# Nucleon Resonance Electroexcitation Amplitudes and Emergent Hadron Mass

^{1}

^{2}

^{3}

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^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Basics for Insight into EHM Using CSMs

#### 2.1. CSMs and the EHM Paradigm

#### 2.2. Some Highlights from the EHM Experiment-Theory Connection

## 3. Nucleon Resonance Electrocouplings and Their Impact on the Insight into EHM

#### 3.1. Extraction of Electrocouplings from Exclusive Meson Electroproduction Data

#### 3.2. Insights into the Dressed-Quark Mass Function from the ${\gamma}_{v}p{N}^{*}$ Electrocouplings

#### 3.3. Novel Tests of CSM Predictions

## 4. Studies of ${\mathit{N}}^{*}$ Structure in Experiments with CLAS12 and Beyond

## 5. Conclusions and Outlook

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CM | center-of-mass |

CSM | continuum Schwinger function method |

DCSB | dynamical chiral symmetry breaking |

$d.p.$ | data point |

EHM | emergence of hadron mass |

EIC | Electron-Ion Collider (at Brookhaven National Laboratory) |

EicC | Electron-ion collider China |

HB | Higgs boson |

JLab | Thomas Jefferson National Accelerator Facility (Jefferson Laboratory) |

JM | JLab-Moscow State University |

lQCD | lattice-regularized quantum chromodynamics |

NG (mode/boson) | Nambu-Goldstone (mode/boson) |

PDFs | Particle Distribution Functions |

PDG | Particle Data Group (and associated publications) |

pQCD | perturbative QCD |

QCD | quantum chromodynamics |

RMS | root mean square |

RPP | Review of Particle Properties (and associated publications) |

sQCD | strong QCD |

SM | Standard Model of particle physics |

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**Figure 1.**Integral equations for the dressed quark and gluon two-point functions [84] (Section 2.2), drawn in terms of the Feynman diagrams that govern the emergence of gluon and quark quasiparticles from the partons used to express the QCD Lagrangian. (Total momentum k flows from left to right in each diagram, being conserved in passing through the loop integrals.) These quasiparticles are the active components in hadron structure at low resolving scales. Their parton content is revealed at higher resolutions. (Unbroken lines—quarks; spring-like lines—gluons; short-dashed lines—ghosts; filled circles—dressed propagators; open circles—two-point = self-energies and three/ four-point = dressed vertices. The vertices satisfy their own Dyson–Schwinger equations, involving higher n-point functions [84]).

**Figure 2.**(

**left**): CSM predictions for the momentum dependence of the dressed-gluon (blue solid) and quark (green dot-dashed) masses [9,10,11]. The associated like-colored bands express the uncertainties in the CSM predictions. (N.B. Since the Poincaré-invariant kinetic energy operator for a vector boson has mass–dimension two and that for a spin-half fermion has mass–dimension unity, then for ${m}_{p}^{2}/{k}^{2}\to 0$, ${M}_{0}\left(k\right)\propto 1/{k}^{2}$ and ${m}_{g}^{2}\left(k\right)\propto 1/{k}^{2}$, up to $ln{k}^{2}$ corrections). (

**Right**) CSM prediction [3] (magenta band) for the process-independent QCD running coupling $\widehat{\alpha}\left(k\right)$ compared with the empirical results [4] for the process-dependent effective charge defined via the Bjorken sum rule, which is prominent in deep inelastic scattering.

**Figure 3.**CSM predictions for observables of the structure for the ground state hadrons in comparison with experimental results (points with error bars) or comparable theory. (

**Upper left**)—pion valence quark PDF [100]; (

**Upper right**)—nucleon axial form factor ${G}_{A}$ [101]; (

**Lower left**)—pion elastic form factor [11]; and (

**Lower right**)—ratio of nucleon elastic electric and magnetic form factors [79]. Sources for comparison curves and points are listed in References [11,79,100,101].

**Figure 4.**Mass budgets for the proton (outermost annulus), $\rho $-meson, kaon, and pion (innermost annulus). Each annulus is drawn using a Poincaré-invariant decomposition. The separation is made at a renormalization scale $\zeta =2\phantom{\rule{0.166667em}{0ex}}$GeV, calculated using information from References [76,109,110,111].

**Figure 5.**Resonant and non-resonant amplitudes contributing to exclusive meson electroproduction channels in the resonance region.

**Figure 6.**$N\left(1440\right)1/{2}^{+}$ and $N\left(1520\right)3/{2}^{-}$ electrocouplings extracted from the $\pi N$ [121,123] and ${\pi}^{+}{\pi}^{-}p$ [67,83,136,137] electroproduction channels. The photocouplings from the Review of Particle Properties (RPP) [76] and from Reference [138] are shown by the blue squares and triangles, respectively.

**Figure 7.**Description of the results for the $N\to \Delta $ magnetic transition form factor G${}_{\mathrm{M}}^{*}$ (

**left**) and the electrocoupling amplitude ${A}_{1/2}$ for the $N\to N\left(1440\right)1/{2}^{+}$ (

**right**) achieved using CSMs [65,66,68,69]. Results obtained with a momentum-independent (frozen) dressed-quark mass [65,66] (dotted red curves) are compared with QCD-kindred results (solid blue curves) obtained with the momentum-dependent quark mass function in Figure 2. The electrocoupling data were taken from References [50,121,123]—$\pi N$ electroproduction, and References [67,83,136,137]—${\pi}^{+}{\pi}^{-}p$ electroproduction. The photocouplings for the $N\left(1440\right)1/{2}^{+}$ are from the RPP [76] and from Reference [138]—blue square and triangle, respectively. The ranges of ${Q}^{2}$ where the contributions from the meson–baryon cloud remain substantial are highlighted in gray.

**Figure 8.**Regarding extraction of $\Delta \left(1600\right)3/{2}^{+}$ electrocouplings, representative examples of the nine independent one-fold differential cross sections available from the ${\pi}^{+}{\pi}^{-}p$ measurements with CLAS [145,146] at two different ${Q}^{2}$ values, along with the data fits within the data-driven meson–baryon JM reaction model [75,82,83].

**Figure 9.**Preliminary $\Delta \left(1600\right)3/{2}^{+}$ electrocouplings with their assigned uncertainties, determined from independent analysis of the ${\pi}^{+}{\pi}^{-}p$ differential cross sections in three overlapping W intervals: 1.46–1.56 GeV (filled blue squares), 1.51–1.61 GeV (filled red triangles), and 1.56–1.66 GeV (filled black triangles) [136,137,144]. CSM predictions [74] are drawn as solid red curves.

**Figure 10.**(

**Top**) Momentum ranges accessible in the exploration of the momentum dependence of the dressed-quark mass function using results on the ${Q}^{2}$-evolution of ${\gamma}_{v}p{N}^{*}$ electrocouplings. The range of k covered by available data is mostly from experiments with CLAS, shown in yellow. The expected reach of CLAS12 experiments is shown in purple, and that achievable after a proposed increase of the JLab beam energy to 22 GeV in cyan. (

**Bottom**) Yields of representative exclusive meson electroproduction channels available from the experiments with the CLAS12 detector.

**Figure 11.**Luminosity versus CM energy in lepton–proton collisions for existing and foreseeable facilities capable of exploring hadron structure in measurements with large-acceptance detectors.

**Table 1.**Comparison between the measured masses of the proton and neutron, ${m}_{p,n}$, and the sum of the current-quark masses of their three u- and d-quark constituents [76]. (Current quark masses are listed at a scale of 2 GeV, but the comparison remains qualitatively unchanged if renormalization group invariant current masses are used.)

Proton | Neutron | |
---|---|---|

Measured masses (MeV) | 938.2720813 ± 0.0000058 | 939.5654133± 0.0000058 |

Sum of the current quark masses (MeV) | 8.09${}_{-0.65}^{+1.45}$ | 11.50${}_{-0.60}^{+1.45}$ |

Contribution of the current quark masses to the measured nucleon mass (%) | <1.1 | <1.4 |

**Table 2.**Summary of the results for the ${\gamma}_{v}p{N}^{*}$ electrocouplings from the $\pi N$, $\eta p$, and ${\pi}^{+}{\pi}^{-}p$ electroproduction channels measured with the CLAS detector in Hall B at JLab.

Meson Electroproduction Channels | Excited Proton States | ${\mathit{Q}}^{2}$ Ranges for Extracted ${\mathit{\gamma}}_{\mathit{v}}{\mathit{p}\mathit{N}}^{*}$ Electrocouplings, GeV${}^{2}$ |
---|---|---|

${\pi}^{0}p$, ${\pi}^{+}n$ | $\Delta \left(1232\right)3/{2}^{+}$ | 0.16–6 |

$N\left(1440\right)1/{2}^{+}$, $N\left(1520\right)3/{2}^{-}$ | 0.30–4.16 | |

$N\left(1535\right)1/{2}^{-}$ | 0.30–4.16 | |

${\pi}^{+}n$ | $N\left(1675\right)5/{2}^{-}$, $N\left(1680\right)5/{2}^{+}$ | 1.6-4.5 |

$N\left(1710\right)1/{2}^{+}$ | ||

$\eta p$ | $N\left(1535\right)1/{2}^{-}$ | 0.2–2.9 |

${\pi}^{+}{\pi}^{-}p$ | $N\left(1440\right)1/{2}^{+}$, $N\left(1520\right)3/{2}^{-}$ | 0.25–1.50 |

$\Delta \left(1600\right)3/{2}^{+}$, $\Delta \left(1620\right)1/{2}^{-}$ | 2.0–5.0 | |

$N\left(1650\right)1/{2}^{-}$, $N\left(1680\right)5/{2}^{+}$, | ||

$\Delta \left(1700\right)3/{2}^{-}$ | 0.50–1.50 | |

$N\left(1720\right)3/{2}^{+}$, ${N}^{\prime}\left(1720\right)3/{2}^{+}$ | 0.50-1.50 |

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**MDPI and ACS Style**

Carman, D.S.; Gothe, R.W.; Mokeev, V.I.; Roberts, C.D.
Nucleon Resonance Electroexcitation Amplitudes and Emergent Hadron Mass. *Particles* **2023**, *6*, 416-439.
https://doi.org/10.3390/particles6010023

**AMA Style**

Carman DS, Gothe RW, Mokeev VI, Roberts CD.
Nucleon Resonance Electroexcitation Amplitudes and Emergent Hadron Mass. *Particles*. 2023; 6(1):416-439.
https://doi.org/10.3390/particles6010023

**Chicago/Turabian Style**

Carman, Daniel S., Ralf W. Gothe, Victor I. Mokeev, and Craig D. Roberts.
2023. "Nucleon Resonance Electroexcitation Amplitudes and Emergent Hadron Mass" *Particles* 6, no. 1: 416-439.
https://doi.org/10.3390/particles6010023