Nucleon Structure Research Articles

Recent Highlights from Spin-Physics Experiments

A snapshot of the current experimental status of spin-dependent nucleon structure prior to the arrival of the Electron Ion Collider will be attempted. Highlights of recent results from fixed-target experiments at JLab, Fermilab, CERN and DESY and collider experiments from RHIC are presented. Many of the recent results address the topics of transverse proton or parton spin and transverse parton momenta (TMDs), and their their (spin-orbit) correlations. Results on generalized parton distributions (GPDs), which provide an alternative pathway to mapping nucleon structure, are also discussed.

Statistical correlations of nuclear quadrupole deformations and charge radii

Shape deformations and charge radii, basic properties of atomic nuclei, are influenced by both the global features of the nuclear force and the nucleonic shell structure. As functions of proton and neutron number, both quantities show regular patterns and, for nuclei away from magic numbers, they change very smoothly from nucleus to nucleus. In this paper, we explain how the local shell effects are impacting the statistical correlations between quadrupole deformations and charge radii in well-deformed even-even Er, Yb, and Hf isotopes. This implies, in turn, that sudden changes in correlations can be useful indicators of underlying shell effects. Our theoretical analysis is performed in the framework of self-consistent mean-field theory using quantified energy density functionals and density-dependent pairing forces. The statistical analysis is carried out by means of the linear least-square regression. The local variations of nuclear quadrupole deformations and charge radii, explained in terms of occupations individual deformed Hartree-Fock orbits, make and imprint on statistical correlations of computed observables. While the calculated deformations or charge radii are, in some cases, correlated with those of their even-even neighbors, the correlations seem to deteriorate rapidly with particle number. The statistical correlations between nuclear deformations and charge radii of different nuclei are affected by the underlying shell structure. Even for well deformed and superfluid nuclei for which these observables change smoothly, the correlation range usually does not exceed $\Delta N=4$ and $\Delta Z=4$, i.e., it is rather short. This result suggests that the frequently made assumption of reduced statistical errors for the differences between smoothly-varying observables cannot be generally justified.

Quantum field theory for spin operator of the photon

All elementary particles in nature can be classified as fermions with half-integer spin and bosons with integer spin. Within quantum electrodynamics (QED), even though the spin of the Dirac particle is well defined, there exist open questions on the quantized description of spin of the gauge field particle -- the photon. Using quantum field theory, we discover the quantum operators for the spin angular momentum (SAM) $\boldsymbol{S}_{M}=(1/c)\int d^{3}x\boldsymbol{\pi}\times\boldsymbol{A}$ and orbital angular momentum (OAM) $\boldsymbol{L}_{M}=-(1/c)\int d^{3}x\pi^{\mu}\boldsymbol{x}\times\boldsymbol{\nabla}A_{\mu}$ of the photon, where $\pi^{\mu}$ is the conjugate canonical momentum of the gauge field $A^{\mu}$. We also reveal a perfect symmetry between the angular momentum commutation relations for Dirac fields and Maxwell fields. We derive the well-known OAM and SAM of classical electromagnetic fields from the above defined quantum operators. Our work shows that the spin and OAM operators commute which is important for simultaneously observing and separating the SAM and OAM. The correct commutation relations of orbital and spin angular momentum of the photon have applications in quantum optics, topological photonics as well as nanophotonics and can be extended in the future for the spin structure of nucleons.

Systematic analysis of the proton mass radius based on photoproduction of vector charmoniums

In this work, the cross-section of the reaction $\gamma p \rightarrow V(J/\psi,\psi(2S)) p$ from the production threshold to medium energy is studied and systematically analyzed within two gluon exchange model. The obtained numerical results are in agreement with experimental data and other theoretical predictions. Under the assumption of the scalar form factor of dipole form, the value of proton mass radius is calculated as $0.55\pm 0.09$ fm and $ 0.77\pm 0.12$ fm from the fit to the predicted $J/\psi $ and $\psi(2S)$ differential cross-section, respectively. Finally the average value of proton mass radius is estimated to be $\sqrt{\left<R^2_m \right>}=0.67\pm 0.11$ fm. Moreover, one find that extracting mass radius from the near-threshold differential cross-section of heavy quarkoniums is always affected by large $|t|_{min}$. These obtained results may provide important theoretical reference for understanding of nucleon structure and future relevant experiments.

Nuclear medium effects in lepton-nucleus DIS in the region of x≳1

The nuclear medium effects in the nuclear structure functions and differential cross sections in the deep inelastic scattering (DIS) of charged lepton and neutrino from nuclear targets are studied in the region of large $x$ including $x\ge 1$. The nuclear medium effects due to the Fermi motion and the binding energy of nucleons and the nucleon correlations are included using nucleon spectral function calculated in a microscopic field theoretical model. The numerical results for the nuclear structure functions and the cross sections are obtained using the nucleon structure function evaluated at the next-to-next-to-leading order (NNLO) with the Martin-Motylinski-Harland Lang-Thorne (MMHT) parameterization of the nucleonic parton distribution functions (PDFs) and are compared with the available experimental data on electron scattering from the Jefferson Lab (JLab) and SLAC Nuclear Physics Facility (NPAS). In the case of neutrino scattering the results are relevant for understanding the DIS contributions to the recent inclusive cross sections measured by the Main Injector Neutrino ExpeRiment to study v-A interactions (MINERvA) as well as theoretical predictions are made for Deep Underground Neutrino Experiment (DUNE). The importance of isoscalarity corrections in heavier nuclear targets as well as the effect of the kinematic cut on the CM energy $W$ in defining the DIS region have also been discussed.

Extending Precision Perturbative QCD with Track Functions

Collider experiments often exploit information about the quantum numbers of final state hadrons to maximize their sensitivity, with applications ranging from the use of tracking information (electric charge) for precision jet substructure measurements, to flavor tagging for nucleon structure studies. For such measurements, perturbative calculations in terms of quarks and gluons are insufficient, and nonperturbative track functions describing the energy fraction of a quark or gluon converted into a subset of hadrons (e.g., charged hadrons) must be incorporated. Unlike fragmentation functions, track functions describe correlations between hadrons and therefore satisfy complicated nonlinear evolution equations whose structure has so far eluded calculation beyond the leading order. In this Letter, we develop an understanding of track functions and their interplay with energy flow observables beyond the leading order, allowing them to be used in state-of-the-art perturbative calculations for the first time. We identify a shift symmetry in the evolution of their moments that fixes their structure, and we explicitly compute the evolution of the first three moments at next-to-leading order, allowing for the description of up to three-point energy correlations. We then calculate the two-point energy correlator on charged particles at O(α_{s}^{2}), illustrating explicitly that infrared singularities in perturbation theory are absorbed by moments of the track functions and also highlighting how these moments seamlessly interplay with modern techniques for perturbative calculations. Our results extend the boundaries of traditional perturbative QCD, enabling precision perturbative predictions for energy flow observables sensitive to the quantum numbers of hadronic states.

Residual mean field model of valence quarks in the nucleon

We develop a non-perturbative model for valence parton distribution functions (PDFs) based on the mean field interactions of valence quarks in the nucleonic interior. The main motivation for the model is to obtain a mean field description of the valence quarks as a baseline to study the short range quark–quark interactions that generate the high x tail of PDFs. The model is based on the separation of the valence three-quark cluster and residual system in the nucleon. Then the nucleon structure function is calculated within the effective light-front diagrammatic approach introducing nonperturbative light-front valence quark and residual wave functions. Within the model a new relation is obtained between the position, x_p, of the peak of xq_V(x) distribution of the valence quark and the effective mass of the residual system, m_R, in the form: x_{p} approx {1over 4} (1-{m_Rover m_N}) at starting Q^2. This relation explains the difference in the peak positions for d- and u-quarks through the expected difference of residual masses for valence d- and u-quark distributions. The parameters of the model are fixed by fitting the calculated valence quark distributions to the phenomenological PDFs. This allowed us to estimate the overall mean field contribution in baryonic and momentum sum rules for valence d- and u-quarks. Finally, the evaluated parameters of the non-perturbative wave functions of valence 3q-cluster and residual system can be used in calculation of other quantities such as nucleon form factors, generalized partonic and transverse momentum distributions.

Parametrization of quark and gluon generalized parton distributions in a dynamical framework

We present a parametrization of the chiral even generalized parton distributions, $H$, $E$, $\stackrel{\texttildelow{}}{H}$, $\stackrel{\texttildelow{}}{E}$, for the quark, antiquark, and gluon, in the perturbative QCD (pQCD)-parton framework. Parametric analytic forms are given as a function of two equivalent sets of variables $x$, $\ensuremath{\xi}$, $t$ (symmetric frame) and $X$, $\ensuremath{\zeta}$, $t$ (asymmetric frame), at an initial scale, ${Q}_{o}^{2}$. In the $X&gt;\ensuremath{\zeta}$ region, a convenient and flexible form is obtained as the product of a Regge term $\ensuremath{\propto}{X}^{\ensuremath{-}\ensuremath{\alpha}+{\ensuremath{\alpha}}^{\ensuremath{'}}t}$, describing the low $X$ behavior, times a spectator model-based functional form depending on various mass parameters; the behavior at $X&lt;\ensuremath{\zeta}$ is determined using the generalized parton distributions symmetry and polynomiality properties. The parameters are constrained using data on the flavor separated nucleon electromagnetic elastic form factors, the axial and pseudoscalar nucleon form factors, and the parton distribution functions from both the deep inelastic unpolarized and polarized nucleon structure functions. For the gluon distributions we use, in particular, constraints provided by recent lattice QCD moments calculations. The parametrization's kinematical range of validity is $0.0001\ensuremath{\le}X\ensuremath{\le}\phantom{\rule{0ex}{0ex}}0.85$, $0.01\ensuremath{\le}\ensuremath{\zeta}\ensuremath{\le}0.85$, $0\ensuremath{\le}\ensuremath{-}t\ensuremath{\le}1\text{ }\text{ }{\mathrm{GeV}}^{2}$, $2\ensuremath{\le}{Q}^{2}\ensuremath{\le}100\text{ }\text{ }{\mathrm{GeV}}^{2}$. With the simultaneous description of the quark, antiquark, and gluon sectors, this parametrization represents a first tool enabling a global QCD analysis of deeply virtual exclusive experiments.

Influence of nucleon structure on spin correlation coefficients in elastic lepton-deuteron scattering

Spin correlation coefficients in elastic scattering of leptons on the deuteron are studied in the one-photon-exchange Born approximation within the limit of zero lepton mass. Numerical estimations for the spin correlation coefficients caused by a polarized beam and a vector polarized target are presented and analyzed at various beam energies and scattering angles in two coordinate systems which are relevant for experiment. The influence of the obtained results on the nucleon structure is investigated, and a considerable dependence at incident beam energies greater than 1[Formula: see text]fm[Formula: see text] and backward scattering angles is found. The estimated results are of particular interest for encouragement of experimental measurements of lepton-deuteron scattering which aim for accurate calculation of the charge radius of the proton to a high degree of precision. These results may also stimulate new generation of measurements of different physical observables very sensitive to the spin structure of the nucleon and light nuclei.

Nuclear medium effects in the deep inelastic ντ/ν¯τ−Ar40 scattering at DUNE energies

The nuclear medium effects are studied in the $\nu_\tau/\bar\nu_\tau$ interactions from nuclei in the deep inelastic scattering (DIS) region and applied to the $^{40}Ar$ nucleus to obtain the scattering cross sections in the energy region of the proposed DUNE experiment. The free nucleon structure functions ($F_{iN}(x,Q^2);~(i=1-5)$) have been calculated at the next-to-leading order (NLO) using Martin-Motylinski-Harland Lang-Thorne 2014 as well as The Coordinated Theoretical-Experimental Project on QCD parameterizations for parton distribution functions (PDFs) and including the effect of perturbative and nonperturbative QCD corrections \cite{Ansari:2020xne}. These free nucleon structure functions are then convoluted with the nucleon spectral function in the nucleus to obtain the nuclear structure functions ($F_{iA}(x,Q^2);~(i=1-5)$). The nucleon spectral function takes into account the Fermi motion and the binding energy of the nucleons as well as the nucleon correlations within the nucleus. These nuclear structure functions are then used to calculate the deep inelastic scattering cross sections. Moreover, the contribution of $\pi$ and $\rho$ mesons as well as the corrections due to the shadowing and antishadowing effects in the relevant kinematic region of the Bjorken variable $x$ are also included. The numerical results for the cross sections have been presented and compared with the results obtained in the phenomenological approach using nuclear structure functions from nCTEQnu.

Structure of Nucleons and Their Interaction in the Concept of Nonperturbative QCD as a Pressing Issue of 21st-Century Physics

Structure of Nucleons and Their Interaction in the Concept of Nonperturbative QCD as a Pressing Issue of 21st-Century Physics

Influence of Nucleon Structure on Tensor Spin Asymmetries in Elastic Lepton–Deuteron Scattering

Influence of Nucleon Structure on Tensor Spin Asymmetries in Elastic Lepton–Deuteron Scattering

Transverse momentum and transverse momentum distributions in the MIT bag model

The typical transverse momentum of a quark in the proton is a basic property of any QCD based model of nucleon structure. However, calculations in phenomenological models typically give rather small values of transverse momenta, which are difficult to reconcile with the larger values observed in high energy experiments such as Drell-Yan reactions and Semi-inclusive deep inelastic scattering. In this letter we calculate the leading twist transverse momentum dependent distribution functions (TMDs) using a generalization of the Adelaide group's relativistic formalism that has previously given good fits to the parton distributions. This enables us to examine the kT dependence of the TMDs in detail, and determine typical widths of these distributions. These are found to be significantly larger than those of previous calculations. We then use TMD factorization in order to evolve these distributions up to experimental scales where we can compare with data on 〈kT〉 and 〈kT2〉. Our distributions agree well with this data.

Time-like Proton Form Factors with Initial State Radiation Technique

Electromagnetic form factors are fundamental quantities describing the internal structure of hadrons. They can be measured with scattering processes in the space-like region and annihilation processes in the time-like region. The two regions are connected by crossing symmetry. The measurements of the proton electromagnetic form factors in the time-like region using the initial state radiation technique are reviewed. Recent experimental studies have shown that initial state radiation processes at high luminosity electron-positron colliders can be effectively used to probe the electromagnetic structure of hadrons. The BABAR experiment at the B-factory PEP-II in Stanford and the BESIII experiment at BEPCII (an electron positron collider in the τ-charm mass region) in Beijing have measured the time-like form factors of the proton using the initial state radiation process e+e−→pp¯γ. The two kinematical regions where the photon is emitted from the initial state at small and large polar angles have been investigated. In the first case, the photon is in the region not covered by the detector acceptance and is not detected. The Born cross section and the proton effective form factor have been measured over a wide and continuous range of the the momentum transfer squared q2 from the threshold up to 42 (GeV/c)2. The ratio of electric and magnetic form factors of the proton has been also determined. In this report, the theoretical aspect and the experimental studies of the initial state radiation process e+e−→pp¯γ are described. The measurements of the Born cross section and the proton form factors obtained in these analyses near the threshold region and in the relatively large q2 region are examined. The experimental results are compared to the predictions from theory and models. Their impact on our understanding of the nucleon structure is discussed.

Probing Nucleon Spin Structure in Deep-inelastic Scattering, Proton–Proton Collisions, and Drell–Yan Processes

A pedagogical summary of current and past experimental results of spin-dependent nucleon structure prior to the arrival of the Electron-Ion Collider is attempted. After an introduction, results from fixed-target experiments at SLAC, Fermilab, Jefferson Lab, CERN and DESY and collider experiments from RHIC are presented, starting with the longitudinal spin structure of the nucleon, followed by generalized parton distributions (GPDs), which map the proton in transverse position space. The final part discusses transverse proton or parton spin and transverse parton momenta (TMDs), and their their (spin-orbit) correlations, which are addressed by a multitude of recent experimental results. The GPDs and TMDs provide complementary pathways to mapping multi-dimensional nucleon structure.

Nucleon structure from basis light-front quantization

We produce the light-front wave functions (LFWFs) of the nucleon from a basis light-front approach in the leading Fock-sector representation. We solve for the mass eigenstates from a light-front effective Hamiltonian, which includes a confining potential adopted from light-front holography in the transverse direction, a longitudinal confinement, and a one-gluon exchange interaction with fixed coupling. We then employ the LFWFs to obtain the electromagnetic and axial form factors, the parton distribution functions (PDFs), and the generalized parton distribution functions for the nucleon. The electromagnetic and axial form factors of the proton agree with the experimental data, whereas the neutron form factors deviate somewhat from the experiments in the low-momentum transfer region. The unpolarized, the helicity, and the transversity valence quark PDFs, after QCD scale evolution, are fairly consistent with the global fits to the data at the relevant experimental scales. The helicity asymmetry for the down quark also agrees well with the measurements; however, the asymmetry for the up quark shows a deviation from the data, especially in the small $x$ region. We also find that the tensor charge agrees well with the extracted data and the lattice QCD predictions, while the axial charge is somewhat outside the experimental error bar. The electromagnetic radii of the protons, the magnetic radius of the neutron, and the axial radius are in excellent agreement with the measurements, while the neutron charge radius deviates from the experiment.

From Gargamelle to MINERvA: exploring the structure of the nucleon with neutrinos

Neutrino scattering experiments have been exploring the structure of the nucleon with Deep-Inelastic Scattering for over 50 years. Although hints of nucleon structure were available using CERN’s first neutrino beam in the 1960s, the study actually started quantitatively in the early 1970s with the Gargamelle heavy liquid bubble chamber that produced first neutrino confirmation of scaling. This study of nucleon structure continued with both bubble chambers and more massive electronic detectors that, with higher energy neutrino beams and \(Q^2\) reach, examined the breaking of this scaling while testing Quantum Chromodynamics. A significant factor when including neutrino explorations of nucleon structure in the overall picture is that to gather significant statistics, neutrino experiments have had to use heavier nuclear targets. It has been experimentally demonstrated that the structure of the nucleon in the nuclear environment is indeed different than the free nucleon structure. Understanding this modified structure has played an important part in the experimental and theoretical exploration of nucleon structure by neutrinos and is one of the important goals of the on-going MINERvA experiment at Fermilab.

Soft-photon radiative corrections to the e−p→e−pl−l+ process

We calculate the leading-order QED radiative corrections to the process $e^- p\rightarrow e^- p l^- l^+ $ in the soft-photon approximation, in two different energy regimes which are of relevance to extract nucleon structure information. In the low-energy region, this process is studied to better constrain the hadronic corrections to precision muonic Hydrogen spectroscopy. In the high-energy region, the beam-spin asymmetry for double virtual Compton scattering allows to directly access the Generalized Parton Distributions. We find that the soft-photon radiative corrections have a large impact on the cross sections and are therefore of paramount importance to extract the nucleon structure information from this process. For the forward-backward asymmetry the radiative corrections are found to affect the asymmetry only around or below the 1\% level, whereas the beam-spin asymmetry is not affected at all in the soft-photon approximation, which makes them gold-plated observables to extract nucleon structure information in both the low- and high-energy regimes.

Neutron spin structure from e-3He scattering with double spectator tagging at the electron-ion collider

The spin structure function of the neutron is traditionally determined by measuring the spin asymmetry of inclusive electron deep inelastic scattering (DIS) off polarized 3He nuclei. In such experiments, nuclear corrections are significant and must be treated carefully in the interpretation of experimental data. Here we study the feasibility of suppressing model dependencies by tagging both spectator protons in the process of DIS off neutrons in 3He at the forthcoming Electron-Ion Collider (EIC). This allows for a reconstruction of the momentum of the struck neutron to ensure it was nearly at rest in the initial state, thereby reducing sensitivity to nuclear corrections and suppressing contributions from electron DIS off protons in 3He. Using realistic accelerator and detector configurations, we demonstrate that the EIC can probe the neutron spin structure from xB of 0.003 to 0.651. We find that the double spectator tagging method results in reduced uncertainties by a factor of 2 on the extracted neutron spin asymmetries over all kinematics and by a factor of 10 in the low-xB region, thereby providing valuable insight into the spin and flavor structure of nucleons.

Analytical perturbation theory and nucleon structure function in infrared region

We employ the analytic QCD (anQCD) approach to analyze the unpolarized nucleon structure function in deep inelastic scattering processes at next-to-leading-order accuracy. Considering the unreliable results of the underlying perturbative QCD (pQCD) at energy scales near to QCD cut off parameter ${Q}^{2}\ensuremath{\sim}{\mathrm{\ensuremath{\Lambda}}}^{2}$ and below, we modify the calculations at these scales using the anQCD approach and compare them with results from underlying pQCD and the available experimental data. The massive perturbation theory model is also used where an effective mass is attributed to gluons. Finally, we use the Jacobi polynomials formalism to transfer the calculations from Mellin moment space to Bjorken-$x$ space. To confirm the validity of the anQCD approach the Gottfried sum rule is also investigated. The achieved numerical results at low energy scales are compatible with what is expected and correspond to an admissible behavior of parton densities.