Elastic Nucleon Research Articles

Dip-Bump Structure in Proton’s Single Diffractive Dissociation at the Large Hadron Collider

By extending the dipole Pomeron (DP) model, successful in describing elastic nucleon–nucleon scattering, to proton single diffractive dissociation (SD), we predict a dip-bump structure in the squared four-momentum transfer (t) distribution of proton’s SD. Structures in the t distribution of single diffractive dissociation are predicted around t=−4GeV2 at LHC energies in the range of 3 GeV2≲|t|≲ 7 GeV2. Apart from the dependence on s (total energy squared) and t (squared momentum transfer), we predict also a dependence on missing masses. We include the minimum set of Regge trajectories, namely the Pomeron and the Odderon, indispensable at the LHC. Further generalization, e.g., by the inclusion of non-leading Regge trajectories, is straightforward. The present model contains two types of Regge trajectories: those connected with t-channel exchanges (the Pomeron, the Odderon, and non-leading (secondary) reggeons) appearing at small and moderate −t, where they are real and nearly linear, as well as direct-channel trajectories α(M2) related to missing masses. In this paper, we concentrate on structures in t neglecting (for the time being) resonances in M2.

Status of Lattice QCD Determination of Nucleon Form Factors and Their Relevance for the Few-GeV Neutrino Program

Calculations of neutrino–nucleus cross sections begin with the neutrino–nucleon interaction, making the latter critically important to flagship neutrino oscillation experiments despite limited measurements with poor statistics. Alternatively, lattice quantum chromodynamics (LQCD) can be used to determine these interactions from the Standard Model with quantifiable theoretical uncertainties. Recent LQCD results of gA are in excellent agreement with data, and results for the (quasi-)elastic nucleon form factors with full uncertainty budgets are expected within a few years. We review the status of the field and LQCD results for the nucleon axial form factor, FA( Q2), a major source of uncertainty in modeling sub-GeV neutrino–nucleon interactions. Results from different LQCD calculations are consistent but collectively disagree with existing models, with potential implications for current and future neutrino oscillation experiments. We describe a road map to solidify confidence in the LQCD results and discuss future calculations of more complicated processes, which are important to few-GeV neutrino oscillation experiments.

Wave-packet continuum discretisation for nucleon–nucleon scattering predictions

In this paper we analyse the efficiency, precision, and accuracy of computing elastic nucleon–nucleon (NN) scattering amplitudes with the wave-packet continuum discretisation method (WPCD). This method provides approximate scattering solutions at multiple scattering energies simultaneously. We therefore utilise a graphics processing unit to explore the benefits of this inherent parallelism. From a theoretical perspective, the WPCD method promises a speedup compared to a standard matrix-inversion method. We use the chiral NNLOopt interaction to demonstrate that WPCD enables efficient computation of NN scattering amplitudes provided one can tolerate an averaged method error of 1–5 mb in the total cross section at scattering energies 0–350 MeV in the laboratory frame of reference. Considering only scattering energies ∼40–350 MeV, we find a smaller method error of ≲ 1–2 mb. By increasing the number of wave-packets we can further reduce the overall method error. However, the parallel leverage of the WPCD method will be offset by the increased size of the resulting discretisation mesh. In practice, a GPU-implementation is mainly advantageous for matrices that fit in the fast on-chip shared memory. We find that WPCD is a promising method for computationally efficient, statistical analyses of nuclear interactions from effective field theory, where we can utilise Bayesian inference methods to incorporate relevant uncertainties.

Nucleon Structure and Spin Effects in Elastic Hadron Scattering

Soft diffraction phenomena in elastic nucleon scattering are considered from the viewpoint of the spin dependence of the interaction potential. Spin-dependent pomeron effects are analyzed for elastic pp scattering, and spin-dependent differential cross sections and spin correlation parameters are calculated. The spin correlation parameter AN is examined on the basis of experimental data from s=4.9 GeV up to 23.4GeV in the framework of the extended High Energy Generalized Structure (HEGS) model. It is shown that the existing experimental data of proton-proton and proton-antiproton elastic scattering at high energy in the region of the diffraction minimum and at large momentum transfer give the support of the existence of the energy-independent part of the hadron spin flip amplitude.

Rearrangement term in the folding model of the nucleon optical potential

Based on the mean-field determination of the single-particle energy in nuclear matter that contains naturally a rearrangement term (RT) implied by the Hugenholtz–van Hove theorem, the folding model of the nucleon optical potential (OP) is extended to take into account the RT using the effective, density dependent CDM3Yn interaction. With the exchange part of the nucleon folded OP treated exactly in the Hartree–Fock manner, a compact nonlocal version of the folding model is suggested in the present work to determine explicitly the isospin-dependent, nonlocal central term of the nucleon OP. To solve the optical model (OM) equation with a complex nonlocal OP, the calculable R-matrix method is used to analyse the elastic neutron and proton scattering on 40,48Ca, 90Zr, and 208Pb targets at low energies. The inclusion of the RT into the folding model calculation of the nonlocal nucleon OP was shown to be essential for the overall good OM description of elastic nucleon scattering. To validate the nonlocal version of the folding model, the OM results given by the nonlocal folded nucleon OP are also compared with those given by the global parametrization of the nonlocal OP using the analytical nonlocal form factor suggested by Perey and Buck.

Neutrino quantum kinetics in compact objects

Neutrinos play a critical role of transporting energy and changing the lepton density within core-collapse supernovae and neutron star mergers. The quantum kinetic equations (QKEs) combine the effects of neutrino-matter interactions treated in classical Boltzmann transport with the neutrino flavor-changing effects treated in neutrino oscillation calculations. We present a method for extending existing neutrino interaction rates to full QKE source terms for use in numerical calculations. We demonstrate the effects of absorption and emission by nucleons and nuclei, electron scattering, electron-positron pair annihilation, nucleon-nucleon bremsstrahlung, neutrino-neutrino scattering. For the first time, we include all these collision terms self-consistently in a simulation of the full isotropic QKEs in conditions relevant to core-collapse supernovae and neutron star mergers. For our choice of parameters, the long-term evolution of the neutrino distribution function proceeds similarly with and without the oscillation term, though with measurable differences. We demonstrate that electron scattering, nucleon-nucleon bremsstrahlung processes, and four-neutrino processes dominate flavor decoherence in the protoneutron star (PNS), absorption dominates near the shock, and all of the considered processes except elastic nucleon scattering are relevant in the decoupling region. Finally, we propose an effective decoherence opacity that at most energies predicts decoherence rates to within a factor of 10 in our model PNS and within 20% outside of the PNS.

Two-Pion Exchange Three-Nucleon Force Effects in Elastic Nucleon–Deuteron Scattering Cross Sections

Differential cross sections for elastic nucleon–deuteron scattering at intermediate energies are calculated by solving the Faddeev equation in coordinate space using interaction models consisting of a two-nucleon potential and three-nucleon potentials based on the exchange of two pions among three nucleons. Attractive effects of the pion-exchange at medium range region in the three-nucleon potential are taken into account by using a larger cutoff mass parameter and by adding phenomenological repulsive three-nucleon potentials as a counterpart. It is shown that these effects increase the cross sections at backward angles, which tends to reduce discrepancies between theoretical calculations and experimental data.

Two-photon exchange corrections to γ*NΔ form factors for Q2≤4(GeV/c)2

We evaluate the corrections of the two-photon exchange (TPE) process on the $\gamma^* N\Delta$ transition form factors. The contributions of the TPE process to the $eN\rightarrow e\Delta(1232)\rightarrow eN\pi$ are calculated in a hadronic model with the inclusion of only the elastic nucleon intermediate states, to estimate its effects on the multipoles $M^{(3/2)}_{1^+}, E^{(3/2)}_{1^+}, S^{(3/2)}_{1^+}$ at the $\Delta$ peak. We find that TPE effects on $G^*_M$ is very small. $G^*_E,$ and $ G^*_C$ are also little affected at small $Q^2 $. For $G^*_E$, the TPE effects reach about $3 - 8\%$ near $Q^2\sim 4$ GeV$^2$, depending on the model, MAID or SAID, used to emulate the data. For $G^*_C$, the TPE effects decrease rapidly with increasing $\epsilon$ while growing with increasing $Q^2$ to reach $\sim 6 - 15\%$ with $Q^2 \sim 4$ GeV$^2$ at $\epsilon = 0.2$. Sizeable TPE corrections to $G^*_E$ and $G^*_C$ found here points to the need of including TPE effects in the multipole analysis in the region of high $Q^2$ and small $\epsilon$. The TPE corrections to $R_{EM}$ and $R_{SM}$ obtained in our hadronic calculation are compared with those obtained in a partonic calculation for moderate momentum transfer of $2<Q^2<4$ GeV$^2$.

Microscopic description of elastic and direct inelastic nucleon scattering off spherical nuclei

The purpose of this study is to improve the modeling of nucleon direct inelastic scattering to the continuum using a microscopic and parameter-free approach. For the first time, direct elastic scattering, inelastic scattering to discrete excitations and to the continuum are described within a microscopic approach without adjustable parameters. Proton scattering off 90Zr and 208Pb are the reactions used as test case examples of the calculations. The model uses the Melbourne g-matrix and the Random Phase Approximation description of nuclear states, implemented with the Gogny D1S interaction. The relevant optical and transition potentials in a finite nucleus are calculated within a local density approximation. As we use the nuclear matter approach we limit our study to incident energies above 40 MeV. We first checked that this model provides an accurate account of measured cross sections for elastic scattering and inelastic scattering to discrete states. It is then applied to the direct inelastic scattering to the continuum considering all one-phonon excitations predicted within the RPA approach. This accounts for a part of the direct pre-equilibrium emission, often labeled as the one-step direct process in quantum-based approaches. Our approach provides a very accurate description of angular distributions where the one-step process dominates. The impact of collective excitations is shown to be non negligible for energy transfer to the target up to 20 MeV, decreasing as the incident energy increases. For incident energies above 80 MeV, our modeling provides a good account of direct proton emission for an energy transfer to the target up to 30 MeV. However, the proton emission we predict underestimates the measured cross sections for incident energies below 80 MeV. We compare our prediction to those of the phenomenological exciton model to help interpret this result. Directions that may improve our modeling are discussed.

Experimental Studies of Nuclear Interactions in Few-Nucleon Systems

Systems of three nucleons (3N) can be treated as a testing ground for modern approaches to describe nuclear interactions. At intermediate energies, observables for 3N systems are sensitive to subtle effects of the dynamics beyond the pairwise nucleon–nucleon force, so-called 3N-force (3NF). For years the search for 3NF has been motivating precise measurements of observables of elastic nucleon–deuteron scattering and for the deuteron breakup reaction. Breakup of a deuteron in collision with a proton leads to the final state of three free nucleons, with variety of possible kinematic configurations, revealing locally enhanced sensitivity to particular aspects of the interaction dynamics, like 3NF, Coulomb force between protons, or relativistic effects. This feature makes the breakup reaction a very versatile tool for validation of the theoretical description. Reactions involving four nucleons pose immense challenges with regard to exact theoretical calculations for such systems. Nonetheless, they attract attention due to expected enhanced sensitivity to certain aspects of the nuclear dynamics, manifesting themselves in various channels and configurations. The most important results of recent experimental studies of 3N and 4N systems at intermediate energies are discussed. A brief survey of the ongoing projects is given.

Elastic total cross-sections in an RSIIp scenario

The total elastic nucleon–nucleus cross-section is calculated at energies up to 600 MeV for the elements [Formula: see text]C, [Formula: see text]Pb, [Formula: see text]O, 9Be, 4He, 2H, [Formula: see text]U and [Formula: see text]Al using a finite electromagnetic potential, which is obtained considering a Randall–Sundrum II scenario modified by the inclusion of [Formula: see text] compact extra dimensions. The length scale bound is set in the potential to compare with known experimental data.

Nucleon Resonance Electrocouplings from Light-Front Quark Models at $${\varvec{Q}}^2 $$ Q 2 up to 12 $$\hbox {GeV}^2$$ GeV 2

A relativistic light-front quark model is used to describe both the elastic nucleon and nucleon-Roper transition form factors in a large \(Q^2\) range, up to 35 \(\hbox {GeV}^2\) for the elastic and up to 12 \(\hbox {GeV}^2\) for the resonance case. Relativistic three-quark configurations satisfying the Pauli exclusion principle on the light-front are used for the derivation of the current matrix elements. The Roper resonance is considered as a mixed state of a three-quark core configuration and a molecular \(N+\sigma \) hadron component. Based on this ansatz we obtain a realistic description of both processes, elastic and inelastic, in the sector of positive parity and show that existing experimental data are indicative of a composite structure of the Roper resonance. A useful generalization of this technique is suggested for description of negative parity nucleon resonances \(1/2^-,\,3/2^-,\,5/2^-\).

Forging the link between nuclear reactions and nuclear structure

A review of the recent applications of the dispersive optical model (DOM) is presented. Emphasis is on the nonlocal implementation of the DOM that is capable of describing ground-state properties accurately when data like the nuclear charge density are available. The DOM, conceived by Claude Mahaux, provides a unified description of both elastic nucleon scattering and structure information related to single-particle properties below the Fermi energy. We have recently introduced a nonlocal dispersive optical potential for both the real and imaginary part. Nonlocal absorptive potentials yield equivalent elastic differential cross sections for ${}^{40}$Ca as compared to local ones but change the $\ell$-dependent absorption profile suggesting important consequences for the analysis of nuclear reactions. Below the Fermi energy, nonlocality is essential for an accurate representation of particle number and the nuclear charge density. Spectral properties implied by $(e,e'p)$ and $(p,2p)$ reactions are correctly described, including the energy distribution of about 10\% high-momentum protons obtained at Jefferson Lab. The nonlocal DOM allows a complete description of experimental data both above (up to 200 MeV) and below the Fermi energy in $^{40}$Ca. It is further demonstrated that elastic nucleon-nucleus scattering data constrain the spectral strength in the continuum of orbits that are nominally bound in the independent-particle model. Extension of this analysis to $^{48}$Ca allows a prediction of the neutron skin of this nucleus that is larger than most predictions made so far.

Theoretical constraints and systematic effects in the determination of the proton form factors

We calculate the two-photon exchange corrections to electron-proton scattering with nucleon and $\mathrm{\ensuremath{\Delta}}$ intermediate states. The results show a dependence on the elastic nucleon and nucleon-$\mathrm{\ensuremath{\Delta}}$-transition form factors used as input which leads to significant changes compared to previous calculations. We discuss the relevance of these corrections and apply them to the most recent and precise data set and world data from electron-proton scattering. Using this, we show how the form factor extraction from these data is influenced by the subsequent inclusion of physical constraints. The determination of the proton charge radius from scattering data is shown to be dominated by the enforcement of a realistic spectral function. Additionally, the third Zemach moment from the resulting form factors is calculated. The obtained radius and Zemach moment are shown to be consistent with Lamb shift measurements in muonic hydrogen.

The hypercentral Constituent Quark Model and its application to baryon properties

The hypercentral Constituent Quark Model (hCQM) for the baryon structure is reviewed and its applications are systematically discussed. The model is based on a simple form of the quark potential, which contains a Coulomb-like interaction and a confinement, both expressed in terms of a collective space coordinate, the hyperradius. The model has only three free parameters, determined in order to describe the baryon spectrum. Once the parameters have been fixed, the model, in its nonrelativistic version, is used to predict various quantities of physical interest, namely the elastic nucleon form factors, the photocouplings and the helicity amplitudes for the electromagnetic excitation of the baryon resonances. In particular, the Q^2 dependence of the helicity amplitude is quite well reproduced, thanks to the Coulomb-like interaction. The model is reformulated in a relativistic version by means of the Point Form hamilton dynamics. While the inclusion of relativity does not! alter the results for the helicity amplitudes, a good description of the nucleon elastic form factors is obtained.

Elastic nucleon-nucleus scattering as a direct probe of correlations beyond the independent-particle model

Employing a recently-developed dispersive optical model (DOM) which allows a complete description of experimental data both above (up to 200 MeV) and below the Fermi energy in $^{40}$Ca, we demonstrate that elastic nucleon-nucleus scattering data constrain the spectral strength in the continuum of orbits that are nominally bound in the independent-particle model. In the energy domain between 0 and 200 MeV, the integrated strength or depletion number is highly sensitive to the separation of the IPM orbit to the scattering continuum. This sensitivity is determined by the influence of the surface-absorption properties of the DOM self-energy. For an ab initio calculation employing the self-energy of the charge-dependent Bonn (CDBonn) interaction which only includes the effect of short-range correlations, no such sensitivity is obtained and a depletion of 4% is predicted between 0 and 200 MeV irrespective of the orbit. The ab initio spectral strength generated with the CDBonn interaction approaches the empirical DOM spectral strength at 200 MeV. Both spectral distributions allow for an additional 3-5% of the strength at even higher energies which is associated with the influence of short-range correlations. We suggest that the non-local form of the DOM allows for an analysis of elastic-nucleon-scattering data that directly determines the depletion of bound orbits. While obviously relevant for the analysis of elastic nucleon scattering on stable targets, this conclusion holds equally well for experiments involving rare isotopes in inverse kinematics as well as experiments with electrons on atoms or molecules.

Isospin dependence of nucleon correlations in ground-state nuclei

The dispersive optical model (DOM) as presently implemented can investigate the isospin (nucleon asymmetry) dependence of the Hartree-Fock-like potential relevant for nucleons near the Fermi energy. Data constraints indicate that a Lane-type potential adequately describes its asymmetry dependence. Correlations beyond the mean-field can also be described in this framework, but this requires an extension that treats the non-locality of the Hartree-Fock-like potential properly. The DOM has therefore been extended to properly describe ground-state properties of nuclei as a function of nucleon asymmetry in addition to standard ingredients like elastic nucleon scattering data and level structure. Predictions of nucleon correlations at larger nucleon asymmetries can then be made after data at smaller asymmetries constrain the potentials that represent the nucleon self-energy. A simple extrapolation for Sn isotopes generates predictions for increasing correlations of minority protons with increasing neutron number. Such predictions can be investigated by performing experiments with exotic beams. The predicted neutron properties for the double closed-shell 132Sn nucleus exhibit similar correlations as those in 208Pb. Future relevance of these studies for understanding the properties of all nucleons, including those with high momentum, and the role of three-body forces in nuclei are briefly discussed. Such an implementation will require a proper treatment of the non-locality of the imaginary part of the potentials and a description of high-momentum nucleons as experimentally constrained by the (e,e'p) reactions performed at Jefferson Lab.

Electromagnetic structure of the nucleon and the Roper resonance in a light-front quark approach

A relativistic light-front quark model is used to describe both the elastic nucleon and nucleon-Roper transition form factors in a large Q2 range, up to 35 GeV2 for the elastic and up to 12 GeV2 for the resonance case. Relativistic three-quark configurations satisfying the Pauli exclusion principle on the light-front are used for the derivation of the current matrix elements. The Roper resonance is considered as a mixed state of a three-quark core configuration and a molecular N+sigma hadron component. Based on this ansatz we obtain a realistic description of both processes, elastic and inelastic, and show that existing experimental data are indicative of a composite structure of the Roper resonance.

The proton form factor measurements at Jefferson Lab, past and future

Use of the double-polarization technique to obtain the elastic nucleon form factors has resulted in a dramatic improvement of the quality of two of the four nucleon electromagnetic form factors, GEp and GEn. It has also changed our understanding of the proton structure, having resulted in a distinctly different Q2-dependence for both GEp and GMp, contradicting the prevailing wisdom of the 1990’s based on cross section measurements, namely that GEp and GMp obey a “scaling” relation μGEp ∼ GMp. A related consequence of the faster decrease of GEp revealed by the Jefferson Lab (JLab) polarization results was the disappearance of the early scaling F2/F1 ∼ 1/Q2 predicted by perturbative QCD. In three experiments, GEp(1), GEp(2) and GEp(3), in Halls A and C at JLab, the ratio of the proton’s electromagnetic elastic form factors, GEp/GMp, was measured up to four momentum transfer Q2 of 8.5 GeV2 with high precision, using the recoil polarization technique. The initial discovery that the proton form factor ratio measured in these three experiments decreases approximately linearly with four-momentum transfer, Q2, for values above ∼ 1 GeV2, was modified by the GEp(3) results, which suggests a slowing down of this decrease. There is an approved experiment, GEp(5), to continue these measurements to 15 GeV2. A dedicated experimental setup, the super bigbite spectrometer (SBS), will be built for this purpose. It will be equipped with a new focal plane polarimeter to measure the polarization of the recoil protons. In this presentation, I will review the status of the proton elastic electromagnetic form factors, mention succinctly a number of theoretical approaches to describe results and show some features required for the future GEp(5) experiment.

Elegent—An elastic event generator

Although elastic scattering of nucleons may look like a simple process, it presents a long-lasting challenge for theory. Due to missing hard energy scale, the perturbative QCD cannot be applied. Instead, many phenomenological/theoretical models have emerged. In this paper we present a unified implementation of some of the most prominent models in a C++ library, moreover extended to account for effects of the electromagnetic interaction. The library is complemented with a number of utilities. For instance, programs to sample many distributions of interest in four-momentum transfer squared, t, impact parameter, b, and collision energy s. These distributions at ISR, Spp̄S, RHIC, Tevatron and LHC energies are available for download from the project web site. Both in the form of ROOT files and PDF figures providing comparisons among the models. The package includes also a tool for Monte-Carlo generation of elastic scattering events, which can easily be embedded in any other program framework. Program summaryProgram title: ElegentCatalogue identifier: AERT_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AERT_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 10551No. of bytes in distributed program, including test data, etc.: 126316Distribution format: tar.gzProgramming language: C++.Computer: Any in principle, tested on x86-64 architecture.Operating system: Any in principle, tested on GNU/Linux.RAM: Strongly depends on the task, but typically below 20MBClassification: 11.6.External routines: ROOT, HepMCNature of problem:Monte-Carlo simulation of elastic nucleon–nucleon collisionsSolution method:Implementation of some of the most prominent phenomenological/theoretical models providing cumulative distribution function that is used for random event generation.Running time:Strongly depends on the task, but typically below 1 h.