A new numerical method for obtaining gluon distribution functions G(x,Q 2)=xg(x,Q 2), from the proton structure function F_{2}^{\gamma p}(x,Q^{2})

An exact expression for the leading-order (LO) gluon distribution function G(x,Q ²)=xg(x,Q ²) from the DGLAP evolution equation for the proton structure function $F_{2}^{\gamma p}(x,Q^{2})$ for deep inelastic γ ^* p scattering has recently been obtained (Block et al., Phys. Rev. D 79:014031, 2009) for massless quarks, using Laplace transformation techniques. Here, we develop a fast and accurate numerical inverse Laplace transformation algorithm, required to invert the Laplace transforms needed to evaluate G(x,Q ²), and compare it to the exact solution. We obtain accuracies of less than 1 part in 1000 over the entire x and Q ² spectrum. Since no analytic Laplace inversion is possible for next-to-leading order (NLO) and higher orders, this numerical algorithm will enable one to obtain accurate NLO (and NNLO) gluon distributions, using only experimental measurements of $F_{2}^{\gamma p}(x,Q^{2})$ .     

Double-spin asymmetries in electron-nucleon scattering in Halls B and C at JLab

Three experiments at JLab have measured the double polarization asymmetries ) in the nucleon resonance region, using polarized electron beams incident on polarized proton and deuteron targets. The analysis for the first experiment, eg1a in Hall B, is nearly finished and preliminary values of the spin structure function g₁(x, Q²) and the first moment (x) have been extracted. The other two experiments, one in Hall B and one in Hall C, are still analysing data. Some results are presented.Received: 1 November 2002, Published online: 15 July 2003PACS: 24.70.+s Polarization phenomena in reactions     

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.     

Measurement of the diffractive cross section in deep inelastic scattering using ZEUS 1994 data

The DIS diffractive cross section, ds^diff_{g^* p ? XN}/dM_Xd\sigma^{di\!f\!f}_{\gamma^* p \to XN}/dM_X, has been measured in the mass range M_X < 15M_X < 15 GeV for g^*p\gamma^*p c.m. energies 60 < W < 20060 < W < 200 GeV and photon virtualities Q² = 7Q^2 = 7 to 140 GeV²^2. For fixed Q²Q^2 and M_XM_X, the diffractive cross section rises rapidly with WW, ds^diff_{g^*p ? XN}(M_X,W,Q²)/dM_X μ W^adiffd\sigma^{di\!f\!f}_{\gamma^*p \to XN}(M_X,W,Q^2)/dM_X \propto W^{a^{diff}} with a^diff = 0.507 ±0.034 (stat)^+0.155_-0.046(syst)a^{diff} = 0.507 \pm 0.034 (stat)^{+0.155}_{-0.046}(syst) corresponding to a t-averaged pomeron trajectory of [`( a<sub>\mathbb P</sub> )] = 1.127 ±0.009 (stat)<sup>+0.039</sup><sub>-0.012</sub> (syst)\overline{ \alpha_{_{{\mathbb P}}} } = 1.127 \pm 0.009 (stat)^{+0.039}_{-0.012} (syst) which is larger than [`( a_{\mathbb P} )]\overline{ \alpha_{_{{\mathbb P}}} } observed in hadron-hadron scattering. The W dependence of the diffractive cross section is found to be the same as that of the total cross section for scattering of virtual photons on protons. The data are consistent with the assumption that the diffractive structure function F^D(3)₂F^{D(3)}_2 factorizes according to x_{\mathbb P} F^D(3)₂ (x_{\mathbb P},b,Q²) = (x₀/ x_{\mathbb P})ⁿ F^D(2)₂(b,Q²)x_{_{{\mathbb P}}} F^{D(3)}_2 (x_{_{{\mathbb P}}},\beta,Q^2) = (x_0/ x_{_{{\mathbb P}}})^n F^{D(2)}_2(\beta,Q^2). They are also consistent with QCD based models which incorporate factorization breaking. The rise of x_{\mathbb P} F^D(3)₂x_{_{{\mathbb P}}} F^{D(3)}_2 with decreasing x_{\mathbb P}x_{_{{\mathbb P}}} and the weak dependence of F^D(2)₂F^{D(2)}_2 on Q²Q^2 suggest a substantial contribution from partonic interactions.     

Decoupling of the Leading Order DGLAP Evolution Equation with Spin Dependent Structure Functions

We propose an analytical solution for DGLAP evolution equations with polarized splitting functions at the Leading Order (LO) approximation based on the Laplace transform method. It is shown that the DGLAP evolution equations can be decoupled completely into two second order differential equations which then are solved analytically by using the initial conditions \(\delta F^{\mathrm {S}}(x,Q^{2})=\mathcal {F}[\partial \delta F^{\mathrm {S}}_{0}(x), \delta F^{\mathrm {S}}_{0}(x)]\) and \({\delta G}(x,Q^{2})=\mathcal {G}[\partial \delta G_{0}(x), \delta G_{0}(x)]\). We used this method to obtain the polarized structure function of the proton as well as the polarized gluon distribution function inside the proton and compared the numerical results with experimental data of COMPASS, HERMES, and AAC’08 Collaborations. It was found that there is a good agreement between our predictions and the experiments.     

Parton Distribution Functions from QCD Analysis of HERA Combined Inclusive e ± p Scattering Cross Section Data

Utilizing very recent deep inelastic scattering measurements, a QCD analysis of proton structure function ${F_{2}^{p} (x,Q^2)}$ is presented. A wide range of the inclusive neutral-current deep-inelastic-scattering (NC DIS) data used in order to extract an updated set of parton distribution functions (PDFs). The HERA ‘combined’ data set on ${\sigma_{r,NC}^\pm (x,Q^2)}$ together with all available published data for heavy quarks ${F_2^{c,b}(x,Q^2)}$ , longitudinal F _L (x, Q ²) and also very recent reduced DIS cross section ${\sigma_{r,NC}^\pm (x,Q^2)}$ data from HERA experiments are the input in the present next-to-leading order (NLO) QCD analysis which determines a new set of parton distributions, called ${{\tt KKT11C}}$ . The extracted PDFs in the ‘fixed flavour number scheme’ (FFNS) are in very good agreement with the available theoretical models.     

Shadowing correction to the gluon distribution behavior at small x

We determined the saturation exponent of the gluon distribution using the solution of the QCD nonlinear Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (NLDGLAP) evolution equation at small x . The very small-x behavior of the gluon distribution is obtained by solving the Gribov, Levin, Ryskin, Mueller and Qiu (GLR-MQ) evolution equation with the nonlinear shadowing term incorporated. The form of the initial condition for the equation is determined. We find, with decreasing x , the emergence of a singular behavior and the eventual taming (at R = 5 GeV^-1) and the essential taming (at R = 2 GeV^-1) of this singular behavior by the shadowing term. The nonlinear gluon density functions are calculated and compared with the results for the integrated gluon density from the Balitsky-Kovchegov (BK) equation for the different values of Q². It is shown that the results for the gluon density function are comparable with the results obtained from the BK equation solution. Also we show that for each x , the Q²-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 \( {\frac{{\partial F^{s}_{2} (x,Q^{2})}}{{\partial \ln Q^{2}}}}\) and to compare the results with H1 data and a QCD analysis fit.     

Small-x behaviour of the polarized photon structure function F_3^\gamma(x,Q^2)

We study the small-x behaviour of the polarized photon structure function F₃^gF_3^{\gamma}, measuring the gluon transversity distribution, in the leading logarithmic approximation of perturbative QCD. There are two contributions, both arising from two-gluon exchange. The leading contribution to small-x is related to the BFKL pomeron and behaves like x^-1-w₂x^{-1-\omega_2}, w₂ = O(a_S)\omega_2 ={\cal O}(\alpha_S). The other contribution includes in particular the ones summed by the DGLAP equation and behaves like x^1-w₀(+)x^{1-\omega_0^{(+)}}, w₀⁽⁺⁾ = O(?{a_S})\omega_0^{(+)} = {\cal O}(\sqrt{\alpha_S}).     

Structure Function of the Bound Proton in the Dressed-quark Exchange Formalism

The idea of the dressed quark scenario, in the context of the quark exchange model has been used to obtain the bound parton distribution functions as well as the bound proton structure function. The Glück, Reya, and Vogt (GRV)’s parton distribution functions, where they have a good agreement with the experimental data in the whole range of (x, Q²)-region, have been used to extract the amount of the momentum carried by each parton inside the proton; therefore, by using these values, the different parton densities inside the constituent quark have been calculated. To extract the parton distribution functions in the bound proton, the phenomenological constituent quark exchange formalism and the nuclear convolution theorem have been used. By considering these densities, the structure function of the bound proton, \({F_{2}^{P}}(x,Q^{2})\), inside the light nuclei like helium-3 or tritium has been calculated at low- and high-Q² values. The correctness of the calculations has been tested through computing the difference of the proton-neutron structure functions as well as the ratio of the neutron to the proton structure functions, in which they are well known from experimental data. The results of the present study are in an agreement with both the available and relevant experimental data and the theoretical predictions.

     

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.     

b-tagging in DELPHI at LEP

The numbers of events suitable for the determination of the gluon distribution function f ^p _g (x,Q ²) in a proton at the LHC for various intervals of x and Q ² are estimated. The contributions of background events of different sources are studied and estimated in the considered intervals of x and Q ². The PYTHIA event generator was used to produce physical events for this analysis.Received: 17 June 2003, Revised: 8 July 2004, Published online: 24 August 2004N.B. Skachkov: Present address: Joliot-Curie 6, JINR, 141980, Dubna, Moscow region, Russia     

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.     

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).     

κ-generalized statistics in personal income distribution

Starting from the generalized exponential function , with exp ₀(x)=exp (x), proposed in reference [G. Kaniadakis, Physica A 296, 405 (2001)], the survival function P_>(x)=exp _κ(-βx^α), where x∈R⁺, α,β>0, and , is considered in order to analyze the data on personal income distribution for Germany, Italy, and the United Kingdom. The above defined distribution is a continuous one-parameter deformation of the stretched exponential function P_> ⁰(x)=exp (-βx^α) to which reduces as κ approaches zero behaving in very different way in the x→0 and x→∞ regions. Its bulk is very close to the stretched exponential one, whereas its tail decays following the power-law P_>(x)∼(2βκ)^-1/κx^-α/κ. This makes the κ-generalized function particularly suitable to describe simultaneously the income distribution among both the richest part and the vast majority of the population, generally fitting different curves. An excellent agreement is found between our theoretical model and the observational data on personal income over their entire range.     

A Construction of Blow up Solutions for Co-rotational Wave Maps

The existence of co-rotational finite time blow up solutions to the wave map problem from ${\mathbb{R}^{2+1} \to N}The existence of co-rotational finite time blow up solutions to the wave map problem from \mathbbR²⁺¹ ? N{\mathbb{R}^{2+1} \to N} , where N is a surface of revolution with metric d ρ ² + g(ρ)² dθ², g an entire function, is proven. These are of the form u(t,r)=Q(l(t)t)+R(t,r){u(t,r)=Q(\lambda(t)t)+\mathcal{R}(t,r)} , where Q is a time independent solution of the co-rotational wave map equation −u _tt  + u _rr  + r ⁻¹ u _r  = r ⁻² g(u)g′(u), λ(t) = t ^−1-ν, ν > 1/2 is arbitrary, and R{\mathcal{R}} is a term whose local energy goes to zero as t → 0.     

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/?? photoproduction.     

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.     

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.