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Hadron Structure Using Continuum Schwinger Function Methods

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Abstract

The vast bulk of visible mass emerges from nonperturbative dynamics within quantum chromodynamics—the strong interaction sector of the Standard Model. The past decade has revealed the three pillars that support this emergent hadron mass (EHM); namely, a nonzero gluon mass-scale, a process-independent effective charge, and dressed-quarks with constituent-like masses. Theory is now working to expose their manifold and diverse expressions in hadron observables and highlighting the types of measurements that can be made in order to validate the paradigm. In sketching some of these developments, this discussion stresses the role of EHM in forming nucleon electroweak structure and the wave functions of excited baryons through the generation of dynamical diquark correlations; producing and constraining the dilation of the leading-twist pion distribution amplitude; shaping pion and nucleon parton distribution functions—valence, glue and sea, including the asymmetry of antimatter; and moulding pion and proton charge and mass distributions.

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References

  1. F. Englert, Nobel lecture: the BEH mechanism and its scalar boson. Rev. Mod. Phys. 86, 843 (2014)

    ADS  MATH  Google Scholar 

  2. P.W. Higgs, Nobel lecture: evading the goldstone theorem. Rev. Mod. Phys. 86, 851 (2014)

    ADS  Google Scholar 

  3. C.D. Roberts, S.M. Schmidt, Reflections upon the emergence of hadronic mass. Eur. Phys. J. ST 229(22–23), 3319–3340 (2020)

    Google Scholar 

  4. C.D. Roberts, Empirical consequences of emergent mass. Symmetry 12, 1468 (2020)

    ADS  Google Scholar 

  5. C.D. Roberts, On mass and matter. AAPPS Bull. 31, 6 (2021)

    Google Scholar 

  6. C.D. Roberts, D.G. Richards, T. Horn, L. Chang, Insights into the emergence of mass from studies of pion and kaon structure. Prog. Part. Nucl. Phys. 120, 103883 (2021)

    Google Scholar 

  7. D. Binosi, Emergent hadron mass in strong dynamics. Few Body Syst. 63(2), 42 (2022)

    ADS  MathSciNet  Google Scholar 

  8. Roberts, C.D.: Origin of the Proton Mass. arXiv:2301.07777 [hep-ph]. EPJ Web of Conferences (2023)

  9. J. Papavassiliou, Emergence of mass in the gauge sector of QCD. Chin. Phys. C 46(11), 112001 (2022)

    ADS  Google Scholar 

  10. M. Ding, C.D. Roberts, S.M. Schmidt, Emergence of hadron mass and structure. Particles 6(1), 57–120 (2023)

    Google Scholar 

  11. M.N. Ferreira, J. Papavassiliou, Gauge sector dynamics in QCD. Particles 6(1), 312–363 (2023)

    Google Scholar 

  12. D.S. Carman, R.W. Gothe, V.I. Mokeev, C.D. Roberts, Nucleon resonance electroexcitation amplitudes and emergent hadron mass. Particles 6(1), 416–439 (2023)

    Google Scholar 

  13. V.V. Flambaum, A. Höll, P. Jaikumar, C.D. Roberts, S.V. Wright, Sigma terms of light-quark hadrons. Few Body Syst. 38, 31–51 (2006)

    ADS  Google Scholar 

  14. J. Elvira, M. Hoferichter, B. Kubis, U.-G. Meißner, Extracting the -term from low-energy pion-nucleon scattering. J. Phys. G 45(2), 024001 (2018)

    ADS  Google Scholar 

  15. S. Aoki et al., FLAG review 2019. Eur. Phys. J. C 80, 113 (2020)

    ADS  Google Scholar 

  16. R.L. Workman et al., Review of particle physics. PTEP 2022, 083-01 (2022)

  17. A.C. Aguilar, F. De Soto, M.N. Ferreira, J. Papavassiliou, J. Rodríguez-Quintero, S. Zafeiropoulos, Gluon propagator and three-gluon vertex with dynamical quarks. Eur. Phys. J. C 80, 154 (2020)

    ADS  Google Scholar 

  18. T. Blum et al., Domain wall QCD with physical quark masses. Phys. Rev. D 93, 074505 (2016)

    ADS  Google Scholar 

  19. P.A. Boyle et al., Low energy constants of SU(2) partially quenched chiral perturbation theory from N\(_f\)=2+1 domain wall QCD. Phys. Rev. D 93, 054502 (2016)

    ADS  Google Scholar 

  20. P.A. Boyle, L. Del Debbio, A. Jüttner, A. Khamseh, F. Sanfilippo, J.T. Tsang, The decay constants \({\mathbf{f_D} }\) and \({\mathbf{f_{D_{s}}} }\) in the continuum limit of \({\mathbf{N_f=2+1} }\) domain wall lattice QCD. JHEP 12, 008 (2017)

    ADS  Google Scholar 

  21. D. Binosi, L. Chang, J. Papavassiliou, S.-X. Qin, C.D. Roberts, Natural constraints on the gluon-quark vertex. Phys. Rev. D 95, 031501 (2017)

    ADS  Google Scholar 

  22. T. Horn, C.D. Roberts, The pion: an enigma within the standard model. J. Phys. G. 43, 073001 (2016)

    ADS  Google Scholar 

  23. Aguilar, A.C., et al.: Pion and Kaon structure at the electron-ion collider. Eur. Phys. J. A 55, 190 (2019) [nucl-ex]

  24. X. Chen, F.-K. Guo, C.D. Roberts, R. Wang, Selected science opportunities for the EicC. Few Body Syst. 61, 43 (2020)

    ADS  Google Scholar 

  25. D.P. Anderle et al., Electron-ion collider in China. Front. Phys. (Beijing) 16(6), 64701 (2021)

    ADS  Google Scholar 

  26. J. Arrington et al., Revealing the structure of light pseudoscalar mesons at the electron–ion collider. J. Phys. G 48, 075106 (2021)

    ADS  Google Scholar 

  27. C. Quintans, The new AMBER experiment at the CERN SPS. Few Body Syst. 63(4), 72 (2022)

    ADS  Google Scholar 

  28. J.S. Schwinger, Gauge invariance and mass. Phys. Rev. 125, 397–398 (1962)

    ADS  MathSciNet  MATH  Google Scholar 

  29. J.S. Schwinger, Gauge Invariance and Mass. 2. Phys. Rev. 128, 2425–2429 (1962)

    ADS  MathSciNet  MATH  Google Scholar 

  30. J.M. Cornwall, Dynamical mass generation in continuum QCD. Phys. Rev. D 26, 1453 (1982)

    ADS  Google Scholar 

  31. J.E. Mandula, M. Ogilvie, The gluon is massive: a lattice calculation of the gluon propagator in the Landau gauge. Phys. Lett. B 185, 127–132 (1987)

    ADS  Google Scholar 

  32. D. Binosi, C. Mezrag, J. Papavassiliou, C.D. Roberts, J. Rodríguez-Quintero, Process-independent strong running coupling. Phys. Rev. D 96, 054026 (2017)

    ADS  Google Scholar 

  33. Z.-F. Cui, J.-L. Zhang, D. Binosi, F. Soto, C. Mezrag, J. Papavassiliou, C.D. Roberts, J. Rodríguez-Quintero, J. Segovia, S. Zafeiropoulos, Effective charge from lattice QCD. Chin. Phys. C 44, 083102 (2020)

    ADS  Google Scholar 

  34. C. Itzykson, J.-B. Zuber, Quantum Field Theory (McGraw-Hill Inc., New York, 1980)

    MATH  Google Scholar 

  35. J.C. Ward, An identity in quantum electrodynamics. Phys. Rev. 78, 182 (1950)

    ADS  MathSciNet  MATH  Google Scholar 

  36. Y. Takahashi, On the generalized ward identity. Nuovo Cim. 6, 371–375 (1957)

    ADS  MathSciNet  MATH  Google Scholar 

  37. Deur, A., Brodsky, S.J., Roberts, C.D.: QCD running couplings and effective charges. arXiv:2303.00723 [hep-ph] (2023)

  38. A. Deur, V. Burkert, J.P. Chen, W. Korsch, Experimental determination of the QCD effective charge \(\alpha _{g_1}(Q)\). Particles 5(2), 171–179 (2022)

    Google Scholar 

  39. J.M. Cornwall, J. Papavassiliou, Gauge invariant three gluon vertex in QCD. Phys. Rev. D 40, 3474 (1989)

    ADS  Google Scholar 

  40. A. Pilaftsis, Generalized pinch technique and the background field method in general gauges. Nucl. Phys. B 487, 467–491 (1997)

    ADS  Google Scholar 

  41. D. Binosi, J. Papavassiliou, Pinch technique: theory and applications. Phys. Rep. 479, 1–152 (2009)

    ADS  MathSciNet  Google Scholar 

  42. J.M. Cornwall, J. Papavassiliou, D. Binosi, The Pinch Technique and Its Applications to Non-Abelian Gauge Theories (Cambridge University Press, UK, 2010)

    MATH  Google Scholar 

  43. L.F. Abbott, The background field method beyond one loop. Nucl. Phys. B 185, 189 (1981)

    ADS  Google Scholar 

  44. L.F. Abbott, Introduction to the background field method. Acta Phys. Polon. B 13, 33 (1982)

    MathSciNet  Google Scholar 

  45. S.J. Brodsky, R. Shrock, Maximum wavelength of confined quarks and gluons and properties of quantum chromodynamics. Phys. Lett. B 666, 95–99 (2008)

    ADS  Google Scholar 

  46. F. Gao, S.-X. Qin, C.D. Roberts, J. Rodríguez-Quintero, Locating the Gribov horizon. Phys. Rev. D 97, 034010 (2018)

    ADS  Google Scholar 

  47. Y.L. Dokshitzer, Perturbative QCD theory (includes our knowledge of \(\alpha (s)\)) - hep-ph/9812252. In: High-energy Physics. in Proceedings, 29th International Conference, ICHEP’98, Vancouver, Canada, July 23–29, 1998. vol. 1, 2, pp. 305–324 (1998)

  48. E. Rutherford, Collision of alpha particles with light atoms I. hydrogen. Philos. Mag. xxxvii, 537–561 (1919)

  49. E. Rutherford, Collision of alpha particles with light atoms ii. velocity of the hydrogen atoms. Philos. Mag. xxxvii, 562–571 (1919)

  50. E. Rutherford, Collision of alpha particles with light atoms III. nitrogen and oxygen atoms. Philos. Mag. xxxvii, 571–580 (1919)

  51. E. Rutherford, Collision of alpha particles with light atoms iv. an anomalous effect in nitrogen. Philos. Mag. xxxvii, 581–587 (1919)

  52. R.T. Cahill, C.D. Roberts, J. Praschifka, Baryon structure and QCD. Austral. J. Phys. 42, 129–145 (1989)

    ADS  Google Scholar 

  53. H. Reinhardt, Hadronization of quark flavor dynamics. Phys. Lett. B 244, 316–326 (1990)

    ADS  Google Scholar 

  54. G.V. Efimov, M.A. Ivanov, V.E. Lyubovitskij, Quark-diquark approximation of the three quark structure of baryons in the quark confinement model. Z. Phys. C 47, 583–594 (1990)

    ADS  Google Scholar 

  55. G. Eichmann, R. Alkofer, A. Krassnigg, D. Nicmorus, Nucleon mass from a covariant three-quark Faddeev equation. Phys. Rev. Lett. 104, 201601 (2010)

    ADS  Google Scholar 

  56. G. Eichmann, H. Sanchis-Alepuz, R. Williams, R. Alkofer, C.S. Fischer, Baryons as relativistic three-quark bound states. Prog. Part. Nucl. Phys. 91, 1–100 (2016)

    ADS  Google Scholar 

  57. H. Sanchis-Alepuz, R. Alkofer, C.S. Fischer, Electromagnetic transition form factors of baryons in the space-like momentum region. Eur. Phys. J. A 54, 41 (2018)

    ADS  Google Scholar 

  58. S.-X. Qin, C.D. Roberts, S.M. Schmidt, Poincaré-covariant analysis of heavy-quark baryons. Phys. Rev. D 97, 114017 (2018)

    ADS  Google Scholar 

  59. Q.-W. Wang, S.-X. Qin, C.D. Roberts, S.M. Schmidt, Proton tensor charges from a Poincaré-covariant Faddeev equation. Phys. Rev. D 98, 054019 (2018)

    ADS  Google Scholar 

  60. S.-X. Qin, C.D. Roberts, S.M. Schmidt, Spectrum of light- and heavy-baryons. Few Body Syst. 60, 26 (2019)

    ADS  Google Scholar 

  61. R.T. Cahill, C.D. Roberts, J. Praschifka, Calculation of diquark masses in QCD. Phys. Rev. D 36, 2804 (1987)

    ADS  Google Scholar 

  62. M.Y. Barabanov et al., Diquark correlations in hadron physics: origin, impact and evidence. Prog. Part. Nucl. Phys. 116, 103835 (2021)

    Google Scholar 

  63. M. Oettel, M. Pichowsky, L. Smekal, Current conservation in the covariant quark-diquark model of the nucleon. Eur. Phys. J. A 8, 251–281 (2000)

    ADS  Google Scholar 

  64. C. Chen, C.S. Fischer, C.D. Roberts, J. Segovia, Form factors of the nucleon axial current. Phys. Lett. B 815, 136150 (2021)

    Google Scholar 

  65. C. Chen, C.S. Fischer, C.D. Roberts, J. Segovia, Nucleon axial-vector and pseudoscalar form factors and PCAC relations. Phys. Rev. D 105(9), 094022 (2022)

    ADS  Google Scholar 

  66. U. Mosel, Neutrino interactions with nucleons and nuclei: importance for long-baseline experiments. Ann. Rev. Nucl. Part. Sci. 66, 171–195 (2016) [nucl-th]

  67. L. Alvarez-Ruso, et al.: NuSTEC white paper: status and challenges of neutrino–nucleus scattering. Prog. Part. Nucl. Phys. 100, 1–68 (2018) [hep-ph]

  68. R.J. Hill, P. Kammel, W.J. Marciano, A. Sirlin, Nucleon axial radius and muonic hydrogen: a new analysis and review. Rep. Prog. Phys. 81, 096301 (2018) [hep-ph]

  69. A. Lovato, J. Carlson, S. Gandolfi, N. Rocco, R. Schiavilla, Ab initio study of \(\varvec {(\nu _\ell ,\ell ^-)}\) and \(\varvec {(\overline{\nu }_\ell ,\ell ^+)}\) inclusive scattering in \(^{12}\)C: confronting the MiniBooNE and T2K CCQE data. Phys. Rev. X 10, 031068 (2020) [nucl-th]

  70. G.B. King, L. Andreoli, S. Pastore, M. Piarulli, R. Schiavilla, R.B. Wiringa, J. Carlson, S. Gandolfi, Chiral effective field theory calculations of weak transitions in light nuclei. Phys. Rev. C 102(2), 025501 (2020) [nucl-th]

  71. A.S. Meyer, M. Betancourt, R. Gran, R.J. Hill, Deuterium target data for precision neutrino-nucleus cross sections. Phys. Rev. D 93, 113015 (2016)

    ADS  Google Scholar 

  72. A. Del Guerra, A. Giazotto, M.A. Giorgi, A. Stefanini, D.R. Botterill, H.E. Montgomery, P.R. Norton, G. Matone, Threshold \(\pi ^+\) electroproduction at high momentum transfer: a determination of the nucleon axial vector form-factor. Nucl. Phys. B 107, 65–81 (1976)

    ADS  Google Scholar 

  73. Y.-C. Jang, R. Gupta, B. Yoon, T. Bhattacharya, Axial vector form factors from lattice QCD that satisfy the PCAC relation. Phys. Rev. Lett. 124, 072002 (2020)

    ADS  Google Scholar 

  74. C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, A. Vaquero Aviles-Casco, Nucleon axial form factors using \(N_f\) = 2 twisted mass fermions with a physical value of the pion mass. Phys. Rev. D 96, 054507 (2017)

    ADS  Google Scholar 

  75. I.V. Anikin, V.M. Braun, N. Offen, Axial form factor of the nucleon at large momentum transfers. Phys. Rev. D 94(3), 034011 (2016)

    ADS  Google Scholar 

  76. I.V. Anikin, V.M. Braun, N. Offen, Nucleon form factors and distribution amplitudes in QCD. Phys. Rev. D 88, 114021 (2013)

    ADS  Google Scholar 

  77. C. Chen, C.D. Roberts, Nucleon axial form factor at large momentum transfers. Eur. Phys. J. A 58, 206 (2022)

    ADS  Google Scholar 

  78. K. Park et al., Measurement of the generalized form factors near threshold via \(\gamma ^* p \rightarrow n\pi ^+\) at high \(Q^2\). Phys. Rev. C 85, 035208 (2012)

    ADS  Google Scholar 

  79. L. Chang, C.D. Roberts, S.M. Schmidt, Dressed-quarks and the nucleon’s axial charge. Phys. Rev. C 87, 015203 (2013)

    ADS  Google Scholar 

  80. P. Cheng, F.E. Serna, Z.-Q. Yao, C. Chen, Z.-F. Cui, C.D. Roberts, Contact interaction analysis of octet baryon axial-vector and pseudoscalar form factors. Phys. Rev. D 106(5), 054031 (2022)

    ADS  Google Scholar 

  81. P. Maris, C.D. Roberts, \(\pi \) and \(K\) meson Bethe–Salpeter amplitudes. Phys. Rev. C 56, 3369–3383 (1997)

    ADS  Google Scholar 

  82. M.S. Bhagwat, A. Krassnigg, P. Maris, C.D. Roberts, Mind the gap. Eur. Phys. J. A 31, 630–637 (2007)

    ADS  Google Scholar 

  83. Z.-N. Xu, Z.-Q. Yao, S.-X. Qin, Z.-F. Cui, C.D. Roberts, Bethe–Salpeter kernel and properties of strange-quark mesons. Eur. Phys. J. A 59(3), 39 (2023)

    ADS  Google Scholar 

  84. L. Liu, C. Chen, Y. Lu, C.D. Roberts, J. Segovia, Composition of low-lying \(J=\tfrac{3}{2}^\pm \)\(\Delta \)-baryons. Phys. Rev. D 105(11), 114047 (2022)

    ADS  Google Scholar 

  85. L. Liu, C. Chen, C.D. Roberts, Wave functions of \((I, J^P)=(\tfrac{1}{2},\tfrac{3}{2}^\mp )\) baryons. Phys. Rev. D 107(1), 014002 (2023)

    ADS  Google Scholar 

  86. G. Eichmann, D. Nicmorus, Nucleon to Delta electromagnetic transition in the Dyson–Schwinger approach. Phys. Rev. D 85, 093004 (2012)

    ADS  Google Scholar 

  87. J. Segovia, I.C. Cloet, C.D. Roberts, S.M. Schmidt, Nucleon and \(\Delta \) elastic and transition form factors. Few Body Syst. 55, 1185–1222 (2014)

    ADS  Google Scholar 

  88. Y. Lu, C. Chen, Z.-F. Cui, C.D. Roberts, S.M. Schmidt, J. Segovia, H.S. Zong, Transition form factors: \(\gamma ^\ast + p \rightarrow \Delta (1232)\), \(\Delta (1600)\). Phys. Rev. D 100, 034001 (2019)

    ADS  Google Scholar 

  89. G. Eichmann, C.S. Fischer, H. Sanchis-Alepuz, Light baryons and their excitations. Phys. Rev. D 94, 094033 (2016)

    ADS  Google Scholar 

  90. V.D. Burkert, R. De Vita, M. Battaglieri, M. Ripani, V. Mokeev, Single quark transition model analysis of electromagnetic nucleon resonance transitions in the [70,1-] supermultiplet. Phys. Rev. C 67, 035204 (2003)

    ADS  Google Scholar 

  91. Dugger, M., et al.:\(\pi ^+\) photoproduction on the proton for photon energies from 0.725 to 2.875-GeV. Phys. Rev. C 79, 065206 (2009)

  92. V.I. Mokeev, I.G. Aznauryan, Studies of \(N^*\) structure from the CLAS meson electroproduction data. Int. J. Mod. Phys. Conf. Ser. 26, 1460080 (2014)

    Google Scholar 

  93. P. Maris, C.D. Roberts, P.C. Tandy, Pion mass and decay constant. Phys. Lett. B 420, 267–273 (1998)

    ADS  Google Scholar 

  94. S.-X. Qin, C.D. Roberts, S.M. Schmidt, Ward–Green–Takahashi identities and the axial-vector vertex. Phys. Lett. B 733, 202–208 (2014)

    ADS  MATH  Google Scholar 

  95. A.P. Bakulev, S.V. Mikhailov, A.V. Pimikov, N.G. Stefanis, Comparing antithetic trends of data for the pion-photon transition form factor. Phys. Rev. D 86, 031501 (2012)

    ADS  Google Scholar 

  96. K. Raya, L. Chang, A. Bashir, J.J. Cobos-Martinez, L.X. Gutiérrez-Guerrero, C.D. Roberts, P.C. Tandy, Structure of the neutral pion and its electromagnetic transition form factor. Phys. Rev. D 93, 074017 (2016)

    ADS  Google Scholar 

  97. H.J. Behrend et al., A measurement of the \(\pi ^0\), \(\eta \) and \(\eta ^\prime \) electromagnetic form-factors. Z. Phys. C 49, 401–410 (1991)

    Google Scholar 

  98. J. Gronberg et al., Measurements of the meson photon transition form factors of light pseudoscalar mesons at large momentum transfer. Phys. Rev. D 57, 33–54 (1998)

    ADS  Google Scholar 

  99. S. Uehara et al., Measurement of \(\gamma \gamma ^* \rightarrow \pi ^0\) transition form factor at Belle. Phys. Rev. D 86, 092007 (2012)

    ADS  Google Scholar 

  100. L. Chang, I.C. Cloet, J.J. Cobos-Martinez, C.D. Roberts, S.M. Schmidt, P.C. Tandy, Imaging dynamical chiral symmetry breaking: pion wave function on the light front. Phys. Rev. Lett. 110, 132001 (2013)

    ADS  Google Scholar 

  101. G.P. Lepage, S.J. Brodsky, Exclusive processes in quantum chromodynamics: evolution equations for hadronic wave functions and the form-factors of mesons. Phys. Lett. B 87, 359–365 (1979)

    ADS  Google Scholar 

  102. A.V. Efremov, A.V. Radyushkin, Factorization and asymptotical behavior of pion form-factor in QCD. Phys. Lett. B 94, 245–250 (1980)

    ADS  Google Scholar 

  103. G.P. Lepage, S.J. Brodsky, Exclusive processes in perturbative quantum chromodynamics. Phys. Rev. D 22, 2157–2198 (1980)

    ADS  Google Scholar 

  104. V.L. Chernyak, A.R. Zhitnitsky, Asymptotic behavior of exclusive processes in QCD. Phys. Rep. 112, 173 (1984)

    ADS  Google Scholar 

  105. S.J. Brodsky, F.-G. Cao, G.F. Teramond, Evolved QCD predictions for the meson-photon transition form factors. Phys. Rev. D 84, 033001 (2011)

    ADS  Google Scholar 

  106. A. Höll, A. Krassnigg, P. Maris, C.D. Roberts, S.V. Wright, Electromagnetic properties of ground and excited state pseudoscalar mesons. Phys. Rev. C 71, 065204 (2005)

    ADS  Google Scholar 

  107. B.-L. Li, L. Chang, F. Gao, C.D. Roberts, S.M. Schmidt, H.-S. Zong, Distribution amplitudes of radially-excited \(\pi \) and \(K\) mesons. Phys. Rev. D 93(11), 114033 (2016)

    ADS  Google Scholar 

  108. S.J. Brodsky, G.F. Teramond, Hadronic spectra and light-front wavefunctions in holographic QCD. Phys. Rev. Lett. 96, 201601 (2006)

    ADS  Google Scholar 

  109. H.-M. Choi, C.-R. Ji, Distribution amplitudes and decay constants for (\(\pi \), \(K\), \(\rho \), \(K^\ast \)) mesons in light-front quark model. Phys. Rev. D 75, 034019 (2007)

    ADS  Google Scholar 

  110. T. Zhong, Z.-H. Zhu, H.-B. Fu, X.-G. Wu, T. Huang, Improved light-cone harmonic oscillator model for the pionic leading-twist distribution amplitude. Phys. Rev. D 104(1), 016021 (2021)

    ADS  Google Scholar 

  111. H.-N. Li, Dispersive derivation of the pion distribution amplitude. Phys. Rev. D 106(3), 034015 (2022)

    ADS  MathSciNet  Google Scholar 

  112. V.M. Braun, S. Collins, M. Göckeler, P. Pérez-Rubio, A. Schäfer, R.W. Schiel, A. Sternbeck, Second moment of the pion light-cone distribution amplitude from lattice QCD. Phys. Rev. D 92(1), 014504 (2015)

    ADS  Google Scholar 

  113. G.S. Bali, V.M. Braun, S. Bürger, M. Göckeler, M. Gruber, F. Hutzler, P. Korcyl, A. Schäfer, A. Sternbeck, P. Wein, Light-cone distribution amplitudes of pseudoscalar mesons from lattice QCD. JHEP 08, 065 (2019). [Addendum: JHEP 11, 037 (2020)]

  114. H.-N. Li, Y.-L. Shen, Y.-M. Wang, Joint resummation for pion wave function and pion transition form factor. JHEP 01, 004 (2014)

    ADS  Google Scholar 

  115. G.S. Bali, V.M. Braun, B. Gläßle, M. Göckeler, M. Gruber, F. Hutzler, P. Korcyl, A. Schäfer, P. Wein, J.-H. Zhang, Pion distribution amplitude from Euclidean correlation functions: exploring universality and higher-twist effects. Phys. Rev. D 98(9), 094507 (2018)

    ADS  MathSciNet  Google Scholar 

  116. S.-S. Xu, L. Chang, C.D. Roberts, H.-S. Zong, Pion and kaon valence-quark parton quasidistributions. Phys. Rev. D 97, 094014 (2018)

    ADS  Google Scholar 

  117. Y. Lu, L. Chang, K. Raya, C.D. Roberts, J. Rodríguez-Quintero, Proton and pion distribution functions in counterpoint. Phys. Lett. B 830, 137130 (2022)

    Google Scholar 

  118. S.J. Brodsky, M. Burkardt, I. Schmidt, Perturbative QCD constraints on the shape of polarized quark and gluon distributions. Nucl. Phys. B 441, 197–214 (1995)

    ADS  Google Scholar 

  119. F. Yuan, Generalized parton distributions at \(x \rightarrow 1\). Phys. Rev. D 69, 051501 (2004)

    ADS  Google Scholar 

  120. Z.-F. Cui, M. Ding, J.M. Morgado, K. Raya, D. Binosi, L. Chang, J. Papavassiliou, C.D. Roberts, J. Rodríguez-Quintero, S.M. Schmidt, Concerning pion parton distributions. Eur. Phys. J. A 58(1), 10 (2022)

    ADS  Google Scholar 

  121. Z.-F. Cui, M. Ding, J.M. Morgado, K. Raya, D. Binosi, L. Chang, F. De Soto, C.D. Roberts, J. Rodríguez-Quintero, S.M. Schmidt, Emergence of pion parton distributions. Phys. Rev. D 105(9), 091502 (2022)

    ADS  Google Scholar 

  122. L. Chang, F. Gao, C.D. Roberts, Parton distributions of light quarks and antiquarks in the proton. Phys. Lett. B 829, 137078 (2022) [hep-ph]

  123. Y.L. Dokshitzer, Calculation of the structure functions for deep inelastic scattering and \(e^+\)\(e^-\) annihilation by perturbation theory in quantum chromodynamics. (in Russian). Sov. Phys. JETP 46, 641–653 (1977)

  124. V.N. Gribov, L.N. Lipatov, Deep inelastic electron scattering in perturbation theory. Phys. Lett. B 37, 78–80 (1971)

    ADS  Google Scholar 

  125. L.N. Lipatov, The parton model and perturbation theory. Sov. J. Nucl. Phys. 20, 94–102 (1975)

    Google Scholar 

  126. G. Altarelli, G. Parisi, Asymptotic freedom in parton language. Nucl. Phys. B 126, 298–318 (1977)

    ADS  Google Scholar 

  127. Y. Yu, M. Ding, C.D. Roberts, Contact interaction analysis of proton polarised parton distribution functions—-in progress (2023)

  128. L. Chang, C.D. Roberts, Regarding the distribution of glue in the pion. Chin. Phys. Lett. 38(8), 081101 (2021)

    ADS  Google Scholar 

  129. Z. Fan, H.-W. Lin, Gluon parton distribution of the pion from lattice QCD. Phys. Lett. B 823, 136778 (2021)

    Google Scholar 

  130. R.D. Ball, et al.: Parton distributions from high-precision collider data. Eur. Phys. J. C 77(10), 663 (2017) [hep-ph]

  131. S.J. Brodsky, P. Hoyer, C. Peterson, N. Sakai, The Intrinsic Charm of the Proton. Phys. Lett. B 93, 451–455 (1980)

    ADS  Google Scholar 

  132. R.D. Ball, A. Candido, J. Cruz-Martinez, S. Forte, T. Giani, F. Hekhorn, K. Kudashkin, G. Magni, J. Rojo, Evidence for intrinsic charm quarks in the proton. Nature 608(7923), 483–487 (2022)

    ADS  Google Scholar 

  133. C.D. Roberts, R.J. Holt, S.M. Schmidt, Nucleon spin structure at very high \(x\). Phys. Lett. B 727, 249–254 (2013)

    ADS  Google Scholar 

  134. S. Tkachenko, et al.: Measurement of the structure function of the nearly free neutron using spectator tagging in inelastic \(^2\)H(e, e’p)X scattering with CLAS. Phys. Rev. C 89, 045206 (2014). [Addendum: Phys. Rev. C 90 (2014) 059901]

  135. D. Abrams et al., Measurement of the nucleon \(F^n_2/F^p_2\) structure function ratio by the Jefferson Lab MARATHON Tritium/helium-3 deep inelastic scattering experiment. Phys. Rev. Lett. 128(13), 132003 (2022)

    ADS  Google Scholar 

  136. A. Bodek, M. Breidenbach, D.L. Dubin, J.E. Elias, J.I. Friedman, H.W. Kendall, J.S. Poucher, E.M. Riordan, M.R. Sogard, D.H. Coward, Comparisons of deep inelastic \(e p\) and \(e n\) cross-sections. Phys. Rev. Lett. 30, 1087 (1973)

    ADS  Google Scholar 

  137. J.S. Poucher et al., High-energy single-arm inelastic \(e p\) and \(e d\) scattering at 6-degrees and 10-degrees. Phys. Rev. Lett. 32, 118 (1974)

    ADS  Google Scholar 

  138. L.W. Whitlow, E.M. Riordan, S. Dasu, S. Rock, A. Bodek, Precise measurements of the proton and deuteron structure functions from a global analysis of the SLAC deep inelastic electron scattering cross-sections. Phys. Lett. B 282, 475–482 (1992)

    ADS  Google Scholar 

  139. I.R. Afnan, F.R.P. Bissey, J. Gomez, A.T. Katramatou, W. Melnitchouk, G.G. Petratos, A.W. Thomas, Neutron structure function and \(A = 3\) mirror nuclei. Phys. Lett. B 493, 36–42 (2000)

    ADS  Google Scholar 

  140. E. Pace, G. Salme, S. Scopetta, A. Kievsky, Neutron structure function \(F_2^n(x)\) from deep inelastic electron scattering off few nucleon systems. Phys. Rev. C 64, 055203 (2001)

    ADS  Google Scholar 

  141. E.P. Segarra, A. Schmidt, T. Kutz, D.W. Higinbotham, E. Piasetzky, M. Strikman, L.B. Weinstein, O. Hen, Neutron valence structure from nuclear deep inelastic scattering. Phys. Rev. Lett. 124, 092002 (2020)

    ADS  Google Scholar 

  142. E. Pace, M. Rinaldi, G. Salmè, S. Scopetta, The European muon collaboration effect in light-front hamiltonian dynamics. Phys. Lett. B 839, 137810 (2023)

    Google Scholar 

  143. S. Amoroso et al., Snowmass 2021 whitepaper: proton structure at the precision frontier. Acta Phys. Polon. B 53(12), 1 (2022)

    ADS  MathSciNet  Google Scholar 

  144. G.R. Farrar, D.R. Jackson, Pion and nucleon structure functions near \(x=1\). Phys. Rev. Lett. 35, 1416 (1975)

    ADS  Google Scholar 

  145. S.J. Brodsky, G.P. Lepage, Perturbative quantum chromodynamics. Prog. Math. Phys. 4, 255–422 (1979)

    Google Scholar 

  146. S.-S. Xu, C. Chen, I.C. Cloet, C.D. Roberts, J. Segovia, H.-S. Zong, Contact-interaction Faddeev equation and, inter alia, proton tensor charges. Phys. Rev. D 92, 114034 (2015) [nucl-th]

  147. J. Dove et al., The asymmetry of antimatter in the proton. Nature 590(7847), 561–565 (2021)

    ADS  Google Scholar 

  148. Z.-F. Cui, F. Gao, D. Binosi, L. Chang, C.D. Roberts, S.M. Schmidt, Valence quark ratio in the proton. Chin. Phys. Lett. Express 39(04), 041401 (2022)

    ADS  Google Scholar 

  149. R.D. Field, R.P. Feynman, Quark elastic scattering as a source of high transverse momentum mesons. Phys. Rev. D 15, 2590–2616 (1977)

    ADS  Google Scholar 

  150. K. Gottfried, Sum rule for high-energy electron-proton scattering. Phys. Rev. Lett. 18, 1174 (1967)

    ADS  Google Scholar 

  151. R. Brock, et al. Handbook of perturbative QCD: Version 1.0. Rev. Mod. Phys. 67, 157–248 (1995)

  152. P. Amaudruz et al., The Gottfried sum from the ratio \(F_2^n/F_2^p\). Phys. Rev. Lett. 66, 2712–2715 (1991)

    ADS  Google Scholar 

  153. M. Arneodo et al., A reevaluation of the Gottfried sum. Phys. Rev. D 50, 1–3 (1994)

    ADS  Google Scholar 

  154. A. Baldit et al., Study of the isospin symmetry breaking in the light quark sea of the nucleon from the Drell–Yan process. Phys. Lett. B 332, 244–250 (1994)

    ADS  Google Scholar 

  155. R.S. Towell et al., Improved measurement of the \({\bar{d}}/{\bar{u}}\) asymmetry in the nucleon sea. Phys. Rev. D 64, 052002 (2001)

    ADS  Google Scholar 

  156. T.-J. Hou et al., New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC. Phys. Rev. D 103(1), 014013 (2021)

    ADS  Google Scholar 

  157. J.S. Conway et al., Experimental study of muon pairs produced by 252-GeV pions on tungsten. Phys. Rev. D 39, 92–122 (1989)

    ADS  Google Scholar 

  158. S.R. Amendolia et al., A measurement of the pion charge radius. Phys. Lett. B 146, 116 (1984)

    ADS  Google Scholar 

  159. S.R. Amendolia et al., A measurement of the space-like pion electromagnetic form-factor. Nucl. Phys. B 277, 168 (1986)

    ADS  Google Scholar 

  160. J. Volmer, et al. Measurement of the charged pion electromagnetic form-factor. Phys. Rev. Lett. 86, 1713–1716 (2001) [nucl-ex]

  161. T. Horn, et al. Determination of the charged pion form factor at \(Q^2=1.60\) and \(2.45 \,({\rm GeV/c})^2\). Phys. Rev. Lett. 97, 192001 (2006)

  162. V. Tadevosyan, et al. Determination of the pion charge form factor for \(Q^2=0.60- 1.60\,{\rm GeV}^2\). Phys. Rev. C 75, 055205 (2007)

  163. H.P. Blok, et al. Charged pion form factor between \(Q^2\)=0.60 and 2.45 GeV\(^2\). I. Measurements of the cross section for the \({^1}\)H(\(e,e^{\prime }\pi ^+\))\(n\) reaction. Phys. Rev. C 78, 045202 (2008) [nucl-ex]

  164. G.M. Huber, et al. Charged pion form-factor between \(Q^2 = 0.60\,\)GeV\(^2\) and \(2.45\,\)GeV\(^2\). II. Determination of, and results for, the pion form-factor. Phys. Rev. C 78, 045203 (2008) [nucl-ex]

  165. Y.-Z. Xu, K. Raya, Z.-F. Cui, C.D. Roberts, J. Rodríguez-Quintero, Empirical determination of the pion mass distribution. Chin. Phys. Lett. Express 40(4), 041201 (2023)

    ADS  Google Scholar 

  166. Z.-N. Xu, Z.-F. Cui, C.D. Roberts, C. Xu, Heavy + light pseudoscalar meson semileptonic transitions. Eur. Phys. J. C 81(12), 1105 (2021)

    ADS  Google Scholar 

  167. H.-Y. Xing, Z.-N. Xu, Z.-F. Cui, C.D. Roberts, C. Xu, Heavy + heavy and heavy + light pseudoscalar to vector semileptonic transitions. Eur. Phys. J. C 82(10), 889 (2022)

    ADS  Google Scholar 

  168. Z.-Q. Yao, D. Binosi, Z.-F. Cui, C.D. Roberts, Semileptonic \(B_c \rightarrow \eta _c, J/\psi \) transitions. Phys. Lett. B 818, 136344 (2021)

    Google Scholar 

  169. Z.-Q. Yao, D. Binosi, Z.-F. Cui, C.D. Roberts, Semileptonic transitions: \(B_{(s)} \rightarrow \pi (K)\); \(D_s \rightarrow K\); \(D\rightarrow \pi, K\); and \(K\rightarrow \pi \). Phys. Lett. B 824, 136793 (2022)

    Google Scholar 

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Acknowledgements

This contribution is based on results obtained and insights developed through collaborations with many people, to all of whom I am greatly indebted. Work supported by National Natural Science Foundation of China (Grant No. 12135007).

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  1. School of Physics, Nanjing University, Nanjing, 210093, Jiangsu, China

    Craig D. Roberts

  2. Institute for Nonperturbative Physics, Nanjing University, Nanjing, 210093, Jiangsu, China

    Craig D. Roberts

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Roberts, C.D. Hadron Structure Using Continuum Schwinger Function Methods. Few-Body Syst 64, 51 (2023). https://doi.org/10.1007/s00601-023-01837-6

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