Structure Function F2 Research Articles

Decoupling of the structure functions in momentum space based on the Laplace transformation

Using Laplace transform techniques, we describe the determination of the longitudinal structure function FL(x,Q2), at the leading-order approximation in momentum space, from the structure function F2(x,Q2) and its derivative with respect to lnQ2 in a kinematical region of low values of the Bjorken variable x. Since the x dependence of F2(x,Q2) and its evolution with Q2 are determined much better by the data than FL(x,Q2), this method provides both a direct check on FL(x,Q2) where measured, and a way of extending FL(x,Q2) into regions of x and Q2 where there are currently no data. In our calculations, we ultilize the Block-Durand-Ha parametrization for the structure function F2(x,Q2) [M. M. Block , .]. We find that the Laplace transform method in momentum space provides correct behaviors of the extracted longitudinal structure function FL(x,Q2) and that our obtained results are in line with data from the H1 Collaboration and other results for FL(x,Q2) obtained using Mellin transform method. Published by the American Physical Society 2024

Superfast quarks in deuterium

An extension to our previous study on nuclear parton distribution functions (nPDFs) [Kim and Miller, ] using light-front holographic quantum chromodynamics (LFHQCD) [Brodsky, de Teramond, Dosch, and Erlich, ] is presented. We apply the effects of nucleon motion inside the nucleus (Fermi motion/smearing) to deuterium, extending our deuterium nPDFs to the , x>1, region [Frankfurt and Strikman, ] where we estimate our results to be reasonable up to x≈1.7. We utilize four different deuteron wave functions (AV18, NijmI, NijmII, Nijm93). We find that our model, with no additional new parameters, shows very good agreement with deuterium EMC ratio data obtained from the BONuS experiment [Fenker , ; Baillie (CLAS Collaboration), ; ; Tkachenko (CLAS Collaboration), ; ; Griffioen , ]. Looking beyond conventional nuclear physics, and in anticipation of 12 GeV experiments at Jefferson Lab, we use a LFHQCD ansatz to predict the contributions of an exotic six-quark state to the deuteron F2 structure function, F2D, in the superfast region. We find that the effects of using other potentials are about the same magnitude as six-quark effects—both have small effects in x<1, but have significant contributions at x>1. Published by the American Physical Society 2024

Inferring the initial condition for the Balitsky-Kovchegov equation

We apply Bayesian inference to determine the posterior likelihood distribution for the parameters describing the initial condition of the small-x Balitsky-Kovchegov evolution equation at leading logarithmic accuracy. The HERA structure function data is found to constrain most of the model parameters well. In particular, we find that the HERA data prefers an anomalous dimension γ≈1 unlike in previous fits where γ>1 which led to, e.g., the unintegrated gluon distribution and the quark-target cross sections not being positive definite. The determined posterior distribution can be used to propagate the uncertainties in the nonperturbative initial condition when calculating any other observable in the color glass condensate framework. We demonstrate this explicitly for the inclusive quark production cross section in proton-proton collisions and by calculating predictions for the nuclear modification factor for the F2 structure function in the EIC and LHeC/FCC-he kinematics. Published by the American Physical Society 2024

Self-consistent description of HERA data at low Q2 and soft hadron production at LHC

Using the analytical expression for transverse momentum dependent (TMD) gluon density in a proton, a self-consistent simultaneous description of low Q2 data on proton structure function F2(x,Q2), reduced cross section for the electron-proton deep inelastic scattering at HERA and soft hadron production in pp collisions at the LHC is achieved in the framework of color dipole approach and modified quark-gluon string model.

Solutions to the Balitsky-Kovchegov equation including the dipole orientation

Solutions of the target-rapidity Balitsky-Kovchegov (BK) equation are studied considering, for the first time, the complete impact-parameter dependence, including the orientation of the dipole with respect to the impact-parameter vector. In our previous work [1] it has been demonstrated that the spurious Coulomb tails could be tamed using the collinearly-improved kernel and an appropriate initial condition in the projectile-rapidity BK equation. Introducing a different interpretation of the evolution variable, the target-rapidity formulation of the BK equation brings non-locality in rapidity and a kernel modification, removing the term that previously helped to suppress the Coulomb tails. To address this newly emerged non-locality, three different prescriptions are explored here to take into account the rapidities preceding the initial condition. Two of these approaches induce mild Coulomb tails, while the other is free from this effect within the studied rapidity range. The range is chosen to correspond to that of interest for existing and future experiments. To demonstrate that this set up can be used for phenomenological studies, the obtained solutions are used to compute the F2 structure function of the proton and the diffractive photo- and electro-production of J/ψ off protons. The predictions agree well with HERA data, confirming that the target-rapidity Balitsky-Kovchegov equation with the full impact-parameter dependence is a viable tool to study the small Bjorken-x limit of perturbative QCD at current facilities like RHIC and LHC as well as in future colliders like the EIC.

A model for $F_L$ structure function at low values of $Q^2$ and $x$ - revisited

A reanalysis of the model for the longitudinal structure function F_L (x,Q^2)FL(x,Q2) at low xx and low Q^2Q2 was undertaken, in view of the advent of the EIC. The model includes the kinematic constraint F_L\sim Q^4FL∼Q4 as Q^2\rightarrowQ2→ 0. It is based on the photon-gluon fusion mechanism suitably extrapolated to the region of low Q^2Q2. Revised model was critically updated, extended to the EIC kinematic region, and e.g. contains new parameterisations of the parton distribution functions.

Magnetic moments of the octet, decuplet, low-lying charm, and low-lying bottom baryons in a nuclear medium

Abstract We study the magnetic moments of the octet, decuplet, low-lying charm, and low-lying bottom baryons with nonzero light quarks in symmetric nuclear matter using the quark–meson coupling (QMC) model, which satisfies the constraint for the allowed maximum change (swelling) of the in-medium nucleon size derived from the y-scaling data for 3He(e, e′) and 56Fe(e, e′). This is the first study to estimate the in-medium magnetic moments of the low-lying charm and bottom baryons with nonzero light quarks. The present QMC model also satisfies the expected allowed maximum enhancement of the nucleon magnetic moments in nuclear matter. Moreover, it has been proven that the calculated in-medium to free proton electromagnetic form factor (EMFF) ratios calculated within the QMC model reproduce well the proton EMFF super ratio extracted from $^4{\rm He}(\vec{e},e^{\prime }\vec{p})^3{\rm H}$ at Jefferson Laboratory. The medium modifications of the magnetic moments are estimated by evaluating the in-medium to free space baryon magnetic moment ratios to compensate the MIT bag deficiency to describe the free space octet baryon magnetic moments, where ratios are often measured directly in experiments even without knowing the absolute values, such as the free and bound proton electromagnetic form factors, as well as the European Muon Collaboration effect to extract the structure function F2 ratio of the bound to free nucleons by the corresponding cross section ratio. We also present the results calculated with the different current quark mass values for the strange and bottom quarks to see the possible impact. Furthermore, for practical use we give the explicit density-dependent parametrizations for the vector potentials of the baryons and light-(u, d) quarks, as well as for the effective masses of the baryons treated in this study, and of the mesons ω, ρ, K, K*, η, $\eta^{\prime}$, D, D*, B, and B*.

Computing real time correlation functions on a hybrid classical/quantum computer

Quantum devices may overcome limitations of classical computers in studies of nuclear structure functions and parton Wigner distributions of protons and nuclei. In this talk, we discuss a worldline approach to compute nuclear structure functions in the high energy Regge limit of QCD using a hybrid quantum computer, by expressing the fermion determinant in the QCD path integral as a quantum mechanical path integral over 0 + 1-dimensional fermionic and bosonic world-lines in background gauge fields. Our simplest example of computing the well-known dipole model result for the structure function F2 in the high energy Regge limit is feasible with NISQ era technology using few qubits and shallow circuits. This example can be scaled up in complexity and extended in scope to compute structure functions, scattering amplitudes and other real-time correlation functions in QCD, relevant for example to describe non-equilibrium transport of quarks and gluons in a Quark-Gluon-Plasma.

Searching for top quark pair production cross-section at LHeC and FCC-eh

The deep inelastic scattering mode of pair production at the proposed LHeC and FCC-eh is considered. We present a method to extract the top reduced cross-section related to the transversal structure function F2(x, Q2) parameterization. Numerical calculations with known kinematics of the LHeC and FCC-eh colliders are presented. The results obtained for charm and beauty pair production are comparable with the experimental data. We show that for a wide range of the momentum transfer into the top quark pair, the reduced cross-section is well described by center-of-mass energies.

QCD effective charges and the structure function F2 at small-x: Higher twist effects

We consider the effect of higher twist operators of the Wilson operator product expansion in the structure function F2(x,Q2) at small-x, taking into account QCD effective charges whose infrared behavior is constrained by a dynamical mass scale. The higher twist corrections are obtained from the renormalon formalism. Our analysis is performed within the conventional framework of next-to-leading order, with the factorization and renormalization scales chosen to be Q2. The infrared properties of QCD are treated in the context of the generalized double-asymptotic-scaling approximation. We show that the corrections to F2 associated with twist-four and twist-six are both necessary and sufficient for a good description of the deep infrared experimental data.

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.

Small-x analysis on the effect of gluon recombinations inside hadrons in light of the GLR-MQ-ZRS equation

We present a study of the contribution of antishadowing effects on the gluon distribution functions G(x,Q2) in light of the Gribov-Levin-Ryskin-Mueller-Qiu, Zhu-Ruan-Shen (GLR-MQ-ZRS) nonlinear equation at small-x, where x is the momentum fraction or Bjorken variable and Q2 is the four momentum transfer squared or photon virtuality. In this work, we have solved the GLR-MQ-ZRS nonlinear equation using Regge like behaviour of gluons in the kinematic range of 10−2≤x≤10−6 and 5GeV2≤Q2≤100GeV2 respectively. We have obtained the solution of G(x,Q2) by considering two particular cases: (a) αs fixed; and (b) the leading order QCD dependency of αs on Q2. A comparative analysis is also performed where we compare the gluon distribution function due to inclusion of the antishadowing effect with that of the gluon distribution without including the antishadowing effect. Our obtained results of G(x,Q2) are compared with NNPDF3.0, CT14 and PDF4LHC. We also compare our results with the result obtained from the IMParton C++ package. Using the solutions of G(x,Q2), we have also predicted x and Q2 evolution of the logarithmic derivative of proton's F2 structure function i.e. dF2(x,Q2)/dln⁡Q2. We incorporated both the leading order(LO) and next-to-leading order (NLO) QCD contributions of the gluon-quark splitting kernels, in dF2(x,Q2)/dln⁡Q2. Our result of dF2(x,Q2)/dln⁡Q2 agrees reasonably well with the experimental data recorded by HERA's H1 detector.

Extracting the longitudinal structure function FL(x,Q2) at small x from a Froissart-bounded parametrization of F2(x,Q2)

We present a method to extract, in the leading and next-to-leading order approximations, the longitudinal deep-inelastic scattering structure function FL(x,Q2) from the experimental data by relying on a Froissart-bounded parametrization of the transversal structure function F2(x,Q2) and, partially, on the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equations. Particular attention is paid on kinematics of low and ultra low values of the Bjorken variable x. Analytical expressions for FL(x,Q2) in terms of the effective parameters of the parametrization of F2(x,Q2) are presented explicitly. We argue that the obtained structure functions FL(x,Q2) within both, the leading and next-to-leading order approximations, manifestly obey the Froissart boundary conditions. Numerical calculations and comparison with available data from ZEUS and H1-Collaborations at HERA demonstrate that the suggested method provides reliable structure functions FL(x,Q2) at low x in a wide range of the momentum transfer (1 GeV2 < Q2 < 3000 GeV2) and can be applied as well in analyses of ultra-high energy processes with cosmic neutrinos.

The variable flavor number scheme at next-to-leading order

We present the matching relations of the variable flavor number scheme at next-to-leading order, which are of importance to define heavy quark partonic distributions for the use at high energy colliders such as Tevatron and the LHC. The consideration of the two-mass effects due to both charm and bottom quarks, having rather similar masses, are important. These effects have not been considered in previous investigations. Numerical results are presented for a wide range of scales. We also present the corresponding contributions to the structure function F2(x,Q2).

An improved analysis of proton structure function F2(x,t) at small x

We report an improved analysis of Taylor approximated coupled DGLAP equations for singlet $F_{2}^{S}(x,t)$ and gluon G(x,t) distributions at small x pursued in recent years. To that end, we assume a plausible t-dependent relation between the singlet and gluon distribution valid at small x and omit the boundary condition $F_{2}^{S} (1, t) = 0$ , for any t which needs large x extrapolation of small x solution. We observe that in general two inequivalent t-evolutions of $ F_{2}^{S} (x, t)$ and G(x, t) are possible. Theoretical advantages of one over the other are discussed and compared with the recently compiled data in order to choose the best one. Phenomenological range of validity of solutions is also reported.

Gottfried Sum Rule in QCD Nonsinglet Analysis of DIS Fixed-Target Data

Deep-inelastic-scattering data from fixed-target experiments on the structure function F2 were analyzed in the valence-quark approximation at the next-to-next-to-leading-order accuracy level in the strong-coupling constant. In this analysis, parton distributions were parametrized by employing information from the Gottfried sum rule. The strong-coupling constant was found to be α s (M2Z) = 0.1180 ± 0.0020 (total expt. error), which is in perfect agreement with the world-averaged value from an updated Particle Data Group (PDG) report, αPDG s (M2Z) = 0.1181 ± 0.0011. Also, the value of 〈x〉u−d = 0.187 ± 0.021 found for the second moment of the difference in the u- and d-quark distributions complies very well with the most recent lattice result 〈x〉LATTICEu−d = 0.208 ± 0.024.

The two-mass contribution to the three-loop pure singlet operator matrix element

We present the two-mass QCD contributions to the pure singlet operator matrix element at three loop order in x-space. These terms are relevant for calculating the structure function F2(x,Q2) at O(αs3) as well as for the matching relations in the variable flavor number scheme and the heavy quark distribution functions at the same order. The result for the operator matrix element is given in terms of generalized iterated integrals that include square root letters in the alphabet, depending also on the mass ratio through the main argument. Numerical results are presented.

Comparison of the structure function F2 as measured by charged lepton and neutrino scattering from iron targets

A comparison study of world data for the structure function F2 for Iron, as measured by both charged lepton and neutrino scattering experiments, is presented. Consistency of results for both charged lepton and neutrino scattering is observed for the full global data set in the valence regime. Consistency is also observed at low x for the various neutrino data sets, as well as for the charged lepton data sets, independently. However, data from the two probes exhibit differences on the order of 15% in the shadowing/anti-shadowing transition region where the Bjorken scaling variable x is < 0.15. This observation is indicative that neutrino probes of nucleon structure might be sensitive to different nuclear effects than charged lepton probes. Details and results of the data comparison are here presented.

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.