Quark Structure Functions Research Articles

Heavy quark structure functions from unifying the color dipole picture and double asymptotic scaling approaches

We present an analysis of the heavy quark structure functions from the kt factorization scheme, using unifying the color dipole picture and double asymptotic scaling approaches at small x. The gluon distribution is obtained from the Golec-Biernat-Wüsthoff and Bartels, Golec-Biernat and Kowalski models. The main elements are based on the color dipole picture and the generalized double asymptotic scaling approach for usual parton distribution functions. The comparisons with the HERA data are made and predictions for the proposed LHeC and FCC-he colliders are also provided in a wide range of the transverse separation r. In particular, the ratio Rh=FLh/F2h,h=c,b,t is well described by the dipole models and is sensitive to the collider energies from HERA until FCC-he. We derive correlated bounds on the ratio F2c/F2 and F2b/F2 and compared them with the BGK and impact-parameter dependent saturation models. The uncertainties are due to the renormalization and factorization scales at large and low r values. The Sudakov form factor into the heavy quark structure functions is incorporated and the results are considered, which are dependent on the hard scale in a wide range of the transverse separation r. Published by the American Physical Society 2024

Production cross section of heavy quarks in e−p interaction at the NLO approximation

We present the production cross section of heavy quarks σcc¯, σbb¯ and σtt¯ at the next-to-leading order in the electron-proton interaction by using the quarks and gluon distribution functions at the initial scale Q02. To do this, we use a fitted form of the heavy quark coefficient functions for deep-inelastic lepton-hadron scattering to obtain the structure functions of heavy quarks. Then, we calculate the reduced cross section of heavy quarks by using the structure functions and subsequently present the single differential and the integrated cross section of heavy quarks at the center-of-mass energies of s=319GeV, 1.3TeV and 3.5TeV in the electron-proton collision. The obtained numerical results of the cross section of the charm and beauty quarks are compared with the HERA data, which is a combination from the results of the H1 and ZEUS detectors. Furthermore, we present the production cross section of top quark as a direct prediction from our calculations.

Geometrical scaling of heavy-quark contributions in the low x region

We describe the determination of the heavy quarks structure functions $F_{2,L}^{\mathcal{Q}\bar{\mathcal{Q}}}$ with help of the scaling properties. We observe that the structure functions for inclusive charm and bottom production exhibits geometric scaling at low $x$. The geometrical scaling means that the heavy quark structure function is a function of only one dimensionless variable $\tau{\equiv}Q^{2}/Q^{2}_{sat}(x)$ including quark mass. These results are valid for any value of $\delta$, being $x^{-\delta}$ the behavior of the parton densities at low $x$. The determination of the heavy quark structure function is presented as a parameterization of the proton structure function $F_{2}(x,Q^{2})$ and its derivative. Analytical expressions for $F_{2,L}^{\mathcal{Q}\bar{\mathcal{Q}}}$ in terms of the effective parameters of the parameterization of $F_{2}(x,Q^{2})$ ,with respect to the BDH and ASW models, are presented. To study the heavy quark production processes, we use the collinear results in DAS approach. Numerical calculations and comparison with HERA data demonstrate that the suggested method provides reliable $F_{2}^{\mathcal{Q}\bar{\mathcal{Q}}}$, $R^{c\bar{c}}$ and $\sigma^{\mathcal{Q}\bar{\mathcal{Q}}}$ at low $x$ in a wide range of the low absolute four-momentum transfers squared ($4~\mathrm{GeV}^{2}<Q^{2}<2000~\mathrm{GeV}^{2}$). Also, in the HERA kinematic range, the ratio of $F_{2}^{c\bar{c}}/F_{2}^{b\bar{b}}$, $F_{2}^{c\bar{c}}/F_{2}^{BDHM}$ and $F_{2}^{b\bar{b}}/F_{2}^{BDHM}$ are obtained. Expanding the method to low and ultra low values of $x$ can be considered in the process analysis of the LHeC and FCC-eh colliders.

Charged lepton flavor violation associated with heavy quark production in deep inelastic lepton-nucleon scattering via scalar exchange

We study charged lepton flavor violation (CLFV) associated with heavy quark pair production in lepton-nucleon deep-inelastic scattering ℓiN → ℓjq overline{q} X. Here ℓi and ℓj denote the initial and final leptons; N and X are respectively the initial nucleon and arbitrary final hadronic system. We employ a model Lagrangian in which a scalar and pseudoscalar mediator generates the CLFV. We derive heavy quark structure functions for scalar and pseudoscalar currents and compute momentum distributions of the final lepton for the process. Our focus is on the heavy quark mass effects in the final lepton momentum distribution. We clarify the necessity of inclusion of the heavy quark mass to obtain reliable theory predictions for the CLFV signal searches in the deep-inelastic scattering.

Deep inelastic scattering as a probe of entanglement: Confronting experimental data

Parton distributions can be defined in terms of the entropy of entanglement between the spatial region probed by deep inelastic scattering (DIS) and the rest of the proton. For very small $x$, the proton becomes a maximally entangled state. This approach leads to a simple relation $S = \ln N $ between the average number $N$ of color-singlet dipoles in the proton wave function and the entropy of the produced hadronic state $S$. At small $x$, the multiplicity of dipoles is given by the gluon structure function, $N = x G(x,Q^2)$. Recently, the H1 Collaboration analyzed the entropy of the produced hadronic state in DIS, and studied its relation to the gluon structure function; poor agreement with the predicted relation was found. In this letter we argue that a more accurate account of the number of color-singlet dipoles in the kinematics of H1 experiment (where hadrons are detected in the current fragmentation region) is given not by $xG(x,Q^2)$ but by the sea quark structure function $x\Sigma(x,Q^2)$. Sea quarks originate from the splitting of gluons, so at small $x$ $x\Sigma(x,Q^2)\,\sim\, xG(x,Q^2)$, but in the current fragmentation region this proportionality is distorted by the contribution of the quark-antiquark pair produced by the virtual photon splitting. In addition, the multiplicity of color-singlet dipoles in the current fragmentation region is quite small, and one needs to include $\sim 1/N$ corrections to $S= \ln N$ asymptotic formula. Taking both of these modifications into account, we find that the data from the H1 Collaboration in fact agree well with the prediction based on entanglement.

QCD traveling waves phenomenology revisited

In this paper we review and update the Amaral-Gay Ducati-Betemps-Soyez saturation model, by testing it against the recent H1-ZEUS combined data on deep inelastic scattering, including heavy quarks in the dipole amplitude. We obtain that this model, which is based on traveling wave solutions of the Balitsky-Kovchegov equation and built in the momentum space framework, yields very accurate descriptions of the reduced cross section, $\sigma_{r}(x,y,Q^{2})$, as well as DIS structure functions such as $F_{2}(x,Q^{2})$ and $F_{L}(x,Q^{2})$, all measured at HERA. Additionally, it provides good descriptions of heavy quark structure functions, $F_{2}^{cc}$ and $F_{2}^{bb}$ at small-$x$ and $Q^{2}\lesssim 60$ GeV$^{2}$. We also use the improved model to make predictions for structure functions to be measured in the near future at LHeC.

Probing proton structure at the Large Hadron electron Collider

For the foreseeable future, the exploration of the high-energy frontier will be the domain of the Large Hadron Collider (LHC). Of particular significance will be its high-luminosity upgrade (HL-LHC), which will operate until the mid-2030s. In this endeavour, for the full exploitation of the HL-LHC physics potential an improved understanding of the parton distribution functions (PDFs) of the proton is critical. The HL-LHC program would be uniquely complemented by the proposed Large Hadron electron Collider (LHeC), a high-energy lepton-proton and lepton-nucleus collider based at CERN. In this work, we build on our recent PDF projections for the HL-LHC to assess the constraining power of the LHeC measurements of inclusive and heavy quark structure functions. We find that the impact of the LHeC would be significant, reducing PDF uncertainties by up to an order of magnitude in comparison to state-of-the-art global fits. In comparison to the HL-LHC projections, the PDF constraints from the LHeC are in general more significant for small and intermediate values of the momentum fraction x. At higher values of x, the impact of the LHeC and HL-LHC data is expected to be of a comparable size, with the HL-LHC constraints being more competitive in some cases, and the LHeC ones in others. Our results illustrate the encouraging complementarity of the HL-LHC and the LHeC in terms of charting the quark and gluon structure of the proton.

Dipole model analysis of high precision HERA data

We analyse, within a dipole model, the inclusive DIS cross section data, obtained from the combination of the H1 and ZEUS HERA measurements. We show that these high precision data are very well described within the dipole model framework, which is complemented with a valence quark structure functions. We discuss the properties of the gluon density obtained in this way.

General mass scheme for jet production in DIS

We propose a method for calculating DIS jet production cross sections in QCD at NLO accuracy with consistent treatment of heavy quarks. The scheme relies on the dipole subtraction method for jets, which we extend to all possible initial state splittings with heavy partons, so that the Aivazis-Collins-Olness-Tung massive collinear factorization scheme can be applied. As a first check of the formalism we recover the Aivazis-Collins-Olness-Tung result for the heavy quark structure function using a dedicated Monte Carlo program.

SPIN-DEPENDENT STRUCTURE FUNCTION OF 3He and 3H

We present the results of our QCD analysis for quark distributions and structure functions for polarized deep inelastic scattering of leptons on nucleons and nuclei using the Jacobi polynomials up to NLO approximation. Finally by having proton and also neutron structure functions, we have enough motivation to extract 3 He and 3 H polarized structure functions.

Analytical approach for the approximate solution of the longitudinal structure function with respect to the GLR-MQ equation at small x

We show that the nonlinear corrections to the longitudinal structure function can be tamed the singularity behavior at low x values, with respect to GLR-MQ equations. This approach can determined the shadowing longitudinal structure function based on the shadowing corrections to the gluon and singlet quark structure functions. Comparing our results with HERA data show that at very low x this behavior completely tamed by these corrections.

Heavy quarks in deep-inelastic scattering

We discuss a general framework for the inclusion of heavy quark mass contributions to deep-inelastic structure functions and their perturbative matching to structure functions computed in variable-mass schemes. Our approach is based on the so-called FONLL method, previously introduced and applied to heavy quark hadroproduction and photoproduction. We define our framework, provide expressions up to second order in the strong coupling, and use them to construct matched expressions for structure functions up to NNLO. After checking explicitly the consistency of our results, we perform a study of the phenomenological impact of heavy quark terms, and compare results obtained at various perturbative orders, and with various prescriptions for the treatment of subleading terms, specifically those related to threshold behaviour. We also consider the heavy quark structure function F 2 c and discuss issues related to the presence of mass singularities in their coefficient functions.

Nonsinglet parton distribution functions from the precise next-to-next-to-next-to leading order QCD fit

We present the results of our QCD analysis for nonsinglet unpolarized quark distributions and structure function ${F}_{2}(x,{Q}^{2})$ up to next-to-next-to-next-to leading order (${\mathrm{N}}^{3}\mathrm{LO}$). In this regards 4-loop anomalous dimension can be obtained from the Pad\'e approximations. The analysis is based on the Jacobi polynomials expansion of the structure function. New parameterizations are derived for the nonsinglet quark distributions for the kinematic wide range of $x$ and ${Q}^{2}$. Our calculations for nonsinglet unpolarized quark distribution functions up to ${\mathrm{N}}^{3}\mathrm{LO}$ are in good agreement with available theoretical models. The higher twist contributions of ${F}_{2}^{p,d}(x,{Q}^{2})$ are extracted in the large $x$ region in ${\mathrm{N}}^{3}\mathrm{LO}$ analysis. The values of ${\ensuremath{\Lambda}}_{\mathrm{QCD}}$ and ${\ensuremath{\alpha}}_{s}({M}_{z}^{2})$ are determined.

Polarized EMC effect in the Thermodynamical Bag Model

We determine the polarized quark distributions and structure functions for nuclear media such as Li7 and Al27 by using a phenomenological model known as Thermodynamical Bag Model. The evaluation of nuclear medium modifications to single nucleon structure function discusses the predictions of the polarized EMC effect. The deviation of polarized EMC from unpolarized case shows quenching of polarized quark distributions and proves the significance of this study as adding the spin observables will explore more about medium modification of nuclear structure and the nature of the strong interaction.

Analytic treatment of leading-order parton evolution equations: Theory and tests

We recently derived an explicit expression for the gluon distribution function G(x, Q^2) = xg(x, Q^2) in terms of the proton structure function F_2^{\gamma p} (x, Q^2) in leading-order (LO) QCD by solving the the LO DGLAP equation for the Q^2 evolution of F_2^{\gamma p} (x, Q^2) analytically, using a differential-equation method. We re-derive and extend the results here using a Laplace-transform technique, and show that the singlet quark structure function F_S(x,Q^2) can be determined directly in terms of G from the DGLAP gluon evolution equation. To illustrate the method and check the consistency of existing LO quark and gluon distributions, we used the published values of the LO quark distributions from the CTEQ5L and MRST2001LO analyses to form F_2^{\gamma p} (x, Q^2), and then solved analytically for G(x,Q^2). We find that the analytic and fitted gluon distributions from MRST2001LO agree well with each other for all x and Q^2, while those from CTEQ5L differ significantly from each other for large x values, x>~0.03 - 0.05 at all Q^2. We conclude that the published CTEQ5L distributions are incompatible in this region. Using a non-singlet evolution equation, we obtain a sensitive test of quark distributions which holds in both LO and NLO perturbative QCD. We find in either case that the CTEQ5 quark distributions satisfy the tests numerically for small x, but fail the tests for x>~0.03 - 0.05 - their use could potentially lead to significant shifts in predictions of quantities sensitive to large x. We encountered no problems with the MRST2001LO distributions or later CTEQ distributions. We suggest caution in the use of the CTEQ5 distributions.

3-Loops QCD analysis for non-singlet sector of [formula omitted

In this paper we present the results of our QCD analysis for non-singlet quark distributions and structure function F 2 ( x , Q 2 ) . New parameterizations are derived for the parton distribution functions for the kinematic wide range of x and Q 2 . The analysis is based on the Jacobi polynomials expansion of the structure function. Higher twist effects is take in to account in our QCD analysis also. Our calculations for non-singlet quark distribution functions based on the Jacobi polynomials method are in good agreement with the other theoretical models.

Next-to-next-to-leading order QCD contributions to the flavor nonsinglet sector ofF2(x,Q2)

We present the results of our QCD analysis for nonsinglet unpolarized quark distributions and structure function ${F}_{2}(x,{Q}^{2})$. New parameterizations are derived for the nonsinglet quark distributions for the kinematic wide range of $x$ and ${Q}^{2}$. The analysis is based on the Jacobi polynomials expansion of the structure function. The higher twist contributions of proton and deuteron structure function are obtained in the large $x$ region. Our calculations for nonsinglet unpolarized quark distribution functions based on the Jacobi polynomials method are in good agreement with the other theoretical models. The values of ${\ensuremath{\Lambda}}_{\mathrm{QCD}}$ and ${\ensuremath{\alpha}}_{s}({M}_{z}^{2})$ are determined.

Overview of nucleon structure studies

A brief overview of the recent activity in the measurement of the elastic electromagnetic proton and neutron form factors is presented. It is discussed how the quality of the data has been greatly improved by performing double polarization experiments, and the role of two-photon exchange processes will be highlighted. The spatial information on the quark charge distribibutions in the nucleon resulting from the form factors measurements will be discussed, as well as the steady rate of improvements made in the lattice QCD calculations. It is discussed how generalized parton distributions have emerged as a unifying theme in hadron physics linking the spatial densities extracted from form factors with the quark momentum distribution information residing in quark structure functions. The recent progress in the electromagnetic excitation of the Δ(1232) resonance will also briefly be discussed.

Polarized structure functions of nucleons and nuclei

We determine the quark distributions and structure functions for both unpolarized and polarized DIS of leptons on nucleons and nuclei. The scalar and vector mean fields in the nucleus modify the motions of the quarks inside the nucleons. By taking into account this medium modification, we are able to reproduce the experimental data on the unpolarized EMC effect, and to make predictions for the polarized EMC effect. We discuss examples of nuclei where the polarized EMC effect could be measured. We finally present an extension of our model to describe fragmentation functions.

The Jacobi polynomials QCD analysis for the polarized structure function

We present the results of our QCD analysis for polarized quark distribution and structure function $xg_1 (x,Q^2)$. We use very recently experimental data to parameterize our model. New parameterizations are derived for the quark and gluon distributions for the kinematic range $x \epsilon [10^{-8},1]$, $Q^2 \epsilon [1,10^6]$ GeV^2. The analysis is based on the Jacobi polynomials expansion of the polarized structure functions. Our calculations for polarized parton distribution functions based on the Jacobi polynomials method are in good agreement with the other theoretical models. The values of $\Lambda_{QCD}$ and $\alpha_s(M_z)$ are determined.