Colour transparency and scaling properties of nuclear shadowing in deep inelastic scattering

We present the perturbative QCD analysis of nuclear shadowing in the deep inelastic scattering at smallx in terms of the spatial wave function ofq \(\bar q\) fluctuations of virtual photons. The wave function formalism makes it quite obvious that shadowing is the scaling, rather than the higher twist, 1/Q ², effect, contrary to a numerous recent claims. We demonstrate explicitly how the scaling shadowing comes from the large, hadronic, size quarkantiquark pairs even in the limit ofQ ²→∞, and why it should very slowly, ∞1/log (Q ²/m ²) decrease at very largeQ ². We argue in favor of the scaling triple-pomeron contribution to the nuclear shadowing and present predictions for a cross section of diffraction dissociation of virtual photons and for the mass spectrum of diffraction excitation, which can be checked at HERA and Fermilab. We predict strikingly different scaling properties of diffraction dissociation and nuclear shadowing for the longitudinal and transverse photons. Our, numerial predictions for shadowing are in good agreement with the recent EMC data.     

Parametrization of nuclear parton distributions

Optimum nuclear parton distributions are obtained by analysing available experimental data on electron and muon deep inelastic scattering (DIS). The distributions are given at Q ²=1 GeV² with a number of parameters, which are determined by a X ² analysis of the data. Valencequark distributions are relatively well determined at medium x, but they are slightly dependent on the assumed parametrization form particularly at small x. Although antiquark distributions are shadowed at small x, their behavior is not obvious at medium x from the F ₂ data. The gluon distributions could not be restricted well by the inclusive DIS data; however, the analysis tends to support the gluon shadowing at small x. We provide analytical expressions and computer subroutines for calculating the nuclear parton distributions, so that other researchers could use them for applications to other high-energy nuclear reactions.     

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.     

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     

The next-to-leading order (NLO) gluon distribution from DGLAP equation and the logarithmic derivatives of the proton structure functionF 2(x, Q 2) at lowx

At lowx, an analytic solution of the DGLAP equation for gluon in the next-to-leading order (NLO) is obtained by applying the method of characteristics. Its compatibility with double leading logarithmic approximation (DLLA) asymptotics is discussed and comparison with the exact ones like GRV98NLO is made. The solution is then utilized to calculate the derivatives∂F ₂ (x,Q ²)/∂ lnQ ² and ∂ lnF ₂(x,Q ²)/∂ ln (1/x) and compared with the recent HERA data. Our solution is found to reproduce most of the essential features of the data on the derivatives.     

The exponent λ (x, Q2) of the proton structure functionF2(x, Q2) at lowx

The exponent λ of the structure function F₂ ∼x ^−λ is calculated using the solution of the DGLAP equation for gluon at lowx reported recently by the present authors. The quantity λ is calculated both as a function ofx at fixedQ ² and as a function ofQ ² at fixedx and compared with the most recent data from H1     

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.     

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).     

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.     

Analysis of the electroproduction structure functions in the lowQ 2 region,combining the vector meson dominance and the parton model with possible scaling violation

The nucleon structure functionF ₂ is constructed and analysed for low values ofQ ² using the generalised vector meson dominance representation with the largeQ ² spectral function calculated from the analytic continuation of the parton model structure function. Various parametrisations of the parton distributions are considered. Possible effects of the largeQ ² scaling violation on the lowQ ² part of the structure function are investigated. The magnitude of the total contribution given by this asymptotic part can be as high as 50% of the vector meson contribution in the low-Q ², low-χ region. The contribution of the valence quarks alone to the structure function at lowQ ² turns out to be at least as important as the corresponding non-Pomeron Regge-pole-like terms coming from the vector meson part. Increase of the structure function with ν coming from the increase of the quark sea in the limit of small χ implied by QCD turns out to be relatively weak at lowQ ². Predictions of the model are compared with available experimental data. The photoproduction cross section and the nuclear effects in the structure function are also briefly discussed.     

Studies of the neutron spin structure at Jefferson Lab

The polarized ³He program of Hall A at Jefferson Lab will be described. Results on the generalized Gerasimov-Drell-Hearn integral for the neutron in a Q² range between 0.02 GeV²/c² < Q² < 0.9 GeV²/c² will be presented. Preliminary results of the virtual photon asymmetry A₁ⁿ(x, Q²) and the spin structure function g₂ⁿ(x, Q²) at large values of Bjorken x and low Q², respectively, will be discussed.Received: 1 November 2002, Published online: 15 July 2003PACS: 11.55.Hx Sum rules - 13.40.Gp Electromagnetic form factors - 13.88.+e Polarization in interactions and scattering - 29.25.Pj Polarized and other targets     

Effect of the Fermi motion on nuclear structure functions and the EMC effect

The effect of the Fermi motion on nuclear structure function at largeQ ² (the EMC effect) is discussed in detail. We especially study its dependences on the kinematics of interaction, the 4-momentum transferQ ², the nucleon momentum distribution inside nucleus and the nuclear mass number. Moreover the structure function of¹²C at very largex such asx>1 is evaluated by only the nuclear Fermi motion, and related problems are discussed. We also point out a new dynamical mechanism expected to produce the depletion of valence quark inside nucleus for intermediatex, which does not encroach upon the conventional nuclear physics, and discuss it along the line of scaling hypothesis proposed by Daté et al.     

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 Q-dependence of the generalised Gerasimov-Drell-Hearn integral for the deuteron,proton and neutron

The Gerasimov-Drell-Hearn (GDH) sum rule connects the anomalous contribution to the magnetic moment of the target nucleus with an energy-weighted integral of the difference of the helicity-dependent photoabsorption cross sections. Originally conceived for real photons, the GDH integral can be generalised to the case of photons with virtuality Q². For spin-1/2 targets such as the nucleon, it then represents the non-perturbative limit of the first moment of the spin structure function g ₁ (x,Q ² ) in deep inelastic scattering (DIS). The data collected by HERMES with a deuterium target are presented together with a re-analysis of previous measurements on the proton. This provides an unprecedented and complete measurement of the generalised GDH integral for photon-virtuality ranging over 1.2&lt;Q ² &lt;12.0 GeV² and for photon-nucleon invariant mass squared W² ranging over 1&lt;W ² &lt;45 GeV², thus covering simultaneously the nucleon-resonance and the deep inelastic scattering regions. These data allow the study of the Q²-dependence of the full GDH integral, which is sensitive to both the Q²-evolution of the resonance form factors and contributions of higher twist. The contribution of the nucleon-resonance region is seen to decrease rapidly with increasing Q². The DIS contribution is sizeable over the full measured range, even down to the lowest measured Q². As expected, at higher Q² the data are found to be in agreement with previous measurements of the first moment of g₁. From data on the deuteron and proton, the GDH integral for the neutron has been derived and the proton-neutron difference evaluated. This difference is found to satisfy the fundamental Bjorken sum rule at Q ² = 5 GeV². Received: 20 October 2002 / Published online: 15 January 2003     

A phenomenological analysis of the longitudinal structure function at small <Emphasis Type="Italic">x</Emphasis> and low <Emphasis Type="Italic">Q</Emphasis><Superscript>2</Superscript>

The longitudinal structure function in deep-inelastic scattering is one of the observables from which the gluon distribution can be unfolded. Consequently, this observable can be used to constrain the QCD dynamics at small x. In this work we compare the predictions of distinct QCD models with the recent experimental results for F _L(x,Q ²) at small x and low Q ² obtained by the H1 Collaboration. We focus mainly on the color dipole approach, selecting those models which include saturation effects. Such models are suitable at this kinematical region and also resum a wide class of higher-twist contributions to the observables. Therefore, we investigate the influence of these corrections to F _L in the present region of interest.Received: 23 June 2004, Revised: 13 July 2004, Published online: 14 September 2004     

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.     

The running BFKL: Resolution of Caldwell’s puzzle

The HERA data on the proton structure function F ₂(x,Q ²) at very small x and Q ₂ show a dramatic departure of the logarithmic slope ∂F ₂/∂log Q ² from theoretical predictions based on the DGLAP evolution. We show that the running BFKL approach provides a quantitative explanation for the observed x and/or Q ² dependence of ∂F ₂/∂log Q ². Pis’ma Zh. éksp. Teor. Fiz. 69, No. 2, 92–97 (25 January 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Roles of survival probability and barrier reduction in heavy ion fusion induced by break up

Shadowing effects in deep-inelastic lepton-nucleus scattering probe the mass spectrum of diffractive leptoproduction from individual nucleons. We explore this relationship using current experimental information on both processes. In recent data from the NMC and E665 collaboration, taken at small x ? 0.1 and Q ² ? 1 GeV ², shadowing is dominated by the diffractive excitation and coherent interaction of low mass vector mesons. If shadowing is explored at small x ? 0.1 but large Q ² ? 1 GeV ² as discussed at HERA, the situation is different. Here dominant contributions come from the coherent interaction of diffractively produced heavy mass states. Furthermore we observe that the energy dependence of shadowing is directly related to the mass dependence of the diffractive production cross section for free nucleon targets.