Deep Inelastic Structure Functions Research Articles

Detailed study of quark-hadron duality in spin structure functions of the proton and neutron

Background: The response of hadrons, the bound states of the strong force (QCD), to external probes can be described in two different, complementary frameworks: as direct interactions with their fundamental constituents, quarks and gluons, or alternatively as elastic or inelastic coherent scattering that leaves the hadrons in their ground state or in one of their excited (resonance) states. The former picture emerges most clearly in hard processes with high momentum transfer, where the hadron response can be described by the perturbative expansion of QCD, while at lower energy and momentum transfers, the resonant excitations of the hadrons dominate the cross section. The overlap region between these two pictures, where both yield similar predictions, is referred to as quark-hadron duality and has been extensively studied in reactions involving unpolarized hadrons. Some limited information on this phenomenon also exists for polarized protons, deuterons, and $^{3}\mathrm{He}$ nuclei, but not yet for neutrons.Purpose: In this paper, we present comprehensive and detailed results on the correspondence between the extrapolated deep inelastic structure function ${g}_{1}$ of both the proton and the neutron with the same quantity measured in the nucleon resonance region. Thanks to the fine binning and high precision of our data, and using a well-controlled perturbative QCD (pQCD) fit for the partonic prediction, we can make quantitative statements about the kinematic range of applicability of both local duality and global duality.Method: We use the most updated QCD global analysis results at high $x$ from the Jefferson Lab Angular Momentum Collaboration to extrapolate the spin structure function ${g}_{1}$ into the nucleon resonance region and then integrate over various intervals in the scaling variable $x$. We compare the results with the large data set collected in the quark-hadron transition region by the CLAS Collaboration, including, for the first time, deconvoluted neutron data, integrated over the same intervals. We present this comparison as a function of the momentum transfer ${Q}^{2}$.Results: We find that, depending on the integration interval and the minimum momentum transfer chosen, a clear transition to quark-hadron duality can be observed in both nucleon species. Furthermore, we show, for the first time, the approach to scaling behavior for ${g}_{1}$ measured in the resonance region at sufficiently high momentum transfer.Conclusions: Our results can be used to quantify the deviations from the applicability of pQCD for data taken at moderate energies and can help with extraction of quark distribution functions from such data.

Test of the TMD gluon density in a proton with the longitudinal structure function FL(x,Q2)

We investigate the dependence of the deep inelastic proton structure function FL(x,Q2) on different forms of the transverse momentum dependent (TMD, or unintegrated) gluon distribution. We present a comparison of theoretical results with the latest experimental data taken by the H1 and ZEUS Collaborations at HERA. We demonstrate that these data, despite of having quite large uncertainties, can test different kinds of the gluon TMD in a proton. Moreover, different phenomenological models at low x could be tested by future experiments on deep inelastic scattering.

Production of e+e− in proton-lead collision: Photon-photon fusion

We analyze the photon-initiated processes for production of $ e ^ + e ^- $ pairs in proton-nucleus collisions at LHC energy, taking into account both elastic processes and proton dissociations in the low-mass region (LMR) and intermediate-mass region (IMR) as defined by the ALICE collaboration. The calculations are performed within the $k_{\rm T}$-factorisation approach, including transverse momenta of intermediate photons. We discuss several differential distributions in invariant mass of both the leptons $M_{ll}$, pair rapidity $Y_{ll}$ and transverse momenta of the lepton pair $p_{t,ll}$. In addition, we present the two-dimensional distributions in $\log_{10} x_{Bj}$ and $\log_{10} Q^{2}$ and ($\log_{10} W,\log_{10} Q^{2}$) the arguments of the deep-inelastic structure functions. All presented results were obtained with modern parametrizations of proton structure functions. Limiting to small invariant masses of dielectrons one tests structure functions in the nonperturbative region of small $Q^2$ and/or small $W$. We quantify difference for different parametrizations from the literature.

Massive quarks in NLO dipole factorization for DIS: Longitudinal photon

In this work, we will present the first complete calculation of the one-loop longitudinal photon-to-quark-antiquark light cone wave function, with massive quarks. The quark masses are renormalized in the pole mass scheme. The result is used to calculate the next-to-leading order correction to the high energy Deep Inelastic Scattering longitudinal structure function on a dense target in the dipole factorization framework. For massless quarks the next-to-leading order correction was already known to be sizeable, and our result makes it possible to evaluate it also for massive quarks.

The Parton Distribution Functions and Calculation of Dirac Neutron Form Factor Considering the EMC and Shadowing Effects

Due to the absence of the free neutron target, the form factors or the structure function of the neutron directly cannot be obtained from the elastic scattering data. Therefore, in this paper, we obtained the neutron structure function using the deep inelastic structure function of the proton and by applying the EMC and the shadowing effects on the structure function of the deuteron. Then, using the quantum chromo dynamics theorem and re-analysis of data from the SLAC experiments E49, E61, E87, and E89 in electron–proton deep inelastic scattering at Q2 = 1–20 GeV2, we have extracted the up and down quark distribution functions, $${{u}_{{v}}}$$ (x), $${{d}_{{v}}}$$ (x), and also, using an appropriate model for the general parton distribution function, we have plotted the Dirac form factor of the neutron. At the end, the curves obtained in this work were compared with the curves obtained in other models.

Gluon Evolution for the Berger-Block-Tan Form of the Structure Function F2

We present a nonlinear modification of the evolution of the gluon density, obtained at small x from the Berger-Block-Tan form of the deep inelastic structure function F2 in the leading order of perturbation theory.

Production of Heavy Particle Pairs via Photon–Photon Processes at the LHC

We discuss production of $W^+ W^-$ pairs and $t \bar t$ quark-antiquark pairs in proton-proton collisions induced by two-photon fusion including transverse momenta of incoming photons. The unintegrated inelastic fluxes (related to proton dissociation) of photons are calculated based on modern parametrizations of deep inelastic structure functions in a broad range of $x$ and $Q^2$. We focus on processes with single and double proton dissociation. Highly excited remnant systems hadronise producing particles that can be vetoed in the calorimeter. We calculate associated gap survival factors. The gap survival factors depend on the process, mass of the remnant system and collision energy. The rapidity gap survival factor due to remnant fragmentation for double dissociative (DD) collisions is smaller than that for single dissociative (SD) process. We observe approximate factorisation: $S_{R,DD} \approx S_{R,SD}^2$ when imposing rapidity veto. For the $W^+W^-$ final state, the remnant fragmentation leads to a taming of the cross section when the rapidity gap requirement is imposed. Also for $t \bar t$ quark-antiquark pairs such a condition reverses the hierarchy observed for the case when such condition is taken into account. Our results imply that for the production of such heavy objects as $t$ quark and $\bar t$ antiquark the virtuality of the photons attached to the dissociative system are very large ($Q^2 <$ 10$^{4}$ GeV$^2$). A similar effect is observed for the $W^+ W^-$ system.

Can New Physics Hide inside the Proton?

Modern global analyses of the structure of the proton include collider measurements which probe energies well above the electroweak scale. While these provide powerful constraints on the parton distribution functions (PDFs), they are also sensitive to beyond the standard model (BSM) dynamics if these affect the fitted distributions. Here we present a first simultaneous determination of the PDFs and BSM effects from deep-inelastic structure function data by means of the NNPDF framework. We consider representative four-fermion operators from the SM effective field theory (SMEFT), quantify to which extent their effects modify the fitted PDFs, and assess how the resulting bounds on the SMEFT degrees of freedom are modified. Our results demonstrate how BSM effects that might otherwise be reabsorbed into the PDFs can be systematically disentangled.

Nucleon parton distributions from hadronic quantum fluctuations

A physical model is presented for the nonperturbative parton distributions in the nucleon. This is based on quantum fluctuations of the nucleon into baryon-meson pairs convoluted with Gaussian momentum distributions of partons in hadrons. The hadronic fluctuations, here developed in terms of hadronic chiral perturbation theory, occur with high probability and generate sea quarks as well as dynamical effects also for valence quarks and gluons. The resulting parton momentum distributions $f(x,{Q}_{0}^{2})$ at low momentum transfers are evolved with conventional Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equations from perturbative QCD to larger scales. This provides parton density functions $f(x,{Q}^{2})$ for the gluon and all quark flavors with only five physics-motivated parameters. By tuning these parameters, experimental data on deep-inelastic structure functions can be reproduced and interpreted. The contribution to sea quarks from hadronic fluctuations explains the observed asymmetry between $\overline{u}$ and $\overline{d}$ in the proton. The strange-quark sea is strongly suppressed at low ${Q}^{2}$, as observed.

Low Q2 boundary conditions for DGLAP equations dictated by quantum statistical mechanics

We discuss the role of quantum statistical mechanics in the description of the parton distribution functions in the proton. It provides the low Q2 boundary conditions for DGLAP equations in terms of Fermi–Dirac and Bose–Einstein functions of the fractional momentum variable x. The successful comparison with experimental data on both the unpolarised and polarised deep inelastic structure functions is reviewed. We argue that the statistical approach for the nucleon parton distributions functions has the nice feature that the free model parameters are fixed from data with high statistics and small systematic uncertainties, providing a strong constraint on the information not supplied by the experiments.

Nuclear parton distributions from lepton-nucleus scattering and the impact of an electron-ion collider

We present a first determination of the nuclear parton distribution functions (nPDF) based on the NNPDF methodology: nNNPDF1.0. This analysis is based on neutral-current deep-inelastic structure function data and is performed up to NNLO in QCD calculations with heavy quark mass effects. For the first time in the NNPDF fits, the chi ^2 minimization is achieved using stochastic gradient descent with reverse-mode automatic differentiation (backpropagation). We validate the robustness of the fitting methodology through closure tests, assess the perturbative stability of the resulting nPDFs, and compare them with other recent analyses. The nNNPDF1.0 distributions satisfy the boundary condition whereby the NNPDF3.1 proton PDF central values and uncertainties are reproduced at A=1, which introduces important constraints particularly for low-A nuclei. We also investigate the information that would be provided by an Electron-Ion Collider (EIC), finding that EIC measurements would significantly constrain the nPDFs down to xsimeq 5times 10^{-4}. Our results represent the first-ever nPDF determination obtained using a Monte Carlo methodology consistent with that of state-of-the-art proton PDF fits, and provide the foundation for a subsequent global nPDF analyses including also proton-nucleus data.

Production of W+W\u2212 pairs via \u03b3*\u03b3* \u2192 W+W\u2212 subprocess with photon transverse momenta

We discuss production of W+W− pairs in proton-proton collisions induced by two-photon fusion including, for a first time, transverse momenta of incoming photons. The unintegrated inelastic fluxes (related to proton dissociation) of photons are calculated based on modern parametrizations of deep inelastic structure functions in a broad range of their arguments (x and Q2). In our approach we can get separate contributions of different W helicities states. Several one- and two-dimensional differential distributions are shown and discussed. The present results are compared to the results of previous calculations within collinear factorization approach. Similar results are found except of some observables such as e.g. transverse momentum of the pair of W+ and W−. We find large contributions to the cross section from the region of large photon virtualities. We show decomposition of the total cross section as well as invariant mass distribution into the polarisation states of both W bosons. The role of the longitudinal FL structure function is quantified. Its inclusion leads to a 4–5% decrease of the cross section, almost independent of MWW.

Parton distributions with small-x resummation: evidence for BFKL dynamics in HERA data

We present a determination of the parton distribution functions of the proton in which NLO and NNLO fixed-order calculations are supplemented by NLLx small-x resummation. Deep-inelastic structure functions are computed consistently at hbox {NLO+NLL}x or hbox {NNLO+NLL}x, while for hadronic processes small-x resummation is included only in the PDF evolution, with kinematic cuts introduced to ensure the fitted data lie in a region where the fixed-order calculation of the hard cross-sections is reliable. In all other respects, the fits use the same methodology and are based on the same global dataset as the recent NNPDF3.1 analysis. We demonstrate that the inclusion of small-x resummation leads to a quantitative improvement in the perturbative description of the HERA inclusive and charm-production reduced cross-sections in the small x region. The impact of the resummation in our fits is greater at NNLO than at NLO, because fixed-order calculations have a perturbative instability at small x due to large logarithms that can be cured by resummation. We explore the phenomenological implications of PDF sets with small-x resummation for the longitudinal structure function F_L at HERA, for parton luminosities and LHC benchmark cross-sections, for ultra-high-energy neutrino–nucleus cross-sections, and for future high-energy lepton–proton colliders such as the LHeC.

Gluon distribution function from the Berger–Block–Tan form of the structure function F 2

We present a set of formulas to extract the gluon distribution function from the Berger–Block–Tan form of the deep inelastic structure function F 2 and its derivative dF 2/d ln Q 2 at small x in the leading order of perturbation theory.

Gluon density from the Berger–Block–Tan form of the structure function F 2

We present a set of formulas to extract the gluon density from the Berger-Block-Tan form of the deep inelastic structure function F2 at small x in the leading order of perturbation theory.

CGC/saturation approach: A new impact-parameter-dependent model in the next-to-leading order of perturbative QCD

This paper is the first attempt to build CGC/saturation model based on the next-to-leading order corrections to linear and non-linear evolution in QCD. We assume that the renormalization scale is the saturation momentum and found that the scattering amplitude has geometric scaling behaviour deep in the saturation domain with the explicit formula of this behaviour at large $\tau = r^2 Q^2_s$. We built a model that include this behaviour, as well as the ingredients that has been known: (i) the behaviour of the scattering amplitude in the vicinity of the saturation momentum, using the NLO BFKL kernel, (ii) the pre-asymptotic behaviour of $\ln\Lb Q^2_s\Lb Y \Rb\Rb$, as function of $Y$ and (iii) the impact parameter behaviour of the saturation momentum, which has exponential behaviour $\propto \exp\Lb -\, m\, b\Rb$ at large $b$.We demonstrated that the model is able to describe the experimental data for the deep inelastic structure function. Despite this, our model has difficulties that are related to the small value of the QCD coupling at $Q_s\Lb Y_0\Rb$ and the large values of the saturation momentum, which indicate the theoretical inconsistency of our description.

A determination of the charm content of the proton: The NNPDF Collaboration.

We present an unbiased determination of the charm content of the proton, in which the charm parton distribution function (PDF) is parametrized on the same footing as the light quarks and the gluon in a global PDF analysis. This determination relies on the NLO calculation of deep-inelastic structure functions in the FONLL scheme, generalized to account for massive charm-initiated contributions. When the EMC charm structure function dataset is included, it is well described by the fit, and PDF uncertainties in the fitted charm PDF are significantly reduced. We then find that the fitted charm PDF vanishes within uncertainties at a scale Qsim 1.6 GeV for all xlesssim 0.1, independent of the value of m_c used in the coefficient functions. We also find some evidence that the charm PDF at large xgtrsim 0.1 and low scales does not vanish, but rather has an “intrinsic” component, very weakly scale dependent and almost independent of the value of m_c, carrying less than 1% of the total momentum of the proton. The uncertainties in all other PDFs are only slightly increased by the inclusion of fitted charm, while the dependence of these PDFs on m_c is reduced. The increased stability with respect to m_c persists at high scales and is the main implication of our results for LHC phenomenology. Our results show that if the EMC data are correct, then the usual approach in which charm is perturbatively generated leads to biased results for the charm PDF, though at small x this bias could be reabsorbed if the uncertainty due to the charm mass and missing higher orders were included. We show that LHC data for processes, such as high p_T and large rapidity charm pair production and Z+c production, have the potential to confirm or disprove the implications of the EMC data.

A determination of m c (m c ) from HERA data using a matched heavy-flavor scheme

The charm quark mass is one of the fundamental parameters of the Standard Model Lagrangian. In this work we present a determination of the $$ \overline{\mathrm{MS}} $$ charm mass from a fit to the inclusive and charm HERA deep-inelastic structure function data. The analysis is performed within the xFitter framework, with structure functions computed in the FONLL general-mass scheme as implemented in APFEL. In the case of the FONLL-C scheme, we obtain m c (m c ) = 1.335 ± 0.043(exp) − 0.000 + 0.019 (param) − 0.008 + 0.011 (mod) − 0.008 + 0.003 (th) GeV. We also perform an analogous determination in the fixed-flavor-number scheme at next-to-leading order, finding m c (m c ) = 1.318 ± 0.054(exp) − 0.010 + 0.011 (param) − 0.004 + 0.015 (mod) − 0.004 + 0.045 (th) GeV, compatible with the FONLL-C value. Our results are consistent with previous determinations from DIS data as well as with the PDG world average.

Color dipole cross section and inelastic structure function

Instead of starting from a theoretically motivated form of the color dipole cross section in the dipole picture of deep inelastic scattering, we start with a parametrization of the deep inelastic structure function for electromagnetic scattering with protons, and then extract the color dipole cross section. Using the parametrizations of $F_2(\xi=x \ {\rm or}\ W^2,Q^2)$ by Donnachie-Landshoff and Block et al., we find the dipole cross section from an approximate form of the presumed dipole cross section convoluted with the perturbative photon wave function for virtual photon splitting into a color dipole with massless quarks. The color dipole cross section determined this way reproduces the original structure function within about 10\% for $0.1$ GeV$^2\leq Q^2\leq 10$ GeV$^2$. We discuss the large and small form of the dipole cross section and compare with other parameterizations.

Neutron spin structure with polarized deuterons and spectator proton tagging at EIC

The neutron's deep-inelastic structure functions provide essential information for the flavor separation of the nucleon parton densities, the nucleon spin decomposition, and precision studies of QCD phenomena in the flavor-singlet and nonsinglet sectors. Traditional inclusive measurements on nuclear targets are limited by dilution from scattering on protons, Fermi motion and binding effects, final-state interactions, and nuclear shadowing at x ≪ 0.1. An Electron-Ion Collider (EIC) would enable next-generation measurements of neutron structure with polarized deuteron beams and detection of forward-moving spectator protons over a wide range of recoil momenta (0 < pR < several 100MeV in the nucleus rest frame). The free neutron structure functions could be obtained by extrapolating the measured recoil momentum distributions to the on-shell point. The method eliminates nuclear modifications and can be applied to polarized scattering, as well as to semi-inclusive and exclusive final states. We review the prospects for neutron structure measurements with spectator tagging at EIC, the status of R&D efforts, and the accelerator and detector requirements.