Gluon Distribution Function Research Articles

Perturbative QCD and Regge theory: closing the circle

We explain how Regge theory and perturbative evolution may be made compatible at small x. The result not only gives striking support to the two-pomeron description of small-x behaviour, but gives a rather clean test of perturbative QCD itself. When x is very small, the proton's gluon distribution function is significantly larger than is commonly believed. Perturbative evolution is invalid below Q2≈5 GeV2.

Extraction of the structure function F2(x,Q2) at low-x from the cross section derivative

Abstract A method to obtain an approximate relation between the reduced cross section derivative and the F 2 ( x , Q 2 ) scaling violations at low- x is presented. The resulting formula can be used to determine the structure function F 2 from the HERA reduced cross section data taken at low- x . This new approach can determine the structure function F 2 with reasonable precision even for low value x which has not been investigated. To test the validity of our determined structure functions,the obtained F 2 data are used to find the gluon distribution function as well as the longitudinal structure function which have been found by leading order approximation and from which comparison are made to QCD parton distribution functions.

Uncertainties of predictions from parton distribution functions. II. The Hessian method

We develop a general method to quantify the uncertainties of parton distribution functions and their physical predictions, with emphasis on incorporating all relevant experimental constraints. The method uses the Hessian formalism to study an effective chi-squared function that quantifies the fit between theory and experiment. Key ingredients are a recently developed iterative procedure to calculate the Hessian matrix in the difficult global analysis environment, and the use of parameters defined as components along appropriately normalized eigenvectors. The result is a set of 2D eigenvector basis parton distributions (where $d\ensuremath{\approx}16$ is the number of parton parameters) from which the uncertainty on any physical quantity due to the uncertainty in parton distributions can be calculated. We illustrate the method by applying it to calculate uncertainties of gluon and quark distribution functions, W boson rapidity distributions, and the correlation between W and Z production cross sections.

Heavy-ion collision multiplicities and gluon distribution functions

Atomic number ( A) and energy ( s ) scaling exponents of multiplicity and transverse energy in heavy-ion collisions are analytically derived in the perturbative QCD+saturation model. The exponents depend on the small- x behaviour of gluon distribution functions at an x-dependent scale. The relation between initial-state and final-state saturation is also discussed.

Ioffe time in double logarithmic approximation

We analyze the light cone (Ioffe) time structure of the gluon distribution function in the double logarithmic approximation. We show that due to QCD evolution Ioffe equation is modified. The characteristic light cone time of the gluons does not increase as fast with increasing energy (decreasing Bjorken x) as predicted by the parton distributions exhibiting Bjorken scaling due to the increase of the transverse momenta of the gluons in the DGLAP ladder.

Electroproduction of ψ′ and polarized gluon distribution in a proton

In order to get information about the polarized gluon distribution in a proton, we studied the electroproduction of \psi' in polarized electron and polarized proton collisions in the framework of the NRQCD factorization approach. The value of the cross section d\Delta\sigma / dp_T^2 for color-octet mechanism is about 5 times larger than that for color-singlet one, and it might be another test of the color-octet model if the polarized gluon distribution \Delta g(x) is well known. Furthermore, we found that this reaction is quite effective for testing the model of gluon polarization by measuring the double spin asymmetry A_{LL} for the initial electron and proton. Though the shape of p_T^2 distribution of A_{LL} is quite similar for the production mechanism with color-octet and color-singlet, we can see a big difference in A_{LL} among various models of the polarized gluon distribution function \Delta g(x).

Diffractiveφand ρ production in a perturbative QCD model

The elastic leptoproduction of $\ensuremath{\varphi}$ measured by the H1 Collaboration at DESY HERA is described by a perturbative QCD model, based on open $s\overline{s}$ production and parton hadron duality, proposed by Martin et al. We observe that both the total cross section and the ratio of the longitudinal and transverse cross sections are well reproduced with an effective strange-quark mass ${m}_{s}\ensuremath{\sim}370\ensuremath{-}400 \mathrm{MeV}$ for various gluon distribution functions. The possible connection of the effective mass and the momentum dependent dynamical mass associated with dynamical breaking of chiral symmetry is discussed. The $\ensuremath{\rho}$ leptoproduction data are also well reproduced with an effective quark mass $\ensuremath{\sim}315 \mathrm{MeV}.$

Polarized gluon distribution function of nucleon in diffractive lepto-production of charmonium

We investigate the longitudinal double spin asymmetry in the diffractive lepto-production of charmonium off the proton, and study its relation to the polarized gluon distribution Δ G of the proton. It is found that the transverse momenta of the exchanged gluons and Fermi motion of charm quarks give a finite spin asymmetry. Our calculation for the photo-production indicates the asymmetry is small and of O(10 −3) ∼ O(10 −2) at HERA collider energy. On the other hand, at fixed-target energy (HERMES), the asymmetry is of O(10 −2) for J/ψ production. The asymmetry for the ψ′ is even large, of O(10 −1), due to the large Fermi motion of c c in ψ′.

Screening corrections in photo and DIS production of J/ Ψ

Photo and DIS production of J/ Ψ are investigated and compared with calculations based on pQCD in the LLA approximation without and with screening corrections. The calculation includes corrections induced by the real part of the production amplitude, the skewed (off diagonal) gluon distribution function and the relativistic Fermi motion within the charmonium system. Our pQCD results are also compared with the predictions obtained from a Regge type two-pomeron model. Our results show that the screened pQCD model gives a better reproduction of the data than the nonscreened model. However, the available data does not enable us to exclude any of the three models we have examined. The predictions of these models, when extrapolated to both low and very high energies may provide a more effective discrimination between the different parameterizations.

Longitudinal resolution in a large relativistic nucleus: Adding a dimension to the McLerran-Venugopalan model

We extend the McLerran-Venugopalan model for the gluon distribution functions of very large nuclei to larger values of the longitudinal momentum fraction xF. Because gluons with larger values of xF begin to resolve the longitudinal structure of the nucleus, we find that it is necessary to set up a fully three-dimensional formalism for performing the calculation. We obtain a relatively compact expression for the gluon number density provided that the nucleus is sufficiently large and consists of color-neutral nucleons. Our expressions for the gluon number density saturate at small transverse momenta. The nuclear dependence we obtain is such that the number of gluons increases more slowly than the number of nucleons is increased.

Resolved $\gamma^*_L$ in hard collisions of virtual photons: QCD effects

The manifestations of QCD effects on quark and gluon distribution functions of longitudinally polarized virtual photons involved in hard collisions are investigated. It is shown that for moderate photon virtualities and in the kinematical region accessible at HERA and LEP these effects are sizable and significantly enhance theoretical predictions based on contributions of transversally polarized virtual photon only.

Single spin asymmetry in heavy flavor photoproduction as a test of pQCD

We analyze in the framework of pQCD the properties of the single spin asymmetry in heavy flavor production by linearly polarized photons. At leading order, the parallel- perpendicular asymmetry in azimuthal distributions of both charm and bottom quark is predicted to be about 20% in a wide region of initial energy. Our analysis shows that the next-to-leading order corrections practically do not affect the Born predictions for the azimuthal asymmetry at energies of the fixed target experiments. Both leading and next-to-leading order predictions for the asymmetry are insensitive to within few percent to theoretical uncertainties in the QCD input parameters: $m_{Q}$, $\mu_{R}$, $\mu_{F}$, $\Lambda_{QCD}$ and in the gluon distribution function. We estimate also nonperturbative contributions to azimuthal distributions due to the gluon transverse motion in the target and the final quark fragmentation. Our calculations show that nonperturbative corrections to a $B$-meson azimuthal asymmetry are negligible. We conclude that measurements of the single spin asymmetry would provide a good test of pQCD applicability to heavy flavor production at fixed target energies.

Regge behaviour of structure function and gluon distribution at low-x in leading order

We present a method to find the gluon distribution from the $F_2$ proton structure function data at low-x assuming the Regge behaviour of the gluon distribution function at this limit. We use the leading order (LO) Altarelli–Parisi (AP) evolution equation in our analysis and compare our result with those of other authors. We also discuss the limitations of the Taylor expansion method in extracting the gluon distribution from the $F_2$ structure function used by those authors.

Parton degrees of freedom from the path-integral formalism

We formulate the hadronic tensor $W_{\mu\nu}$ of deep inelastic scattering in the path-integral formalism. It is shown that there are 3 gauge invariant and topologically distinct contributions. Besides the valence contribution, there are two sources for the sea -- one in the connected insertion and the other in the disconnected insertion. The operator product expansion is carried out in this formalism. The operator rescaling and mixing reveal that the connected sea partons evolve the same way as the valence, i.e. their evolution is decoupled from the disconnected sea and the gluon distribution functions. We explore the phenomenological consequences of this classification in terms of the small x behavior, Gottfried sum rule violation, and the flavor dependence. In particular, we point out that in the nucleon $\bar{u}$ and $\bar{d}$ partons have both the connected and disconnected sea contributions; whereas, $\bar{s}$ parton has only the disconnected sea contribution. This difference between $\bar{u} + \bar{d}$ and $\bar{s}$, as far as we know, has not been taken into account in the fitting of parton distribution functions to experiments.

InelasticJ/ψproduction in polarized photon-hadron collisions

We present a calculation of inelastic $J/\ensuremath{\psi}$ production in polarized photon-hadron collisions. We consider the photoproduction of $J/\ensuremath{\psi}$ in the energy range relevant to the DESY HERA collider. The Weizs\"acker-Williams approximation is adopted in the evaluation of the cross sections for $\mathrm{ep}$ collisions. We find that this process can give another independent test for the color-octet mechanism, and the different features for the two color-octet processes may provide further information on the mechanism for inelastic $J/\ensuremath{\psi}$ photoproduction. The discrepancy in the production asymmetry $A$ between various sets of polarized gluon distribution functions is also found to be distinctive.

$ J/\psi +{\mathrm{jet}} $ diffractive production in the direct photon process at HERA

We present a study of $J/\psi + jet$ diffractive production in the direct photon process at HERA based on the factorization theorem for lepton-induced hard diffractive scattering and the factorization formalism of the nonrelativistic QCD (NRQCD) for quarkonia production. Using the diffractive gluon distribution function extracted from HERA data on diffractive deep inelastic scattering and diffractive dijet photon production, we show that this process can be studied at HERA with present integrated luminosity, and can give valuable insights in the color-octet mechanism for heavy quarkonia production.

Prompt atmospheric neutrinos and muons: Dependence on the gluon distribution function

We compute the next-to-leading order QCD predictions for the vertical flux of atmospheric muons and neutrinos from decays of charmed particles, for different PDF's (MRS-R1, MRS-R2, CTEQ-4M and MRST) and different extrapolations of these at a small partonic momentum fraction x. We find that the predicted fluxes vary up to almost two orders of magnitude at the largest energies studied, depending on the chosen extrapolation of the PDF's. We show that the spectral index of the atmospheric leptonic fluxes depends linearly on the slope of the gluon distribution function at very small x. This suggests the possibility of obtaining some bounds on this slope in ``neutrino telescopes,'' at values of x not reachable at colliders, provided the spectral index of atmospheric leptonic fluxes could be determined.

The semiclassical gluon distribution at next-to-leading order

The interaction of the partonic fluctuation of the virtual photon in deep inelastic scattering with soft color fields describing the hadron is treated in an eikonal approximation. It is known that, in this approach, the small- x limit of the leading-order gluon distribution xg( x, Q 2) is a constant characterizing the averaged local field strength in the target. Matching the next-to-leading order calculation in this semiclassical framework with the one-loop parton model result, we obtain the next-to-leading order contribution to xg( x, Q 2). It shows a ln(1/ x) enhancement at small x and is sensitive to the large distance structure of the target. The final expression is a simple integral over non-abelian eikonal factors measuring the target color field. We derive a quantitative relation between the short-distance cutoff of this integral and the scale of the gluon distribution function in the MS scheme. Our calculation demonstrates that higher order contributions can be systematically included in the semiclassical approach.

Prompt atmospheric neutrinos and muons: NLO versus LO QCD predictions

We compare the leading and next-to-leading order QCD predictions for the flux of atmospheric muons and neutrinos from decays of charmed particles. We find that the full NLO lepton fluxes can be approximated to within $\ensuremath{\sim}10%$ by the Born-level fluxes multiplied by an overall factor of $2.2\ensuremath{-}2.4,$ which depends slightly on the PDF. This supports the approach of Thunman, Ingelman and Gondolo. We also find that their very low lepton fluxes are due to the mild slope they used for the gluon distribution function at small momentum fractions, and that substantially larger lepton fluxes result when the slope of the gluon distribution function at small momentum fractions is larger.

Color neutrality and the gluon distribution in a very large nucleus

We improve the McLerran-Venugopalan model for the gluon distribution functions in very large nuclei by imposing the condition that the nucleons should be color neutral. We find that enforcing color neutrality cures the infrared divergences in the transverse coordinates which are present in the McLerran-Venugopalan model. Since we obtain well-defined expressions for the distribution functions, we are able to draw unambiguous conclusions about various features of the model. In particular, we show that the gluon distribution functions in the absence of quantum corrections behave as ${1/x}_{F}$ to all orders in the coupling constant. Furthermore, our distribution functions exhibit saturation at small transverse momenta. The normalization of the distribution function we obtain is not arbitrary but specified in terms of the nucleon structure. We derive a sum rule for the integral of the gluon distribution function over transverse momenta, and show that the non-Abelian contributions serve only to modify the shape of the transverse momentum distribution. We obtain a relatively simple expression for the mean value of the transverse momentum-squared. The connection between the McLerran-Venugopalan model and the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equation is discussed quantitatively. Finally, we illustrate our results in terms of a simple nuclear model due to Kovchegov.