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Non-perturbative calculations for the effective potential of the PT symmetric and non-Hermitian (-gφ4) field theoretical model

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Abstract

We investigate the effective potential of the PT symmetric (-gφ4) field theory, perturbatively as well as non-perturbatively. For the perturbative calculations, we first use normal ordering to obtain the first order effective potential, from which the predicted vacuum condensate vanishes exponentially as G→0+, in agreement with previous calculations. For the higher orders, we employ the invariance of the bare parameters under the change of the mass scale t to fix the transformed form totally equivalent to the original theory. The form so obtained up to G3 is new and shows that all the 1PI amplitudes are perturbative for both the \(G\ll1\) and the \(G\gg1\) region. For the intermediate region, we modified the fractal self-similar resummation method to have a unique resummation formula for all G values. This unique formula is necessary, because the effective potential is the generating functional for all the one-particle irreducible (1PI) amplitudes that can be obtained via ∂nE/∂bn, and thus we can obtain an analytic calculation of the 1PI amplitudes. Again, the resummed form of the effective potential is new and interpolates the effective potential between the perturbative regions. Moreover, the resummed effective potential agrees in spirit to a previous calculation concerning bound states.

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References

  1. C.M. Bender, Peter N. Meisinger, H. Yang, Phys. Rev. D 63, 045001 (2001)

    Article  ADS  Google Scholar 

  2. C.M. Bender, S. Boettcher, P.N. Meisinger, J. Math. Phys. 40, 2201 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. C.M. Bender, F. Cooper, P.N. Meisinger, V.M. Savage, Phys. Lett. A 259, 224 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. C.M. Bender, G.V. Dunne, J. Math. Phys. 40, 4616 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. E. Delabaere, F. Pham, Phys. Lett. A 250, 25 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  7. E. Delabaere, F. Pham, Phys. Lett. A 250, 29 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  8. E. Delabaere, D.T. Trinh, J. Phys. A 33, 8771 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. G.A. Mezincescu, J. Phys. A 33, 4911 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. C.M. Bender, S. Boettcher, V.M. Savage, J. Math. Phys. 41, 6381 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. C.M. Bender, Q. Wang, J. Phys. A 34, 9835 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. K.C. Shin, University of Illinois, preprint

  13. C.M. Bender, E.J. Weniger, J. Math. Phys. 42, 2167 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. C.M. Bender, G.V. Dunne, P.N. Meisinger, M. Simsek, Phys. Lett. A 281, 311 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. P. Dorey, C. Dunning, R. Tateo, hep-th/0103051

  16. C.M. Bender, M. Berry, P.N. Meisinger, V.M. Savage, M. Simsek, J. Phys. A 34, L31 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. C.M. Bender, J.-H. Chen, K.A. Milton, J. Phys. A 39, 1657 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. C.M. Bender, S. Boettcher, H.F. Jones, P.N. Meisinger, M. Simsek, Phys. Lett. A 291, 197 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. S. Gluzman, V.I. Yukalov, Phys. Rev. E 55, 3983 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  20. E.P. Yukalova, V.I. Yukalov, Phys. Lett. A 115, 27 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  21. V.I. Yukalov, Int. J. Theor. Phys. 28, 1237 (1989)

    Article  MATH  Google Scholar 

  22. V.I. Yukalov, Phys. Rev. A 42, 3324 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  23. V.I. Yukalov, Physica A 167, 833 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  24. V.I. Yukalov, Nuovo Cim. A 103, 1577 (1990)

    ADS  Google Scholar 

  25. V.I. Yukalov, Proc. Lebedev Phys. Inst. 188, 297 (1991)

    Google Scholar 

  26. V.I. Yukalov, J. Math. Phys. 32, 1235 (1991)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. S. Gluzman, V.I. Yukalov, Phys. Rev. E 55, 6552 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  28. S. Coleman, Phys. Rev. D 11, 2088 (1975)

    Article  ADS  Google Scholar 

  29. M. Dineykhan, G.V. Efimov, G. Ganbold, S.N. Nedelko, Lect. Notes Phys. 26, 1 (1995)

    Article  Google Scholar 

  30. M.E. Peskin, D.V. Schroeder, An Introduction to the Quantum Field Theory (Addison-Wesley, Reading, MA, 1995)

    Google Scholar 

  31. C. Caratheodory, Theory of Functions of a Complex Variable (A.M.S. Chelsea Publishing, American Mathematical Society, Providence, Rhode Island, 1954)

    Google Scholar 

  32. M. Kaku, Quantum Field Theory, a Modern Introduction (Oxford University Press, New York, Oxford, 1993)

    Google Scholar 

  33. A.M. Shalaby, Czech. J. Phys. 56, 1033 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. P.M. Stevenson, Phys. Rev. D 32, 1389 (1985)

    Article  ADS  Google Scholar 

  35. W.-F. Lu, C.K. Kim, J. Phys. A 35, 393 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  36. S.J. Chang, Phys. Rev. D 12, 1071 (1975)

    Article  ADS  Google Scholar 

  37. J.C. Collins, Renormalization (Cambridge UniversityPress, Cambridge, 1984)

    MATH  Google Scholar 

  38. E. Brezin, J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. D 15, 1544 (1977)

    Article  ADS  Google Scholar 

  39. E. Brezin, J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. D 15, 1558 (1977)

    Article  ADS  Google Scholar 

  40. H. Kleinert, S. Thoms, W. Janke, Phys. Rev. A 55, 915 (1996)

    Article  ADS  Google Scholar 

  41. G.A. Baker, P. Graves-Moris, Padé Approximants (Cambridge University, Cambridge, 1996)

    MATH  Google Scholar 

  42. G.A. Baker, G.S. Rashbrooke, H.E. Gilbert, Phys. Rev. A 135, 1272 (1964)

    ADS  Google Scholar 

  43. R.E. Shafer, SIAM J. Numer. Anal. 11, 447 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  44. A.V. Sergeev, J. Phys. A 28, 4157 (1995)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  45. V.I. Yukalov, Mod. Phys. Lett. B 14, 791 (2000)

    Article  MATH  ADS  Google Scholar 

  46. B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983)

    Google Scholar 

  47. H. Kröger, Phys. Rep. 323, 81 (2000)

    Article  MathSciNet  Google Scholar 

  48. V.I. Yukalov, S. Gluzman, Phys. Rev. E 55, 6552 (1997)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Centre for Theoretical Physics, The British University in Egypt, El Sherouk City, Misr Ismalia Desert Road, Postal No. 11837, P.O. Box 43, Egypt

    A.M. Shalaby

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Correspondence to A.M. Shalaby.

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PACS

11.10.Kk; 02.30.Mv; 11.10.Lm; 11.30.Er; 11.30.Qc; 11.15.Tk

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Shalaby, A. Non-perturbative calculations for the effective potential of the PT symmetric and non-Hermitian (-gφ4) field theoretical model. Eur. Phys. J. C 50, 999–1006 (2007). https://doi.org/10.1140/epjc/s10052-007-0236-4

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  • DOI: https://doi.org/10.1140/epjc/s10052-007-0236-4

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