Longitudinal Proton Structure Function Research Articles

Analytic Derivation of the Longitudinal Proton Structure Function FL(x, Q2) and the Reduced Cross Section σr(x, Q2) at the Leading Order and the Next-to-leading Order Approximations

Analytic Derivation of the Longitudinal Proton Structure Function FL(x, Q2) and the Reduced Cross Section σr(x, Q2) at the Leading Order and the Next-to-leading Order Approximations

Longitudinal structure function from logarithmic slopes of F2 at low x

Using Laplace transform techniques, I calculate the longitudinal structure function ${F}_{L}(x,{Q}^{2})$ from the scaling violations of the proton structure function ${F}_{2}(x,{Q}^{2})$ and make a critical study of this relationship between the structure functions at leading order (LO) up to next-to-next-to leading order (NNLO) analysis at small $x$. Furthermore, I consider heavy quark contributions to the relation between the structure functions, which leads to compact formula for ${N}_{f}=3+\mathrm{Heavy}$. The nonlinear corrections to the longitudinal structure function at LO up to NNLO analysis are shown in the ${N}_{f}=4$ (light quark flavor) based on the nonlinear corrections at $R=2$ and $R=4\phantom{\rule{4pt}{0ex}}{\text{GeV}}^{\ensuremath{-}1}$. The results are compared with experimental data of the longitudinal proton structure function ${F}_{L}$ in the range of $6.5\ensuremath{\le}{Q}^{2}\ensuremath{\le}800\phantom{\rule{4.pt}{0ex}}{\text{GeV}}^{2}$.

Evidence of quasi-partonic higher-twist effects in deep inelastic scattering at HERA at moderate $$Q^2$$ Q 2

The combined HERA data for the inclusive deep inelastic scattering (DIS) cross sections for the momentum transfer Q^2 > 1,mathrm {GeV}^2 are fitted within the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) framework at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) accuracy, complemented by a QCD-inspired parameterisation of twist 4 corrections. A modified form of the input parton density functions is also included, motivated by parton saturation mechanism at small Bjorken x and at a low scale. These modifications lead to a significant improvement of the data description in the region of low Q^2. For the whole data sample, the new benchmark NNLO DGLAP fit yields chi ^2/mathrm{d.o.f. } simeq 1.19 to be compared to 1.46 resulting from the standard NNLO DGLAP fit. We discuss the results in the context of the parton saturation picture and describe the impact of the higher-twist corrections on the derived parton density functions. The resulting description of the longitudinal proton structure function F_{mathrm {L}} is consistent with the HERA data. Our estimates of higher-twist contributions to the proton structure functions are comparable to the leading-twist contributions at low Q^2 simeq 2 GeV^2 and x simeq 10^{-5}. The x-dependence of the twist 4 corrections obtained from the best fit is consistent with the leading twist 4 quasi-partonic operators, corresponding to an exchange of four interacting gluons in the t-channel.

A new phenomenological investigation of KMR and MRW unintegrated parton distribution functions

The longitudinal proton structure function, $F_L(x,Q^2)$, from the $k_t$ factorization formalism by using the unintegrated parton distribution functions (UPDF) which are generated through the KMR and MRW procedures. The LO UPDF of the KMR prescription is extracted, by taking into account the PDF of Martin et al, i.e. MSTW2008-LO and MRST99-NLO and next, the NLO UPDF of the MRW scheme is generated through the set of MSTW2008-NLO PDF as the inputs. The different aspects of $F_L(x,Q^2)$ in the two approaches, as well as its perturbative and non-perturbative parts are calculated. Then the comparison of $F_L(x,Q^2)$ is made with the data given by the ZEUS and H1 collaborations. It is demonstrated that the extracted $F_L(x,Q^2)$ based on the UPDF of two schemes, are consistent to the experimental data, and by a good approximation, they are independent to the input PDF. But the one developed from the KMR prescription, have better agreement to the data with respect to that of MRW. As it has been suggested, by lowering the factorization scale or the Bjorken variable in the related experiments, it may be possible to analyze the present theoretical approaches more accurately.

Analysis of Small x Behaviour of Longitudinal and Heavy Favour Structure Functions of Proton

The behaviour of structure functions F L , ${F_{k}^{h}} (k=L,2; h=c, b)$ with respect to Bjorken variable x are studied using Taylor series expansion method at small x. At small values of x, all these quantities are dominated by the gluon content of the proton. Here, we use the input distribution of gluon from Donnachie-Landshoff (DL) model to determine the longitudinal and heavy flavour structure function of proton. We compare our results with the recent HERA data and results of DL and Colour Dipole models which shows good agreement with data and fit. We use our results of heavy flavour structure function to analyze the behaviour of DIS cross section ratio R h (x,Q 2) and reduced cross section ${\sigma _{r}^{h}}$ in heavy quark lepto-production at small values of x. We have also studied the behaviour of the heavy quark content of the F L structure functions with respect to x.

A phenomenological solution small x to the longitudinal structure function dynamical behavior

In this paper, the evolutions of longitudinal proton structure function have been obtained at small x up to next-to-next-to-leading order using a hard Pomeron behavior. In our paper, evolutions of gluonic as well as heavy longitudinal structure functions have been obtained separately and the total contributions have been calculated. The total longitudinal structure functions have been compared with results of Donnachie–Landshoff (DL) model, Color Dipole (CD) model, kT factorization and H1 data.

Measurement of inclusive $$\varvec{ep}$$ e p cross sections at high $${\varvec{Q}^{2}}$$ Q 2 at $$\varvec{\sqrt{s}}=225$$ s = 225 and 252 GeV and of the longitudinal proton structure function $${\varvec{F}_{\varvec{L}}}$$ F L at HERA

Inclusive $$ep$$ double differential cross sections for neutral current deep inelastic scattering are measured with the H1 detector at HERA. The data were taken with a lepton beam energy of $$27.6$$ GeV and two proton beam energies of $$E_p=460$$ and 575 GeV corresponding to centre-of-mass energies of 225 and 252 GeV, respectively. The measurements cover the region of $$6.5\times 10^{-4} \le x \le 0.65$$ for $$35\le Q^2 \le 800$$ GeV $$^2$$ up to $$y=0.85$$ . The measurements are used together with previously published H1 data at $$E_p=920$$ GeV and lower $$Q^2$$ data at $$E_p=460$$ , $$575$$ and $$920$$ GeV to extract the longitudinal proton structure function $$F_L$$ in the region $$1.5\le Q^2 \le 800$$ GeV $$^2$$ .

Longitudinal Proton Structure Function at LO and NLO Approximations

The purpose of this paper is to calculate the longitudinal structure function of proton through the well-known equation FL=F2−2xF1. To determine this structure function, we need to identify parton distribution functions to find F2. In this case, the valon model and DGLAP equations are utilized to obtain the parton distributions. Our calculations are carried out in two approximations LO and NLO. The results at the NLO approximation are in better agreement with the experimental data.

Analytic Study of FL with the New Combined Data from Color Glass Condensate

We perform a global analysis of the new combined data on inclusive structure functions in electron-proton scattering at small value of Bjorken x (x ≤ 0.01) in the framework of the color glass condensate. The new combined HERA data at small-x are studied by an analytic proton structure function. It is shown that the theoretical results of the analytic proton structure function are in agreement with the new combined experimental data. The longitudinal proton structure function is investigated with the analytic expression. It is demonstrated that our approach shows a very good description of the longitudinal structure function experimental data, which is evidence showing the saturation effect is a domination mechanism during the rapidity evolution of the gluon density at high energy.

LHC soft physics and transverse momentum dependent gluon density at lowx

We study the unintegrated, or transverse momentum dependent gluon distribution obtained from the best description of the LHC data on the inclusive spectra of hadrons produced in the midrapidity region and at low transverse momenta at the starting scale ${Q}_{0}^{2}=1\text{ }\text{ }\mathrm{GeV}{}^{2}$. To extend this gluon density to higher ${Q}^{2}$ we apply the Catani-Ciafoloni-Fiorani-Marchesini evolution equation. The influence of the initial (starting) nonperturbative gluon distribution is studied. The application of the obtained gluon density to the analysis of the deep inelastic $ep$ scattering allows us to get the results which describe reasonably well the H1 and ZEUS data on the longitudinal proton structure function ${F}_{L}(x,{Q}^{2})$. So, the connection between the soft processes at the LHC and small-$x$ physics at HERA has been confirmed and extended to a wide kinematical region.

Moments of the longitudinal proton structure function FL from global data in the Q(2) range 0.75-45.0 (GeV/c)(2).

We present an extraction of the lowest three moments of the proton longitudinal structure function FL from world data between Q(2)=0.75 and 45 (GeV/c)(2). The availability of new FL data at low Bjorken x from HERA and at large x from Jefferson Lab allows the first determination of these moments over a large Q(2) range, relatively free from uncertainties associated with extrapolations into unmeasured regions. The moments are found to be underestimated by leading twist structure function parametrizations, especially for the higher moments, suggesting either the presence of significant higher twist effects in FL and/or a larger gluon distribution at high x.

Nonlinear correction to the longitudinal structure function at small x

We computed the longitudinal proton structure function FL, using the nonlinear Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (NLDGLAP) evolution equation approach at small x . For the gluon distribution, the nonlinear effects are related to the longitudinal structure function. As 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, we show that the strong rise that corresponds to the linear QCD evolution equations can be tamed by screening effects. Consequently, the obtained longitudinal structure function shows a tamed growth at small x . We computed the predictions for all details of the nonlinear longitudinal structure function in the kinematic range where it has been measured by the H1 Collaboration and made comparisons with the computation by Moch, Vermaseren and Vogt at the second order with input data from the MRST QCD fit.

FLproton structure function from the unified DGLAP/BFKL approach

We compute the longitudinal proton structure function ${F}_{L}$ from the ${k}_{T}$ factorization scheme, using the unified DGLAP/BFKL resummation approach at small $x$ for the unintegrated gluon density. The differences between the ${k}_{T}$ factorization, collinear factorization, and dipole approaches are analyzed and discussed. The comparisons with the DESY collider HERA data are made and predictions for the proposed large hadron-electron collider are also provided.

Direct measurement of the longitudinal proton structure function at HERA

The first direct measurements of the longitudinal proton structure function F L ( x , Q 2 ) at HERA are reported. The measurements were performed by the H1 and ZEUS collaborations. They are based on inclusive deep inelastic e + p scattering cross sections with a positron beam energy of 27.5 GeV and proton beam energies of 920, 575 and 460 GeV. The F L values agree with higher order QCD calculations based on parton densities obtained using cross section data previously measured at HERA.

Measurement of the Inclusive ep Scattering Cross Section at HERA

Measurements of the inclusive ep scattering cross section are presented in the wide region of four-momentum transfers squared, 0.2 GeV 2 ⩽ Q 2 ⩽ 30000 GeV 2 , and Bjorken x, 4 ⋅ 10 −6 ≲ x ≲ 0.8 . The results are based on data sets collected by the H1 and ZEUS collaborations at HERA. The data include the region of high inelasticity y up to 0.9, which is sensitive to the longitudinal proton structure function F L ( x , Q 2 ) .

Measurement of the proton structure function [formula omitted] at low x

A first measurement is reported of the longitudinal proton structure function FL(x,Q2) at the ep collider HERA. It is based on inclusive deep inelastic e+p scattering cross section measurements with a positron beam energy of 27.5 GeV and proton beam energies of 920, 575 and 460 GeV. Employing the energy dependence of the cross section, FL is measured in a range of squared four-momentum transfers 12⩽Q2⩽90 GeV2 and low Bjorken x0.00024⩽x⩽0.0036. The FL values agree with higher order QCD calculations based on parton densities obtained using cross section data previously measured at HERA.

Measurement of DIS cross section at HERA

This paper presents recent measurements of the inclusive DIS cross section performed by the H1 and ZEUS collaborations at the HERA collider. We discuss relations of the HERA results with the upcoming experiments at the LHC. Importance of the planed measurement of the longitudinal proton structure function F L is commented.

Longitudinal structure function of the proton at low momentum transfer and extended constituents

Recent data on the (inelastic) Nachtmann moments of the unpolarized proton structure function F2p, obtained at low momentum transfer with the CLAS detector at Jefferson Lab., have been interpreted in terms of the dominance of the elastic coupling of the virtual photon with extended substructures inside the proton. Adopting the same constituent form factors and the same light-front wave function describing the motion of the constituents in the proton, the (inelastic) Nachtmann moments of the longitudinal proton structure function FLp are calculated (without further parameters) for values of the squared four-momentum transfer 0.2≲Q2 (GeV/c)2≲2. The different role played by the Pauli form factor of the constituents in the transverse and longitudinal channels is illustrated. Our predictions, including an estimate of the theoretical uncertainties, may be checked against the forthcoming results of the experiment E94110 at Jefferson Lab. A positive comparison with the new data may provide compelling evidence that constituent quarks are intermediate substructures between the hadrons and the current quarks and gluons of QCD.

Measurement of the structure functions F2 and FL at low χ

A preliminary measurement of the proton structure function F 2 is presented for momentum transfers squared 2 ≤ Q 2 ≤ 120 GeV 2 and for Bjorken-χ values 3·10 −5 < χ < 0.1. The longitudinal proton structure function F L is determined for low χ values in the range of 3 ≤ Q 2 ≤ 35 GeV 2, using two independent methods. The analysis is based on data collected by the HERA experiment H1 in 1997, with a luminosity of 15 pb −1 for Q 2 > 10 GeV 2 and of 2 pb −1 taken in a dedicated run for Q 2 < 10 GeV 2. This measurement uses in the low Q 2 region the new backward silicon tracker. The accuracy is by nearly a factor of two better than previous results.