Fermi Motion Effects Research Articles

Nuclear Parton Distribution Functions After the First Decade of LHC Data

We present a review of the conceptual basis, current knowledge, and recent progress regarding global analysis of nuclear parton distribution functions (PDFs). After introducing the theoretical foundations and methodological approaches for the extraction of nuclear PDFs from experimental data, we discuss how different measurements in fixed-target and collider experiments provide increasingly precise constraints on various aspects of nuclear PDFs, including shadowing, antishadowing, the EMC effect, Fermi motion, flavor separation, deuteron binding, and target-mass and other higher-twist effects. Particular emphasis is given to measurements carried out in proton–lead collisions at the Large Hadron Collider, which have revolutionized the global analysis during the past decade. These measurements include electroweak boson, jet, light hadron, and heavy flavor observables. Finally, we outline the expected impact of the future Electron Ion Collider and discuss the role and interplay of nuclear PDFs with other branches of nuclear, particle, and astroparticle physics.

Fermi motion effects in electroproduction of hypernuclei

In a previous analysis of electroproduction of hypernuclei the cross sections were calculated in distorted-wave impulse approximation where the momentum of the initial proton in the nucleus was set to zero (the frozen-proton approximation). In this paper we go beyond this approximation assuming a non zero effective proton momentum due to proton Fermi motion inside of the target nucleus discussing also other kinematical effects. To this end we have derived a more general form of the two-component elementary electroproduction amplitude (Chew-Goldberger-Low-Nambu like) which allows its use in a general reference frame moving with respect to the nucleus-rest frame. The effects of Fermi motion were found to depend on kinematics and elementary amplitudes. The largest effects were observed in the contributions from the longitudinal and interference parts of the cross sections. The extension of the calculations beyond the frozen-proton approximation improved the agreement of predicted theoretical cross sections with experimental data and once we assumed the optimum on-shell approximation, we were able to remove an inconsistency which was previously present in the calculations.

Fermi motion and medium effects on reaction cross sections of 13,14,15B + C, Al at intermediate and high energies

Analysis of reaction cross sections of [Formula: see text] are performed with Glauber model (GM) including medium modifications arising from Fermi motion and in-medium nucleon–nucleon (NN) total reaction cross section. The density distributions of Boron isotopes are described by a deformed-average Fermi shape, whose proton and neutron radii and deformation parameters are derived from relativistic mean field (RMF), harmonic oscillator (HO) and two-parameter Fermi (2pF). These densities can describe the reaction cross section of Boron isotopes except for [Formula: see text] which is better described by an HO core plus extended Gaussian (HOG) or HO plus a modified Yukawa (HOMY) tail, which depends on the 1n separation energy. It is found that Fermi motion effects dominated at intermediate energy, and they are important to describe the data in addition to in-medium effects. In-medium effects, which are incorporated locally, reduced the reaction cross section by about 3–5% at all energies. [Formula: see text] is confirmed as a one-neutron halo nucleus, where a large extended tail is observed in the extracted HOMY and HOG model densities as well as in the extracted radii, where the matter radius of [Formula: see text] is found to be about 2.92 fm, which is larger than that of [Formula: see text] (2.5 fm) and [Formula: see text] (2.82 fm) predicted using the GM model including Fermi motion and in-medium effects.

Comparative study of proton and positive kaon interactions with carbon: hint of new physics like partial restoration of chiral symmetry in nuclei

Concerning the scattering of K+ - mesons on nuclei, the explicit discrepancy between experimental data and optical model calculations is a longstanding problem and none of the conventional medium effect corrections improves significantly the agreement with the data. In this paper the cross sections for the reactions K++12C at incident momenta of the kaon Plab < 800 MeV/c are calculated on the basis of Glauber’s high-energy approximation. The noneikonal corrections, Fermi motion effect and both Pauli and dynamic short-range correlations were incorporated in the Glauber approach for nuclear collisions. The influence of all these corrections on the calculation of the cross sections for the interactions of kaon with nucleus is discussed. The full calculations is compared to available data. The corrections seem to fail in reproducing the experimental data. Our results is compared to optical model calculations. We show that both our model and optical model fails to give a satisfactory description of K+-nuclei cross sections. Different attempts to remove this discrepancy with the in-medium effects taken into account are discussed.

Fragmentation production of ground state spin-1/2 heavy baryons including the Fermi motion of constituents

We analytically compute the process-independent fragmentation functions of heavy quarks to fragment into ground state triply heavy baryons considering the Fermi motion effect of their constituent quarks. The effect of Fermi motion is basically included in the baryon wave functions so in all previous studies, a delta function form was approximated for the baryon wave functions which corresponds to ignoring this effect. Here, we study for the first time the effect of nonperturbative dynamics of baryon bound state considering a typical wave function which is different from the delta function and is the nonrelativistic limit of the solution of Bethe-Salpeter equation with the QCD kernel. Our analytical results will be presented for the fragmentation processes $b\rightarrow\Omega_{bbc}$ and $c\rightarrow\Omega_{ccb}$ through a scalar bc diquark intermediated.

Direct fragmentation of $ \eta_{c}^{}$ η c (1S) and $ \eta_{b}^{}$ η b (1S) considering the transverse momentum effect of constituents

In this work, we calculate the fragmentation function (FF) of pseudoscalar S -wave heavy quarkonia from gluon splitting. We also study, for the first time, the Fermi motion effect of constituent quarks on the fragmentation production of the $ \eta_c(1S)$ and $ \eta_b(1S)$ states. In all previous studies of heavy quarkonium FFs, by ignoring the Fermi motion of constituents a Dirac-delta functional form was approximated for the wave function of bound state. Here, to study the Fermi motion of quarkonium constituents we consider a typical wave function which is different from the delta form and is the nonrelativistic limit of the solution of the Bethe-Salpeter equation with the QCD kernel. We shall present our results for the fragmentation of $ g\rightarrow (\eta_c, \eta_b)$ and show how the Fermi motion effect changes the results.

A study of the nucleus-nucleus total reaction cross section of stable systems at intermediate energies: An application to 12C

A semi-microscopic analytical expression of the nucleus-nucleus total reaction cross section (σR) was proposed based on the strong absorption model. It is suitable for stable nuclei at intermediate energies. The matter density distributions of nuclei and the nucleon-nucleon total cross section were both considered. Particularly, the Fermi motion effect of the nucleons in a nucleus was also taken into account. The parametrization of σR was applied to the colliding systems including 12C. The experimental data at energies from 30 to 1000 MeV/nucleon were well reproduced, according to which an approach of deriving σR without adjustable parameters was developed. The necessity of considering the Fermi motion effect in the parametrization was discussed.

Global study of nuclear modifications on parton distribution functions

A global analysis of nuclear medium modifications of parton distributions is presented using deeply inelastic scattering data of various nuclear targets. Two obtained data sets are provided for quark and gluon nuclear modification factors, referred as nIMParton16. One is from the global fit only to the experimental data of isospin-scalar nuclei (Set A), and the other is from the fit to all the measured nuclear data (Set B). The scale-dependence is described by DGLAP equations with nonlinear corrections in this work. The Fermi motion and off-shell effect, nucleon swelling, and parton–parton recombination are taken into account together for modeling the complicated x-dependence of nuclear modification. The nuclear gluon shadowing in this paper is dynamically generated by the QCD evolution of parton splitting and recombination processes with zero gluon density at the input scale. Sophisticated nuclear dependence of nuclear medium effects is studied with only two free parameters. With the obtained free parameters from the global analysis, the nuclear modifications of parton distribution functions of unmeasured nuclei can be predicted in our model. Nuclear modification of deuteron is also predicted and shown with recent measurement at JLab.

Fermi-motion effect on the intermediate energy nucleus–nucleus collision

The Glauber model is modified with the Fermi-motion effect in the calculation of elastic differential cross-sections and momentum distributions of a fragment from mother nucleus. Different reaction systems at low energies are calculated with the modified Glauber model. It is found that calculations including the Fermi-motion provide a better prescription relating the model to a proper nuclear density distribution by comparing with the experimental data. On the basis of the studies, the influence of the correction on the extracted nuclear radius is quantified. The results further confirm the importance of the Fermi-motion in the nucleus–nucleus collision reactions at low energies.

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.

EXTRACTION OF RESONANCE PARAMETERS AND ROLE OF THE FINAL STATE INTERACTIONS

We present an overview of the SAID group effort to analyze new γn → π-p cross sections vs. the world database to get new multipoles and determine neutron electromagnetic couplings. The differential cross section for the processes γn → π-p was extracted from new measurements at CLAS and MAMI-B accounting for Fermi motion effects in the impulse approximation (IA) as well as NN- and πN-FSI effects beyond the IA. We evaluated results of several pion photoproduction analyses and compared πN PWA results as a constraint for analyses of pion photoproduction data (Watson's theorem).

SAID analysis of meson photoproduction: Determination of neutron and proton EM couplings

We present an overview of the GW SAID group effort to analyze on new pion photoproduction on both proton- and neutron-targets. The main database contribution came from the recent CLAS and MAMI unpolarized and polarized measurements. The differential cross section for the processes gamma n --> pi- p was extracted from new measurements accounting for Fermi motion effects in the impulse approximation (IA) as well as NN and piN effects beyond the IA. The EM coupling results are compared to other recent studies.

ν¯linduced pion production from nuclei at∼1 GeV

We have studied charged current ${\overline{\ensuremath{\nu}}}_{l}$ induced one pion production from $^{12}\mathrm{C}$ and $^{16}\mathrm{O}$ nuclear targets at MiniBooNE and atmospheric antineutrino energies. The calculations have been done for the incoherent pion production process as well as for the pions coming from the hyperons in the quasielastic production of $\ensuremath{\Lambda}$ and $\ensuremath{\Sigma}$. The calculations are done in the local density approximation. For the inelastic processes the calculations have been done in the $\ensuremath{\Delta}$ dominance model, and we take into account the effect of Pauli blocking, Fermi motion of the nucleon, and renormalization of $\ensuremath{\Delta}$ properties in the nuclear medium. The effect of final state interaction (FSI) of pions is also taken into account. For the hyperon production, the nuclear medium effects due to Fermi motion and FSI effects due to hyperon-nucleon scattering have been taken into account. These results may be quite useful in the analysis of SciBooNE, MicroBooNE, $\mathrm{MINER}\ensuremath{\nu}\mathrm{A}$, and ArgoNeuT experiments when the pion analysis is done by using antineutrino beams.

THE ROLE OF FERMI MOTION ON THE STRUCTURE FUNCTIONS OF 3He AND 3H NUCLEI IN THE QUARK EXCHANGE FRAMEWORK

The quark exchange model and the full three-nucleon wave function in the configuration space are used to evaluate the role of Fermi motion on the structure functions (SFs) of helium-3 and tritium nuclei. The three-nucleon wave function is obtained from the solution of the Faddeev equations with the Malfliet–Tjon-type potential, by using the three-dimensional approach as a function of the magnitudes of the Jacobi momenta vectors and the angle between them. In this calculation, the initial valence quarks inputs are taken from the GRV's (Glück, Reya and Vogt) fitting procedure and the next-to-leading order (NLO) QCD calculation on [Formula: see text], which give a very good fit to the available experimental data in the (x, Q2)-plane. The role of Fermi motion on the EMC ratio of the SFs of 3 He and 3 H nuclei are analyzed through the NLO expansion of the nuclear wave function in the coordinate space. A good agreement between the calculated EMC ratios, the corresponding experimental data and the theoretical results is found. Finally, the ratios of the SFs of the neutron to the proton (with the isospin symmetry assumption) with and without the Fermi motion effect, are also calculated, and they are compared with the available experimental data. Our results show that the roles of the Fermi motion in the framework of the quark exchange model for the calculations of the nuclear SFs are important.

Pion-exchange and Fermi-motion effects on the proton-deuteron Drell-Yan process

Within a nuclear model where the deuteron has $NN$ and $\ensuremath{\pi}NN$ components, we derive a convolution formula for investigating the Drell-Yan process in proton-deuteron ($pd$) reactions. The contribution from the $\ensuremath{\pi}NN$ component is expressed in terms of a pion momentum distribution that depends sensitively on the $\ensuremath{\pi}NN$ form factor. With a $\ensuremath{\pi}NN$ form factor determined by fitting the $\ensuremath{\pi}N$ scattering data up to invariant mass $W=1.3\text{ }\text{ }\mathrm{GeV}$, we find that the pion-exchange and nucleon Fermi-motion effects can change significantly the ratios between the proton-deuteron and proton-proton Drell-Yan cross sections, ${R}_{pd/pp}={\ensuremath{\sigma}}^{pd}/(2{\ensuremath{\sigma}}^{pp})$, in the region where the partons emitted from the target deuteron are in the Bjorken ${x}_{2}\ensuremath{\gtrsim}0.4$ region. The calculated ratios ${R}_{pd/pp}$ at 800 GeV agree with the available data. Predictions at 120 GeV for analyzing the forthcoming data from Fermilab are presented.

QCD AND QED DYNAMICS IN THE EMC EFFECT

Applying exact QCD sum rules for the baryon charge and energy–momentum conservation we demonstrate that if the only degrees of freedom in nuclei were nucleons, the structure function of a nucleus would be the additive sum of the nucleon distributions at the same Bjorken x = AQ2/2(pA⋅q)≤0.5 up to very small Fermi motion corrections if 1/2mN x is significantly less than the nucleus radius. Hence QCD implies that the proper quantity to reveal violation of the additivity due to presence of nonnucleonic degrees of freedom in nuclei is the ratio RA(x, Q2) = (2/A)F2A(x, Q2)/F2D(x, Q2). Use of variable xp = Q2/2q0mp in the experimental studies instead of x leads to the deviation of RA(xp, Q2) from one even if the nucleus would consist only of nucleons with small momenta. Implementation of QCD dynamics accounts in the case of the light nuclei for at least a half of the deviation of RA(xp, Q2) from one for x≤0.55. In the case of heavy nuclei account of the QCD dynamics and of light-cone momentum fraction carried by Fermi, Weizsacker, Williams equivalent photons are responsible for ≈ one half the deviation of RA(x, Q2) from one at x≤0.55. We argue that direct observation of large and predominantly nucleonic short-range correlations (SRCs) in nuclei impacts strongly on the understanding of the EMC effect for x≥0.6 posing a serious challenge for most of the proposed models of the EMC effect. The data are consistent with a scenario in which the hadronic EMC effect reflects suppression of rare quark–gluon configurations in nucleons belonging to SRC appears to be the only viable. The dynamic realization of this scenario is presented in which quantum fluctuations of the nucleon wave function with x≥0.5 parton have a weaker interaction with nearby nucleons, leading to suppression of such configurations in bound nucleons and to the significant suppression of nucleon Fermi motion effects at x≥0.55 giving a right magnitude of the EMC effect. Implications of discussed effects for the analyses of the neutron structure function and nuclear parton distributions are presented. The directions for the future studies and challenging questions are outlined.

Quasi-free photoproduction of η-mesons off the deuteron

Precise data for quasi-free photoproduction of $\eta$ mesons off the deuteron have been measured at the Bonn ELSA accelerator with the combined Crystal Barrel/TAPS detector for incident photon energies up to 2.5 GeV. The $\eta$-mesons have been detected in coincidence with recoil protons and neutrons. Possible nuclear effects like Fermi motion and re-scattering can be studied via a comparison of the quasi-free reaction off the bound proton to $\eta$-production off the free proton. No significant effects beyond the folding of the free cross section with the momentum distribution of the bound protons have been found. These Fermi motion effects can be removed by an analysis using the invariant mass of the $\eta$-nucleon pairs reconstructed from the final state four-momenta of the particles. The total cross section for quasi-free $\eta$-photoproduction off the neutron reveals even without correction for Fermi motion a pronounced bump-like structure around 1 GeV of incident photon energy, which is not observed for the proton. This structure is even narrower in the invariant mass spectrum of the $\eta$-neutron pairs. Position and width of the peak in the invariant mass spectrum are $W\approx 1665$ MeV and FWHM $\Gamma\approx 25$ MeV. The data are compared to the results of different models.

Photon parton distributions in nuclei and the EMC effect

Photons, as well as quarks and gluons, are constituents of the infinite momentum frame (IMF) wave function of an energetic particle. They are mostly equivalent photons whose amplitude follows from the Lorentz transformation of the particle rest frame Coulomb field into the IMF and from the conservation of the electromagnetic current. We evaluate in a model independent way the dominant photon contribution $\ensuremath{\propto}{\ensuremath{\alpha}}_{\mathrm{em}}({Z}^{2}/{A}^{4/3})\mathrm{ln}(1/{R}_{A}{m}_{N}x)$ to the nuclear structure functions as well as the term $\ensuremath{\propto}{\ensuremath{\alpha}}_{\mathrm{em}}Z/A$. In addition we show that the definition of $x$ consistent with the exact kinematics of $\mathit{eA}$ scattering (with exact sum rules) works in the same direction as the nucleus field of equivalent photons. Combined, these effects account for the bulk of the hadronic European Muon Collaboration (EMC) effect for $x\ensuremath{\leqslant}0.5$ where Fermi motion effects are small. In particular, for these $x$ the hadronic mechanism contribution to the EMC effect does not exceed $~3%$ for all nuclei. Also, the $A$ dependence of the hadronic mechanism of the EMC effect for $x&gt;0.5$ is significantly modified.

The Constituent Quark Exchange Model for the Bound State Nucleons

The aim of present article is to compare the different bound state parton distributions in the protons with those coming from the free protons experimental data and to investigate the effect of quark structure of protons on each other. So, the constituent quark model (CQM), in which the quarks are assumed to be the complex objects and was originally proposed by Altarelli et al. (ACMP), is used in the frame work of the quark exchange model (QEM) to calculate the parton distributions of bound protons. Unlike our previous works, the effect of sea quarks and gluons are included in the QEM. Our results are in agreement with those of Gluck et al. (GRV) when we ignore the binding and the Fermi motion effects for the constituent quarks. In this case, we get more sea-quarks and gluons with respect to the bound state constituents quarks calculation. It is also shown that the QEM, which is a realistic formalism gives better result with respect to the Isgur and Karl et al. model which is a field theoretical approach. The ratio of the structure functions of neutron to proton is also calculated with the assumption of isospin symmetry and it is compared with the available data and our previous works.

Reaction cross sections at intermediate energies and Fermi-motion effect

Precise measurements of the reaction cross sections (σR) for C12 on Be, C, and Al targets and Be11 on Be targets were performed in the energy range between 30 and 400 MeV/nucleon. The energy dependencies of σR for C12, Be11, and B8 were compared to Glauber-type calculations performed using reliable nucleon-density distributions that are supported by experimental data. It was found that the Glauber-type calculations, which include the higher-order multiple scattering effect, the finite-range effect, and the Fermi-motion effect of nucleons in the nucleus, seem to provide a good prescription relating σR at intermediate energies to a proper nucleon-density distribution. This method provides a powerful tool with which to study the nuclear surface structure through the σR measurements.Received 2 August 2008DOI:https://doi.org/10.1103/PhysRevC.79.061601©2009 American Physical Society