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Nucleon elastic form factors at accessible large spacelike momenta

Zhu-Fang Cui, Chen Chen, Daniele Binosi, Feliciano De Soto, Craig D. Roberts, José Rodríguez-Quintero, Sebastian M. Schmidt, and Jorge Segovia
Phys. Rev. D 102, 014043 – Published 28 July 2020
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  • INTRODUCTION
  • NUCLEON STRUCTURE AND ELECTROMAGNETIC…
  • FORM FACTOR INTERPOLATION AND…
  • ELASTIC FORM FACTORS
  • FLAVOR-SEPARATED FORM FACTORS
  • SUMMARY AND OUTLOOK
  • ACKNOWLEDGMENTS
    • References

    Abstract

    A Poincaré-covariant quark+diquark Faddeev equation, augmented by a statistical implementation of the Schlessinger point method for the interpolation and extrapolation of smooth functions, is used to compute nucleon elastic form factors on 0≤Q2≤18mN2 (mN is the nucleon mass) and elucidate their role as probes of emergent hadronic mass in the Standard Model. The calculations expose features of the form factors that can be tested in new generation experiments at existing facilities, e.g., a zero in GEp/GMp, a maximum in GEn/GMn, and a zero in the proton’s d-quark Dirac form factor, F1d. Additionally, examination of the associated light-front-transverse number and anomalous magnetization densities reveals inter alia: a marked excess of valence u quarks in the neighborhood of the proton’s center of transverse momentum, and that the valence d quark is markedly more active magnetically than either of the valence u quarks. The calculations and analysis also reveal other aspects of nucleon structure that could be tested with a high-luminosity accelerator capable of delivering higher beam energies than are currently available.

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    • Received 25 March 2020
    • Accepted 25 June 2020

    DOI:https://doi.org/10.1103/PhysRevD.102.014043

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Bethe-Salpeter equationBound statesQuantum chromodynamics
    1. Physical Systems
    NeutronsProtons
    1. Properties
    Form factorsMagnetic momentMass
    1. Techniques
    Functional analytical methods
    Particles & Fields

    Authors & Affiliations

    Zhu-Fang Cui1,2,*, Chen Chen3,†, Daniele Binosi4,‡, Feliciano De Soto5,§, Craig D. Roberts1,2,∥, José Rodríguez-Quintero6,¶, Sebastian M. Schmidt7,8,**, and Jorge Segovia9,2,††

    • 1School of Physics, Nanjing University, Nanjing, Jiangsu 210093, China
    • 2Institute for Nonperturbative Physics, Nanjing University, Nanjing, Jiangsu 210093, China
    • 3Institut für Theoretische Physik, Justus-Liebig-Universität Gießen, D-35392 Gießen, Germany
    • 4European Centre for Theoretical Studies in Nuclear Physics and Related Areas; Villa Tambosi, Strada delle Tabarelle 286, I-38123 Villazzano (TN), Italy
    • 5Dpto. Sistemas Físicos, Químicos y Naturales, Univ. Pablo de Olavide, E-41013 Sevilla, Spain
    • 6Department of Integrated Sciences and Center for Advanced Studies in Physics, Mathematics and Computation; University of Huelva, E-21071 Huelva; Spain.
    • 7Helmholtz-Zentrum Dresden-Rossendorf, Dresden D-01314, Germany
    • 8RWTH Aachen University, III. Physikalisches Institut B, Aachen D-52074, Germany
    • 9Dpto. Sistemas Físicos, Químicos y Naturales, Univ. Pablo de Olavide, E-41013 Sevilla, Spain

    • *phycui@nju.edu.cn
    • †Chen.Chen@theo.physik.uni-giessen.de
    • ‡binosi@ectstar.eu
    • §fcsotbor@upo.es
    • ∥cdroberts@nju.edu.cn
    • jose.rodriguez@dfaie.uhu.es
    • **s.schmidt@hzdr.de
    • ††jsegovia@upo.es

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    Vol. 102, Iss. 1 — 1 July 2020

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    • Figure 1
      Figure 1

      Nucleon=quark+diquark Faddeev equation. This is a linear integral equation for the Poincaré-covariant matrix-valued function Ψ, the Faddeev amplitude for a state with total momentum P=pq+pd. It describes the relative momentum correlation between the dressed quarks and diquarks. Legend. Shaded rectangle—kernel of the Faddeev equation; single line—dressed-quark propagator; Γ—diquark correlation amplitude; and double line—diquark propagator. Ground-state nucleons (n neutron, p proton) contain both isoscalar-scalar diquarks, [ud]∈(n,p), and isovector-pseudovector diquarks {dd}∈n, {ud}∈(n,p), {uu}∈p.

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    • Figure 2
      Figure 2

      Vertex that ensures a conserved current for on-shell nucleons that are described by the Faddeev amplitudes obtained from the equation illustrated in Fig. 1: single line, dressed-quark propagator; undulating line, photon; Γ, diquark correlation amplitude; and double line, diquark propagator. Diagrams D3, D5, and D6 are eight-dimensional integrals. Monte Carlo methods are required for their evaluation. (Details are provided in Appendix C of Ref. [43].)

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    • Figure 3
      Figure 3

      Proton: herein, solid blue curves. Upper panels—Dirac form factor, F1p(x), x=Q2/mN2 and lower panels—Pauli form factor, F2p(x) (κp=1.50.) In both cases, the right panels depict xs-weighted form factors, with s chosen to match the associated naïve scaling dimension. The empirical fits from Ref. [84] are displayed in the left panels (dashed green curves). In all panels, the 1σ band for the SPM approximants is shaded in light blue, i.e., 68% of all SPM approximants lie within the light blue band centered on the blue curve.

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    • Figure 4
      Figure 4

      Neutron: herein, solid blue curves. Upper panels—Dirac form factor, F1n(x), x=Q2/mN2 and lower panels—Pauli form factor, F2n(x) (κp=−1.59.) In both cases, the right panels depict xs-weighted form factors, with s chosen to match the associated naïve scaling dimension. The empirical fits from Ref. [84] are displayed in the left panels (dashed green curves). In all panels, the 1σ band for the SPM approximants is shaded in light blue.

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    • Figure 5
      Figure 5

      Nucleon Sachs magnetic form factors. Upper panels—neutron. Left panel, Sachs magnetic form factor, GMn(x), x=Q2/mN2 and right panel, x2GMn(x). Lower panels—proton. Left panel, Sachs magnetic form factor, GMp(x) and right panel, x2GMp(x). All panels, the 1σ band for the SPM approximants is shaded in light blue. N.B., both right panels display x2GM(x); hence, agreement at the level of 0.1% between theory and experiment is expressed in the image as an absolute difference of 0.1 at x=10. The fits to experimental data in Ref. [84] are depicted as dashed green curves, with the quoted uncertainty on the fits drawn as shaded green bands in the right panels.

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    • Figure 6
      Figure 6

      Ratios of Sachs form factors, μNGEN(x)/GMN(x). Upper panels—proton. Left, calculation herein compared with data (red up triangles [2], green squares [3], blue circles [4], black down triangles [5], and cyan diamonds [6]); right, compared with available lQCD results, drawn from Ref. [17]. Lower panels—neutron. Left, comparison with data (blue circles [96] and green squares [97]); right, with available lQCD results, drawn from Ref. [17]. In all panels, the 1σ band for the SPM approximants is shaded in light blue.

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    • Figure 7
      Figure 7

      Flavor separation. Upper panels—x- and x2-weighted Dirac form factors: d quark, solid blue curve and u quark, dot-dashed red curve. Lower panel—x- and x2-weighted Pauli form factors: d quark, solid blue curve and u quark, dot-dashed red curve. In all panels, shaded bands centered on each curve mark the 1σ uncertainty for the SPM approximants and data from Ref. [8]: d quark, blue circles and u quark, red squares.

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    • Figure 8
      Figure 8

      Light-front-transverse valence u- and d-quark densities. Upper panels—one-dimensional profiles defined in Eq. (10): left, ρ^1f(|b^|)/2 and right, |b^|ρ^1f(|b^|). Lower panels—two-dimensional images derived from the curves in the upper-left panel: left, ρ^1u(|b^|)/2 and right, ρ^1d(|b^|). N.B., in all panels, the u-quark density is halved, eliminating the overall 2u:1d feature of the proton, so that a direct comparison between the |b^| dependence of the densities can readily be made.

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    • Figure 9
      Figure 9

      Light-front-transverse valence u- and d-quark anomalous magnetization densities. Upper panels—one-dimensional profiles defined in Eq. (15): left, ρ^2f(|b^|) and right, |b^|ρ^2f(|b^|). The shaded bands centered on each curve mark the 1σ uncertainty for the SPM approximants; and the d-quark density is multiplied by “−1” to simplify visual comparisons. Lower panels—two-dimensional images derived from the curves in the upper-left panel: left ρ^2u(|b^|) and right, −ρ^2d(|b^|).

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