically inverted the function g(s), s being the variable in Laplace space, to G(v), where v is the variable in ordinary space. We have since discovered that the algorithm does not work if g(s)→0 less rapidly than 1/s as s→∞, e.g., as 1/s ^β for 0<β<1. In this note, we derive a new numerical algorithm for such cases, which holds for all positive and non-integer negative values of β. The new algorithm is exact if the original function G(v) is given by the product of a power v ^β−1 and a polynomial in v. We test the algorithm numerically for very small positive β, β=10⁻⁶ obtaining numerical results that imitate the Dirac delta function δ(v). We also devolve the published MSTW2008LO gluon distribution at virtuality Q ²=5 GeV² down to the lower virtuality Q ²=1.69 GeV². For devolution, β is negative, giving rise to inverse Laplace transforms that are distributions and not proper functions. This requires us to introduce the concept of Hadamard Finite Part integrals, which we discuss in detail.     

Q 2 dependence of the measured asymmetry A 1: A test of the Bjorken sum rule

We analyze the proton and deutron data on the spin-dependent asymmetry A ₁(x, Q ²), supposing that the DIS structure functions g ₁(x, Q ²) and F ₃(x, Q ²) have a similar Q ² dependence. As a result, we have found that Λ ₁ ^p −Λ ₁ ⁿ =0.190±0.038 at Q ²=10 GeV² and Λ ₁ ^p −Λ ₁ ⁿ =0.165±0.026 at Q ²=3 GeV²; these values are in the best agreement with the Bjorken sum rule predictions. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 1, 9–14 (10 January 1997) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Gluon distributions in real and virtual photon and the photon structure function

Gluon distributions in real and virtual photons are calculated using evolution equations in the NLO approximation. The quark distributions in the photon determined on the basis of the QCD sum rule approach in [1] are taken as an input. It is shown that gluon distribution in the photon can be reliably determined up tox=0.03÷0.05, much lower than the corresponding values in the case of quark distributions. Two variants of the calculations are considered: (1) it is assumed that there are no intrinsic gluons in the photon at some low normalization pointQ ²=Q ₀ ² ∼1GeV²; (2) it is assumed that gluonic content of the photon at lowQ ₀ ² is described by gluonic content of vector mesonsρ, ω, ϕ. The gluon distributions in these two variants appear to be different. This fact permits one to clarify the origin of nonperturbative gluonic content of the photon by comparing the results with experiment. Structure functionsF ₂(x) for real and virtual photon are calculated and it is shown that in the regionx≥0.2 where QCD approach is valid, there is a good agreement with experiment.     

Possible parametrization of parton distributions incorporating singular smallx behaviour implied by QCD and its phenomenological implications for HERA

The gluon and quark distributions of the nucleon are evaluated using the Altarelli-Parisi equations with the input distributions atQ ₀ ² =5 GeV² for seaquarks and gluons modified by the factor (ax ^?0.5+b). The new parametrization is constrained to satisfy the momentum sum rule and after backward evolution (fromQ ₀ ² =5 GeV² toQ ²=1 GeV²) it is also constrained to give approximately 1/x behaviour of the sea-quark and gluon distributions in the limited region of smallx (10^?3<x<10^?2 or so). The theoretical predictions relevant for HERA for structure functionsF ₂(x, Q ²) andF _L (x, Q ²) in the region of very smallx(10^?4<x<10^?2) and largeQ ² and for the cross-sectionσ(γ^* p→ΨX) are presented. Distributions of heavy quarks (c,b,t) are also discussed.     

Diffraction at HERA, Color Opacity and Nuclear Shadowing in DIS off nuclei

The QCD factorization theorem for diffractive processes in DIS is used to derive formulae for the leading twist contribution to the nuclear shadowing of parton distributions in the low thickness limit (due to the coherent projectile (photon) interactions with two nucleons). Based on the current analyzes of diffraction at HERA we find that the average strength of the interactions which govern diffraction in the gluon sector at x≤ 10⁻³, Q ₀= 2 GeV is ∼50mb. This is three times larger than in the quark sector and suggests that applicability of DGLAP approximation requires significantly larger Q ₀ in the gluon sector. We use this information on diffraction to estimate the higher order shadowing terms due to the photon interactions with N≥ 3 nucleons which are important for the scattering of heavy nuclei and to calculate nuclear shadowing and Q ² dependence of gluon densities. For the heavy nuclei the amount of the gluon shadowing: G _A(x,Q ₀ ²) /AG _N(x,Q ₀ ²)_{|x ≤ 10−3}∼ 0.25–0.4 is sensitive to the probability of the small size configurations within wave function of the gluon “partonometer” at the Q ₀ scale. At this scale for A∼ 200 the nonperturbative contribution to the gluon density is reduced by a factor of 4–5 at x≤ 10⁻³ unmasking PQCD physics in the gluon distribution of heavy nuclei. We point out that the shadowing of this magnitude would strongly modify the first stage of the heavy ion collisions at the LHC energies, and also would lead to large color opacity effects in eA collisions at HERA energies. In particular, the leading twist contribution to the cross section of the coherent J/ψ production off A≥ 12 nuclei at s ⁻²≥ 70 GeV is strongly reduced as compared to the naive color transparency expectations. The Gribov black body limit for F _2A(x,Q ²) is extended to the case of the gluon distributions in nuclei and shown to be relevant for the HERA kinematics of eA collisions. Properties of the final states are also briefly discussed. Received: 12 March 1999     

Nuclear effects in deep inelastic scattering of charged-current neutrino off nuclear target

The nuclear effects in the neutrino–nucleus charged-current inelastic scattering process is studied by analyzing the CCFR and NuTeV data. The structure functions F₂(x,Q²) and xF₃(x,Q²) as well as differential cross sections are calculated by using CTEQ parton distribution functions and the EKRS and HKN nuclear parton distribution functions, and these are compared with the CCFR and NuTeV data. It is found that the corrections of the nuclear effect to the differential cross section for the charged-current antineutrino scattering on the nucleus are negligible, the EMC effect exists in the neutrino structure function F₂(x,Q²) in the large x region, the shadowing and anti-shadowing effect occur in the distribution functions of valence quarks in the small and medium x region, respectively. It is also found that shadowing effects on F₂(x,Q²) in the small x region in the neutrino–nucleus and the charged-lepton–nucleus deep inelastic scattering processes are different. It is clear that the neutrino–nucleus deep inelastic scattering data should further be employed in restricting the nuclear parton distributions. PACS 13.15.+g; 24.85.+p; 25.30.-c     

x-dependence of the quark distribution functions in the χCQM<Subscript>config</Subscript>

Chiral constituent quark model with configuration mixing (χCQM_config) is known to provide a satisfactory explanation of the “proton spin problem” and related issues. In order to enlarge the scope of χCQM_config, we have attempted to phenomenologically incorporate x-dependence in the quark distribution functions. In particular, apart from calculating valence and sea quark distributions q_val(x) and q̄(x), we have carried out a detailed analysis to estimate the sea quark asymmetries d̄(x)-ū(x), d̄(x)/ū(x) and as well as spin independent structure functions F₂ ^p(x)-F₂ ⁿ(x) and F₂ ⁿ(x)/F₂ ^p(x) as functions of x. We are able to achieve a satisfactory fit for all the above mentioned quantities simultaneously. The inclusion of effects due to configuration mixing have also been examined in the case F₂ ^p(x)-F₂ ⁿ(x) and F₂ ⁿ(x)/F₂ ^p(x) where the valence quark distributions dominate and it is found that it leads to considerable improvement in the results. Further, the valence quark structure has also be tested by extrapolating the predictions of our model in the limit x→1 where data is not available.     

Pion form factor in QCD: How to calculate?

The calculation of the pion form factor F _π(Q ²) in QCD is discussed. The main points of the nonlocal condensate QDC sum rule approach are considered and its results for the pion form factor are shown compared with the predictions of the perturbative and lattice QCD. The local duality (LD) approach for the pion FF in QCD is studied. It is shown that the main parameter of the approach for Q ² ≥ 2 GeV², namely, s ₀^LD(Q ²) should grow with an increase in Q ², rather than remain constant.     

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.     

Study of nucleon spin-dependent properties in the resonance region

In this paper, the spin-dependent structure functions of nucleon g ₁, and photoabsorption cross sections σ_1/2, σ_3/2 and σ_T in the resonance region are estimated based on the constituent quark model and the properties of the five phenomenological Breit-Wigner resonances P ₃₃(1232), S ₁₁(1535), D ₁₃(1520), P ₁₁(1440), and F ₁₅(1680). Our results are compared to the recent E143 data of the polarized structure functions g ₁(W ², Q ²) at points Q ²=0.5 GeV² and Q ²=1.2 GeV² and the data of the total inclusive photoabsorption cross sections. Received: 7 October 1997     

On the rise of the proton structure function F2 towards low x

A measurement of the derivative (∂ lnF₂/∂ lnx)_Q²≡−λ(x,Q²) of the proton structure function F₂ is presented in the low x domain of deeply inelastic positron–proton scattering. For 5×10⁻⁵x0.01 and Q²1.5 GeV², λ(x,Q²) is found to be independent of x and to increase linearly with lnQ².     

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 functionF ₂ in the range 3.5×10⁻⁵≤x≤4×10⁻³ and 1.5 GeV²≤Q ²≤15GeV² at theep collider HERA operating at a centre-of-mass energy of √s=300GeV. The rise ofF ₂ with decreasingx observed in the previous HERA measurements persists in this lowerx andQ ² range. TheQ ² evolution ofF ₂, even at the lowestQ ² andx measured, is consistent with perturbative QCD. supported by EU HCM contract ERB-CHRX-CT93-0376     

Measurement and QCD analysis of the diffractive deep-inelastic scattering cross section at HERA

A detailed analysis is presented of the diffractive deep-inelastic scattering process ep→eXY, where Y is a proton or a low mass proton excitation carrying a fraction 1-x_IP>0.95 of the incident proton longitudinal momentum and the squared four-momentum transfer at the proton vertex satisfies |t|<1 GeV². Using data taken by the H1 experiment, the cross section is measured for photon virtualities in the range 3.5≤Q²≤1600 GeV², triple differentially in x_IP, Q² and β=x/x_IP, where x is the Bjorken scaling variable. At low x_IP, the data are consistent with a factorisable x_IP dependence, which can be described by the exchange of an effective pomeron trajectory with intercept α_IP(0)=1.118±0.008(exp.)^+0.029 _-0.010(model). Diffractive parton distribution functions and their uncertainties are determined from a next-to-leading order DGLAP QCD analysis of the Q² and β dependences of the cross section. The resulting gluon distribution carries an integrated fraction of around 70% of the exchanged momentum in the Q² range studied. Total and differential cross sections are also measured for the diffractive charged current process e⁺p→ν̄_eXY and are found to be well described by predictions based on the diffractive parton distributions. The ratio of the diffractive to the inclusive neutral current ep cross sections is studied. Over most of the kinematic range, this ratio shows no significant dependence on Q² at fixed x_IP and x or on x at fixed Q² and β.     

Photon-pion transition form factor at high photon virtualities within the nonlocal chiral quark model

Recently, the BABAR collaboration reported the measurements of the photon-pion transition form factor F _πγγ*(Q ²), which are in strong contradiction to the predictions of the standard factorization approach to perturbative QCD. In the present work, based on a nonperturbative approach to the QCD vacuum and on rather universal assumptions, we show that there exist two asymptotic regimes for the pion transition form factor. One regime with the asymptotic behavior F _πγ*γ(Q ²) ∼ 1/Q ² corresponds to the result of the standard QCD factorization approach, while other violates the standard factorization and leads to asymptotic behavior as F _πγ*γ(Q ²) ∼ ln(Q ²)/Q ². Furthermore, considering specific nonlocal chiral quark models, we find the region of parameters, where the existing CELLO, CLEO and BABAR data for the pion transition form factor are successfully described.     

Deconfinement transition for nonzero baryon density in the field correlator method

Deconfinement phase transition due to the disappearance of confining colorelectric field correlators is described using a nonperturbative equation of state. The resulting transition temperature T _c (μ) at any chemical potential μ is expressed in terms of the change of the gluon condensate ΔG ₂ and absolute value of the Polyakov loop L _fund(T _c ), which is known from the lattice and analytic data, and is in good agreement with the lattice data for ΔG ₂ ≈ 0.0035 GeV⁴; e.g., T _c (0) = 0.27, 0.19, and, 0.17 GeV for n _f = 0, 2, and 3, respectively. The text was submitted by the authors in English.     

On the perturbative stability of the QCD predictions for the ratio R=FL/FT in heavy-quark leptoproduction

We analyze the perturbative and parametric stability of the QCD predictions for the Callan–Gross ratio, R(x,Q ²)=F _L /F _T , in heavy-quark leptoproduction. We consider the radiative corrections to the dominant photon–gluon fusion mechanism. In various kinematic regions, the following contributions are investigated: exact NLO results at low and moderate Q ²≲m ², asymptotic NLO predictions at high Q ²≫m ², and both NLO and NNLO soft-gluon (or threshold) corrections at large Bjorken variable x. Our analysis shows that large radiative corrections to the structure functions F _T (x,Q ²) and F _L (x,Q ²) cancel each other in their ratio R(x,Q ²) with good accuracy. As a result, the NLO contributions to the Callan–Gross ratio are less than 10% in a wide region of the variables x and Q ². We provide compact LO predictions for R(x,Q ²) in the case of low x ≪1. A simple formula connecting the high-energy behavior of the Callan–Gross ratio and low-x asymptotics of the gluon density is derived. It is shown that the obtained hadron-level predictions for R(x→0,Q ²) are stable under the DGLAP evolution of the gluon distribution function. Our analytic results simplify the extraction of the structure functions F ₂ ^c (x,Q ²) and F ₂ ^b (x,Q ²) from measurements of the corresponding reduced cross sections, in particular at DESY HERA.     

The curvature of F<Subscript>2</Subscript><Superscript>p</Superscript>(x,Q<Superscript>2</Superscript>) as a probe of the range of validity of perturbative QCD evolutions in the small-x region

Perturbative NLO and NNLO QCD evolutions of parton distributions are studied, in particular in the (very) small-x region, where they are in very good agreement with all recent precision measurements of F₂ ^p(x,Q²). These predictions turn out to be also rather insensitive to the specific choice of the factorization scheme (MS̄ or DIS). A characteristic feature of perturbative QCD evolutions is a positive curvature of F₂ ^p which increases as x decreases. This perturbatively stable prediction provides a sensitive test of the range of validity of perturbative QCD.     

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.