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Schwinger–Dyson Equation for Quarks in a QCD Inspired Model

  • PHYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
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Abstract

We discuss formulation of QCD in Minkowski–spacetime and effect of an operator product expansion by means of normal ordering of fields in the QCD Lagrangian. The formulation of QCD in the Minkowski–spacetime allows us to solve a constraint equation and decompose the gauge field propagator in the sum of an instantaneous part, which forms a bound state, and a retarded part, which contains the relativistic corrections. In Quantum Field Theory, for a Lagrangian with unordered operator fields, one can make normal ordering by means of the operator product expansion, then the gluon condensate appear. This gives us a natural way of obtaining a dimensional parameter in QCD, which is missing in the QCD Lagrangian. We derive a Schwinger–Dyson equation for a quark, which is studied both numerically and analytically. The critical value of the strong coupling constant \({{\alpha }_{s}} = {4 \mathord{\left/ {\vphantom {4 \pi }} \right. \kern-0em} \pi }\), above which a nontrivial solution appears and a spontaneous chiral symmetry breaking occurs, is found. For the sake of simplicity, the considered model describes only one flavor massless quark, but the methods can be used in more general case. The Fourier-sine transform of a function with log-power asymptotic was performed.

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Notes

  1. Moreover, in gauge (2) the ghosts can be removed by the certain transformation [62].

REFERENCES

  1. C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954).

    Article  ADS  MathSciNet  Google Scholar 

  2. L. D. Faddeev and V. N. Popov, Phys. Lett. B 25, 29 (1967).

    Article  ADS  Google Scholar 

  3. L. D. Faddeev, Theor. Math. Phys. 1, 1 (1969).

    Article  Google Scholar 

  4. G. ’t Hooft, Nucl. Phys. B 33, 173 (1971).

    Article  ADS  Google Scholar 

  5. D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973).

    Article  ADS  Google Scholar 

  6. H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).

    Article  ADS  Google Scholar 

  7. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979).

    Article  ADS  Google Scholar 

  8. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 147, 448 (1979).

    Article  ADS  Google Scholar 

  9. B. L. Ioffe, Nucl. Phys. B 188, 317 (1981).

    Article  ADS  Google Scholar 

  10. M. Volkov and V. Pervushin, Essentially Nonlinear Field Theory, Dynamical Symmetry and Pion Physics (Atomizdat, Moscow, 1978) [in Russian].

    Google Scholar 

  11. M. K. Volkov and V. N. Pervushin, Sov. J. Part. Nucl. 6, 632 (1975).

    Google Scholar 

  12. M. K. Volkov and V. N. Pervushin, Sov. Phys. Usp. 19, 894 (1976).

    Article  ADS  Google Scholar 

  13. J. Gasser and H. Leutwyler, Ann. Phys. 158, 142 (1984).

    Article  ADS  Google Scholar 

  14. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961).

    Article  ADS  Google Scholar 

  15. D. Ebert and M. Volkov, Z. Phys. C 16, 205 (1983).

    Article  ADS  Google Scholar 

  16. M. Volkov, Ann. Phys. 157, 282 (1984).

    Article  ADS  Google Scholar 

  17. M. K. Volkov, Sov. J. Part. Nucl. 17, 433 (1986).

    Google Scholar 

  18. H. Reinhardt and R. Alkofer, Phys. Lett. B 207, 482 (1988).

    Article  ADS  Google Scholar 

  19. V. Bernard, R. Jaffe, and U.-G. Meissner, Nucl. Phys. B 308, 753 (1988).

    Article  ADS  Google Scholar 

  20. A. Le Yaouanc, L. Oliver, S. Ono, O. Pène, and J.‑C. Raynal, Phys. Rev. D 31, 137 (1985).

    Article  ADS  Google Scholar 

  21. P. Bogolubov, Ann. Inst. Henri Poincare Phys. Theor. 8, 163 (1968).

    Google Scholar 

  22. A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, Phys. Rev. D 9, 3471 (1974).

    Article  ADS  MathSciNet  Google Scholar 

  23. A. Chodos, R. L. Jaffe, K. Johnson, and C. B. Thorn, Phys. Rev. D 10, 2599 (1974).

    Article  ADS  Google Scholar 

  24. R. Haymaker, Riv. Nuovo Cimento 14, 1 (1991).

    Article  MathSciNet  Google Scholar 

  25. J. Pawlowski, Ann. Phys. 322, 2831 (2007). arXiv:hep-th/0512261 hep-th.

  26. E. Shuryak, Nucl. Phys. B 203, 93 (1982).

    Article  ADS  Google Scholar 

  27. E. Shuryak, Nucl. Phys. B 203, 116 (1982).

    Article  ADS  Google Scholar 

  28. E. Shuryak, Nucl. Phys. B 203, 140 (1982).

    Article  ADS  Google Scholar 

  29. D. Dyakonov and V. Petrov, Nucl. Phys. B 245, 259 (1984).

    Article  ADS  Google Scholar 

  30. G. Efimov and S. Nedelko, Phys. Rev. D 51, 176 (1995).

    Article  ADS  Google Scholar 

  31. J. Burdanov, G. Efimov, S. Nedelko, and S. Solunin, Phys. Rev. D 54, 4483 (1996). arXiv:hep-ph/9601344 hep-ph.

  32. A. Kalloniatis and S. Nedelko, Phys. Rev. D 64, 114025 (2001). arXiv:hep-ph/0108010 hep-ph.

  33. A. Kalloniatis and S. Nedelko, Phys. Rev. D 69, 074029 (2004), Erratum: Phys. Rev. D 70,119903(2004). arXiv: hep-ph/0311357 hep-ph.

  34. B. Galilo and S. Nedelko, Phys. Part. Nucl. Lett. 8, 67 (2011). arXiv:1006.0248 hep-ph.

  35. S. Nedelko and V. Voronin, Eur. Phys. J. A 51, 45 (2015). arXiv:1403.0415 hep-ph.

  36. C. Roberts and A. Williams, Prog. Part. Nucl. Phys. 33, 477 (1994).

    Article  ADS  Google Scholar 

  37. L. von Smekal, A. Hauck, and R. Alkofer, Ann. Phys. 267, 1 (1998). arXiv:hep-ph/9707327 hep-ph.

  38. R. Alkofer and L. von Smekal, Phys. Rep. 353, 281 (2001). arXiv:hep-ph/0007355 hep-ph.

  39. P. Maris and C. Roberts, Int. J. Mod. Phys. E 12, 297 (2003). arXiv:nucl-th/0301049 nucl-th.

  40. C. Fischer, J. Phys. G: Nucl. Part. Phys. 32, R253 (2006). arXiv:hep-ph/0605173 hep-ph.

  41. D. Epple, H. Reinhardt, and W. Schleifenbaum, Phys. Rev. D 75, 045011 (2007). arXiv:hep-th/0612241 hep-th.

  42. A. Aguilar, D. Binosi, and J. Papavassiliou, Phys. Rev. D 78, 025010 (2008). arXiv:0802.1870 hep-ph.

  43. V. Shilin and V. Pervushin, Phys. At. Nucl. 76, 1289 (2013). arXiv:1205.2860 hep-ph.

  44. I. V. Polubarinov, Phys. Part. Nucl. 34, 738 (2003).

    Google Scholar 

  45. V. N. Pervushin, Phys. Part. Nucl. 34, 678 (2003).

    Google Scholar 

  46. V. Pervushin, Nucl. Phys. B Proc. Sup. 15, 197 (1990).

  47. V. Sauli, Few Body Syst. 39, 45 (2006). arXiv:hep-ph/0412188 hep-ph.

  48. V. Sauli, J. Adam, Jr., and P. Bicudo, Phys. Rev. D 75, 087701 (2007). arXiv:hep-ph/0607196 hep-ph.

  49. A. Cherny, A. Dorokhov, N. Han, V. Pervushin, and V. Shilin, Phys. At. Nucl. 76, 382 (2013). arXiv: 1112.5856 hep-th.

  50. V. Pervushin, A. Arbuzov, B. Barbashov, A. Cherny, A. Dorokhov, A. Borowiec, R. Nazmitdinov, A. Pavlov, V. Shilin, and A. Zakharov, in Proceedings of the 21st Baldin Seminar on High Energy Physics Problems, Dubna, Russia, 2012; PoS 23 (2012). arXiv:1211.4386 hep-ph.

  51. V. N. Pervushin, A. B. Arbuzov, A. Yu. Cherny, V. I. Shilin, R. G. Nazmitdinov, A. E. Pavlov, K. N. Pichugin, and A. F. Zakharov, in Proceedings of the 22nd Baldin Seminar on High Energy Physics Problems, Dubna, Russia, 2014; PoS 136 (2015). arXiv: 1502.00267 gr-qc.

  52. J. Mandula and M. Ogilvie, Phys. Lett. B 185, 127 (1987).

    Article  ADS  Google Scholar 

  53. K. Amemiya and H. Suganuma, Phys. Rev. D 60, 114509 (1999). arXiv:hep-lat/9811035 hep-lat.

  54. J. Cornwall and A. Soni, Phys. Lett. B 120, 431 (1983).

    Article  ADS  Google Scholar 

  55. A. Bogolyubskaya, Y. Kalinovsky, W. Kallies, and V. Pervushin, Acta Phys. Pol. B 21, 139 (1990).

    Google Scholar 

  56. F. Halzen, G. Krein, and A. Natale, Phys. Rev. D 47, 295 (1993).

    Article  ADS  Google Scholar 

  57. S. A. Larin, in Proceedings of the 11th Conference on Quark Confinement and the Hadron Spectrum (Confinement XI); AIP Conf. Proc. 1701, 070003 (2016). arXiv: 1304.8107 hep-ph.

  58. S. Larin, in Proceedings of the 22nd Baldin Seminar on High Energy Physics Problems, Dubna, Russia, 2015; PoS 18 (2015).

  59. S. A. Larin, in Proceedings of the 16th Workshop on High Energy Spin Physics (DSPIN-15); J. Phys. Conf. Ser. 678, 012025 (2016).

    Article  Google Scholar 

  60. P. A. Dirac, Can. J. Phys. 33, 650 (1955).

    Article  ADS  Google Scholar 

  61. N. H. Christ and T. D. Lee, Phys. Rev. D 22, 939 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  62. P. Watson and H. Reinhardt, Phys. Rev. D 85, 025014 (2012). arXiv:1111.6078 hep-ph.

  63. D. Ebert and V. N. Pervushin, in Problems of Gauge Theories, Ed. by B. M. Barbashov and V. V. Nesterenko (JINR, Dubna, 2004), pp. 62–78.

  64. D. Ebert and V. Pervushin, in Proceedings of the 18th Conference on High Energy Physics, Tbilisi; JINR Preprint E2-10020 (1976).

  65. H. Kleinert, Phys. Lett. B 62, 429 (1976).

    Article  ADS  Google Scholar 

  66. V. N. Pervushin, H. Reinhardt, and D. Ebert, Sov. J. Part. Nucl. 10, 1114 (1979).

    Google Scholar 

  67. R. L. Stratonovich, Sov. Phys. Dokl. 2, 416 (1957).

    ADS  Google Scholar 

  68. J. Hubbard, Phys. Rev. Lett. 3, 77 (1959).

    Article  ADS  Google Scholar 

  69. Y. L. Kalinovsky, L. Kaschluhn, and V. N. Pervushin, Fortschr. Phys. 38, 353 (1990).

    Article  Google Scholar 

  70. V. N. Pervushin, Y. L. Kalinovsky, W. Kallies, and N. A. Sarikov, Fortschr. Phys. 38, 333 (1990).

    Article  Google Scholar 

  71. Y. L. Kalinovsky, V. Kallies, B. N. Kuranov, V. N. Pervushin, and N. A. Sarikov, Yad. Fiz. 49, 1709 (1989).

    Google Scholar 

  72. Y. L. Kalinovsky, L. Kaschluhn, and V. N. Pervushin, Phys. Lett. B 231, 288 (1989).

    Article  ADS  Google Scholar 

  73. I. Bogolubsky, E. Ilgenfritz, M. Muller-Preussker, and A. Sternbeck, Phys. Lett. B 676, 69 (2009). arXiv: 0901.0736 hep-lat.

  74. I. V. Puzynin, I. V. Amirkhanov, E. V. Zemlyanaya, V. N. Pervushin, T. P. Puzynina, T. A. Strizh, and V. D. Lakhno, Phys. Part. Nucl. 30, 210 (1999).

    Article  Google Scholar 

  75. V. A. Miransky and K. Yamawaki, Phys. Rev. D 55, 5051 (1997). arXiv:hep-th/9611142 hep-th.

  76. T. Banks and A. Zaks, Nucl. Phys. B 196, 189 (1982).

    Article  ADS  Google Scholar 

  77. T. Appelquist, D. Nash, and L. C. R. Wijewardhana, Phys. Rev. Lett. 60, 2575 (1988).

    Article  ADS  MathSciNet  Google Scholar 

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Shilin, V.I., Pervushin, V.N. Schwinger–Dyson Equation for Quarks in a QCD Inspired Model. Phys. Part. Nuclei Lett. 20, 1308–1325 (2023). https://doi.org/10.1134/S1547477123060341

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