Skip to main content
SpringerLink
Log in

Dynamics of quarks and gauge fields in the lowest-energy states in QED and QCD interactions

  • Regular Article - Theoretical Physics
  • Published:
The European Physical Journal A Aims and scope Submit manuscript

Abstract

We examine the dynamics of quarks and gauge fields in the lowest energy states in the QED and QCD interactions by combining Schwinger’s longitudinal confinement in (1+1)D with Polyakov’s transverse confinement in (2+1)D in a “stretch (2+1)D” flux tube model in (3+1)D. For such QED and QCD systems in the flux tube configuration with cylindrical symmetry, we separate out the transverse and longitudinal degrees of freedom, approximate the non-Abelian QCD in the quasi-Abelian approximation, and solve the derived equations to study the collective excitations. We find stable collective QED and QCD excitations showing up as confined QED and QCD mesons, in support of previous theoretical studies and recent observations of the anomalous hypothetical X17 and E38 particles. Future theoretical lattice gauge calculations of QED in (3+1)D with the inclusion of the Schwinger longitudinal confinement mechanism and experimental confirmation of the hypothetical X17 and E38 particles will shed definitive light on quark confinement in the QED interaction in (3+1)D.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data availability

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: All data generated during this study are contained in this published article.]

Notes

  1. Because quarks and antiquarks cannot be isolated, the production of a \(q{\bar{q}}\) pair can occur only at certain CM eigenstate energies, \(\sqrt{s }(q{\bar{q}})\), of confined and bound \(q{\bar{q}}\) systems, such as in a QED meson or a QCD meson. At other energies different from confined and bound eigenstate energies, the q and \({\bar{q}}\) do not exist. Hence, a quark and an antiquark cannot be produced at energies that are not eigenenergies of bound QCD or QED mesons.

  2. We adopt here the notations that unless stated otherwise, \(g^\lambda \) is actually \(g_{{}_{\textrm{2D}}}^\lambda \), the coupling constant in 1+1 dimensions, and \(g_{{}_{\textrm{4D}}}^\lambda \) is the coupling constants in 3+1 dimensions. The superscript \(\lambda \) specifies the interaction, with \(\lambda =0\) for QED and \(\lambda =1\) for QCD. The strong coupling constant is \(\alpha _{{}^{\textrm{QCD}}}=(g^{{}_{\textrm{QCD}}}_{{}_{\textrm{4D}}})^2/4\pi \) and the fine structure constant is \(\alpha _c=\alpha _{{}^{\textrm{QED}}}=(g_{{}_{\textrm{4D}}}^{{}_{\textrm{QED}}})^2/4\pi =(-e_{{}_{\textrm{4D}}}^{{}_{\textrm{QED}}})^2/(\hbar c)=1/137\). As pointed out in [38, 41, 172, 173] and Eq. (67), \(g_{{}_{\textrm{2D}}}^\lambda \) and \(g_{{}_{\textrm{4D}}}^\lambda \) are approximately related by the flux tube radius \(R_T\) as \((g_{{}_{\textrm{2D}}}^\lambda )^2=(g_{{}_{\textrm{4D}}}^\lambda )^2/(\pi R_T^2)\), when the confining flux tube of radius \(R_T\) is idealized as an open string without a structure.

  3. There is a factor 2 on the left-hand side of the trace relation, \(2\textrm{tr} \{t^a t^b\} \,= \,\delta ^{ab}\), for the generator \(t^a\) and \(t^b\). For the convenience of notation, we shall define the operation “Tr trace” over the color space as \(Tr_\textrm{color} (t^a t^b)\) \( \equiv \)2tr\( \{t^a t^b\} \,= \,\delta ^{ab}\), when we calculate the trace of the product of generators \(t^a t^b\) in the color space.

References

  1. P.V. Chliapnikov et al., Observation of direct soft photon production in \(\pi ^- p\) interactions at 280 GeV/c. Phys. Lett. B 141, 276 (1984)

    ADS  Google Scholar 

  2. F. Botterweck et al., (EHS-NA22 Collaboration), Direct soft photon production in \(K^+p\) and \(\pi ^+p\) interactions at 250 GeV/c. Z. Phys. C 51, 541 (1991)

    ADS  Google Scholar 

  3. S. Banerjee et al., (SOPHIE/WA83 Collaboration), Observation of direct soft photon production in \(\pi ^- p\) interactions at 280 GeV/c. Phys. Lett. B 305, 182 (1993)

    ADS  Google Scholar 

  4. A. Belogianni et al., (WA91 Collaboration), Confirmation of a soft photon signal in excess of QED expectations in \(\pi ^- p\) interactions at 280 GeV/c. Phys. Lett. B 408, 487 (1997)

    ADS  Google Scholar 

  5. A. Belogianni et al., (WA102 Collaboration), Further analysis of a direct soft photon excess in pi- p interactions at 280-GeV/c. Phys. Lett. B 548, 122 (2002)

    ADS  Google Scholar 

  6. A. Belogianni et al., (WA102 Collaboration), Observation of a soft photon signal in excess of QED expectations in \(pp\) interactions. Phys. Lett. B 548, 129 (2002)

    ADS  Google Scholar 

  7. V. Perepelitsa, for the DELPHI Collaboration, Anomalous soft photons in hadronic decays of Z\(^0\), Proceedings of the XXXIX International Symposium on Multiparticle Dynamics, Gomel, Belarus, September 4-9, 2009, Nonlin. Phenom. Complex Syst. 12, 343 (2009)

  8. J. Abdallah et al., (DELPHI Collaboration), Evidence for an excess of soft photons in hadronic decays of Z\(^0\). Eur. Phys. J. C47, 273 (2006). [arXiv:hep-ex/0604038]

    ADS  Google Scholar 

  9. J. Abdallah et al., (DELPHI Collaboration), Observation of the muon inner bremsstrahlung at LEP1. Eur. Phys. J. C 57, 499 (2008). [arXiv:0901.4488]

    ADS  Google Scholar 

  10. J. Abdallah et al., (DELPHI Collaboration), Study of the dependence of direct soft photon production on the jet characteristics in hadronic Z\(^0\) decays. Eur. Phys. J. C 67, 343 (2010). [arXiv:1004.1587]

    ADS  Google Scholar 

  11. A.J. Krasznahorkay et al., Observation of anomalous internal pair creation in 8Be: a possible indication of a light, neutral boson. Phys. Rev. Lett. 116, 042501 (2016). [arXiv:1504.01527]

  12. A. J. Krasznahorkay et al., New evidence supporting the existence of the hypothetical X17 particle, arXiv:1910.10459 (2019)

  13. D.S. Firak, A.J. Krasznahorkay, M. Csatlós, L. Csige, J. Gulyás, Confirmation of the existence of the X17 particle. EPJ Web Conf. 232, 04005 (2020)

    Google Scholar 

  14. A.J. Krasznahorkay et al., New anomaly observed in He4 supports the existence of the hypothetical X17 particle. Phys. Rev. C 104, 044003 (2021). [arxiv:2104.10075]

  15. N.J. Sas, A.J. Krasznahorkay, M. Csatlós, J. Gulyás, B. Kertész, A. Krasznahorkay, J. Molnár, I. Rajta, J. Timár, I. Vajda, M.N. Harakeh, Observation of the X17 anomaly in the 7Li(p,\(e^+e^-\))8Be direct proton-capture reaction. [arXiv:2205.07744]

  16. A. J. Krasznahorkay et al., New anomaly observed in \(^{12}\)C supports the existence and the vector character of the hypothetical X17 boson. [arXiv:2209.10795]

  17. A.J. Krasznahorkay, X17: status of the experiments on 8 Be and 4 He, Talk presented at the Workshop on “Shedding Light on X17”, September 6, 2021, Rome, Italy, in Ref. [60]

  18. A.J. Krasznahorkay, An update on the hypothetical X17 particle, Talk presented at the International Symposium on Multiparticle Dynamics, at Gyöngyös, Hungary, August 20-26, 2023

  19. The-Anh Tran, Confirmation the \(^8\)Be anomaly with a different spectrometer, Talk presented at International Symposium on Multiparticle Dynamics, at Gyöngyös, Hungary, August 20-26, 2023

  20. Abraamyan, A. B. Anisimov, M. I. Baznat, K .K. Gudima, M. A. Nazarenko, S. G. Reznikov, A.S. Sorin, Observation of the E(38)-boson. arxiv:1208.3829v1 (2012)

  21. K. Abraamyan, C. Austin, M. Baznat, K. Gudima, M. Kozhin, S. Reznikov, A. Sorin, Check of the structure in photon pairs spectra at the invariant mass of about 38 MeV/\(c^2\). E PJ Web Conf. 204, 08004 (2019)

    Google Scholar 

  22. K. Abraamyan, C. Austin, M. Baznat, K. Gudima, M. Kozhin, S. Reznikov, A. Sorin, Observation of structures at \(\sim \)17 and \(\sim \)38 MeV/c\(^2\) in the diphoton invariant mass spectra in pC, dC, and dCu collisions at p\(_{lab}\) of a few GeV/c per nucleon, Talk presented at the International Symposium on Multiparticle Dynamics, at Gyöngyös, Hungary, August 20-26, 2023

  23. L. Van Hove, Cold quark-gluon plasma and multiparticle production. Ann. Phys. (N.Y.) 192, 66 (1989)

  24. P. Lichard, L. Van Hove, The cold quark-gluon plasma as a source of very soft photons in high energy collisions. Phys. Lett. B 245, 605 (1990)

    ADS  Google Scholar 

  25. P. Lichard, Consistency of data on soft photon production in hadronic interactions. Phys. Rev. D 50, 6824 (1994)

    ADS  Google Scholar 

  26. E. Kokoulina, A. Kutov, V. Nikitin, Braz. J. Phys. 37, 785 (2007)

    ADS  Google Scholar 

  27. M. Volkov, E. Kokoulina, E. Kuraev, Gluon dominance model and cluster production. Ukr. J. Phys. 49, 1252 (2003)

    Google Scholar 

  28. S. Barshay, Anomalous soft photons from a coherent hadronic phase in high-energy collisions. Phys. Lett. B 227, 279 (1989)

    ADS  Google Scholar 

  29. E. Shuryak, The soft photon puzzle\(\lesssim \) and pion modification in hadronic matter. Phys. Lett. B 231, 175 (1989)

    ADS  Google Scholar 

  30. V. Balek, N. Pisutova, J. Pisut, The puzzle of very soft photon production in hadronic Interactions. Acta. Phys. Pol. B21, 149 (1990)

    Google Scholar 

  31. W. Czyz, W. Florkowski, Soft photon production in the boost invariant color flux tube model. Z. Phys. C 61, 171 (1994)

    ADS  Google Scholar 

  32. O. Nachtmann, Nonperturbative QCD effects in high-energy collisions. arXiv: hep-ph/9411345

  33. G.W. Botz, P. Haberl, O. Nachtmann, Soft photons in hadron hadron collisions: Synchrotron radiation from the QCD vacuum? Z. Phys. C 67, 143 (1995)

    ADS  Google Scholar 

  34. P. Lebiedowicz, O. Nachtmann, A. Szczurek, Soft-photon radiation in high-energy proton-proton collisions within the tensor-Pomeron approach: Bremsstrahlung. Phys. Rev. D 106, 034023 (2022). arXiv:2206.03411

  35. Y. Hatta, T. Ueda, Soft photon anomaly and gauge/string duality. Nucl. Phys. B 837, 22 (2010). arXiv:1002.3452

    ADS  Google Scholar 

  36. S.M. Darbinian, K.A. Ispirian, A.T. Margarian, Unruh radiation of quarks and the soft photon puzzle in hadronic interactions. Sov. J. Nucl. Phys. 54, 364 (1991)

    Google Scholar 

  37. Y.A. Simonov, Di-pion decays of heavy quarkonium in the field correlator method. Phys. Atom. Nucl. 71, 1049 (2008). ArXiv: hep-ph/0711.3626

    ADS  Google Scholar 

  38. C.Y. Wong, Anomalous soft photons in hadron production. Phys. Rev. C 81, 064903 (2010). [arXiv:1001.1691]

    ADS  Google Scholar 

  39. C. Y. Wong, Anomalous soft photons associated with hadron production in string fragmentation, Talk presented at the IX International Conference on Quark Confinement and Hadron Spectrum, Madrid, Spain, Aug 30-Sept 3, 2010, AIP Conf.Proc.1343:447 (2011), [arXiv:1011.6265]

  40. C. Y. Wong, An overview of the anomalous soft photons in hadron production, Talk presented at International Conference on the Structure and the Interactions of the Photon, 20-24 May 2013, Paris, France, PoS Photon 2013, 002 (2014), [arXiv:1404.0040]

  41. C. Y. Wong, Open string QED meson description of the X17 particle and dark matter, JHEP (2020) 165, [arxiv:2001.04864]

  42. C.Y. Wong, On the stability of the open-string QED neutron and dark matter. Euro. Phys. J. A 58, 100 (2022). arxiv:2010.13948

  43. C. Y. Wong, QED mesons, the QED neutron, and the dark matter, in Proceedings of the 19th International Conference on Strangeness in Quark Matter, EPJ Web of Conferences 259, 13016 (2022). arXiv:2108.00959

  44. C. Y. Wong, QED meson description of the X17 and other anomalous particles, presented at the Workshop of “Shedding Light on X17”, September 6, 2021, Rome, Italy, [arXiv:2201.09764]

  45. C. Y. Wong, On the question of quark confinement in the QED interaction, in press in Frontiers of Physics, (2022), [arxiv:2208.09920]

  46. A. Koshelkin and C. Y. Wong, Dynamics of quarks and gauge fields in the lowest-energy states in QCD and QED, in Proceedings of the 41st International Conference in High Energy Physics, 6-13 July, 2022 Bologna, Italy, PoS 414, 302 (2022), [arXiv:2212.11749]

  47. D.E. Kharzeev, F. Loshaj, Anomalous soft photon production from the induced currents in Dirac sea. Phys. Rev. D 89, 074053 (2014)

    ADS  Google Scholar 

  48. J. Schwinger, Gauge invariance and mass II. Phys. Rev. 128, 2425 (1962)

    MathSciNet  ADS  Google Scholar 

  49. J. Schwinger, Gauge theory of vector particles, in Theoretical Physics, Trieste Lectures, (IAEA. Vienna) 1963, 89 (1962)

  50. W.T. Donnelly, S.J. Freedman, R.S. Lytel, R.D. Peccei, M. Schwartz, Do axions exist? Phys. Rev. D 18, 1607 (1978)

    ADS  Google Scholar 

  51. A. Vitéz, A. Krasznahorkay, J. Gulyás, M. Csatlós, L. Csige Z. Gácsi, A. Krasznahorkay Jr., B.M. Nyakó, F. W. N. de Boer, T.J. Ketel, J. van Klinken, Anomalous internal pair creation in \(^8\)Be as a signature of the decay of a new particle, Acta Phys. Polon. B39, 483 (2008)

  52. A. T. D’yachenko, and E. S. Gromova, Detection of particles of dark matter from the spectrum of secondary particles in high-energy proton-proton collisions in a thermodynamic model, Jour. of Physics, Conf. Series 2131, 022 (2021)

  53. A.T. D’yachenko, A.A. Verisokina, M.A. Verisokina, High-energy collisions of protons and nuclei and the possibility of detecting dark matter particles in the spectra of soft photons. Acta Phys. Pol. B Proc. Suppl. 14, 761 (2021)

  54. J. Bernhard and K. Schönning for the COMPASS Collaboration, Test of OZI violation in vector meson production with COMPASS, arxiv:1109.0272v2 (2011)

  55. J. Bernhard, Exclusive vector meson production in \(pp\) collisions at the COMPASS experiment, Ph. D. Thesis, University of Mainz, (2014)

  56. T. Schlüter for the COMPASS Collaboration, The exotic \(\eta \pi ^-\) wave in 190 GeV \(\pi ^- p \rightarrow \pi ^- \eta ^{\prime } p\) at COMPASS, arxiv:1108.6191v2 (2011)

  57. T. Schlüter, The \(\pi ^- \eta \) and \(\pi ^- \eta ^{\prime }\) systems in exclusive 190 GeV/c \(\pi ^- p\) Reactions at COMPASS Ph. D. Thesis, Univ. München, (2012)

  58. J. Bernhard, J. M. Friedrich, T. Schlüter, K. Schönning, Comment on “Material evidence of a 38 MeV boso”, arxiv:1204.2349 (2012)

  59. M. El-Nadi, O.E. Badawy, Production of a new light neutral boson in high-energy collisions. Phys. Rev. Lett. 61, 1271 (1988)

    ADS  Google Scholar 

  60. M. E. El-Nadi \(et.al\), External electron pair production in high-energy collisions, Nuo. Cim. A109A, 1517 (1996)

  61. P.L. Jain, G. Singh, Search for new particles decaying into electron pairs of mass below 100 MeV/c\(^2\). Phys. G: Nucl. Part. Phys. 34, 129 (2007)

    ADS  Google Scholar 

  62. F.W.N. de Boer, C.A. Fields, A re-evaluation of evidence for light neutral bosons in nuclear emulsions. Int. J. Mod. Phys. E 20, 1787 (2011). [arxiv:1001.3897]

    ADS  Google Scholar 

  63. See Proceedings of Workshop on “Shedding Light on X17”, September 6, 2021, Rome, Italy; Editors: M. Raggi, P. Valente, M. Nardecchia, A. Frankenthal, G. Cavoto, and references cited therein

  64. G. Xilin Zhang, A Miller, Can nuclear physics explain the anomaly observed in the internal pair production in the Beryllium-8 nucleus? Phys. Lett. B 773, 159 (2017). arXiv:1703.04588

  65. D.S.M. Alves, N.J. Weiner, A viable QCD axion in the MeV mass range. JHEP 07, 92 (2018)

    ADS  Google Scholar 

  66. J. Feng et al., Protophobic fifth force interpretation of the observed anomaly in \(^8\)Be nuclear transitions, Phys. Rev. Lett. 2016 117, 071803 (2016)

  67. J. Feng et al., Particle physics models for the 17 MeV anomaly in beryllium nuclear decays. Phys. Rev. D 95, 035017 (2017)

    ADS  Google Scholar 

  68. J. Batley et al., (NA48/2 Collaboration), Search for the dark photon in \(\pi ^0\) decays. Phys. Lett. B 746, 178 (2015). arXiv:1504.00607

  69. B. Fornal, Is there a sign of new physics in beryllium transitions? Int. J. Mod. Phys. A 32, 1730020 (2017)

    ADS  Google Scholar 

  70. J. Bordes, H. M. Chan, Tsou Sheung Tsun, Accommodating three low-scale anomalies (g-2, Lamb shift, and Atomki) in the framed standard model, Int.J.Mod.Phys.A 34 (2019) 25, 1830034 (2019) and references cited therein

  71. J. Bordes, H. M. Chan, S. T. Tsou, Accommodating three low-scale anomalies (g-2, Lamb shift, and Atomki) in the framed standard model, Int.J.Mod.Phys.A 34 (2019) 25, 1830034 (2019) and references cited therein

  72. H. M. Chan, S. T. Tsou, Two variations on the theme of Yang and Mills – the SM and the FSM Invited contribution to the “Festschrift for the Yang Centenary”. edited FC Chen et al., [arXiv:2201.12256] (2022)

  73. Luigi Delle Rose et al., Explanation of the 17 MeV Atomki anomaly in a U(1)-extended two Higgs doublet model. Phys. Rev. D 96, 115024 (2017)

  74. Lugi Delle Rose et al., Atomki anomaly in family-dependent U(1) extension of the standard model. Phys. Rev. D 99, 055022 (2019)

  75. Lugi Delle Rose et al., New physics suggested by Atomki anomaly. Frontiers in Physics 7, 73 (2019)

  76. V. Kubarovsky, J. Rittenhouse West, S. J. Brodsky Quantum Chromodynamics Resolution of the ATOMKI Anomaly in 4He Nuclear Transitions, [arXiv:2206.14441]

  77. U. Ellwanger, S. Moretti, Possible explanation of the electron positron anomaly at 17 MeV in \(^8\)Be transitions through a light pseudoscalar. JHEP 11, 39 (2016)

    Google Scholar 

  78. D. Banerjee et al., (NA64 Collaboration), Search for a hypothetical 16.7 MeV gauge boson and dark photons in the NA64 Experiment at CERN, Phys. Rev. Lett. 120, 231802 (2018), arXiv:1803.07748

  79. D. Banerjee et al. (NA64 Collaboration), Search for vector mediator of dark matter production in invisible decay mode Phys. Rev. D 97, 072002 (2018), arXiv:1710.00971

  80. D. Banerjee et al., (NA64 Collaboration), Improved limits on a hypothetical X(16.7) boson and a dark photon decaying into \(e^+e^-\) pairs, arXiv:1912.11389

  81. C. Taruggi, A. Ghoshal, M. Raggi (for the PADME Collaboration), Searching for dark photons with the PADME experiment, (Conference: C18-05-07.4, p.17-21, p.28-34, and 337-344), Frascati Phys. Ser. 67, 17, 28, and 334 (2018)

  82. M. Viviani, L. Girlanda, A. Kievsky, L.E. Marcucci, n + 3 H, p + 3 He, p + 3 H, and n + 3 He scattering with the hyperspherical harmonic method. Phys. Rev. C 102, 034007 (2020)

    ADS  Google Scholar 

  83. M. Viviani, E. Filandri, L. Girlanda, C. Gustavino, A. Kievsky, L.E. Marcucci, R. Schiavilla, X 17 boson and the 3 H (p, e+ e? ) 4 He and 3 He(n, e+ e? ) 4 He processes: A theoretical analysis. Phys. Rev. C 105, 014001 (2022)

    ADS  Google Scholar 

  84. Kh.U. Abraamyan, Ch. Austin, M.I. Baznat, K.K. Gudima, M.A. Kozhin, S.G. Reznikov, and A.S. Sorin, (Dubna Collboration), private communications

  85. Y.S. Cheng, H.Z. Huang and G. Wang (STAR Collaboration), private communications

  86. A. Papa, X17 search with the MEGII apparatus, Talk presented at the Workshop on “Shedding Light on X17”, September 6, 2021, Rome, Italy, in Ref. [60]

  87. H. N. da Luz, Measurements of Internal Pair Creation with a Time Projection Chamber-based setup, Talk presented at the Workshop on “Shedding Light on X17”, September 6, 2021, Rome, Italy, in Ref. [60]

  88. H. N. da Luz, The construction of the X17 spectrometer at CTU in Prague, Talk presented at International Symposium on Multiparticle Dynamics, at Gyöngyös, Hungary, August 20-26, 2023

  89. C. Gustavino, The search for 4 He anomaly at n_TOF experiment, Talk presented at the Workshop on “Shedding Light on X17”, September 6, 2021, Rome, Italy, in Ref. [60]

  90. E. Depero, X17 in the NA64 experiment, Talk presented at the Workshop on “Shedding Light on X17”, September 6, 2021, Rome, Italy, in Ref. [60]

  91. L. Darmé, M. Raggi, E. Nardi, X17 production mechanism at accelerators, Talk presented at the Workshop on “Shedding Light on X17”, September 6, 2021, Rome, Italy, in Ref. [60]

  92. E. Goudzovski, Search for dark photon in \(\pi \)0 decays by NA48/2 at CERN, Talk presented at the Workshop on “Shedding Light on X17”, September 6, 2021, Rome, Italy, in Ref. [60]

  93. A-K Perrevoort, Prospects for Dark Photon Searches in the Mu3e Experiment, Talk presented at the Workshop on “Shedding Light on X17”, September 6, 2021, Rome, Italy, in Ref. [60]

  94. L. Doria, Dark Matter and X17 Searches at MESA 4.4.2 Light Dark Matter, Talk presented at the Workshop on “Shedding Light on X17”, September 6, 2021, Rome, Italy, in Ref. [60]

  95. A. Gasparian, A Direct Detection Search for Hidden Sector New Particles in the 3 - 60 MeV Mass Range, Talk presented at the Workshop on “Shedding Light on X17”, September 6, 2021, Rome, Italy, in Ref. [60]

  96. A. Ahmidouch et al. (for the JLAB-PAC50 proposal), A Direct Detection Search for Hidden Sector New Particles in the 3-60 MeV Mass Range, [arxiv:2108.13276], (2021)

  97. V. Kozhuharov, Searching X17 with positrons at PADME, Talk presented at the Workshop on “Shedding Light on X17”, September 6, 2021, Rome, Italy, in Ref. [60]

  98. M. Raggi, Search for the resonant X17 boson production in PADME Run III, Talk presented at International Symposium on Multiparticle Dynamics, at Gyöngyös, Hungary, August 20-26, 2023

  99. E Cline et al., (for the DarkLight Collaboration), Searching for New Physics with DarkLight at the ARIEL Electron-Linac, arxiv: 2208.04120, (2022)

  100. P. Navrátil, ARIEL experiments and theory, [arxiv:2210.08438]

  101. V. Vivian et al., (for the DarkLight Collaboration), Searching for New Physics with DarkLight at the ARIEL Electron-Linac, arxiv: 2208.04120, (2022)

  102. Shan Huang (for the LUXE Collaboration), Probing new physics at the LUXE experiment, Proceedings of 41st International Conference on High Energy physics - ICHEP2022 6-13 July, 2022, [arXiv:2211.11045]

  103. J. Feng, Collider searches for X17 and other light gauge bosons, Talk presented at International Symposium on Multiparticle Dynamics, at Gyöngyös, Hungary, August 20-26, 2023

  104. Tibor Kibédi, Searching for the x17 using magnetic separation, Talk presented at International Symposium on Multiparticle Dynamics, at Gyöngyös, Hungary, August 20-26, 2023

  105. E. van Beveren, G. Rupp, First indications of the existence of a 38 MeV light scalar boson arxiv:1102.1863 (2011)

  106. E. van Beveren, G. Rupp, Material evidence of a 38 MeV boson, arxiv:1202.1739 (2012)

  107. E. van Beveren, G. Rupp, Reply to Comment on “Material evidence of a 38 MeV boson”, arxiv:1204.3287 (2012)

  108. E. van Beveren, G. Rupp, Z0(57) and E(38): possible surprises in the Standard Model, accepted for publication in Acta Physica Polonica B Proc. Suppl. arxiv:2005.08559 (2020)

  109. B. A Snook, Measurement of the v2 of \(\pi ^0\) mesons Produced in \(\sqrt{s_{\rm NN}}\)=2.76 TeV PbPb Collisions at the Large Hadron Collider, Ph. D. Thesis, Vanderbilt University, (2014), specifically, Fig. 5.8, CMS PbPb events at \(\sqrt{s_{\rm NN}}\)= 2.76 TeV, in the 40 -50% centrality class, with \(p_T\) in the range of 1.6 \({<} p_T {<} \) 2.0 GeV/c

  110. S. Chatrchyan et al. (CMS Collaboration), Measurement of the azimuthal anisotropy of neutral pions in Pb-Pb collisions at \(\sqrt{s_{NN}}\)=2.76 TeV , Phys. Rev. Lett. 110, 042301 (2013)

  111. G. Veneziano, Construction of a crossing-simmetric. Regge-behaved amplitude for linearly rising trajectories, Nuovo Cim 57A, 190 (1968)

    Google Scholar 

  112. Y. Nambu, Quark model of the factorization of the Veneziano Amplitude, in Lectures at the Copenhagen Symposium: Symmetry and Quark Models, edited by R. Chand, Gordon and Breach, 1970, p. 269

  113. Y. Nambu, Strings, monopoles, and gauge fields. Phys. Rev. D 10, 4262 (1974)

    ADS  Google Scholar 

  114. T. Goto, Relativistic quantum mechanics of one-dimensional mechanical continuum and subsidiary condition of dual resonance model arXiv:hep-th/9302104 Prog. Theo. Phys 46, 1560 (1971)

    ADS  Google Scholar 

  115. D.J. Gross, F. Wilczek, Ultraviolet Behavior of Non-Abelian Gauge Theories. Phys. Rev. Lett. 30, 1343 (1973)

    ADS  Google Scholar 

  116. H.D. Politzer, Reliable Perturbative Results for Strong Interactions? Phys. Rev. Lett. 30, 1346 (1973)

    ADS  Google Scholar 

  117. K.G. Wilson, Confinement of quarks. Phys. Rev. D 19, 2445 (1974)

    ADS  Google Scholar 

  118. G. ’t Hooft, A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461 (1974)

  119. G. ’t Hooft, A two-dimensional model for mesons. Nucl. Phys. B 75, 461 (1974)

  120. A. Casher, J. Kogut, L. Susskind, Vacuum polarization and the absence of free quarks. Phys. Rev. D 10, 732 (1974)

    ADS  Google Scholar 

  121. J. Kogut, L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D11, 395 (1975)

    ADS  Google Scholar 

  122. S. Mandelstam, Vortices and quark confinement in non-abelian gauge theories. Phys. Lett. 53B, 476 (1975)

    Google Scholar 

  123. A.M. Polyakov, Quark confinement and topology of gauge theories Nucl. Phys. B120, 429 (1977)

    Google Scholar 

  124. A.M. Polyakov, Quantum geometry of bosonic strings Phys. Lett. B103, 207 (1981)

    ADS  Google Scholar 

  125. A.M. Polyakov, Gauge Fields and Strings (Hardwood Academic Publishers, Switzerland, 1987)

    Google Scholar 

  126. T. Banks, B. Myerson, J. Kogut, Phase transitions in Abelian lattice gauge theories. Nucl. Phys. B 129, 493 (1977)

    ADS  Google Scholar 

  127. J. Glimm, A. Jaffe, Instantons in a U(1) lattice gauge theory: A Coulomb dipole gas. Comm. Math. Phys. 56, 195 (1977)

    MathSciNet  ADS  Google Scholar 

  128. M.E. Peskin, Mandelstam-’t Hooft duality in Abelian lattice models. Ann. Phys. (N. Y. ) 113, 122 (1978)

  129. S.D. Drell, H.R. Quinn, B. Svetitsky, M. Weinstein, Quantum electrodynamics on a lattice: A Hamiltonian variational approach to the physics of the weak-coupling region. Phys. Rev. D 19, 619 (1979)

    ADS  Google Scholar 

  130. A. Guth, Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory. Phys. Rev. D 21, 2291 (1980)

    MathSciNet  ADS  Google Scholar 

  131. K.-I. Kondo, Existence of confinement phase in quantum electrodynamics. Phys. Rev. D 58, 085013 (1998)

    ADS  Google Scholar 

  132. G. Arnold, B. Bunk, T. Lippert, K. Schilling, Compact QED under scrutiny: it’s first order. Nucl. Phys. Proc. Suppl. 119, 864 (2003). [arXiv:hep-lat/0210010]

    ADS  Google Scholar 

  133. L.C. Loveridge, O. Oliveira, P.J. Silva, Lattice pure gauge compact QED in the Landau gauge: photon propagator, phase structure, Dirac string. Phys. Rev. D 104, 114511 (2021). (and references cited therein)

  134. G. Magnifico, T. Felser, P. Silvi, S. Montangero, Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks. Nature Com. 12, 3600 (2021). [arXiv:2011.10658]

  135. H. Georgi, The Schwinger point, JHEP11 057, (2019) arxiv:1905.09632

  136. H. Georgi, B. Noether, Non-perturbative Effects and Unparticle Physics in Generalized Schwinger Models, arxiv:1908.03279v3

  137. H. Georgi, B. Warner, Generalizations of the Sommerfield and Schwinger models. JHEP 01, 047 (2020). arXiv:1907.12705v2

  138. H. Georgi, Automatic Fine-Tuning in the Two-Flavor Schwinger Model. Phys. Rev. Lett. 125, 181601 (2020). arXiv:2007.15965

  139. H. Georgi, Mass Perturbation Theory in the 2-Flavor Schwinger Model with Opposite Masses , JHEP (2022) 119, arxiv:2206.14691

  140. N. Dorey, T.J. Hollowood, D. Tong, The BPS spectra of gauge theories in two and four dimensions. JHEP 9905, 006 (1999). [arXiv:hep-th/9902134]

    ADS  Google Scholar 

  141. A. Hanany, D. Tong, Vortex strings and four-dimensional gauge dynamics. JHEP 0404, 066 (2004). [hep-th/0403158]

    MathSciNet  ADS  Google Scholar 

  142. D. Tong, Quantum Vortex Strings: A Review. Ann. Phys. 324, 30 (2009). [arxiv:0809.5060]

    MathSciNet  ADS  Google Scholar 

  143. M. Lüscher, Symmetry Breaking Aspects Of The roughening Transition In Gauge Theories. Nucl. Phys. B 180, 317 (1981)

    MathSciNet  ADS  Google Scholar 

  144. J. Polchinski, A. Strominger, Effective string theory. Phys. Rev. Lett. 67, 1681 (1991)

    MathSciNet  ADS  Google Scholar 

  145. J. Polchinski, Strings and QCD?, Contribution to: International Symposium on Black holes, Membranes, Wormholes and Superstrings, 220, (1992), [arxiv:hep-th/9210045]

  146. O. Aharony, E. Karzbrun, On the effective action of confining strings. JHEP 0906, 012 (2009). [arXiv:0903.1927 [hep-th]]

    MathSciNet  ADS  Google Scholar 

  147. O. Aharony, M. Field, On the effective theory of long open strings. JHEP 1101, 065 (2011). [arXiv:1008.2636 [hep-th]]

    MathSciNet  ADS  Google Scholar 

  148. O. Aharony, Z. Komargodski, The Effective Theory of Long Strings. JHEP 05, 118 (2013). [arxiv:1302.6157]

    MathSciNet  ADS  Google Scholar 

  149. S. Dubovsky, R. Flauger, V. Gorbenko, Effective String Theory Revisited. JHEP 1209, 044 (2012). [arXiv:1203.1054

    Google Scholar 

  150. R.C. Wang, C.Y. Wong, Finite-size effect in the Schwinger particle-production mechanism. Phys. Rev. D 38, 348 (1988). (and references cited therein)

    ADS  Google Scholar 

  151. K. Sailer, T. Schonfeld, A. Schafer, B. Miiller, W. Greiner, Transversesize effects in the fragmentation of hadronic strings. Phys. Lett. B 240, 381 (1990)

    ADS  Google Scholar 

  152. Th. Schonfeld, A. Schafer, B. Miiller, K. Sailer, Reinhardt, W. Greiner, Pair creation in a flux tube, Phys. Lett. B 247, 5 (1990)

  153. H.P. Pavel, D.M. Brink, \(q{\bar{q}}\) pair creation with flux tube confinement. Z. Phys. C 51, 119 (1991)

    Google Scholar 

  154. C.Y. Wong, R.C. Wang, J.S. Wu, Schwinger particle-production with transverse confinement for a finite-length flux tube. Phys. Rev. D 51, 3940 (1995)

    ADS  Google Scholar 

  155. M. Gell-Mann, The interpretation of the new particles as displaced charge multiplets. Nuo. Cim. 4, 848 (1956)

    MathSciNet  Google Scholar 

  156. R.L. Workman et al. (Particle Data Group 2022), Prog. Theor. Exp. Phys. 2022, 083C01 (2022), and M. Tanabashi et al. (Particle Data Group 2019), Phys. Rev. D98, 030001 (2019)

  157. S. Coleman, R. Jackiw, L. Suskind, Charge shielding and quark confinement in the massive Schwinger model. Ann. Phys. 93, 267 (1975)

    ADS  Google Scholar 

  158. S. Coleman, More about the massive Schwinger model. Ann. Phys. 101, 239 (1976)

    ADS  Google Scholar 

  159. J. Bender, E. Zohar, A gauge redundancy-free formulation of compact QED with dynamical matter for quantum and classical computations. Phys. Rev. D 102, 114517 (2020). [arXiv:2008.01349]

  160. C. W. Bauer, D. M. Grabowska, Efficient representation for simulating U(1) gauge theories on digital quantum computers at all values of the coupling, [arxiv:2111.08015]

  161. D. B. Kaplan, J. R. Stryker Gauss’s law, duality, and the Hamiltonian formulation of U(1) lattice gauge theory, Phys. Rev. D102, 094515 (2020)

  162. G.S. Bali, H. Neff, T. Duessel, T. Lippert, K. Schilling (SESAM), Observing long colour flux tubes in SU(2) lattice gauge theory, Phys.Rev.D 71,114513(2005), arXiv: hep-lat/0505012

  163. A. Amado, N. Cardoso, P. Bicudo Flux tube widening in compact U(1) lattice gauge theory computed at T \({<}\) Tc with the multilevel method and GPUs, [arxiv:1309.3859]

  164. N. Cardoso, M. Cardoso, P. Bicudo, Phys.Rev.D 88 (2013) 054504, Inside the SU(3) quark-antiquark QCD flux tube: screening versus quantum widening, [arxiv:1302.3633]

  165. L. Cosmai, P. Cea, F. Cuteri, A. Papa, Flux tubes in QCD with (2+1) HISQ fermions, Pos, 4th annual International Symposium on Lattice Field Theory 24-30 July 2016 University of Southampton, UK, arxiv:1701.03371 (2017)

  166. P. Bicudo, N. Cardoso, M. Cardoso, Pure gauge QCD Flux Tubes and their widths at Finite Temperature. Nucl. Phys. B 940, 88 (2019). [arxiv:1702.03454]

  167. P. Bicudo and N. Cardoso, Colour field densities of the quark-antiquark excited flux tubes in SU(3) lattice QCD, Phys.Rev.D 98 11, 114507, (2018) [arxiv:1808.08815]

  168. C.N. Yang, Charge quantization, compactness of the gauge group, and flux quantization. Phys. Rev. D 8, 2360 (1970)

    ADS  Google Scholar 

  169. G. ‘t Hooft, Magnetic monopoles in unified gauge theories. Nucl. Phys. B 79, 276 (1974)

  170. A.M. Polyakov, Particle spectrum in quantum field theory. JETP Lett. 20, 194 (1974)

    ADS  Google Scholar 

  171. M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley Publishing Company, 1995)

  172. C.Y. Wong, The Wigner function of produced particles in string fragmentation. Phys. Rev. C 80, 054917 (2009). arXiv:0903.3879

  173. A.V. Koshelkin, C.Y. Wong, The compactification of QCD4 to QCD2 in a flux tube. Phys. Rev. D 86, 125026 (2012). [arXiv:1212.3301]

  174. V.B. Berestetskii, E.M. Lifshitz, L.P. Pitsevskii, Quantum Electrodynamics (Pergamon Press, 1979)

  175. E. Witten, Non-Abelian bosonization in two dimensions. Comm. Math, Phys. 92, 455 (1984)

  176. G. t’Hooft, Topology of the gauge condition and new confinement phases in non-abelian gauge theories. Nucl. Phys. B 190, 455 (1981)

  177. X. Artru, G. Mennessier, String model and multiproduction. Nucl. Phys. B70, 93 (1974)

    ADS  Google Scholar 

  178. B. Andersson, G. Gustafson, G. Ingelman, T. Sjöstrand, Parton fragmentation and string dynamics. Phys. Rep. 97, 31 (1983)

    ADS  Google Scholar 

  179. C.Y. Wong, Introduction to High-Energy Heavy-Ion Collisions (World Scientific, Singapore, 1994)

  180. A. Casher, Chiral Symmetry Breaking in Confining Theories. Phys. Lett. B 83, 395 (1980)

  181. T. Banks, A. Casher, Chiral symmetry breaking in confining theories. Nucl. Phys. B 169, 103 (1980)

    ADS  Google Scholar 

  182. H. Ohata, H. Sugamauma, Chiral correlation between momopoles and the chiral condensate. Phys. Rev. D 103, 054505 (2021)

  183. J. D. Bjorken, Lectures presented in the 1973 Proceedings of the Summer Institute on Particle Physics, edited by Zipt, SLAC-167 (1973)

  184. L.V. Belvedere, J.A. Swieca, K.D. Rothe, B. Schroer, Generlaized two-dimensional Abelian gauge theories and confinement. Nucl. Phys. B 153, 112 (1979)

    ADS  Google Scholar 

  185. T. Sekido, K. Ishiguro, Y. Koma, Y. Mori, T. Suzuki, Abelian dominance and the dual Meissner effect in local unitary gauges in SU(2) gluodynamics. Phys. Rev. C 75, 064906 (2007). [arXiv:hep-ph/0703002]

    Google Scholar 

  186. T. Suzuki, K. Ishiguro, Y. Koma, T. Sekido, Gauge-independent Abelian mechanism of color confinement in gluodynamics. Phys. Rev. D 77, 034502 (2008). [arXiv:0706.4366]

    ADS  Google Scholar 

  187. H. Suganuma, H. Ohata Local correlation among the chiral condensate, monopoles, and color magnetic fields in Abelian projected QCD, [arxiv:2108.08499.]

  188. D. Kutasov, A. Schwimmer, Universality in two-dimensional gauge theory. Nucl. Phys. B 442, 447 (1995)

    MathSciNet  ADS  Google Scholar 

  189. A. Armoni, Y. Frishman, J. Sonnenschein, U. Trittmann, The spectrum of multi-flavor QCD2 and the non-abelian Schwinger equation. Nucl. Phys. B 537, 503–515 (1998)

    ADS  Google Scholar 

  190. U. Trittmann, On the Spectrum of QCD(1+1) with SU(N\(_c\)) Currents. Phys. Rev. D 66, 025001 (2002). arXiv:hep-th/0110058

    MathSciNet  ADS  Google Scholar 

  191. Z. Komargodski, K. Ohmori, K. Roumpedakis, S. Seifnashri, Symmetries and Strings of Adjoint QCD2 JHEP 03 103 (2021), arxiv:2008.07567

  192. Uwe Trittmann, Solving two-dimensional adjoint QCD with a basis-function approach, arXiv:2307.15212

Download references

Acknowledgements

The authors would like to thank Prof. Y. Jack Ng for helpful communications. CYW’s research was supported in part by the Division of Nuclear Physics, U.S. Department of Energy under Contract DE-AC05-00OR22725 with UT-Battelle, LLC.

Author information

Authors and Affiliations

  1. Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 37831, USA

    Cheuk-Yin Wong

  2. National Research Nuclear University MEPhI, Moscow, 115409, Russia

    Andrew V. Koshelkin

Authors
  1. Cheuk-Yin Wong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrew V. Koshelkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cheuk-Yin Wong.

Additional information

Communicated by Evgeni Kolomeitsev

Cheuk-Yin Wong’s research has been supported in part by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan), Oak Ridge, Tennessee 37831, USA.

Appendices

Appendices

Appendix A: Separation of quark transverse and longitudinal equations of motion in (3+1)D

To separate out transverse and the longitudinal degrees of freedom, we follow methods used previously by Wang, Pavel, Brink, Wong, and many others [150,151,152,153,154]. We employ the Weyl representation of the gamma matrices in Eq. (11) and write the quark field in terms of the transverse functions \(G_{\{1,2\}}({\varvec{r}}_\perp )\) and the longitudinal functions \(f_\pm (X )\) as

$$\begin{aligned} \varPsi _{{}_{\textrm{4D}}}=\varPsi _{{}_{\textrm{4D}}}(X){} & {} = \begin{pmatrix} G_1 ({\varvec{r}}_\perp ) f_+ (X ) \\ G_2 ({\varvec{r}}_\perp ) f_- (X ) \\ G_1 ({\varvec{r}}_\perp ) f_- (X ) \\ -G_2 ({\varvec{r}}_\perp ) f_+(X ) \\ \end{pmatrix}. \end{aligned}$$
(A.1)

where \(\varvec{r}_\perp = \{x^1,x^2\}\), \(X = \{x^3,x^0\}\). Using the Weyl representation of the gamma matrices as given in Eq. (10), we obtain

$$\begin{aligned} {\bar{\varPsi }}_{{}_{\textrm{4D}}} \gamma ^0 \varPi ^0 \varPsi _{{}_{\textrm{4D}}}= & {} ( G_1^* G_1 + G_2^* G_2 ) ( f_+^* \varPi ^0f_+ + f_-^* \varPi ^0f_-), \\ {\bar{\varPsi }}_{{}_{\textrm{4D}}} \gamma ^3 \varPi ^3 \varPsi _{{}_{\textrm{4D}}}= & {} ( G_1^* G_1 + G_2^* G_2 )( f_+^*\varPi ^3 f_+ - f_-^*\varPi ^3 f_-), \\ {\bar{\varPsi }}_{{}_{\textrm{4D}}} \varPsi _{{}_{\textrm{4D}}}= & {} (G_1^*G_1 - G_2^* G_2 )( f_+^* f_-+ f_-^* f_+), \\ {\bar{\varPsi }}_{{}_{\textrm{4D}}} \gamma ^1 \varPi ^1 \varPsi _{{}_{\textrm{4D}}}= & {} ( G_1^* \varPi ^1 G_2 + G_2^*\varPi ^1 G_1 )( f_+^* f_- + f_-^* f_+), \\ {\bar{\varPsi }}_{{}_{\textrm{4D}}} \gamma ^2 \varPi ^2 \varPsi _{{}_{\textrm{4D}}}= & {} ( -G_1^* \,i\varPi ^2 G_2 + G_2^*i\varPi ^2G_1 )( f_+^* f_- \!+\! f_-^* f_+). \end{aligned}$$

The quark Lagrangian density becomes

$$\begin{aligned} {{{\mathcal {L}}}}{} & {} = {\bar{\varPsi }}_{{}_{\textrm{4D}}} \!\gamma ^0\! \varPi ^0\! \psi _{{}_{\textrm{4D}}}\! \!-\! {\bar{\varPsi }}_{{}_{\textrm{4D}}} \!\gamma ^3 \varPi ^3 \!\varPsi _{{}_{\textrm{4D}}} \!\!\!- \!{\bar{\varPsi }}_{{}_{\textrm{4D}}} \gamma ^1 \! \varPi ^1 \varPsi _{{}_{\textrm{4D}}} \nonumber \\{} & {} \quad -{\bar{\varPsi }}_{{}_{\textrm{4D}}} \! \gamma ^2 \varPi ^2 \varPsi _{{}_{\textrm{4D}}} - m {\bar{\psi }}_{{}_{\textrm{4D}}} \varPsi _{{}_{\textrm{4D}}} \nonumber \\{} & {} = ( G_1^* G_1 + G_2^* G_2 ) ( f_+^* \varPi ^0f_+ + f_-^* \varPi ^0f_-) \nonumber \\{} & {} \quad - ( G_1^* G_1 + G_2^* G_2 )( f_+^*\varPi ^3 f_+ - f_-^*\varPi ^3 f_-) \nonumber \\{} & {} \quad - ( G_1^* \varPi ^1 G_2 + G_2^*\varPi ^1 G_1 )( f_+^* f_- + f_-^* f_+) \nonumber \\{} & {} \quad - ( -G_1^* ~i\varPi ^2~G_2 + G_2^*~i\varPi ^2~ G_1 )( f_+^* f_- + f_-^* f_+) \nonumber \\{} & {} \quad - m (G_1^*G_1 - G_2^* G_2 )( f_+^* f_-+ f_-^* f_+).\nonumber \\ \end{aligned}$$
(A.2)

where \(\varPi ^\mu =p^\mu + g A^\mu (x)\). The minimization of the action integral by variations with respect to \(f_\pm ^*\) and \(G_{\{1,2\}}^*\) leads to

$$\begin{aligned} \delta ^2 {{{\mathcal {L}}}}/\delta f_+^* \delta G_1^*{} & {} =\!( \varPi ^0 \!\!-\!\varPi ^3) G_1 f_+ \!\!-\! m G_1 f_- \nonumber \\{} & {} \quad -\!(p^1\!\!- \!ip^2) G_2 f_- \!=\!0,~~ \end{aligned}$$
(A.3)
$$\begin{aligned} \delta ^2 {{{\mathcal {L}}}}/\delta f_+^* \delta G_2^*{} & {} =( \varPi ^0 \!\!\!-\!\varPi ^3) G_2 f_+ \!\!+\! m G_2 f_-\nonumber \\{} & {} \quad - \! (\varPi ^1\!+\!i\varPi ^2) G_1 f_- \!=\!0, \nonumber \\ \end{aligned}$$
(A.4)
$$\begin{aligned} \delta ^2 {{{\mathcal {L}}}}/\delta f_-^* \delta G_1^*{} & {} = ( \varPi ^0\!\!+\! \varPi ^3)G_1 f_- \!\! - \! m G_1 f_+ \nonumber \\{} & {} \quad - (\varPi ^1\!\!- \! i\varPi ^2)G_2 f_+ \!=\!0, \nonumber \\ \end{aligned}$$
(A.5)
$$\begin{aligned} \delta ^2 {{{\mathcal {L}}}}/\delta f_-^* \delta G_2^*{} & {} = (\varPi ^0\! \!+\!\varPi ^3) G_2 f_- \! \!+\! m G_2 f_+ \nonumber \\{} & {} \quad - \!(\varPi ^1 \! \! + \! i\varPi ^2) G_1 f_+ \!= \!0. \nonumber \\ \end{aligned}$$
(A.6)

We sum over \(G_1^*(\varvec{r}_\perp )\) \(\times \) (A.3) + \(G_2^*(\varvec{r}_\perp )\) \(\times \) (A.4) and perform an integration over the transverse coordinates. Similarly, we sum over \(G_1^*(\varvec{r}_\perp )\) \(\times \) (A.5) + \(G_2^*(\varvec{r}_\perp )\) \(\times \) (A.6) and perform an integration over the transverse coordinates. We get

$$\begin{aligned}{} & {} \int d\varvec{r}_\perp \left\{ G_1^*( \varPi ^0 -\varPi ^3) G_1 + G_2^*( \varPi ^0 -\varPi ^3) G_2 \right\} f_+ \nonumber \\{} & {} \quad - \int d\varvec{r}_\perp \biggl \{ G_1^*m G_1 - G_2^*m G_2 +G_1^*(\varPi ^1- i\varPi ^2) G_2 \nonumber \\{} & {} \quad + G_2^*(\varPi ^1+i\varPi ^2) G_1 \biggr \} f_-=0,\end{aligned}$$
(A.7a)
$$\begin{aligned} \text {and} \nonumber \\{} & {} \int d\varvec{r}_\perp \left\{ G_1^*( \varPi ^0 +\varPi ^3) G_1 + G_2^*( \varPi ^0 +\varPi ^3) G_2 \right\} f_- \nonumber \\{} & {} - \int d\varvec{r}_\perp \biggl \{ G_1^*m G_1 - G_2^*m G_2 +G_1^*(\varPi ^1- i\varPi ^2) G_2 \nonumber \\{} & {} \quad + G_2^*(\varPi ^1+i\varPi ^2) G_1 \biggr \} f_+=0. \end{aligned}$$
(A.7b)

From the second terms in each of the above two equation, we note that we can separate out the longitudinal and transverse equations by introducing the separation constant \(m_T\) defined by

$$\begin{aligned} m_T{} & {} =\int d \varvec{r}_\perp \biggl \{ m( |G_1 |^2 - |G_2 |^2) \nonumber \\{} & {} \quad + G_1^*(\varPi ^1- i\varPi ^2) G_2) + G_2^* (\varPi ^1+ i\varPi ^2) G_1 \biggr \}. \end{aligned}$$
(A.8)

Because of the normalization condition Eq. (12) for \(G_{\{1,2\}}\), the above equation can be rewritten as

$$\begin{aligned}{} & {} \int d \varvec{r}_\perp \biggl \{ G_1^*[ - m_T G_1+ m G_1 + (\varPi ^1- i\varPi ^2) G_2 ] \\{} & {} \quad +G_2^*[ - m_T G_2 - m G_2 + (\varPi ^1+i\varPi ^2) G_1 ] \biggr \} = 0 \end{aligned}$$

The above is zero if the two terms in the integrand are zero, and we obtain the eigenvalue equations for the transverse functions \(G_{\{1,2\}}(\varvec{r}_\perp )\),

$$\begin{aligned}{} & {} \!-\! m_T G_1 (\varvec{r}_\perp \! )\!+ \! m G_1(\varvec{r}_\perp \! ) \! + (\! \varPi ^1\! - \! i\varPi ^2) G_2 (\varvec{r}_\perp \! ) \!=0, \nonumber \\{} & {} \quad \! - \! m_T G_2 (\varvec{r}_\perp \! )\!-\!m G_2 (\varvec{r}_\perp \!) \!+ (\varPi ^1 + i\varPi ^2) G_1 (\varvec{r}_\perp \!) \!=0.\nonumber \\ \end{aligned}$$
(A.9)

We apply \( (\varPi ^1+i\varPi ^2)\) on the first equation and \( (p^1+ip^2)\) on the second equation and we get

$$\begin{aligned}{} & {} (\varPi ^1\!\!+\!i\varPi ^2)(\varPi ^1\!\!-\! i\varPi ^2) G_2 (\varvec{r}_\perp \!)\! =\! (m_T\!\!-\!m) (m_T\!\!+\!m) G_2 (\varvec{r}_\perp \!) \\{} & {} \quad (\varPi ^1\!\!- \! i\varPi ^2\!) (\varPi ^1\!\!+\! i\varPi ^2) G_1(\varvec{r}_\perp \!) \!= \!( m_T\!\! - \!m) (m_T\!\!+\!m) G_1(\varvec{r}_\perp ). \end{aligned}$$

We have then

$$\begin{aligned}{} & {} \left\{ \varPi _T^2 + i [\varPi ^2,\varPi ^1] + m^2 - m_T^2\right\} G_2 (\varvec{r}_\perp ) = 0 \end{aligned}$$
(A.10a)
$$\begin{aligned}{} & {} \left\{ \varPi _T^2 + i [\varPi ^1,\varPi ^2]+ m^2 - m_T^2 \right\} G_1(\varvec{r}_\perp )= 0 \end{aligned}$$
(A.10b)

where \(\varPi _T^2=(\varPi ^1)^2+(\varPi ^2)^2\) is the square of the transverse momentum. Upon the introduction of the constant of separation \(m_T\), Eqs. (A.7a) and (A.7b) become

$$\begin{aligned}{} & {} \int \! \!d\varvec{r}_\perp (G_1^*(\varvec{r}_\perp \!)G_1 (\varvec{r}_\perp \!) \!+ \!G_2^* (\varvec{r}_\perp \!)G_2(\varvec{r}_\perp \!) )( \varPi ^0 \!-\!\varPi ^3) f_+ (X\!) \nonumber \\{} & {} \quad - m_T f_- (X)=0, \end{aligned}$$
(A.11a)
$$\begin{aligned}{} & {} \int \! \! d\varvec{r}_\perp ( G_1^*(\varvec{r}_\perp \!)G_1 (\varvec{r}_\perp \!)\!+ \!G_2^* (\varvec{r}_\perp \!)G_2(\varvec{r}_\perp \!) )( \varPi ^0 \!+\!\varPi ^3)f_- (X\!) \nonumber \\{} & {} \quad - m_T f_+(X) =0. \end{aligned}$$
(A.11b)

In the above two equations, the operators \(\varPi ^\mu \) with \(\mu =0,3\) are actually \(\varPi _{{}_{\textrm{4D}}}^\mu \)=\(p^\mu + g_{{}_{\textrm{4D}}} A_{{}_{\textrm{4D}}}^\mu (\varvec{r}_\perp ,X)\) where the gauge field \(A_{{}_{\textrm{4D}}}^\mu \) is a function of the (3+1)D coordinates \((\varvec{r}_\perp ,X) \) and \(g_{{}_{\textrm{4D}}}\) is the dimensionless coupling constant in (3+1)D space-time. Upon carrying out the integration over the transverse coordinates \(\varvec{r}_\perp \), the above two equations can be cast into the Dirac equation for a particle with a mass \(m_T\) in (1+1)D space time of X coordinates by identifying the terms integrated over the transverse coordinates as the corresponding gauge fields \(A_{{}_{\textrm{2D}}}^\mu \) in the (1+1)D space-time with the coupling constant \(g_{{}_{\textrm{2D}}}\)

$$\begin{aligned}{} & {} g_{{}_{\textrm{4D}}} \!\int \!d\varvec{r}_\perp (G_1^*(\varvec{r}_\perp \!)G_1(\varvec{r}_\perp \!) \!+ \!G_2^* (\varvec{r}_\perp \!)G_2(\varvec{r}_\perp \!) )A_{{}_{\textrm{4D}}}^\mu (\varvec{r}_\perp ,X\!) \nonumber \\{} & {} \quad \equiv g_{{}_{\textrm{2D}}} A_{{}_{\textrm{2D}}}^\mu (X), \end{aligned}$$
(A.12)

where \(A_{{}_{\textrm{2D}}}(X)\) is the solution of the Maxwell equation for the gauge fields in (1+1)D space-time. By comparing the Maxwell equations in (3+1)D and (1+1)D space time in Appendix B, it can be shown in Appendix B that the coupling constant \(g_{{}_{\textrm{2D}}}\) is then determined by

$$\begin{aligned} g_{{}_{\textrm{2D}}}^2\!\!= \! g_{{}_{\textrm{4D}}}^2 \!\int \!\!d \varvec{r}_\perp (G_1^*(\varvec{r}_\perp \!)G_1(\varvec{r}_\perp \!) + G_2^* (\varvec{r}_\perp \!)G_2(\varvec{r}_\perp \!) )^{2}\!.\nonumber \\ \end{aligned}$$
(A.13)

We note that the coupling constant \(g_{{}_{\textrm{2D}}}\) acquires the dimension of a mass. By such an introduction of \(A_{{}_{\textrm{2D}}}^\mu \) and \(g_{{}_{\textrm{2D}}}\), Eqs. (A13a) and (A13b) become

$$\begin{aligned}{} & {} ( \varPi _{{}_{\textrm{2D}}}^0 - \varPi _{{}_{\textrm{2D}}} ^3) f_+ (X) - m_T f_-(X)=0, \end{aligned}$$
(A.14a)
$$\begin{aligned}{} & {} ( \varPi _{{}_{\textrm{2D}}}^0+ \varPi _{{}_{\textrm{2D}}}^3) f_- (X) - m_T f_+(X)=0, \end{aligned}$$
(A.14b)

where \(\varPi _{{}_{\textrm{2D}}}^\mu =p^\mu +g_{{}_{\textrm{2D}}} A_{{}_{\textrm{2D}}}^\mu (X)\), with \(\mu =0,3\). Equations (A.10), and (A.14) (which correspond to Eqs. (17) and (22) in Section III) are the set of transverse and longitudinal equations of motion for the quark field \(\varPsi \) in (3+1)D space for our problem.

Appendix B: Relation between quantities in (1+1)D and (3+1)D

We shall examine the relation between various quantities in (1+1)D and (3+1)D. The time-like component of the quark current is the quark density, and the density in (3+1)D involves the transverse spatial distribution while the density in (1+1)D does not involve the transverse spatial distribution.

We consider the quasi-Abelian approximation of the non-Abelian QCD gauge field in so that the QED and QCD gauge fields can be represented in terms of only the commuting \(\tau ^0\) and \(\tau ^1\) components, as discussed in Sect. 8.2. To make the problem simple, we consider lowest-energy state systems with cylindrical symmetry so that we can write our wave function of the quark fields in the form of Eq. (11), and the transverse currents \(j^1\) and \(j^2\) are given in terms of the basic functions \(G_{\{1,2\}}\) and \(f_\pm \) by

$$\begin{aligned} j^1_{{}_{\textrm{4D}}}= & {} {\bar{\varPsi }}_{{}_{\textrm{4D}}} \!\gamma ^1 \varPsi _{{}_{\textrm{4D}}} = ( G_1^* G_2 \!+ \!G_2^* G_1 \!)( f_+^* f_- \! + \! f_-^* f_+\!), \end{aligned}$$
(B.1a)
$$\begin{aligned} j^{2}_{{}_{\textrm{4D}}}= & {} {\bar{\varPsi }}_{{}_{\textrm{4D}}} \! \gamma ^2 \varPsi _{{}_{\textrm{4D}}} \! = ( \!-G_1^*G_2 \!+ \!G_2^*G_1\! )( f_+^* f_- \! \!+ \! f_-^* f_+\!).\nonumber \\ \end{aligned}$$
(B.1b)

For our problem, we choose to examine the quark-QED system in which the quarks and antiquarks reside in the zero mode with \(m_T=m\), and with the quark wave functions given by (66). For these zero mode states, \(G_{\{1,2\}}\) are spinors aligned in the longitudinal direction. As a consequence, when \(G_1(\varvec{r}_\perp )\) is non-zero, \(G_2(\varvec{r}_\perp )\) is zero, and vice-versa. The product \(G_1^* G_2\) and \(G_2^*G_1\) are always zero for quark and antiquark in the lowest energy zero mode states. Therefore for these lowest energy states the currents in the direction of \(x^1\) and \(x^2\) are zero. We assume that the transverse gauge fields \(A^{\{1,2\}}\) are determined by the transverse dynamics of their own to yield the solutions of the transverse stationary states \(G_{\{1,2\}}(\varvec{r}_\perp )\). Within the cylindrical flux tube, the gauge fields are only a weak function of the transverse coordinates so that it is necessary to consider the Maxwell equation only for \(\nu , \mu \)=0,3 in (3+1)D with

$$\begin{aligned} \partial _\mu F_{{}_{\textrm{4D}}}^{\mu \nu }{} & {} = \partial _\mu \partial ^\mu A_{{}_{\textrm{4D}}} ^{\nu }(\varvec{r}_\perp ,X) - \partial ^\nu \partial _\mu A_{{}_{\textrm{4D}}}^{\mu }(\varvec{r}_\perp ,X) \nonumber \\{} & {} = - g_{{}_{\textrm{4D}}} j_{{}_{\textrm{4D}}} ^\nu (\varvec{r}_\perp ,X). \end{aligned}$$
(B.2)

With the quark field \(\varPsi _{{}_{\textrm{4D}}}\) as given by Eq. (11), the quark currents in (3+1)D are given by

$$\begin{aligned}{} & {} j_{{}_{\textrm{4D}}} ^0\!\! (\varvec{r}_\perp ,X\!)\! =\!{\bar{\varPsi }}_{{}_{\textrm{4D}}} \!\! \gamma ^0 \varPsi _{{}_{\textrm{4D}}}\!\!=\! ( G_1^* G_1 \!+\! G_2^* G_2 \!) ( f_+^* f_+ \!+\! f_-^* f_-\!),\end{aligned}$$
(B.3)
$$\begin{aligned}{} & {} j_{{}_{\textrm{4D}}} ^3\!\! (\varvec{r}_\perp ,X\!)\!=\!{\bar{\varPsi }}_{{}_{\textrm{4D}}} \!\!\gamma ^3 \varPsi _{{}_{\textrm{4D}}} \!\! = \! ( G_1^* G_1 \!+\! G_2^* G_2 \!)( f_+^* f_+ \! - \! f_-^* f_-\!).\nonumber \\ \end{aligned}$$
(B.4)

From the 2D action \({{{\mathcal {A}}}_{{}_{\textrm{2D}}}}\) in Eq. (43), we can derive the Maxwell equation in (1+1)D as given by

$$\begin{aligned}{} & {} \partial _\mu F_{{}_{\textrm{2D}}}^{\mu \nu } = \partial _\mu \partial ^\mu A_{{}_{\textrm{2D}}} ^{\nu }(X) -\partial ^\nu \partial _\mu A_{{}_{\textrm{2D}}}^{\mu }(X)= - g_{{}_{\textrm{2D}}} j_{{}_{\textrm{2D}}} ^\nu (X),\nonumber \\ \end{aligned}$$
(B.5)

for \(\nu ,\mu =0,1\) in (1+1)D. With the quark field \(\psi _{{}_{\textrm{2D}}}\) as given by Eq. (26), the quark currents \(j_{{}_{\textrm{2D}}} ^\nu (X)\) in (1+1)D are given by

$$\begin{aligned}{} & {} j_{{}_{\textrm{2D}}} ^0 (\varvec{r}_\perp ,X) ={\bar{\psi }}_{{}_{\textrm{2D}}} \gamma ^0 \psi _{{}_{\textrm{2D}}}= ( f_+^* f_+ + f_-^* f_-), \end{aligned}$$
(B.6)
$$\begin{aligned}{} & {} j_{{}_{\textrm{2D}}} ^3 (\varvec{r}_\perp ,X)={\bar{\psi }}_{{}_{\textrm{2D}}} \gamma ^3 \varPsi _{{}_{\textrm{2D}}} = ( f_+^* f_+ - f_-^* f_-). \end{aligned}$$
(B.7)

Comparison of the quark currents in (3+1)D and (1+1)D in Eqs. (B.3), (B.4) (B.6), and (B.7) gives

$$\begin{aligned}{} & {} j_{{}_{\textrm{4D}}} ^\mu (\varvec{r}_\perp ,X) = ( G_1^*(\varvec{r}_\perp ) G_1 (\varvec{r}_\perp )+ G_2^*(\varvec{r}_\perp ) G_2 (\varvec{r}_\perp ) ) j_{{}_{\textrm{2D}}} ^\mu (X).\nonumber \\ \end{aligned}$$
(B.8)

Then, substituting the above equation into Eq. (B.2), we obtain for \(\nu ,\mu =0,3\) in (3+1)D

$$\begin{aligned}{} & {} \partial _\nu \partial ^\nu A_{{}_{\textrm{4D}}} ^{\mu }(\varvec{r}_\perp ,X) - \partial ^\mu \partial _\nu A_{{}_{\textrm{4D}}}^{\nu }(\varvec{r}_\perp ,X) \nonumber \\{} & {} \quad = - g_{{}_{\textrm{4D}}} ( G_1^*(\varvec{r}_\perp ) G_1 (\varvec{r}_\perp )+ G_2^*(\varvec{r}_\perp ) G_2 (\varvec{r}_\perp ) ) j_{{}_{\textrm{2D}}} ^\mu (X).\nonumber \\ \end{aligned}$$
(B.9)

The above equation in (3+1)D would be consistent with the (1+1)D Maxwell equation (B.5) if for \(\mu =0,3\),

$$\begin{aligned} A_{{}_{\textrm{4D}}}^{\mu }\!(\varvec{r}_\perp ,X\!)\!=\!\frac{g_{{}_{\textrm{4D}}}}{g_{{}_{\textrm{2D}}}} \!(G_1^*(\varvec{r}_\perp \!)G_1 (\varvec{r}_\perp \!)\!+\! G_2^*(\varvec{r}_\perp \!) G_2 (\varvec{r}_\perp \!) ) A_{{}_{\textrm{2D}}}^\mu \!(X\!).\nonumber \\ \end{aligned}$$
(B.10)

On the other hand, the equation of motion in Eqs. (A14) would require that \(g_{{}_{\textrm{4D}}}\) and \(g_{{}_{\textrm{2D}}}\) are related by

$$\begin{aligned}{} & {} g_{{}_{\textrm{4D}}} \int d\varvec{r}_\perp (|G_1(\varvec{r}_\perp )|^2 + |G_2(\varvec{r}_\perp )|^2) A_{{}_{\textrm{4D}}}^\mu (\varvec{r}_\perp ,X) \nonumber \\{} & {} \quad = g_{{}_{\textrm{2D}}} A_{{}_{\textrm{2D}}}^\mu (X). \end{aligned}$$
(B.11)

The consistency of both equations (B.10) and (B.11) requires \(g_{{}_{\textrm{4D}}}\) and \(g_{{}_{\textrm{2D}}}\) to satisfy the relation

$$\begin{aligned} g_{{}_{\textrm{2D}}}^2\!\! =\! g_{{}_{\textrm{4D}}}^2\!\! \int \!d\varvec{r}_\perp [G_1^*(\varvec{r}_\perp \!) G_1 (\varvec{r}_\perp \!)\!+\! G_2^*(\varvec{r}_\perp \!) G_2 (\varvec{r}_\perp \!) ]^2, \end{aligned}$$
(B.12)

which is Eq. (21).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wong, CY., Koshelkin, A.V. Dynamics of quarks and gauge fields in the lowest-energy states in QED and QCD interactions. Eur. Phys. J. A 59, 285 (2023). https://doi.org/10.1140/epja/s10050-023-01180-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epja/s10050-023-01180-8

Search

Navigation