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 Leading-Order Charm Quark Contribution to the Next-to-Leading-Order Proton Structure Function Using {\cal A}=3 Mirror Nuclei as Input Valence Quarks

The charm quark contribution to the proton structure function (SF) is investigated in the leading-order (LO) QCD at small x region. A next-to-leading order (NLO) QCD analysis for the proton SF is made within the renormalization scheme of the radiation parton evolution model (DGLAP). The valence quark distribution is obtained from the relativistic quark-exchange calculation for the mirror nuclei, i.e., ³He and ³H, which is based on a realistic model. The inverse Mellin technique is performed to extract the parton distribution in the (x, Q ²)-plane. The calculated F ₂ ^c (x, Q ²) and F ₂ ^p (x, Q ²) as well as the longitudinal SF, F _L ^p (x, Q ²) are compared with the experimental data available at present, namely H1, ZEUS, and HERMES at HERA ring as well as other theoretical models, especially the hard pomeron phenomenological model.     

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.     

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 effect of quark exchange in A = 3 mirror nuclei and neutron/proton structure functions ratio

By using the quark-exchange formalism, the realistic Faddeev wave function and the Fermi motion effect, we investigate the deep inelastic electron scattering from A = 3 mirror nuclei in the deep-valence region. The initial valence quark input is taken from the GRV's (Glück, Reya and Vogt) fitting procedure and the next-to-leading-order QCD evolution on F₂^p(x, Q²) which gives a very good fit to the available data in the (x, Q²)-plane. It is shown that the free neutron to proton structure functions ratio can be extracted from the corresponding EMC ratios for ³He and ³H mirror nuclei by using the self-consistent iteration procedure and the results are in good agreement with the other theoretical models as well as the present available experimental data and especially the projected data expected from the proposed 11GeV Jefferson Laboratory in the near future.     

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.     

α(Q) corrections to particle multiplicity ratios in gluon and quark jets

The order α(Q²) correction to the particle multiplicity ratio in gluon and quark jets is calculated in QCD. Through α(Q²) we find , with r = <n>_{gluon jet}/<n>_{quark jet}. This ratio is independent of the opening angle chosen to define the jets.     

Unpolarized structure functions and the parton distributions for nucleon in an independent quark model

Considering the nucleon as consisting entirely of its valence quarks confined independently in a scalar-vector harmonic potential; unpolarized structure functions F ₁(x, μ ²) and F ₂(x, μ ²) are derived in the Bjorken limit under certain simplifying assumptions; from which valence quark distribution functions u _v(x, μ ²) and d _v(x, μ ²) are appropriately extracted satisfying the normalization constraints. QCD-evolution of these input distributions from a model scale of μ ²=0.07 GeV² to a higher Q ² scale of Q ₀ ² =15 GeV² yields xu _v(x, Q ₀ ² ) and xd _v(x, Q ₀ ² ) in good agreement with experimental data. The gluon and sea-quark distributions such as G(x, Q ₀ ² ) and q _s(x, Q ₀ ² ) are dynamically generated with a reasonable qualitative agreement with the available data; using the leading order renormalization group equations with appropriate valence-quark distributions as the input.     

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.     

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.     

Axial form-factor and induced pseudoscalar form-factor of the nucleons

We calculate the axial and the induced pseudoscalar form-factors G_A(t=-Q²) and G_P(t=-Q²) of the nucleons in the framework of the light-cone QCD sum-rules approach up to twist-6 three valence quark light-cone distribution amplitudes, and observe that the form-factors G_A(t=-Q²) and G_P(t=-Q²) at intermediate and large momentum transfers with Q²>2 GeV² have significant contributions from the end-point (soft) terms. The numerical values for the axial form-factor G_A(t=-Q²) are compatible with the experimental data and theoretical calculations; for example, the chiral quark models and lattice QCD. The numerical values for the induced pseudoscalar form-factor G_P(t=-Q²) are compatible with the calculation from the Bethe–Salpeter equation. PACS 12.38.Aw; 12.38.-t; 14.20.Dh     

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     

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.     

The polarised valence quark distribution from semi-inclusive DIS

The semi-inclusive difference asymmetry A^h+−h− for hadrons of opposite charge has been measured by the COMPASS experiment at CERN. The data were collected in the years 2002–2004 using a 160 GeV polarised muon beam scattered off a large polarised ⁶LiD target in the kinematic range 0.006<x<0.7 and 1<Q²<100 (GeV/c)². In leading order QCD (LO) the deuteron asymmetry A^h+−h− measures the valence quark polarisation and provides an evaluation of the first moment of Δu_v+Δd_v which is found to be equal to 0.40±0.07(stat.)±0.06(syst.) over the measured range of x at Q²=10 (GeV/c)². When combined with the first moment of previously measured on the same data, this result favours a non-symmetric polarisation of light quarks at a confidence level of two standard deviations, in contrast to the often assumed symmetric scenario .     

The approximation method for calculation of the exponents of the gluon distribution, λ g , and the structure function, λ S ,at low x

We present a set of formulas using the solution of the QCD Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation to extract of the exponents of the gluon distribution, λ _g , and structure function, λ _S , from the Regge-like behavior at low x. The exponents are found to be independent of x and to increase linearly with lnQ ² and are compared with the most data from the H1 Collaboration. We also calculated the structure function F ₂(x,Q ²) and the gluon distribution G(x,Q ²) at low x assuming the Regge-like behavior of the gluon distribution function at this limit and compared them with an NLO-QCD fit to theH1 data, two-Pomeron fit, multipole Pomeron exchange fit, and MRST (A.D. Martin, R.G. Roberts, W.J. Stirling, and R.S. Thorne), DL (A. Donnachie and P.V. Landshoff), and NLO GRV (M. Glük, E. Reya, and A. Vogt) fit results. The text was submitted by the authors in English.     

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.     

The inverses-wave scattering problem for a class of potentials depending on energy

The inverse scattering problem is considered for the radials-wave Schrödinger equation with the energy-dependent potentialV ⁺(E,x)=U(x)+2 Q(x). (Note that this problem is closely related to the inverse problem for the radials-wave Klein-Gordon equation of zero mass with a static potential.) Some authors have already studied it by extending the method given by Gel'fand and Levitan in the caseQ=0. Here, a more direct approach generalizing the Marchenko method is used. First, the Jost solutionf ⁺(E,x) is shown to be generated by two functionsF ⁺(x) andA ⁺(x,t). After introducing the potentialV ^–(E,x)=U(x)–2 Q(x) and the corresponding functionsF ^–(x) andA ^–(x,t), fundamental integral equations are derived connectingF ⁺(x),F ^–(x),A ⁺(x,t) andA ^–(x,t) with two functionsz ⁺(x) andz ^–(x);z ⁺(x) andz ^–(x) are themselves easily connected with the binding energiesE _n ⁺ and the scattering matrixS ⁺(E),E>0 (the input data of the inverse problem). The inverse problem is then reduced to the solution of these fundamental integral equations. Some specific examples are given. Derivation of more elaborate results in the case of real potentials, and applications of this work to other inverse problems in physics will be the object of further studies.Physique Mathématique et Théorique, Equipe de recherche associée au C.N.R.S.     

Thermodynamical analysis of the EMC effect

Assuming that the sea quark distribution vanishes for x > 0.3, we analyse the F₂^Fe(x, Q²) and F₂^D(x, Q²) structure functions measured by the European Muon Collaboration in the framework of a thermodynamical model of the valence quarks. The experimental ratio $\text{F}{}_2{}^{\mathrm{<mtext>Fe</mtext>}}\text{(x)}\text{F}{}_2{}^{\mathrm{<mtext>D</mtext>}}\text{(x)}$ is well reproduced over the whole x range by the ratio of two valence quark distributions at different temperatures T and confinement volumes V. We obtain T_D?T_Fe≈3 MeV and $\text{V}{}_{\mathrm{<mtext>Fe</mtext>}}\text{V}{}_{\mathrm{<mtext>D</mtext>}}\text{≈ 1.3}$.     

Regge behavior of DIS structure functions in scalar theories

The Bethe-Salpeter formalism is used to incorporate the valence Regge behavior into the total DIS amplitude. For a special case of scalar quarks with massless scalar exchange, the model is solved both analytically and numerically and exact scaling is found for the valence quark contribution F ₂(x) ∼ (1/x) ^l(0)−1 which mimicks the ρ-trajectory term. The solution solves a long-standing problem by showing that the coefficient in the Regge pole expansion is indeed fine-tuned to give the expected scaling. The method allows for generalization to the region of nonzero momentum transfer and calculation of the DVCS amplitude. The text was submitted by the authors in English.     

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.