# Empirical Consequences of Emergent Mass

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## Abstract

**:**

## 1. Introduction

## 2. Strong Interactions in the Standard Model

#### 2.1. Natural Mass Scale

The establishment by the mid-1970’s of QCD as the correct theory of the strong interactions completed what is now known prosaically as the Standard Model. It offers a description of all known fundamental physics except for gravity, and gravity is something that has no discernible effect when particles are studied a few at a time. However, the situation is a bit like the way that the Navier-Stokes equation accounts for the flow of water. The equations are at some level obviously correct, but there are only a few, limited circumstances in which their consequences can be worked out in any detail. Nevertheless, many leading physicists were inclined to conclude in the late 1970’s that the task of basic physics was nearly complete, and we’d soon be out of jobs. A famous example was the inaugural lecture of Stephen Hawking as Lucasian Professor of Mathematics, a chair first held by Isaac Barrow at Cambridge University. Hawking titled his lecture, ‘Is the End in Sight for Theoretical Physics?’ And he argued strongly for ‘Yes’.

The Higgs field is often said to give mass to everything. That is wrong. The Higgs field only gives mass to some very simple particles. The field accounts for only one or two percent of the mass of more complex things, like atoms, molecules, and everyday objects, from your mobile phone to your pet llama. The vast majority of mass comes from the energy needed to hold quarks together inside atoms.

QCD is quite possibly the most remarkable fundamental theory ever invented.

#### 2.2. Whence Mass?

## 3. Confinement

The Confinement Hypothesis: Colour-charged particles cannot be isolated and therefore cannot be directly observed. They clump together in colour-neutral bound-states.

## 4. Strong QCD

#### 4.1. Dyson–Schwinger Equations

#### 4.2. Gluon Mass

#### 4.3. Effective Charge

#### 4.4. Dynamical Chiral Symmetry Breaking

The Nambu–Goldstone theorem is fundamentally an expression of equivalence between the one-body problem and the two-body problem in QCD’s colour-singlet pseudoscalar channel.

#### 4.5. Pion and the Trace Anomaly

## 5. Empirical Manifestations of Emergent Mass

#### 5.1. Pion Wave Function

#### 5.2. Pion Electromagnetic Form Factor

#### 5.3. Valence-Quark Distributions in the Pion

#### 5.4. Emergence of Diquark Correlations

#### 5.5. Proton Wave Function

#### 5.6. Proton’s First Radial Excitation

#### 5.7. Emergent Features of Nucleon Form Factors

## 6. Epilogue

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CSM | continuum Schwinger-function method |

DA (PDA) | (parton) distribution amplitude |

DF (PDF) | (parton) distribution function |

DCSB | dynamical chiral symmetry breaking |

DSE | Dyson–Schwinger Equation |

EFT | effective field theory |

EHM | emergent hadronic mass |

ERBL | Efremov–Radyushkin–Brodsky–Lepage |

JLab | Thomas Jefferson National Accelerator Facility |

lQCD | lattice-regularised quantum chromodynamics |

PDG | Particle Data Group |

PI | process independent |

PFF | parton fragmentation function |

pQCD | perturbative quantum chromodynamics |

QED | quantum electrodynamics |

QCD | quantum chromodynamics |

RGI | renormalisation group invariant |

SM | Standard Model (of Particle Physics) |

SPM | Schlessinger point method |

2PI | two-particle irreducible |

## References

- Politzer, H.D. The dilemma of attribution. Proc. Nat. Acad. Sci. USA
**2005**, 102, 7789–7793. [Google Scholar] [CrossRef] [PubMed] - ATLAS Collaboration. Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B
**2012**, 716, 1–29. [Google Scholar] [CrossRef] - CMS Collaboration. Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC. Phys. Lett. B
**2012**, 716, 30–61. [Google Scholar] [CrossRef] - Englert, F. Nobel Lecture: The BEH mechanism and its scalar boson. Rev. Mod. Phys.
**2014**, 86, 843. [Google Scholar] [CrossRef] [Green Version] - Higgs, P.W. Nobel Lecture: Evading the Goldstone theorem. Rev. Mod. Phys.
**2014**, 86, 851. [Google Scholar] [CrossRef] [Green Version] - Taylor, R.E. Deep inelastic scattering: The Early years. Rev. Mod. Phys.
**1991**, 63, 573–595. [Google Scholar] [CrossRef] [Green Version] - Kendall, H.W. Deep inelastic scattering: Experiments on the proton and the observation. Rev. Mod. Phys.
**1991**, 63, 597–614. [Google Scholar] [CrossRef] - Friedman, J.I. Deep inelastic scattering: Comparisons with the quark model. Rev. Mod. Phys.
**1991**, 63, 615–629. [Google Scholar] [CrossRef] [Green Version] - Neddermeyer, S.H.; Anderson, C.D. Note on the Nature of Cosmic-Ray Particles. Phys. Rev.
**1937**, 51, 884–886. [Google Scholar] [CrossRef] - Lattes, C.M.G.; Muirhead, H.; Occhialini, G.P.S.; Powell, C.F. Processes involving charged mesons. Nature
**1947**, 159, 694–697. [Google Scholar] [CrossRef] - Marciano, W.J.; Pagels, H. Quantum Chromodynamics: A Review. Phys. Rept.
**1978**, 36, 137. [Google Scholar] [CrossRef] - Marciano, W.J.; Pagels, H. Quantum Chromodynamics. Nature
**1979**, 279, 479–483. [Google Scholar] [CrossRef] - Particle Data Group. Review of Particle Properties. Prog. Theor. Exp. Phys.
**2020**, 083C01. [Google Scholar] - Qin, S.X.; Roberts, C.D.; Schmidt, S.M. Spectrum of light- and heavy-baryons. Few Body Syst.
**2019**, 60, 26. [Google Scholar] [CrossRef] [Green Version] - Durr, S.; Fodor, Z.; Frison, J.; Hoelbling, C.; Hoffmann, R.; Katz, S.D.; Krieg, S.; Kurth, T.; Lellouch, L.; Lippert, T. Ab-Initio Determination of Light Hadron Masses. Science
**2008**, 322, 1224–1227. [Google Scholar] [CrossRef] [Green Version] - Pascual, P.; Tarrach, R. QCD: Renormalization for the Practitioner; Springer: Berlin, Germany, 1984. [Google Scholar]
- Cornwall, J.M. Dynamical Mass Generation in Continuum QCD. Phys. Rev. D
**1982**, 26, 1453. [Google Scholar] [CrossRef] - Gribov, V.N. The theory of quark confinement. Eur. Phys. J. C
**1999**, 10, 91–105. [Google Scholar] [CrossRef] - Dudal, D.; Verschelde, H.; Gracey, J.A.; Lemes, V.E.R.; Sarandy, M.S.; Sobreiro, R.F.; Sorella, S.P. Dynamical gluon mass generation from 〈A
^{2}(μ)〉 in linear covariant gauges. JHEP**2004**, 01, 044. [Google Scholar] [CrossRef] [Green Version] - Bowman, P.O.; Heller, U.M.; Leinweber, D.B.; Parappilly, M.B.; Williams, A.G. Unquenched gluon propagator in Landau gauge. Phys. Rev. D
**2004**, 70, 034509. [Google Scholar] [CrossRef] [Green Version] - Luna, E.; Martini, A.; Menon, M.; Mihara, A.; Natale, A. Influence of a dynamical gluon mass in the pp and p anti-p forward scattering. Phys. Rev. D
**2005**, 72, 034019. [Google Scholar] [CrossRef] [Green Version] - Aguilar, A.; Binosi, D.; Papavassiliou, J. Gluon and ghost propagators in the Landau gauge: Deriving lattice results from Schwinger-Dyson equations. Phys. Rev. D
**2008**, 78, 025010. [Google Scholar] [CrossRef] [Green Version] - Rodríguez-Quintero, J. On the massive gluon propagator, the PT-BFM scheme and the low-momentum behaviour of decoupling and scaling DSE solutions. JHEP
**2011**, 1101, 105. [Google Scholar] [CrossRef] [Green Version] - Boucaud, P.; Leroy, J.P.; Le Yaouanc, A.; Micheli, J.; Pene, O.; Rodríguez-Quintero, J. The Infrared Behaviour of the Pure Yang-Mills Green Functions. Few Body Syst.
**2012**, 53, 387–436. [Google Scholar] [CrossRef] [Green Version] - Strauss, S.; Fischer, C.S.; Kellermann, C. Analytic structure of the Landau gauge gluon propagator. Phys. Rev. Lett.
**2012**, 109, 252001. [Google Scholar] [CrossRef] [Green Version] - Binosi, D.; Chang, L.; Papavassiliou, J.; Roberts, C.D. Bridging a gap between continuum-QCD and ab initio predictions of hadron observables. Phys. Lett. B
**2015**, 742, 183–188. [Google Scholar] [CrossRef] [Green Version] - Aguilar, A.C.; Binosi, D.; Papavassiliou, J. The Gluon Mass Generation Mechanism: A Concise Primer. Front. Phys. China
**2016**, 11, 111203. [Google Scholar] [CrossRef] [Green Version] - Siringo, F. Analytical study of Yang–Mills theory in the infrared from first principles. Nucl. Phys. B
**2016**, 907, 572–596. [Google Scholar] [CrossRef] [Green Version] - Cyrol, A.K.; Fister, L.; Mitter, M.; Pawlowski, J.M.; Strodthoff, N. Landau gauge Yang-Mills correlation functions. Phys. Rev. D
**2016**, 94, 054005. [Google Scholar] [CrossRef] [Green Version] - Gao, F.; Qin, S.X.; Roberts, C.D.; Rodríguez-Quintero, J. Locating the Gribov horizon. Phys. Rev. D
**2018**, 97, 034010. [Google Scholar] [CrossRef] [Green Version] - Binosi, D.; Tripolt, R.A. Spectral functions of confined particles. Phys. Lett. B
**2020**, 801, 135171. [Google Scholar] [CrossRef] - Binosi, D.; Mezrag, C.; Papavassiliou, J.; Roberts, C.D.; Rodríguez-Quintero, J. Process-independent strong running coupling. Phys. Rev. D
**2017**, 96, 054026. [Google Scholar] [CrossRef] [Green Version] - Rodríguez-Quintero, J.; Binosi, D.; Mezrag, C.; Papavassiliou, J.; Roberts, C.D. Process-independent effective coupling. From QCD Green’s functions to phenomenology. Few Body Syst.
**2018**, 59, 121. [Google Scholar] [CrossRef] [Green Version] - Cui, Z.F.; Zhang, J.L.; Binosi, D.; de Soto, F.; Mezrag, C.; Papavassiliou, J.; Rodríguez-Quintero, J.; Roberts, C.D.; Segovia, J.; Zafeiropoulos, S. Effective charge from lattice QCD. Chin. Phys. C
**2020**, 44, 083102. [Google Scholar] [CrossRef] - Kharzeev, D. Quarkonium interactions in QCD. Proc. Int. Sch. Phys. Fermi
**1996**, 130, 105–131. [Google Scholar] - Close, F. The Quark Parton Model. Rept. Prog. Phys.
**1979**, 42, 1285–1335. [Google Scholar] [CrossRef] - Nambu, Y. Quasiparticles and Gauge Invariance in the Theory of Superconductivity. Phys. Rev.
**1960**, 117, 648–663. [Google Scholar] [CrossRef] - Goldstone, J. Field Theories with Superconductor Solutions. Nuovo Cim.
**1961**, 19, 154–164. [Google Scholar] [CrossRef] - Aguilar, A.C.; Ahmed, Z.; Aidala, C.; Ali, S.; Andrieux, V.; Arrington, J.A.; Bashir, A.; Berdnikov, V.; Binosi, D.; Chang, L.; et al. Pion and Kaon Structure at the Electron-Ion Collider. Eur. Phys. J. A
**2019**, 55, 190. [Google Scholar] [CrossRef] - Roberts, C.D. Resonance Electroproduction and the Origin of Mass—arXiv:1909.11102 [nucl-th]. In Proceedings of the 12th International Workshop on the Physics of Excited Nucleons (NSTAR 2019), Bonn, Germany, 10–14 June 2019. [Google Scholar]
- Brodsky, S.J.; Burkert, V.D.; Carman, D.S.; Chen, J.P.; Cui, Z.F.; Döring, M.; Dosch, H.G.; Draayer, J.; Elouadrhiri, L.; Glazier, D.I.; et al. Strong QCD from Hadron Structure Experiments—arXiv:2006.06802 [hep-ph]. Intern. J. Mod. Phys. E
**2020**, in press. [Google Scholar] - Roberts, C.D.; Schmidt, S.M. Reflections upon the Emergence of Hadronic Mass. arXiv
**2020**, arXiv:2006.08782. [Google Scholar] - Aznauryan, I.; Bashir, A.; Braun, V.; Brodsky, S.J.; Burkert, V.D.; Chang, L.; Chen, C.; El-Bennich, B.; Cole, P.; Edwards, R.G.; et al. Studies of Nucleon Resonance Structure in Exclusive Meson Electroproduction. Int. J. Mod. Phys. E
**2013**, 22, 1330015. [Google Scholar] [CrossRef] - Gross, D.J. The discovery of asymptotic freedom and the emergence of QCD. Proc. Nat. Acad. Sci. USA
**2005**, 102, 9099–9108. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wilczek, F. Asymptotic freedom: From paradox to paradigm. Proc. Nat. Acad. Sci. USA
**2005**, 102, 8403–8413. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jaffe, A.M. The Millennium Grand Challenge in Mathematics. Not. Am. Math. Soc.
**2006**, 53, 652–660. [Google Scholar] - Casher, A. Chiral Symmetry Breaking in Quark Confining Theories. Phys. Lett. B
**1979**, 83, 395. [Google Scholar] [CrossRef] - Banks, T.; Casher, A. Chiral Symmetry Breaking in Confining Theories. Nucl. Phys. B
**1980**, 169, 103. [Google Scholar] [CrossRef] - McNeile, C. Lattice status of gluonia/glueballs. Nucl. Phys. Proc. Suppl.
**2009**, 186, 264–267. [Google Scholar] [CrossRef] [Green Version] - Wilson, K.G. Confinement of quarks. Phys. Rev. D
**1974**, 10, 2445–2459. [Google Scholar] [CrossRef] - Isgur, N.; Paton, J.E. A Flux Tube Model for Hadrons. Phys. Lett. B
**1983**, 124, 247–251. [Google Scholar] [CrossRef] - Bali, G.S.; Neff, H.; Duessel, T.; Lippert, T.; Schilling, K. Observation of string breaking in QCD. Phys. Rev. D
**2005**, 71, 114513. [Google Scholar] [CrossRef] [Green Version] - Prkacin, Z.; Bali, G.S.; Dussel, T.; Lippert, T.; Neff, H.; Schilling, K. Anatomy of string breaking in QCD. PoS
**2006**, LAT2005, 308. [Google Scholar] - Chang, L.; Cloet, I.C.; El-Bennich, B.; Klähn, T.; Roberts, C.D. Exploring the light-quark interaction. Chin. Phys. C
**2009**, 33, 1189–1196. [Google Scholar] - Glimm, J.; Jaffee, A. Quantum Physics. A Functional Point of View; Springer: New York, NY, USA, 1981. [Google Scholar]
- Munczek, H.J.; Nemirovsky, A.M. The Ground State qq¯ Mass Spectrum in QCD. Phys. Rev. D
**1983**, 28, 181–186. [Google Scholar] [CrossRef] - Cahill, R.T.; Roberts, C.D. Soliton Bag Models of Hadrons from QCD. Phys. Rev. D
**1985**, 32, 2419. [Google Scholar] [CrossRef] [PubMed] - Stingl, M. Propagation Properties and Condensate Formation of the Confined Yang-Mills Field. Phys. Rev. D
**1986**, 34, 3863–3881, Erratum in**1987**, 36, 651. [Google Scholar] [CrossRef] - Roberts, C.D.; Williams, A.G.; Krein, G. On the implications of confinement. Int. J. Mod. Phys. A
**1992**, 7, 5607–5624. [Google Scholar] [CrossRef] - Burden, C.J.; Roberts, C.D.; Williams, A.G. Singularity structure of a model quark propagator. Phys. Lett. B
**1992**, 285, 347–353. [Google Scholar] [CrossRef] - Hawes, F.T.; Roberts, C.D.; Williams, A.G. Dynamical chiral symmetry breaking and confinement with an infrared vanishing gluon propagator. Phys. Rev. D
**1994**, 49, 4683–4693. [Google Scholar] [CrossRef] [Green Version] - Maris, P. Analytic structure of the full fermion propagator in quenched and unquenched QED. Phys. Rev. D
**1994**, 50, 4189–4193. [Google Scholar] [CrossRef] - Roberts, C.D.; Williams, A.G. Dyson-Schwinger equations and their application to hadronic physics. Prog. Part. Nucl. Phys.
**1994**, 33, 477–575. [Google Scholar] [CrossRef] [Green Version] - Bhagwat, M.; Pichowsky, M.; Tandy, P.C. Confinement phenomenology in the Bethe-Salpeter equation. Phys. Rev. D
**2003**, 67, 054019. [Google Scholar] [CrossRef] [Green Version] - Roberts, C.D. Hadron Properties and Dyson-Schwinger Equations. Prog. Part. Nucl. Phys.
**2008**, 61, 50–65. [Google Scholar] [CrossRef] [Green Version] - Bashir, A.; Raya, A.; Sánchez-Madrigal, S.; Roberts, C.D. Gauge invariance of a critical number of flavours in QED3. Few Body Syst.
**2009**, 46, 229–237. [Google Scholar] [CrossRef] [Green Version] - Bashir, A.; Raya, A.; Rodríguez-Quintero, J. QCD: Restoration of Chiral Symmetry and Deconfinement for Large N
_{f}. Phys. Rev. D**2013**, 88, 054003. [Google Scholar] [CrossRef] [Green Version] - Qin, S.X.; Rischke, D.H. Quark Spectral Function and Deconfinement at Nonzero Temperature. Phys. Rev. D
**2013**, 88, 056007. [Google Scholar] [CrossRef] [Green Version] - Lowdon, P. Conditions on the violation of the cluster decomposition property in QCD. J. Math. Phys.
**2016**, 57, 102302. [Google Scholar] [CrossRef] - Lucha, W.; Schöberl, F.F. Analytic Bethe-Salpeter Description of the Lightest Pseudoscalar Mesons. Phys. Rev. D
**2016**, 93, 056006. [Google Scholar] [CrossRef] [Green Version] - Binosi, D.; Roberts, C.D.; Rodríguez-Quintero, J. Scale-setting, flavour dependence and chiral symmetry restoration. Phys. Rev. D
**2017**, 95, 114009. [Google Scholar] [CrossRef] [Green Version] - Krein, G.; Nielsen, M.; Puff, R.D.; Wilets, L. Ghost poles in the nucleon propagator: Vertex corrections and form-factors. Phys. Rev. C
**1993**, 47, 2485–2491. [Google Scholar] [CrossRef] [Green Version] - Bracco, M.E.; Eiras, A.; Krein, G.; Wilets, L. Selfconsistent solution of the Schwinger-Dyson equations for the nucleon and meson propagators. Phys. Rev. C
**1994**, 49, 1299–1308. [Google Scholar] [CrossRef] [Green Version] - Gattringer, C.; Lang, C.B. Quantum chromodynamics on the lattice. Lect. Notes Phys.
**2010**, 788, 1–343. [Google Scholar] - Philipsen, O. The QCD equation of state from the lattice. Prog. Part. Nucl. Phys.
**2013**, 70, 55–107. [Google Scholar] [CrossRef] [Green Version] - Bañuls, M.C.; Cichy, K. Review on Novel Methods for Lattice Gauge Theories. Rept. Prog. Phys.
**2020**, 83, 024401. [Google Scholar] [CrossRef] [Green Version] - Roberts, C.D.; Schmidt, S.M. Dyson-Schwinger equations: Density, temperature and continuum strong QCD. Prog. Part. Nucl. Phys.
**2000**, 45, S1–S103. [Google Scholar] [CrossRef] - Maris, P.; Roberts, C.D. Dyson-Schwinger equations: A tool for hadron physics. Int. J. Mod. Phys. E
**2003**, 12, 297–365. [Google Scholar] [CrossRef] [Green Version] - Chang, L.; Roberts, C.D.; Tandy, P.C. Selected highlights from the study of mesons. Chin. J. Phys.
**2011**, 49, 955–1004. [Google Scholar] - Roberts, C.D. Strong QCD and Dyson-Schwinger Equations. IRMA Lect. Math. Theor. Phys.
**2015**, 21, 356–458. [Google Scholar] - Roberts, C.D. Three Lectures on Hadron Physics. J. Phys. Conf. Ser.
**2016**, 706, 022003. [Google Scholar] [CrossRef] - Horn, T.; Roberts, C.D. The pion: An enigma within the Standard Model. J. Phys. G
**2016**, 43, 073001. [Google Scholar] [CrossRef] - Eichmann, G.; Sanchis-Alepuz, H.; Williams, R.; Alkofer, R.; Fischer, C.S. Baryons as relativistic three-quark bound states. Prog. Part. Nucl. Phys.
**2016**, 91, 1–100. [Google Scholar] [CrossRef] [Green Version] - Burkert, V.D.; Roberts, C.D. Roper resonance: Toward a solution to the fifty year puzzle. Rev. Mod. Phys.
**2019**, 91, 011003. [Google Scholar] [CrossRef] [Green Version] - Fischer, C.S. QCD at finite temperature and chemical potential from Dyson–Schwinger equations. Prog. Part. Nucl. Phys.
**2019**, 105, 1–60. [Google Scholar] [CrossRef] [Green Version] - Qin, S.X.; Roberts, C.D. Impressions of the Continuum Bound State Problem in QCD. arXiv
**2020**, arXiv:2008.07629. [Google Scholar] - Munczek, H.J. Dynamical chiral symmetry breaking, Goldstone’s theorem and the consistency of the Schwinger-Dyson and Bethe-Salpeter Equations. Phys. Rev. D
**1995**, 52, 4736–4740. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bender, A.; Roberts, C.D.; von Smekal, L. Goldstone Theorem and Diquark Confinement Beyond Rainbow- Ladder Approximation. Phys. Lett. B
**1996**, 380, 7–12. [Google Scholar] [CrossRef] [Green Version] - Maris, P.; Roberts, C.D.; Tandy, P.C. Pion mass and decay constant. Phys. Lett. B
**1998**, 420, 267–273. [Google Scholar] [CrossRef] [Green Version] - Chang, L.; Roberts, C.D. Sketching the Bethe-Salpeter kernel. Phys. Rev. Lett.
**2009**, 103, 081601. [Google Scholar] [CrossRef] [Green Version] - Qin, S.X.; Roberts, C.D.; Schmidt, S.M. Ward-Green-Takahashi identities and the axial-vector vertex. Phys. Lett. B
**2014**, 733, 202–208. [Google Scholar] [CrossRef] [Green Version] - Williams, R.; Fischer, C.S.; Heupel, W. Light mesons in QCD and unquenching effects from the 3PI effective action. Phys. Rev. D
**2016**, 93, 034026. [Google Scholar] [CrossRef] [Green Version] - Binosi, D.; Chang, L.; Qin, S.X.; Papavassiliou, J.; Roberts, C.D. Symmetry preserving truncations of the gap and Bethe-Salpeter equations. Phys. Rev. D
**2016**, 93, 096010. [Google Scholar] [CrossRef] [Green Version] - Accardi, A.; Albacete, J.L.; Anselmino, M.; Armesto, N.; Aschenauer, E.C.; Bacchetta, A.; Boer, D.; Brooks, W.K.; Burton, T.; Chang, N.B.; et al. Electron Ion Collider: The Next QCD Frontier. Eur. Phys. J. A
**2016**, 52, 268. [Google Scholar] [CrossRef] [Green Version] - Brodsky, S.J.; Deshpande, A.L.; Gao, H.; McKeown, C.A.; Meziani, Z.E.; Milner, R.G.; Qiu, J.W.; Richards, D.G.; Roberts, C.D. QCD and Hadron Physics. arXiv
**2015**, arXiv:1502.05728. [Google Scholar] - Field, R.D.; Feynman, R.P. A Parametrization of the Properties of Quark Jets. Nucl. Phys. B
**1978**, 136, 1–76. [Google Scholar] [CrossRef] - Gell-Mann, M.; Low, F.E. Quantum electrodynamics at small distances. Phys. Rev.
**1954**, 95, 1300–1312. [Google Scholar] [CrossRef] - Cornwall, J.M.; Papavassiliou, J. Gauge Invariant Three Gluon Vertex in QCD. Phys. Rev. D
**1989**, 40, 3474. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pilaftsis, A. Generalized pinch technique and the background field method in general gauges. Nucl. Phys. B
**1997**, 487, 467–491. [Google Scholar] [CrossRef] [Green Version] - Binosi, D.; Papavassiliou, J. Pinch Technique: Theory and Applications. Phys. Rept.
**2009**, 479, 1–152. [Google Scholar] [CrossRef] [Green Version] - Abbott, L.F. The Background Field Method Beyond One Loop. Nucl. Phys. B
**1981**, 185, 189. [Google Scholar] [CrossRef] [Green Version] - Deur, A.; Burkert, V.; Chen, J.P.; Korsch, W. Experimental determination of the effective strong coupling constant. Phys. Lett. B
**2007**, 650, 244–248. [Google Scholar] [CrossRef] [Green Version] - Deur, A.; Burkert, V.; Chen, J.P.; Korsch, W. Determination of the effective strong coupling constant α
_{g1}(s) from CLAS spin structure function data. Phys. Lett. B**2008**, 665, 349–351. [Google Scholar] [CrossRef] - Deur, A.; Brodsky, S.J.; de Teramond, G.F. The QCD Running Coupling. Prog. Part. Nucl. Phys.
**2016**, 90, 1–74. [Google Scholar] [CrossRef] [Green Version] - Grunberg, G. Renormalization Scheme Independent QCD and QED: The Method of Effective Charges. Phys. Rev. D
**1984**, 29, 2315. [Google Scholar] [CrossRef] - Dokshitzer, Y.L. Perturbative QCD theory (includes our knowledge of α(s))—hep-ph/9812252. In Proceedings of the 29th International Conference on High Energy Physics: ICHEP ’98, Vancouver, CU, Canada, 23–29 July 1998; Volume 1, pp. 305–324. [Google Scholar]
- Appelquist, T.; Terning, J.; Wijewardhana, L.C.R. The zero temperature chiral phase transition in SU(N) gauge theories. Phys. Rev. Lett.
**1996**, 77, 1214–1217. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sannino, F. Conformal Dynamics for TeV Physics and Cosmology. Acta Phys. Polon. B
**2009**, 40, 3533–3743. [Google Scholar] - LSD Collaboration. Toward TeV Conformality. Phys. Rev. Lett.
**2010**, 104, 071601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hayakawa, M.; Ishikawa, K.I.; Osaki, Y.; Takeda, S.; Uno, S.; Yamada, N. Running coupling constant of ten-flavor QCD with the Schródinger functional method. Phys. Rev. D
**2011**, 83, 074509. [Google Scholar] [CrossRef] [Green Version] - Cheng, A.; Hasenfratz, A.; Petropoulos, G.; Schaich, D. Scale-dependent mass anomalous dimension from Dirac eigenmodes. JHEP
**2013**, 07, 061. [Google Scholar] [CrossRef] [Green Version] - LatKMI Collaboration. Walking signals in N
_{f}=8 QCD on the lattice. Phys. Rev. D**2013**, 87, 094511. [Google Scholar] [CrossRef] [Green Version] - DeGrand, T. Lattice tests of beyond Standard Model dynamics. Rev. Mod. Phys.
**2016**, 88, 015001. [Google Scholar] [CrossRef] - Cui, Z.F.; Ding, M.; Gao, F.; Raya, K.; Binosi, D.; Chang, L.; Roberts, C.D.; Rodríguez-Quintero, J.; Schmidt, S.M. Kaon parton distributions: Revealing Higgs modulation of emergent mass. arXiv
**2020**, arXiv:2006.14075. [Google Scholar] - Cui, Z.F.; Ding, M.; Gao, F.; Raya, K.; Binosi, D.; Chang, L.; Roberts, C.D.; Rodríguez-Quintero, J.; Schmidt, S.M. Kaon and pion parton distributions.
**2020**. in progress. [Google Scholar] - Nambu, Y. From BCS to NJL: An old story retold. AIP Conf. Proc.
**2011**, 1388, 86–92. [Google Scholar] - Brodsky, S.J.; Shrock, R. Condensates in Quantum Chromodynamics and the Cosmological Constant. Proc. Natl. Acad. Sci. USA
**2011**, 108, 45–50. [Google Scholar] [CrossRef] [Green Version] - Brodsky, S.J.; Roberts, C.D.; Shrock, R.; Tandy, P.C. New perspectives on the quark condensate. Phys. Rev. C
**2010**, 82, 022201(R). [Google Scholar] [CrossRef] [Green Version] - Chang, L.; Roberts, C.D.; Tandy, P.C. Expanding the concept of in-hadron condensates. Phys. Rev. C
**2012**, 85, 012201(R). [Google Scholar] [CrossRef] [Green Version] - Brodsky, S.J.; Roberts, C.D.; Shrock, R.; Tandy, P.C. Confinement contains condensates. Phys. Rev. C
**2012**, 85, 065202. [Google Scholar] [CrossRef] [Green Version] - Bhagwat, M.S.; Pichowsky, M.A.; Roberts, C.D.; Tandy, P.C. Analysis of a quenched lattice QCD dressed quark propagator. Phys. Rev. C
**2003**, 68, 015203. [Google Scholar] [CrossRef] [Green Version] - Bhagwat, M.S.; Tandy, P.C. Analysis of full-QCD and quenched-QCD lattice propagators. AIP Conf. Proc.
**2006**, 842, 225–227. [Google Scholar] - Bowman, P.O.; Heller, U.M.; Leinweber, D.B.; Parappilly, M.B.; Williams, A.G.; Zhang, Z.B. Unquenched quark propagator in Landau gauge. Phys. Rev. D
**2005**, 71, 054507. [Google Scholar] [CrossRef] [Green Version] - Roberts, C.D. Perspective on the origin of hadron masses. Few Body Syst.
**2017**, 58, 5. [Google Scholar] [CrossRef] - Höll, A.; Krassnigg, A.; Roberts, C.D. Pseudoscalar meson radial excitations. Phys. Rev. C
**2004**, 70, 042203(R). [Google Scholar] [CrossRef] [Green Version] - Höll, A.; Krassnigg, A.; Maris, P.; Roberts, C.D.; Wright, S.V. Electromagnetic properties of ground and excited state pseudoscalar mesons. Phys. Rev. C
**2005**, 71, 065204. [Google Scholar] [CrossRef] [Green Version] - Bhagwat, M.S.; Chang, L.; Liu, Y.X.; Roberts, C.D.; Tandy, P.C. Flavour symmetry breaking and meson masses. Phys. Rev. C
**2007**, 76, 045203. [Google Scholar] [CrossRef] [Green Version] - Ding, M.; Raya, K.; Bashir, A.; Binosi, D.; Chang, L.; Chen, M.; Roberts, C.D. γ
^{*}γ→η,η^{′}transition form factors. Phys. Rev. D**2019**, 99, 014014. [Google Scholar] [CrossRef] [Green Version] - Ding, M.; Gao, F.; Chang, L.; Liu, Y.X.; Roberts, C.D. Leading-twist parton distribution amplitudes of S-wave heavy-quarkonia. Phys. Lett. B
**2016**, 753, 330–335. [Google Scholar] [CrossRef] [Green Version] - Chen, C.; Chang, L.; Roberts, C.D.; Wan, S.; Zong, H.S. Valence-quark distribution functions in the kaon and pion. Phys. Rev. D
**2016**, 93, 074021. [Google Scholar] [CrossRef] [Green Version] - Gao, F.; Chang, L.; Liu, Y.X.; Roberts, C.D.; Tandy, P.C. Exposing strangeness: Projections for kaon electromagnetic form factors. Phys. Rev. D
**2017**, 96, 034024. [Google Scholar] [CrossRef] [Green Version] - Flambaum, V.V.; Höll, A.; Jaikumar, P.; Roberts, C.D.; Wright, S.V. Sigma terms of light-quark hadrons. Few Body Syst.
**2006**, 38, 31–51. [Google Scholar] [CrossRef] [Green Version] - Guth, A.H.; Huang, K.; Jaffe, R.L. (Eds.) Asymptotic Realms of Physics; MIT Press: Cambridge, MA, USA, 1983; 262p. [Google Scholar]
- Chang, L.; Cloet, I.C.; Cobos-Martinez, J.J.; Roberts, C.D.; Schmidt, S.M.; Tandy, P.C. Imaging dynamical chiral symmetry breaking: Pion wave function on the light front. Phys. Rev. Lett.
**2013**, 110, 132001. [Google Scholar] [CrossRef] [Green Version] - Segovia, J.; Chang, L.; Cloet, I.C.; Roberts, C.D.; Schmidt, S.M.; Zong, H.S. Distribution amplitudes of light-quark mesons from lattice QCD. Phys. Lett. B
**2014**, 731, 13–18. [Google Scholar] [CrossRef] [Green Version] - Shi, C.; Chang, L.; Roberts, C.D.; Schmidt, S.M.; Tandy, P.C.; Zong, H.S. Flavour symmetry breaking in the kaon parton distribution amplitude. Phys. Lett. B
**2014**, 738, 512–518. [Google Scholar] [CrossRef] [Green Version] - Lepage, G.P.; Brodsky, S.J. Exclusive Processes in Quantum Chromodynamics: Evolution Equations for Hadronic Wave Functions and the Form-Factors of Mesons. Phys. Lett. B
**1979**, 87, 359–365. [Google Scholar] [CrossRef] - Efremov, A.V.; Radyushkin, A.V. Factorization and Asymptotical Behavior of Pion Form- Factor in QCD. Phys. Lett. B
**1980**, 94, 245–250. [Google Scholar] [CrossRef] - Lepage, G.P.; Brodsky, S.J. Exclusive Processes in Perturbative Quantum Chromodynamics. Phys. Rev. D
**1980**, 22, 2157–2198. [Google Scholar] [CrossRef] [Green Version] - Chernyak, V.L.; Zhitnitsky, A.R. Asymptotic Behavior of Exclusive Processes in QCD. Phys. Rept.
**1984**, 112, 173. [Google Scholar] [CrossRef] - Maris, P.; Roberts, C.D. π and K meson Bethe-Salpeter amplitudes. Phys. Rev. C
**1997**, 56, 3369–3383. [Google Scholar] [CrossRef] [Green Version] - Maris, P.; Tandy, P.C. Bethe-Salpeter study of vector meson masses and decay constants. Phys. Rev. C
**1999**, 60, 055214. [Google Scholar] [CrossRef] [Green Version] - Bhagwat, M.S.; Maris, P. Vector meson form factors and their quark-mass dependence. Phys. Rev. C
**2008**, 77, 025203. [Google Scholar] [CrossRef] [Green Version] - Krassnigg, A. Survey of J=0,1 mesons in a Bethe-Salpeter approach. Phys. Rev. D
**2009**, 80, 114010. [Google Scholar] [CrossRef] - Qin, S.X.; Chang, L.; Liu, Y.x.; Roberts, C.D.; Wilson, D.J. Investigation of rainbow-ladder truncation for excited and exotic mesons. Phys. Rev. C
**2012**, 85, 035202. [Google Scholar] [CrossRef] [Green Version] - Mikhailov, S.; Radyushkin, A. Nonlocal condensates and QCD sum rules for pion wave function. JETP Lett.
**1986**, 43, 712–715. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Petrov, V.Y.; Polyakov, M.V.; Ruskov, R.; Weiss, C.; Goeke, K. Pion and photon light cone wave functions from the instanton vacuum. Phys. Rev. D
**1999**, 59, 114018. [Google Scholar] [CrossRef] [Green Version] - Braun, V.; Gockeler, M.; Horsley, R.; Perlt, H.; Pleiter, D.; Rakow, P.E.L.; Schierholz, G.; Schiller, A.; Schroers, W.; Stuben, H.; et al. Moments of pseudoscalar meson distribution amplitudes from the lattice. Phys. Rev. D
**2006**, 74, 074501. [Google Scholar] [CrossRef] [Green Version] - Brodsky, S.J.; de Teramond, G.F. Hadronic spectra and light-front wavefunctions in holographic QCD. Phys. Rev. Lett.
**2006**, 96, 201601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jefferson Lab F
_{π}Collaboration. Measurement of the Charged Pion Electromagnetic Form-Factor. Phys. Rev. Lett.**2001**, 86, 1713–1716. [Google Scholar] [CrossRef] [Green Version] - Jefferson Lab F
_{π}-2 Collaboration. Determination of the Charged Pion Form Factor at Q^{2}=1.60 and 2.45(GeV/c)^{2}. Phys. Rev. Lett.**2006**, 97, 192001. [Google Scholar] [CrossRef] [Green Version] - Jefferson Lab F
_{π}Collaboration. Determination of the pion charge form factor for Q^{2}=0.60-1.60GeV^{2}. Phys. Rev. C**2007**, 75, 055205. [Google Scholar] [CrossRef] [Green Version] - Horn, T.; Qian, X.; Arrington, J.A.; Asaturyan, R.; Benmokthar, F.; Boeglin, W.; Bosted, P.; Bruell, A.; Christy, M.E.; Chudakov, E.; et al. Scaling study of the pion electroproduction cross sections and the pion form factor. Phys. Rev. C
**2008**, 78, 058201. [Google Scholar] [CrossRef] [Green Version] - Jefferson Lab F
_{π}Collaboration. Charged pion form-factor between Q^{2}=0.60GeV^{2}and 2.45GeV^{2}. II. Determination of, and results for, the pion form-factor. Phys. Rev. C**2008**, 78, 045203. [Google Scholar] [CrossRef] [Green Version] - Jefferson Lab F
_{π}Collaboration. Charged pion form factor between Q^{2}= 0.60 and 2.45 GeV^{2}. I. Measurements of the cross section for the^{1}H(e,e^{′}π^{+})n reaction. Phys. Rev. C**2008**, 78, 045202. [Google Scholar] [CrossRef] [Green Version] - Maris, P.; Tandy, P.C. The π, K
^{+}, and K^{0}electromagnetic form factors. Phys. Rev. C**2000**, 62, 055204. [Google Scholar] [CrossRef] [Green Version] - Chang, L.; Cloet, I.C.; Roberts, C.D.; Schmidt, S.M.; Tandy, P.C. Pion electromagnetic form factor at spacelike momenta. Phys. Rev. Lett.
**2013**, 111, 141802. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, M.; Ding, M.; Chang, L.; Roberts, C.D. Mass-dependence of pseudoscalar meson elastic form factors. Phys. Rev. D
**2018**, 98, 091505. [Google Scholar] [CrossRef] [Green Version] - Huber, G.M.; Gaskell, D.; Papandreou, Z.; Bosted, P.; Bruell, A.; Ent, R.; Fenker, H.C.; Gaskel, D.; Horn, T.; Jones, M.K.; et al. Measurement of the Charged Pion Form Factor to High Q2; Jefferson Lab Experiment E12-06-101. 2006. Available online: http://www.jlab.org/exp_prog/proposals/06/PR12-06-101.pdf (accessed on 7 July 2006).
- Horn, T.; Huber, G.M. Jefferson Lab Experiment E12-07-105. 2007. Available online: https://www.jlab.org/exp_prog/experiments/summaries/E12-07-105_summary.pdf (accessed on 22 August 2020).
- Ellis, R.K.; Stirling, W.J.; Webber, B.R. QCD and Collider Physics; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Ezawa, Z.F. Wide-Angle Scattering in Softened Field Theory. Nuovo Cim. A
**1974**, 23, 271–290. [Google Scholar] [CrossRef] - Farrar, G.R.; Jackson, D.R. Pion and Nucleon Structure Functions Near x=1. Phys. Rev. Lett.
**1975**, 35, 1416. [Google Scholar] [CrossRef] [Green Version] - Berger, E.L.; Brodsky, S.J. Quark Structure Functions of Mesons and the Drell-Yan Process. Phys. Rev. Lett.
**1979**, 42, 940–944. [Google Scholar] [CrossRef] - Gribov, V.N.; Lipatov, L.N. Deep inelastic e p scattering in perturbation theory. Sov. J. Nucl. Phys.
**1972**, 15, 438–450. [Google Scholar] - Lipatov, L.N. The parton model and perturbation theory. Sov. J. Nucl. Phys.
**1975**, 20, 94–102. [Google Scholar] - Altarelli, G.; Parisi, G. Asymptotic Freedom in Parton Language. Nucl. Phys. B
**1977**, 126, 298. [Google Scholar] [CrossRef] - Dokshitzer, Y.L. Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamics. Sov. Phys. JETP
**1977**, 46, 641–653. (In Russian) [Google Scholar] - Keppel, C.; Wojtsekhowski, B.; King, P.; Dutta, D.; Annand, J.; Zhang, J. Measurement of Tagged Deep Inelastic Scattering (TDIS); Jefferson Lab experiment PR12-15-006; 2015, approved. Available online: https://www.jlab.org/exp_prog/proposals/15/PR12-15-006.pdf (accessed on 22 August 2020).
- Park, K.; Montgomery, R.; Horn, T. Measurement of Kaon Structure Function through Tagged Deep Inelastic Scattering (TDIS); Jefferson Lab experiment C12-15-006A. 2017. approved. Available online: https://www.jlab.org/exp_prog/proposals/17/C12-15-006A.pdf (accessed on 22 August 2020).
- COMPASS++/AMBER Collaboration. Letter of Intent (Draft 2.0): A New QCD facility at the M2 beam line of the CERN SPS. arXiv
**2018**, arXiv:1808.00848. [Google Scholar] - Xu, S.S.; Chang, L.; Roberts, C.D.; Zong, H.S. Pion and kaon valence-quark parton quasidistributions. Phys. Rev. D
**2018**, 97, 094014. [Google Scholar] [CrossRef] [Green Version] - Zhang, J.H.; Chen, J.W.; Jin, L.; Lin, H.W.; Schäfer, A.; Zhao, Y. First direct lattice-QCD calculation of the x-dependence of the pion parton distribution function. Phys. Rev. D
**2019**, 100, 034505. [Google Scholar] [CrossRef] - Karthik, N.; Izubichi, T.; Jin, L.; Kallidonis, C.; Mukherjee, S.; Petreczky, P.; Shugert, C.; Syritsyn, S. Renormalized quasi parton distribution function of pion. PoS
**2018**, LATTICE2018, 109. [Google Scholar] - Sufian, R.S.; Karpie, J.; Egerer, C.; Orginos, K.; Qiu, J.W.; Richards, D.G. Pion Valence Quark Distribution from Matrix Element Calculated in Lattice QCD. Phys. Rev. D
**2019**, 99, 074507. [Google Scholar] [CrossRef] [Green Version] - Izubuchi, T.; Jin, L.; Kallidonis, C.; Karthik, N.; Mukherjee, S.; Petreczky, P.; Shugert, C.; Syritsyn, S. Valence parton distribution function of pion from fine lattice. Phys. Rev. D
**2019**, 100, 034516. [Google Scholar] [CrossRef] [Green Version] - Oehm, M.; Alexandrou, C.; Constantinou, M.; Jansen, K.; Koutsou, G.; Kostrzewa, B.; Steffens, F.; Urbach, C.; Zafeiropoulos, S. 〈x〉 and 〈x
^{2}〉 of the pion PDF from lattice QCD with N_{f}=2+1+1 dynamical quark flavors. Phys. Rev. D**2019**, 99, 014508. [Google Scholar] [CrossRef] [Green Version] - Joó, B.; Orginos, K.; Radyushkin, A.V.; Richards, D.G.; Sufian, R.S.; Zafeiropoulos, S. Pion valence structure from Ioffe-time parton pseudodistribution functions. Phys. Rev. D
**2019**, 100, 114512. [Google Scholar] [CrossRef] [Green Version] - Hecht, M.B.; Roberts, C.D.; Schmidt, S.M. Valence-quark distributions in the pion. Phys. Rev. C
**2001**, 63, 025213. [Google Scholar] [CrossRef] [Green Version] - Chang, L.; Mezrag, C.; Moutarde, H.; Roberts, C.D.; Rodríguez-Quintero, J.; Tandy, P.C. Basic features of the pion valence-quark distribution function. Phys. Lett. B
**2014**, 737, 23–29. [Google Scholar] [CrossRef] - Ding, M.; Raya, K.; Binosi, D.; Chang, L.; Roberts, C.D.; Schmidt, S.M. Drawing insights from pion parton distributions. Chin. Phys. C
**2020**, 44, 031002. [Google Scholar] [CrossRef] [Green Version] - Ding, M.; Raya, K.; Binosi, D.; Chang, L.; Roberts, C.D.; Schmidt, S.M. Symmetry, symmetry breaking, and pion parton distributions. Phys. Rev. D
**2020**, 101, 054014. [Google Scholar] [CrossRef] [Green Version] - Raya, K.; Chang, L.; Bashir, A.; Cobos-Martinez, J.J.; Gutiérrez-Guerrero, L.X.; Roberts, C.D.; Tandy, P.C. Structure of the neutral pion and its electromagnetic transition form factor. Phys. Rev. D
**2016**, 93, 074017. [Google Scholar] [CrossRef] [Green Version] - Raya, K.; Ding, M.; Bashir, A.; Chang, L.; Roberts, C.D. Partonic structure of neutral pseudoscalars via two photon transition form factors. Phys. Rev. D
**2017**, 95, 074014. [Google Scholar] [CrossRef] [Green Version] - Barry, P.C.; Sato, N.; Melnitchouk, W.; Ji, C.R. First Monte Carlo Global QCD Analysis of Pion Parton Distributions. Phys. Rev. Lett.
**2018**, 121, 152001. [Google Scholar] [CrossRef] [Green Version] - Aicher, M.; Schäfer, A.; Vogelsang, W. Soft-Gluon Resummation and the Valence Parton Distribution Function of the Pion. Phys. Rev. Lett.
**2010**, 105, 252003. [Google Scholar] [CrossRef] [Green Version] - Westmark, D.; Owens, J.F. Enhanced threshold resummation formalism for lepton pair production and its effects in the determination of parton distribution functions. Phys. Rev. D
**2017**, 95, 056024. [Google Scholar] [CrossRef] [Green Version] - Novikov, I.; Abdolmaleki, H.; Britzger, D.; Cooper-Sarkar, A.; Giuli, F.; Glazov, A.; Kusina, A.; Luszczak, A.; Olness, F.; Starovoitov, P.; et al. Parton Distribution Functions of the Charged Pion Within The xFitter. Phys. Rev. D
**2020**, 102, 014040. [Google Scholar] [CrossRef] - Conway, J.S.; Adolphsen, C.E.; Alexander, J.P.; Anderson, K.J.; Heinrich, J.G.; Pilcher, J.E.; Possoz, A.; Rosenberg, E.I.; Biino, C.; Greenhalgh, J.F.; et al. Experimental study of muon pairs produced by 252-GeV pions on tungsten. Phys. Rev. D
**1989**, 39, 92–122. [Google Scholar] [CrossRef] - Barabanov, M.Y.; Bedolla, M.A.; Brooks, W.K.; Cates, G.D.; Chen, C.; Chen, Y.; Cisbani, E.; Ding, M.; Eichmann, G.; Ent, R.; et al. Diquark Correlations in Hadron Physics: Origin, Impact and Evidence. arXiv
**2020**, arXiv:2008.07630. [Google Scholar] - Bhagwat, M.S.; Höll, A.; Krassnigg, A.; Roberts, C.D.; Tandy, P.C. Aspects and consequences of a dressed-quark-gluon vertex. Phys. Rev. C
**2004**, 70, 035205. [Google Scholar] [CrossRef] [Green Version] - Cahill, R.T.; Roberts, C.D.; Praschifka, J. Calculation of diquark masses in QCD. Phys. Rev. D
**1987**, 36, 2804. [Google Scholar] [CrossRef] [PubMed] - Maris, P. Effective masses of diquarks. Few Body Syst.
**2002**, 32, 41–52. [Google Scholar] [CrossRef] [Green Version] - Eichmann, G.; Cloet, I.C.; Alkofer, R.; Krassnigg, A.; Roberts, C.D. Toward unifying the description of meson and baryon properties. Phys. Rev. C
**2009**, 79, 012202(R). [Google Scholar] [CrossRef] - Segovia, J.; El-Bennich, B.; Rojas, E.; Cloet, I.C.; Roberts, C.D.; Xu, S.S.; Zong, H.S. Completing the picture of the Roper resonance. Phys. Rev. Lett.
**2015**, 115, 171801. [Google Scholar] [CrossRef] [Green Version] - Segovia, J.; Roberts, C.D.; Schmidt, S.M. Understanding the nucleon as a Borromean bound-state. Phys. Lett. B
**2015**, 750, 100–106. [Google Scholar] [CrossRef] [Green Version] - Eichmann, G.; Fischer, C.S.; Sanchis-Alepuz, H. Light baryons and their excitations. Phys. Rev. D
**2016**, 94, 094033. [Google Scholar] [CrossRef] [Green Version] - Lu, Y.; Chen, C.; Roberts, C.D.; Segovia, J.; Xu, S.S.; Zong, H.S. Parity partners in the baryon resonance spectrum. Phys. Rev. C
**2017**, 96, 015208. [Google Scholar] [CrossRef] [Green Version] - Chen, C.; El-Bennich, B.; Roberts, C.D.; Schmidt, S.M.; Segovia, J.; Wan, S. Structure of the nucleon’s low-lying excitations. Phys. Rev. D
**2018**, 97, 034016. [Google Scholar] [CrossRef] [Green Version] - Roberts, C.D.; Holt, R.J.; Schmidt, S.M. Nucleon spin structure at very high x. Phys. Lett. B
**2013**, 727, 249–254. [Google Scholar] [CrossRef] [Green Version] - Segovia, J.; Chen, C.; Cloet, I.C.; Roberts, C.D.; Schmidt, S.M.; Wan, S. Elastic and transition form factors of the Δ(1232). Few Body Syst.
**2014**, 55, 1–33. [Google Scholar] [CrossRef] [Green Version] - Segovia, J.; Roberts, C.D. Dissecting nucleon transition electromagnetic form factors. Phys. Rev. C
**2016**, 94, 042201(R). [Google Scholar] [CrossRef] [Green Version] - Mezrag, C.; Segovia, J.; Chang, L.; Roberts, C.D. Parton distribution amplitudes: Revealing correlations within the proton and Roper. Phys. Lett. B
**2018**, 783, 263–267. [Google Scholar] [CrossRef] - Maris, P. Electromagnetic properties of diquarks. Few Body Syst.
**2004**, 35, 117–127. [Google Scholar] [CrossRef] [Green Version] - Roberts, H.L.L.; Bashir, A.; Gutiérrez-Guerrero, L.X.; Roberts, C.D.; Wilson, D.J. π- and ρ-mesons, and their diquark partners, from a contact interaction. Phys. Rev. C
**2011**, 83, 065206. [Google Scholar] [CrossRef] [Green Version] - Anselmino, M.; Predazzi, E.; Ekelin, S.; Fredriksson, S.; Lichtenberg, D.B. Diquarks. Rev. Mod. Phys.
**1993**, 65, 1199–1234. [Google Scholar] [CrossRef] - Aznauryan, I.; Burkert, V.; Lee, T.S.; Mokeev, V. Results from the N* program at JLab. J. Phys. Conf. Ser.
**2011**, 299, 012008. [Google Scholar] [CrossRef] - Burkert, V.D.; Lee, T.S.H. Electromagnetic meson production in the nucleon resonance region. Int. J. Mod. Phys. E
**2004**, 13, 1035–1112. [Google Scholar] [CrossRef] [Green Version] - Segovia, J.; Cloet, I.C.; Roberts, C.D.; Schmidt, S.M. Nucleon and Δ elastic and transition form factors. Few Body Syst.
**2014**, 55, 1185–1222. [Google Scholar] [CrossRef] [Green Version] - Chen, C.; Lu, Y.; Binosi, D.; Roberts, C.D.; Rodríguez-Quintero, J.; Segovia, J. Nucleon-to-Roper electromagnetic transition form factors at large Q
^{2}. Phys. Rev. D**2019**, 99, 034013. [Google Scholar] [CrossRef] [Green Version] - Chen, C.; Krein, G.I.; Roberts, C.D.; Schmidt, S.M.; Segovia, J. Spectrum and structure of octet and decuplet baryons and their positive-parity excitations. Phys. Rev. D
**2019**, 100, 054009. [Google Scholar] [CrossRef] [Green Version] - Lu, Y.; Chen, C.; Cui, Z.F.; Roberts, C.D.; Schmidt, S.M.; Segovia, J.; Zong, H.S. Transition form factors: γ
^{*}+p→Δ(1232), Δ(1600). Phys. Rev. D**2019**, 100, 034001. [Google Scholar] [CrossRef] [Green Version] - Cui, Z.F.; Chen, C.; Binosi, D.; de Soto, F.; Roberts, C.D.; Rodríguez-Quintero, J.; Schmidt, S.M.; Segovia, J. Nucleon elastic form factors at accessible large spacelike momenta. Phys. Rev. D
**2020**, arXiv:2003.11655. [Google Scholar] [CrossRef] - Eichmann, G.; Alkofer, R.; Krassnigg, A.; Nicmorus, D. Nucleon mass from a covariant three-quark Faddeev equation. Phys. Rev. Lett.
**2010**, 104, 201601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Xu, S.S.; Cui, Z.F.; Chang, L.; Papavassiliou, J.; Roberts, C.D.; Zong, H.S. New perspective on hybrid mesons. Eur. Phys. J. A (Lett.)
**2019**, 55, 113. [Google Scholar] [CrossRef] - Souza, E.V.; Ferreira, M.N.; Aguilar, A.C.; Papavassiliou, J.; Roberts, C.D.; Xu, S.S. Pseudoscalar glueball mass: A window on three-gluon interactions. Eur. Phys. J. A (Lett.)
**2020**, 56, 25. [Google Scholar] [CrossRef] [Green Version] - Yin, P.L.; Chen, C.; Krein, G.; Roberts, C.D.; Segovia, J.; Xu, S.S. Masses of ground-state mesons and baryons, including those with heavy quarks. Phys. Rev. D
**2019**, 100, 034008. [Google Scholar] [CrossRef] [Green Version] - Edwards, R.G.; Dudek, J.J.; Richards, D.G.; Wallace, S.J. Excited state baryon spectroscopy from lattice QCD. Phys. Rev. D
**2011**, 84, 074508. [Google Scholar] [CrossRef] [Green Version] - CLAS Collaboration. Measurement of ep→e
^{′}pπ^{+}π^{-}and baryon resonance analysis. Phys. Rev. Lett.**2003**, 91, 022002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Burkert, V.D. Evidence of new nucleon resonances from electromagnetic meson production. EPJ Web Conf.
**2012**, 37, 01017. [Google Scholar] [CrossRef] [Green Version] - Kamano, H.; Nakamura, S.X.; Lee, T.S.H.; Sato, T. Nucleon resonances within a dynamical coupled-channels model of πN and γN reactions. Phys. Rev. C
**2013**, 88, 035209. [Google Scholar] [CrossRef] [Green Version] - Crede, V.; Roberts, W. Progress towards understanding baryon resonances. Rept. Prog. Phys.
**2013**, 76, 076301. [Google Scholar] [CrossRef] [PubMed] - Mokeev, V.I.; Aznauryan, I.; Burkert, V.; Gothe, R. Recent results on the nucleon resonance spectrum and structure from the CLAS detector. EPJ Web Conf.
**2016**, 113, 01013. [Google Scholar] [CrossRef] [Green Version] - Anisovich, A.V.; Burkert, V.; Hadžimehmedović, H.; Ireland, D.G.; Klempt, E.; Nikonov, V.A.; Omerović, R.; Osmanović, H.; Sarantsev, A.V.; Švarc, A.; et al. Strong Evidence for Nucleon Resonances near 1900 MeV. Phys. Rev. Lett.
**2017**, 119, 062004. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Braun, V.M.; Collins, S.; Gläßle, B.; Göckeler, M.; Schäfer, A.; Schiel, R.W.; Södner, W.; Sternbeck, A.; Wein, P. Light-cone Distribution Amplitudes of the Nucleon and Negative Parity Nucleon Resonances from Lattice QCD. Phys. Rev. D
**2014**, 89, 094511. [Google Scholar] [CrossRef] [Green Version] - Bali, G.S.; Braun, V.M.; Göckeler, M.; Gruber, M.; Hutzler, F.; Schäfer, A.; Schiel, R.W.; Simeth, J.; Södner, W.; Sternbeck, A.; et al. Light-cone distribution amplitudes of the baryon octet. JHEP
**2016**, 02, 070. [Google Scholar] [CrossRef] [Green Version] - Braun, V.M.; Collins, S.; Göckeler, M.; Pérez-Rubio, P.; Schäfer, A.; Schiel, R.W.; Sternbeck, A. Second Moment of the Pion Light-cone Distribution Amplitude from Lattice QCD. Phys. Rev. D
**2015**, 92, 014504. [Google Scholar] [CrossRef] [Green Version] - Gao, F.; Chang, L.; Liu, Y.X. Bayesian extraction of the parton distribution amplitude from the Bethe-Salpeter wave function. Phys. Lett. B
**2017**, 770, 551–555. [Google Scholar] [CrossRef] - Zhang, J.H.; Chen, J.W.; Ji, X.; Jin, L.; Lin, H.W. Pion Distribution Amplitude from Lattice QCD. Phys. Rev. D
**2017**, 95, 094514. [Google Scholar] [CrossRef] [Green Version] - Zhang, J.H.; Jin, L.; Lin, H.W.; Schäfer, A.; Sun, P.; Yang, Y.B.; Zhang, R.; Zhao, Y.; Chen, J.W. Kaon Distribution Amplitude from Lattice QCD and the Flavor SU(3) Symmetry. Nucl. Phys. B
**2019**, 939, 429–446. [Google Scholar] [CrossRef] - Roper, L.D. Evidence for a P-11 Pion-Nucleon Resonance at 556 MeV. Phys. Rev. Lett.
**1964**, 12, 340–342. [Google Scholar] [CrossRef] - Bareyre, P.; Bricman, C.; Valladas, G.; Villet, G.; Bizard, J.; Seguinot, J. Pion-nucleon interactions between Tlab = 300 and Tlab = 700 MeV. Phys. Lett.
**1964**, 8, 137–141. [Google Scholar] [CrossRef] - Auvil, P.; Lovelace, C.; Donnachie, A.; Lea, A. Pion-nucleon phase shifts and resonances. Phys. Lett.
**1964**, 12, 76–80. [Google Scholar] [CrossRef] - Adelman, S.L. Evidence for an N
^{*}Resonance at 1425 MeV. Phys. Rev. Lett.**1964**, 13, 555–557. [Google Scholar] [CrossRef] - Roper, L.D.; Wright, R.M.; Feld, B.T. Energy-Dependent Pion-Nucleon Phase-Shift Analysis. Phys. Rev.
**1965**, 138, B190–B210. [Google Scholar] [CrossRef] - Li, B.L.; Chang, L.; Gao, F.; Roberts, C.D.; Schmidt, S.M.; Zong, H.S. Distribution amplitudes of radially-excited π and K mesons. Phys. Rev. D
**2016**, 93, 114033. [Google Scholar] [CrossRef] [Green Version] - Li, B.L.; Chang, L.; Ding, M.; Roberts, C.D.; Zong, H.S. Leading-twist distribution amplitudes of scalar- and vector-mesons. Phys. Rev. D
**2016**, 94, 094014. [Google Scholar] [CrossRef] [Green Version] - Jefferson Lab Hall A Collaboration. G
_{Ep}/G_{Mp}ratio by polarization transfer in e→p→ep→. Phys. Rev. Lett.**2000**, 84, 1398–1402. [Google Scholar] [CrossRef] [Green Version] - Cates, G.; de Jager, C.; Riordan, S.; Wojtsekhowski, B. Flavor decomposition of the elastic nucleon electromagnetic form factors. Phys. Rev. Lett.
**2011**, 106, 252003. [Google Scholar] [CrossRef] [Green Version] - Wilson, D.J.; Cloet, I.C.; Chang, L.; Roberts, C.D. Nucleon and Roper electromagnetic elastic and transition form factors. Phys. Rev. C
**2012**, 85, 025205. [Google Scholar] [CrossRef] [Green Version] - Sachs, R. High-Energy Behavior of Nucleon Electromagnetic Form Factors. Phys. Rev.
**1962**, 126, 2256–2260. [Google Scholar] [CrossRef] - Perdrisat, C.F.; Punjabi, V.; Vanderhaeghen, M. Nucleon electromagnetic form factors. Prog. Part. Nucl. Phys.
**2007**, 59, 694–764. [Google Scholar] [CrossRef] [Green Version] - Akhiezer, A.; Rekalo, M. Polarization effects in the scattering of leptons by hadrons. Sov. J. Part. Nucl.
**1974**, 4, 277. [Google Scholar] - Arnold, R.; Carlson, C.E.; Gross, F. Polarization Transfer in Elastic electron Scattering from Nucleons and Deuterons. Phys. Rev. C
**1981**, 23, 363. [Google Scholar] [CrossRef] [Green Version] - Jefferson Lab Hall A Collaboration. Measurement of G(E(p))/G(M(p)) in e→p→ep→ to Q
^{2}=5.6GeV^{2}. Phys. Rev. Lett.**2002**, 88, 092301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Punjabi, V.; Perdrisat, C.F.; Aniol, K.A.; Baker, F.T.; Berthot, J.; Bertin, P.Y.; Bertozzi, W.; Besson, A.; Bimbot, L.; Boeglin, W.U.; et al. Proton elastic form factor ratios to Q
^{2}=3.5GeV^{2}by polarization transfer. Phys. Rev. C**2005**, 71, 055202. [Google Scholar] [CrossRef] [Green Version] - Puckett, A.J.R.; Brash, E.J.; Gayou, O.; Jones, M.K.; Pentchev, L.; Perdrisat, C.F.; Punjabi, V.; Aniol, V.; Averett, T.; Benmokhtar, F.; et al. Final Analysis of Proton Form Factor Ratio Data at Q
^{2}= 4.0, 4.8 and 5.6 GeV^{2}. Phys. Rev. C**2012**, 85, 045203. [Google Scholar] [CrossRef] [Green Version] - Puckett, A.J.R.; Brash, E.J.; Jones, M.K.; Luo, W.; Meziane, M.; Pentchev, L.; Perdrisat, C.F.; Punjabi, V.; Wesselmann, F.R.; Ahmidouch, A.; et al. Recoil Polarization Measurements of the Proton Electromagnetic Form Factor Ratio to Q
^{2}=8.5GeV^{2}. Phys. Rev. Lett.**2010**, 104, 242301. [Google Scholar] [CrossRef] [Green Version] - Kelly, J.J. Simple parametrization of nucleon form factors. Phys. Rev. C
**2004**, 70, 068202. [Google Scholar] [CrossRef] - Bradford, R.; Bodek, A.; Budd, H.S.; Arrington, J. A New parameterization of the nucleon elastic form-factors. Nucl. Phys. Proc. Suppl.
**2006**, 159, 127–132. [Google Scholar] [CrossRef] [Green Version] - Gutiérrez-Guerrero, L.X.; Bashir, A.; Cloet, I.C.; Roberts, C.D. Pion form factor from a contact interaction. Phys. Rev. C
**2010**, 81, 065202. [Google Scholar] [CrossRef] [Green Version] - Frank, M.; Jennings, B.; Miller, G. The Role of color neutrality in nuclear physics: Modifications of nucleonic wave functions. Phys. Rev. C
**1996**, 54, 920–935. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chang, L.; Liu, Y.X.; Roberts, C.D. Dressed-quark anomalous magnetic moments. Phys. Rev. Lett.
**2011**, 106, 072001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kallidonis, C.; Syritsyn, S.; Engelhardt, M.; Green, J.; Meinel, S.; Negele, J.; Pochinsky, A. Nucleon electromagnetic form factors at high Q
^{2}from Wilson-clover fermions. PoS**2018**, LATTICE2018, 125. [Google Scholar] - E93-038 Collaboration. Measurements of GEn/GMn from the
^{2}H(e→,e^{′}n→) reaction to Q^{2}=1.45GeV/c)^{2}. Phys. Rev. Lett.**2003**, 91, 122002. [Google Scholar] [CrossRef] [Green Version] - Riordan, S.; Abrahamyan, S.; Craver, B.; Kelleher, A.; Kolarkar, A.; Miller, J.; Cates, G.D.; Liyanage, N.; Wojtsekhowski, B.; Acha, A.; et al. Measurements of the Electric Form Factor of the Neutron up to Q
^{2}= 3.4 GeV^{2}using the Reaction^{3}He→(e→,e^{′}n)pp. Phys. Rev. Lett.**2010**, 105, 262302. [Google Scholar] [CrossRef] [Green Version] - Schlessinger, L.; Schwartz, C. Analyticity as a Useful Computation Tool. Phys. Rev. Lett.
**1966**, 16, 1173–1174. [Google Scholar] [CrossRef] - Schlessinger, L. Use of Analyticity in the Calculation of Nonrelativistic Scattering Amplitudes. Phys. Rev.
**1968**, 167, 1411–1423. [Google Scholar] [CrossRef] - Tripolt, R.A.; Haritan, I.; Wambach, J.; Moiseyev, N. Threshold energies and poles for hadron physical problems by a model-independent universal algorithm. Phys. Lett. B
**2017**, 774, 411–416. [Google Scholar] [CrossRef] - Binosi, D.; Chang, L.; Ding, M.; Gao, F.; Papavassiliou, J.; Roberts, C.D. Distribution Amplitudes of Heavy-Light Mesons. Phys. Lett. B
**2019**, 790, 257–262. [Google Scholar] [CrossRef] - Xu, Y.Z.; Binosi, D.; Cui, Z.F.; Li, B.L.; Roberts, C.D.; Xu, S.S.; Zong, H.S. Elastic electromagnetic form factors of vector mesons. Phys. Rev. D
**2019**, 100, 114038. [Google Scholar] [CrossRef] [Green Version] - HAPPEX Collaboration. Parity-violating electron scattering from
^{4}He and the strange electric form-factor of the nucleon. Phys. Rev. Lett.**2006**, 96, 022003. [Google Scholar] [CrossRef] [PubMed] - G0 Collaboration. Strange quark contributions to parity-violating asymmetries in the forward G0 electron-proton scattering experiment. Phys. Rev. Lett.
**2005**, 95, 092001. [Google Scholar] [CrossRef] [PubMed] - Arrington, J.; Melnitchouk, W.; Tjon, J.A. Global analysis of proton elastic form factor data with two-photon exchange corrections. Phys. Rev. C
**2007**, 76, 035205. [Google Scholar] [CrossRef] [Green Version] - Wojtsekhowski, B.; Cates, G.D.; Riordan, S. Measurement of the Neutron Electromagnetic Form Factor Ratio Gen/GMn at High Q2; Jefferson Lab 12 GeV Experiment: E12-09-016; 2009; Approved. Available online: https://www.jlab.org/exp_prog/proposals/09/PR12-09-016.pdf (accessed on 22 August 2020).

**Figure 1.**Masses of pseudoscalar and vector mesons, and ground-state positive-parity octet and decuplet baryons calculated using continuum (Cont${}^{\mathrm{m}}$—squares, red) [14] and lattice (lQCD—circles, blue) [15] methods in QCD compared with experiment [13] (PDG—black bars, with decay-widths of unstable states shaded in grey). The continuum study did not include isospin symmetry breaking effects, which are evidently small, as highlighted by the empirically determined $\mathsf{\Sigma}$-$\mathsf{\Lambda}$ mass difference (<7%).

**Figure 2.**Solid black curve within grey band—RGI PI running-coupling, $\widehat{\alpha}\left({k}^{2}\right)/\pi $, computed in [34] (Cui et al. 2020), and dot-dashed green curve—earlier result (R-Q et al. 2018) [33]. (The grey band bordered by dashed curves indicates the uncertainty in the result arising from that in both continuum and lattice-QCD inputs and is detailed in [34].) For comparison, world data on the process-dependent charge, ${\alpha}_{{g}_{1}}$, defined via the Bjorken sum rule, are also depicted. (The data sources are listed elsewhere [34]. For additional details, see e.g., in [102,103,104].) The k-axis scale is linear to the left of the vertical partition, and logarithmic otherwise. The vertical line, $k={m}_{G}$, marks the gauge sector screening mass, Equation (12).

**Figure 3.**Renormalisation-group-invariant dressed-quark mass function, $M\left(p\right)$ in Equation (13): solid curves—gap equation solutions [121,122], “data”—numerical simulations of lQCD [123], available for current-quark masses $m=30,70$ MeV. QCD’s current-quark evolves into a constituent-quark as its momentum becomes smaller. The constituent-quark mass arises from a cloud of low-momentum gluons attaching themselves to the current-quark. This is DCSB, the essentially non-perturbative effect that generates a quark mass from nothing; namely, it occurs even in the chiral limit. Notably, the size of $M\left(0\right)$ is a measure of the magnitude of the QCD scale anomaly in $n=1$-point Schwinger functions [124]. Moreover, experiments on ${Q}^{2}\in [0,12]\phantom{\rule{0.166667em}{0ex}}$GeV${}^{2}$ at the modern Thomas Jefferson National Accelerator Facility (JLab) will be sensitive to the momentum dependence of $M\left(p\right)$ within a domain that is here indicated approximately by the shaded region.

**Figure 4.**Twist-two pion PDA at the hadronic scale, ${\zeta}_{H}$. Solid blue curve, Equation (22)—determined using the most sophisticated available symmetry preserving DSE kernels; dot-dashed green curve—original prediction from the work in [134]. These PDA results are consistent with contemporary lQCD results [135,136]. The dashed orange curve is ${\phi}_{\pi}^{\mathrm{asy}}\left(x\right)=6x(1-x)$, the limiting form under QCD evolution [137,138,139]. The PDAs are dimensionless.

**Figure 5.**${Q}^{2}{F}_{\pi}\left({Q}^{2}\right)$. Solid curve (black)—theoretical prediction [131,157,158]; dashed curve (blue)—pQCD prediction computed with the modern, dilated pion PDA described in Section 5.1; and dotted (red) curve—pQCD prediction computed with the asymptotic profile, ${\phi}_{\pi}^{\mathrm{asy}}\left(x\right)$, which had previously been used to guide expectations for the asymptotic behaviour of ${Q}^{2}{F}_{\pi}\left({Q}^{2}\right)$. The filled-circles and -squares represent existing JLab data [154] and the filled diamonds and triangle, whose normalisations are arbitrary, indicate the projected ${Q}^{2}$-reach and accuracy of forthcoming experiments [159,160].

**Figure 6.**Pion valence-quark momentum distribution function, $x{u}^{\pi}(x;{\zeta}_{5})$: dot-dot-dashed (grey) curve within shaded band—lQCD result [175]; long-dashed (black) curve—early continuum analysis [179]; and solid (blue) curve within shaded band—modern, continuum calculation [181,182]. From the works in [181,182]: gluon momentum distribution in pion, $x{g}^{\pi}(x;{\zeta}_{5})$—dashed (green) curve within shaded band; and sea-quark momentum distribution, $x{S}^{\pi}(x;{\zeta}_{5})$—dot-dashed (red) curve within shaded band. (The shaded bands indicate the size of calculation-specific uncertainties, detailed in the source material [175,181,182].) Data (purple) from in [189], rescaled according to the analysis in [186].

**Figure 7.**Poincaré covariant Faddeev equation: a homogeneous linear integral equation for $\Psi $, the matrix-valued Faddeev amplitude for a baryon of total momentum $P={p}_{q}+{p}_{d}$, which expresses the relative momentum correlation between the dressed-quarks and -diquarks within the baryon. The shaded rectangle demarcates the kernel of the Faddeev equation: single line, dressed-quark propagator; $\mathsf{\Gamma}$, diquark correlation amplitude; and double line, diquark propagator.

**Figure 8.**Baryon leading-twist dressed-quark distribution amplitudes depicted using barycentric plots. Left panel—asymptotic profile, baryon PDA, ${\phi}_{N}^{\mathrm{asy}}\left(\left[x\right]\right)=120{x}_{1}{x}_{2}{x}_{3}$; middle panel—computed proton PDA evolved to $\zeta =2\phantom{\rule{0.166667em}{0ex}}$GeV, which peaks at $\left(\right[x\left]\right)=(0.55,0.23,0.22)$; and right panel—Roper resonance PDA. The white circle in each panel marks the centre of mass for ${\phi}_{N}^{\mathrm{asy}}\left(\left[x\right]\right)$, whose peak lies at $\left(\right[x\left]\right)=(1/3,1/3,1/3)$. Here, ${x}_{1},{x}_{2},{x}_{3}$ indicate the fraction of the bound-state’s light-front momentum carried by the associated quark; naturally, ${x}_{1}+{x}_{2}+{x}_{3}=1$. The amplitudes are dimensionless; hence, the height is simply a real number.

**Figure 9.**Ratios of Sach’s form factors, ${\mu}_{N}{G}_{E}^{N}\left({Q}^{2}\right)/{G}_{M}^{N}\left({Q}^{2}\right)$. Upper panels— Proton. Left, [213] calculation compared with data (red up-triangles [238]; green squares [245]; blue circles [246]; black down-triangles [247]; and cyan diamonds [248]); right, compared with available lQCD results, drawn from Ref. [254]. Lower panels—Neutron. Left, comparison with data (blue circles [255] and green squares [256]); right, with available lQCD results, drawn from the work in [254]. In all panels, the $1\sigma $ band for the SPM approximants is shaded in light blue.

**Figure 10.**Data: neutron electric form factor—blue circles [256], and proton electric form factor—green squares [264]. Associated curves depict least-squares fits obtained via a jackknife analysis of the depicted ${G}_{E}^{n}$, ${G}_{E}^{p}$ data. Vertical black line within grey bands marks the boundary of the domain described by Equation (38).

**Table 1.**Computed values of the first four moments of the proton and Roper-resonance PDAs. The error on ${f}_{N}$, a dynamically-determined quantity which measures the proton’s “wave function at the origin”, reflects a nucleon scalar diquark content of $65\pm 5$%, and values in rows marked with “$\not\supset \mathrm{av}$” were obtained assuming the baryon is constituted solely from a scalar diquark. Including axial-vector diquark correlations, the mean absolute relative difference between continuum and lattice results improves by 36%. (All results listed at a renormalisation scale $\zeta =2\phantom{\rule{0.166667em}{0ex}}$GeV.)

${10}^{3}{\mathit{f}}_{\mathit{N}}/{\mathbf{GeV}}^{2}$ | ${\langle {\mathit{x}}_{1}\rangle}_{\mathit{u}}$ | ${\langle {\mathit{x}}_{2}\rangle}_{\mathit{u}}$ | ${\langle {\mathit{x}}_{3}\rangle}_{\mathit{d}}$ | |
---|---|---|---|---|

asymptotic PDA | $0.333\phantom{\left(99\right)}$ | $0.333\phantom{\left(9\right)}$ | $0.333\phantom{\left(9\right)}$ | |

lQCD [225] | $2.84\phantom{\rule{3.33333pt}{0ex}}\left(33\right)$ | $0.372\phantom{\rule{3.33333pt}{0ex}}\left(7\right)\phantom{9}$ | 0.314 (3) | 0.314 (7) |

lQCD [226] | $3.60\phantom{\rule{3.33333pt}{0ex}}\left(6\right)\phantom{9}$ | $0.358\phantom{\rule{3.33333pt}{0ex}}\left(6\right)\phantom{9}$ | $0.319\phantom{\rule{3.33333pt}{0ex}}\left(4\right)$ | 0.323 (6) |

DSE proton [203] | $3.78\phantom{\rule{3.33333pt}{0ex}}\left(14\right)$ | $0.379\phantom{\rule{3.33333pt}{0ex}}\left(4\right)\phantom{9}$ | 0.302 (1) | 0.319 (3) |

DSE proton $\not\supset \mathrm{av}$ | $2.97\phantom{\left(17\right)}$ | $0.412\phantom{\left(17\right)}$ | $0.295\phantom{\left(7\right)}$ | $0.293\phantom{\left(7\right)}$ |

DSE Roper [203] | $5.17\phantom{\rule{3.33333pt}{0ex}}\left(32\right)$ | $0.245\phantom{\rule{3.33333pt}{0ex}}\left(13\right)$ | $0.363\phantom{\rule{3.33333pt}{0ex}}\left(6\right)$ | $0.392\phantom{\rule{3.33333pt}{0ex}}\left(6\right)$ |

DSE Roper $\not\supset \mathrm{av}$ | $2.63\phantom{\left(14\right)}$ | $0.010\phantom{\left(19\right)}$ | $0.490\phantom{\left(9\right)}$ | $0.500\phantom{\left(9\right)}$ |

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Roberts, C.D.
Empirical Consequences of Emergent Mass. *Symmetry* **2020**, *12*, 1468.
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Empirical Consequences of Emergent Mass. *Symmetry*. 2020; 12(9):1468.
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2020. "Empirical Consequences of Emergent Mass" *Symmetry* 12, no. 9: 1468.
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