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Problem of the Landau Poles in Quantum Field Theory: from N. N. Bogolyubov to the Present Day

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Russian Physics Journal Aims and scope

A review of problems associated with the unphysical Landau pole in propagators of quantum particles is given. Approaches to eliminating this pole within the framework of electrodynamics and effective theories of strongly interacting particles are investigated. The asymptotic behavior at large momenta in the scalar theory ϕ4 in the two-particle (bubble) approximation is investigated. To formulate a calculational model in the two-particle approximation, we use an iterative scheme for solving the Schwinger–Dyson equation in the formalism of a bilocal field source. The main problem is to develop a recipe for numerical analysis of the solutions of the obtained nonlinear equation for the amplitude at small interaction distances (large values of the momentum) for different values of the constant. The nontrivial behavior of the amplitude in the deeply inelastic region of momenta is determined. The positions of the unphysical poles (the Landau poles) in the expression for the amplitude in the deeply inelastic region of momenta are identified.

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References

  1. L. D. Landau, A. A. Abrikosov, and I. M. Khalatnikov, Dokl. Akad. Nauk SSSR, 95, 1177 (1954).

    Google Scholar 

  2. L. D. Landau and I. Ya. Pomeranchuk, Dokl. Akad. Nauk SSSR, 102, 489 (1955).

    MathSciNet  Google Scholar 

  3. L. D. Landau, Theoretical Physics in the 20th Century: A Memorial Volume to W. Pauli, Interscience, New York (1960).

    Google Scholar 

  4. P. J. Redmond, Phуs. Rev., 112, 1404 (1958).

    Article  ADS  Google Scholar 

  5. N. N. Bogolyubov, A. A. Logunov, and D. V. Shirkov, Zh. Eksp. Teor. Fiz., 37, 805 (1959).

    Google Scholar 

  6. D. V. Shirkov and I. L. Slovtsov, Teor. Mat. Fiz., 150, 152 (2007).

    Article  Google Scholar 

  7. O. A. Battistel et al., Phys. Rev., D77, 065025 (2008).

    ADS  Google Scholar 

  8. V. E. Rochev, J. Phys., A42, 195403 (2009).

    ADS  MathSciNet  Google Scholar 

  9. T.-P. Cheng and L.-F. Li, Gauge Theory of Elementary Particle Physics, Oxford University Press, Oxford (1988).

    Google Scholar 

  10. M. E. Peskin and D. V. Shreder, Introduction to Quantum Field Theory [in Russian], NITs “Regulyarnaya i Khaoticheskaya Dinamika,” Izhevsk (2001).

  11. D. Callaway, Phys. Rep., 167, 241 (1988).

    Article  ADS  Google Scholar 

  12. F. Kleefeld, J. Phys., A39, L9 (2006).

    ADS  MathSciNet  Google Scholar 

  13. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford (1989).

    MATH  Google Scholar 

  14. L. N. Lipatov, Zh. Eksp. Teor. Fiz., 72, 411 (1977).

    Google Scholar 

  15. J. C. Collins and D. Soper, Ann. Phys., 112, 209 (1978).

    Article  ADS  Google Scholar 

  16. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4, Operator Analysis, Academic Press, New York (1978).

    MATH  Google Scholar 

  17. J. Glimm and A. Jaffe, Mathematical Methods of Quantum Physics, Springer, New York (1981).

    MATH  Google Scholar 

  18. E. P. Solodovnikova, A. N. Tavkhelidze, and O. A. Khrustalev, Teor. Mat. Fiz., 10, 162 (1972); 11, 317 (1972).

  19. V. A. Matveev, A. N. Tavkhelidze, and M. E. Shaposhnikov, Teor. Mat. Fiz., 59, 323 (1984).

    Google Scholar 

  20. A. N. Tavkhelidze and V. F. Tokarev, Elem. Chast. Atomn. Yadra, 21, 1126 (1990).

    Google Scholar 

  21. A. N. Sisakyan and I. L. Solovtsov, Elem. Chast. Atomn. Yadra, 25, 1127 (1994).

    Google Scholar 

  22. V. E. Rochev, J. Phys., A26, 1235 (1993).

    ADS  Google Scholar 

  23. V. E. Rochev, J. Phys., A30, 3671 (1997).

    ADS  MathSciNet  Google Scholar 

  24. V. E. Rochev and P. A. Saponov, Int. J. Mod. Phys., A13, 3649 (1998).

    Article  ADS  Google Scholar 

  25. V. E. Rochev, J. Phys., A33, 7379 (2000).

    ADS  MathSciNet  Google Scholar 

  26. V. E. Rochev, Teor. Mat. Fiz., 159, 81 (2009).

    Article  MathSciNet  Google Scholar 

  27. R. G. Jafarov and V. E. Rochev, Yad. Fiz., 76, 1149 (2013).

    Google Scholar 

  28. H. D. Dahmen and G. Jona-Lasinio, Nuovo Cim., A52, 807 (1967).

    Article  ADS  Google Scholar 

  29. A. N. Vasil’ev and A. K. Kazanskii, Teor. Mat. Fiz., 14, 289 (1973).

    Google Scholar 

  30. V. E. Rochev, Teor. Mat. Fiz., 51, 22 (1982).

    Article  Google Scholar 

  31. V. E. Rochev, J. Phys., A44, 305403 (2011).

    MathSciNet  Google Scholar 

  32. V. E. Rochev, J. Phys., A45, 205401 (2012).

    ADS  MathSciNet  Google Scholar 

  33. V. E. Rochev, J. Phys., A46, 185401 (2013).

    ADS  MathSciNet  Google Scholar 

  34. U. Wolff, Phys. Rev., D79, 105002 (2009).

    ADS  Google Scholar 

  35. P. Weisz and U. Wolff, arXiv:1012.0404.[hep-lat] (2010).

  36. I. M. Suslov, Zh. Eksp. Teor. Fiz., 134, 490 (2010); Ibid., 138, 508 (2008).

  37. D. I. Podolsky, arXiv:1003.3670.[hep-th] (2010).

  38. J. Fröhlich, Nucl. Phys., B200, 281 (1982).

    Article  ADS  Google Scholar 

  39. A. V. Manzhirov and A. D. Polyanin, Methods of Solving Integral Equations: Handbook [in Russian], Faktorial, Moscow (1999).

    MATH  Google Scholar 

  40. N. N. Kalitkin, Numerical Methods [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  41. L. V. Mosentsova, Mat. Comp. Model. Ser. Sci., No. 4, 156 (2010).

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Authors and Affiliations

  1. Institute for Physical Problems of Baku State University, Baku, Republic of Azerbaijan

    R. G. Jafarov

  2. Baku State University, Baku, Republic of Azerbaijan

    R. G. Jafarov, L. A. Agham-Alieva, P. É. Agha-Kishieva & S. G. Rahim-zade

  3. Nakhichevan University, Nakhichevan, Republic of Azerbaijan

    S. N. Mamedova

  4. Institute of Applied Mathematics, Baku State University, Baku, Republic of Azerbaijan

    M. M. Mutallimov

Authors
  1. R. G. Jafarov
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  2. L. A. Agham-Alieva
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  3. P. É. Agha-Kishieva
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  4. S. G. Rahim-zade
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  5. S. N. Mamedova
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  6. M. M. Mutallimov
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Correspondence to R. G. Jafarov.

Additional information

Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 204–212, November, 2016.

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Jafarov, R.G., Agham-Alieva, L.A., Agha-Kishieva, P.É. et al. Problem of the Landau Poles in Quantum Field Theory: from N. N. Bogolyubov to the Present Day. Russ Phys J 59, 1971–1980 (2017). https://doi.org/10.1007/s11182-017-1003-0

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  • DOI: https://doi.org/10.1007/s11182-017-1003-0

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