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Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Feng, Jonathan

Managing Editor: Lesna-Szreter, Paulina

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Volume 1, Issue 1 (Mar 2003)

Issues

Lectures on the functional renormalization group method

Janos Polonyi
  • Laboratory of Theoretical Physics, Louis Pasteur University, Strasbourg, France
  • Department of Atomic Physics, L. Eötvös University, Budapest, Hungary
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  • Other articles by this author:
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Published Online: 2003-03-01 | DOI: https://doi.org/10.2478/BF02475552

Abstract

These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained by a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scaler theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.

Keywords: strongly coupled theories; non-perturbative approximation; universality

Keywords: 25.75.-q

  • [1] N.N. Bogoliubov, D.V. Shirkov, Introduction to the Theory of Quantized Fields, 3rd ed., John Wiley and Sons, New York, 1973. Google Scholar

  • [2] K.G. Wilson, J. Kogut, Phys. Rep. c 12, 77 (1974); K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975); Rev. Mod. Phys. 55, 583 (1983). http://dx.doi.org/10.1016/0370-1573(74)90023-4CrossrefGoogle Scholar

  • [3] P.W. Anderson, G.J. Yuval, J. Phys. c 4, 607 (1971). http://dx.doi.org/10.1088/0022-3719/4/5/011CrossrefGoogle Scholar

  • [4] M. Gellman, F. Low, Phys. Rev. 95 1300 (1954). http://dx.doi.org/10.1103/PhysRev.95.1300CrossrefGoogle Scholar

  • [5] N. Goldenfeld, O. Martin, V. Oono, and F. Liu, Phys. Rev. Lett. 64, 1361 (1990); T. Kunihiro, Progr. of Theor. Phys. 95, 503 (1995). http://dx.doi.org/10.1103/PhysRevLett.64.1361CrossrefGoogle Scholar

  • [6] C. G. Callan Phys. Rev. d 2, 1541 (1970); K. Symanzik, Comm. Math. Phys. 18, 227 (1970). http://dx.doi.org/10.1103/PhysRevD.2.1541CrossrefGoogle Scholar

  • [7] J. Alexandre, V. Branchina, J. Polonyi, Phys. Rev. d 58, 16002 (1988). http://dx.doi.org/10.1103/PhysRevD.58.016002CrossrefGoogle Scholar

  • [8] F.J. Wegner, A. Houghton, Phys. Rev. a 8, 401 (1973). http://dx.doi.org/10.1103/PhysRevA.8.401CrossrefGoogle Scholar

  • [9] J. Polchinski, Nucl. Phys. b 231, 269 (1984). http://dx.doi.org/10.1016/0550-3213(84)90287-6CrossrefGoogle Scholar

  • [10] T. Morris, Int. J. Mod. Phys. A9 (1994) 2411. http://dx.doi.org/10.1142/S0217751X94000972CrossrefGoogle Scholar

  • [11] J. Collins, Renormalization, Cambridge University Press, 1984. Google Scholar

  • [12] J. Polonyi, K. Sailer, Phys. Rev. d 63, 105006 (2001). http://dx.doi.org/10.1103/PhysRevD.63.105006CrossrefGoogle Scholar

  • [13] J. Hughes and J. Liu, Nucl. Phys. b 307, 183 (1988); J Hughes, Nucl. Phys. b 312, 125 (1989); G.M. Shore, Nucl. Phys. b 362, 85 (1991); G. Keller and C. Kopper, Comm. Math. Phys. 148, 445 (1992); Comm. Math. Phys. 153, 245 (1993); C. Becchi, On the Construction of Renormalized Gauge Theories Using Renormalization Group Techniques, hep-th/9607188. http://dx.doi.org/10.1016/0550-3213(88)90528-7CrossrefGoogle Scholar

  • [14] B.P. Dolan, Int. J. of Mod. Phys. 10, 2439 (1995); Int. J. of Mod. Phys. 10, 2703 (1995); Int. J. of Mod. Phys. 12, 2413 (1995); D.C. Brody, A. Ritz, Nucl. Phys. b 522, 588 (1998); B.P. Dolan, A. Lewis, Phys. Lett. b 460 302 (1999). http://dx.doi.org/10.1142/S0217751X95001170CrossrefGoogle Scholar

  • [15] G. Altarelli, G. Parisi, Nucl. Phys. b 126, 278 (1977); A. H. Mueller, Phys. Rep. 73, 237 (1981). http://dx.doi.org/10.1016/0550-3213(77)90384-4CrossrefGoogle Scholar

  • [16] C. Wetterich, Phys. Lett. b 301, 90, (1993). http://dx.doi.org/10.1016/0370-2693(93)90726-XCrossrefGoogle Scholar

  • [17] T. Morris, Phys. Lett. b 329, 241 (1944). Google Scholar

  • [18] D. Litim, Optimized renormalization group flow, hep-th/0103195. Google Scholar

  • [19] M. Reuter and C. Wetterich, Nucl. Phys. b 391, 147 (1993); N. Tetradis and D.F. Litim, Nucl. Phys. b 464, 492 (1996). http://dx.doi.org/10.1016/0550-3213(93)90145-FCrossrefGoogle Scholar

  • [20] U. Ellwanger, Phys. Lett. b 335, 364 (1994). http://dx.doi.org/10.1016/0370-2693(94)90365-4CrossrefGoogle Scholar

  • [21] J. Adams, J. Berges, S. Bornholdt, F. Freire, N. Tetradis, C. Wetterich, Mod. Phys. Lett. a 10, 2367 (1995). http://dx.doi.org/10.1142/S0217732395002520CrossrefGoogle Scholar

  • [22] J. Alexandre, J. Polonyi, Ann. Phys. 288, 37 (2001). http://dx.doi.org/10.1006/aphy.2000.6109CrossrefGoogle Scholar

  • [23] S. Coleman, E. Weinberg, Phys. Rev. d 7, 1888 (1973). http://dx.doi.org/10.1103/PhysRevD.7.1888CrossrefGoogle Scholar

  • [24] A. Parola, L. Reatto, Phys. Rev. Lett. 53, 2417 (1984); Phys. Rev. a 31, 3309 (1985); V. I. Tikar, Phys. Lett. a 104, 135 (1984); J. Comellas, A. Travesset, Nucl. Phys. b 498, 539 (1997). http://dx.doi.org/10.1103/PhysRevLett.53.2417CrossrefGoogle Scholar

  • [25] K. I. Aoki et al, Progr. of Theor. Phys. 95, 409 (1996). http://dx.doi.org/10.1143/PTP.95.409CrossrefGoogle Scholar

  • [26] S. B. Liao, M. Strickland, Nucl. Phys. b 497, 611 (1997). http://dx.doi.org/10.1016/S0550-3213(97)00212-5CrossrefGoogle Scholar

  • [27] A. Margaritis, G. Ódor, A. Patkos, Z. Phys. c 39, 109 (1998) http://dx.doi.org/10.1007/BF01560398CrossrefGoogle Scholar

  • [28] P. E. Haagensen, Y. Kubyshin, J. I. Latorre, E. Moreno, Phys. Lett. b 323, 330 (1994) http://dx.doi.org/10.1016/0370-2693(94)91228-9CrossrefGoogle Scholar

  • [29] T. Morris, Phys. Lett. b 334, 355 (1994). http://dx.doi.org/10.1016/0370-2693(94)90700-5CrossrefGoogle Scholar

  • [30] N. Tetradis, C. Wetterich, Nucl. Phys. b 422, 541 (1994). http://dx.doi.org/10.1016/0550-3213(94)90446-4CrossrefGoogle Scholar

  • [31] M. Alford, Phys. Lett. b 336, 237 (1994). http://dx.doi.org/10.1016/0370-2693(94)01005-6CrossrefGoogle Scholar

  • [32] S. Liao, J. Polonyi, M. Strickland, Nucl. Phys. b 567, 493 (2000). http://dx.doi.org/10.1016/S0550-3213(99)00496-4CrossrefGoogle Scholar

  • [33] D. F. Litim, Phys. Lett. b 486, 92 (2000); Int. J. of Mod. Phys. 16, 2081 (2001). http://dx.doi.org/10.1016/S0370-2693(00)00748-6CrossrefGoogle Scholar

  • [34] S. Seide, C. Wetterich, Nucl. Phys. b 562, 524 (1999). http://dx.doi.org/10.1016/S0550-3213(99)00545-3CrossrefGoogle Scholar

  • [35] J. Berges, C. Wetterich, Nucl. Phys. b 487, 675 (1997). http://dx.doi.org/10.1016/S0550-3213(96)00670-0CrossrefGoogle Scholar

  • [36] M. Maggiore, Z. Phys. c 41, 687 (1989); T. E. Clarke, B. Haeri, S. T. Love, Nucl. Phys. b 402, 628 (1993); M. D’Attanasio, T. Morris, Phys. Lett. b 409, 363 (1997); J. Berges, D. U. Jungnickel, C. Wetterich, Phys. Rev. d 59, 034010 (1999); http://dx.doi.org/10.1007/BF01564716CrossrefGoogle Scholar

  • [37] D. Zanchi, H. J. Schultz, Phys. Rev. b 54, 9509 (1996); Phys. Rev. b 61, 13609 (2000); C. J. Halboth, W. Metzner, Phys. Rev. b 61, 4364 (2000); C. Honerkamp, M. Salmhofer, N. Furukawa, T. M. Rrice, Phys. Rev. b 63][45114, 2001 (;) M. Salmhofer. C. Honerkamp, Progr. of Theor. Phys. 105, 1 (2001); S. Correia, J. Polonyi, J. Richert, Functional Callan-Symanzik Equation for the Coulomb Gas, cond-mat/0108048. http://dx.doi.org/10.1103/PhysRevB.54.9509CrossrefGoogle Scholar

  • [38] J. Polonyi, Current-density functional for disordered systems, cond-mat/0203090. Google Scholar

  • [39] A. Bonanno, M. Reuter, Phys. Rev. d 62, 043008 (2000). http://dx.doi.org/10.1103/PhysRevD.62.043008CrossrefGoogle Scholar

  • [40] H. Gies, C. Wetterich, Renormalization group flow of bound states, hep-th/0107221. Google Scholar

  • [41] J. Polonyi, K. Sailer, Effective actions for the density functional theory, cond-mat/0108179. Google Scholar

  • [42] C. Bagnuls, C. Bervillier, hep-th/0002034; Berges, N. Tetradis, C. Wetterich, hep-th/0005122. Google Scholar

  • [43] Proceedings of Workshop on the Exact Renormalization Group, Faro, Portugal, 1998, ed. Krasnitz et al. (World Scientific, Singapore, 1999); 2nd Conference on Exact RG, Rome 2000, ed. S. Arnone, at al. Int. J. of Mod. Phys. 16, 1809 (2001). Google Scholar

  • [44] C. Harvey-Fros, PhD Thesis, university of Southampton, hep-th/0108018; M. D. Turner, PhD Thesis, University of Southampton, hep-th/0108031. Google Scholar

  • [45] F. Wegner, J. Phys. c 7, 2098 (1974). http://dx.doi.org/10.1088/0022-3719/7/12/004CrossrefGoogle Scholar

  • [46] J. I. Latorre, T. Morris, Int. J. of Mod. Phys. 16, 2071 (2001). http://dx.doi.org/10.1142/S0217751X01004724CrossrefGoogle Scholar

  • [47] L. Schulmann, Techniques and Applications of Path Integrations, John Wiley and Sons, New York, 1981. Google Scholar

  • [48] D. McLaughlin, L. Schulman, J. Math. Phys. 12, 2520 (1971). http://dx.doi.org/10.1063/1.1665567CrossrefGoogle Scholar

  • [49] J. Polonyi, Ann. Phys. 252, 300 (1996). http://dx.doi.org/10.1006/aphy.1996.0133CrossrefGoogle Scholar

  • [50] R. D. Ball, P. E. Haagensen, J. I. Latorre, E. Moreno, Phys. Lett. b 347, 80 (1995); http://dx.doi.org/10.1016/0370-2693(95)00025-GCrossrefGoogle Scholar

  • [51] J. Comellas, Phys. Lett. b 509, 662 (1998). Google Scholar

  • [52] T. Morris, Nucl. Phys. b 495, 477 (1997); Phys. Rev. Lett. 77, 1658 (1996). http://dx.doi.org/10.1016/S0550-3213(97)00233-2CrossrefGoogle Scholar

  • [53] A. Hasenfratz, P. Hasenfratz, Nucl. Phys. b 270, 687 (1986). http://dx.doi.org/10.1016/0550-3213(86)90573-0CrossrefGoogle Scholar

  • [54] K. Halpern, K. Huang, Phys. Rev. d 35, 3252 (1996); Phys. Rev. Lett. 77, 1659 (1996); V. Periwal, Mod. Phys. Lett. a 11, 2915 (1996); K. Halpern, Phys. Rev. d 57, 6337 (1998). http://dx.doi.org/10.1103/PhysRevD.53.3252CrossrefGoogle Scholar

  • [55] G. Felder, Comm. Math. Phys. 111, 101 (1987); A. E. Filippov, S. A. Breus, Phys. Lett. a 158, 300 (1981); S. A. Breus, A. E. Filippov, Physica a 192, 486 (1993); T. Morris, Phys. Rev. Lett. 1658, 1996 (.) http://dx.doi.org/10.1007/BF01239018CrossrefGoogle Scholar

  • [56] T. Morris, Phys. Lett. b 329, 241 (1994). http://dx.doi.org/10.1016/0370-2693(94)90767-6CrossrefGoogle Scholar

  • [57] C. Bagnulus, C. Bervillier, Phys. Rev. b 41, 402 (1990). http://dx.doi.org/10.1103/PhysRevB.41.402CrossrefGoogle Scholar

  • [58] T. Morris, Phys. Lett. b 345, 139 (1995); http://dx.doi.org/10.1016/0370-2693(94)01603-ACrossrefGoogle Scholar

  • [59] R. Jackiw, S. Templeton, Phys. Rev. d 23, 2291 (1981). http://dx.doi.org/10.1103/PhysRevD.23.2291CrossrefGoogle Scholar

  • [60] J. Polchinski, in Proceedings of the 1992 TASI Elementary Particle Physics, Eds. J. Polchinski and J. Harvey, World Sci., 1992. Google Scholar

  • [61] S. Weinberg, Nucl. Phys. b 413, 567 (1994). http://dx.doi.org/10.1016/0550-3213(94)90001-9CrossrefGoogle Scholar

  • [62] R. Shankar, Rev. Mod. Phys. 66, 129 (1994). http://dx.doi.org/10.1103/RevModPhys.66.129CrossrefGoogle Scholar

  • [63] J. Polonyi, Nucl. Phys. (Proc. Suppl.)B20 (1991) 32; Confinement and Renormalization, hep-ph/9511243, in the Proceedings of the International School of Physics, Enrico Fermi, Course CXXX, on Selected Topics in Nonperturbative QCD, Soc. It. di Fisica; K. Johnson, L. Lellouch, J. Polonyi, Nucl. Phys. b 367, 675 (1991). http://dx.doi.org/10.1016/0920-5632(91)90876-GCrossrefGoogle Scholar

  • [64] V. A. Miransky, Nuovo Cimento a 90, 149 (1985); P.I. Fomin, V. P. Gusynin, V. A. Miransky, Yu. A. Sitenko, Riv. Nuovo Cimento 6, 1 (1983); C. N. Leung, S. T. Love, W. A. Bardeen, Nucl. Phys. b 273, 649 (1986); W. A. Bardeen, S. T. Love, C. N. Leung, Nucl. Phys. b 323, 493 (1989). http://dx.doi.org/10.1007/BF02724229CrossrefGoogle Scholar

  • [65] T. Appelquist, T. Takeuchi, M. B. Einhorn, L. C. R. Wijewardhana, Phys. Lett. b 220, 223 (1989); K. Matumoto, Progr. of Theor. Phys. 81, 277 (1989); T. Takeuchi, Phys. Rev. d 40, 2697 (1989). http://dx.doi.org/10.1016/0370-2693(89)90041-5CrossrefGoogle Scholar

  • [66] V. A. Miransky, M. Tanabashi, K. Yamawaki, Phys. Lett. b 221, 177 (1989); Mod. Phys. Lett. a 4, 1043 (1989). http://dx.doi.org/10.1016/0370-2693(89)91494-9CrossrefGoogle Scholar

  • [67] V. A. Miransky, Int. J. of Mod. Phys. 6, 1641 (1991). http://dx.doi.org/10.1142/S0217751X91000861CrossrefGoogle Scholar

  • [68] Y. Nambu, Proceedings of the 1989 Workshop on Dynamical Symmetry Breaking, eds. T. Muta, K. Yamawaki (Nagoya University, 1990); W. J. marciano, Phys. Rev. Lett. 62, 2793 (1989); Phys. Rev. d 41, 1647 (1990); W. A. Bardeen, C. T. Hill, M. Lindner, Phys. Rev. d 41, 1647 (1990). Google Scholar

  • [69] V. A. Alessandrini, H. de Vega, F. Schaposnik, Phys. Rev. b 10, 3906 (1974); Phys. Rev. b 12, 5034 (1975); T. Appelquist, J. Carazzone, Phys. Rev. d 11, 2865 (1975); T. Appelquist, J. Carazzone, H. Kluberg-Stern, M. Roth, Phys. Rev. Lett. 36, 768 (1976), Phys. Rev. Lett. 36, 1161 (1976); T. Appelquist, J. Carazzone, Nucl. Phys. b 120, 77 (1977). http://dx.doi.org/10.1103/PhysRevB.10.3906CrossrefGoogle Scholar

  • [70] J. S. Bell, R. Jackiw, Nuovo Cimento a 60, 47 (1969); S. Adler, Phys. Rev. 177, 2426 (1969). http://dx.doi.org/10.1007/BF02823296CrossrefGoogle Scholar

  • [71] S. Adler, J. Collins, A. Duncan, Phys. Rev. d 15, 1712 (1977); J. Collins, A. Duncan, S. Joeglekar, Phys. Rev. d 16, 438 (1977); N. K. Nielsen, Nucl. Phys. b 120, 212 (1977). http://dx.doi.org/10.1103/PhysRevD.15.1712CrossrefGoogle Scholar

  • [72] V. Branchina, J. Polonyi, Nucl. Phys. b 433, 99 (1995). http://dx.doi.org/10.1016/0550-3213(94)00367-NCrossrefGoogle Scholar

  • [73] M. E. Fischer, in Proceedings of the Conference Renormalization Group in Critical Phenomena and Quantum Field Theory, edt. J. D. Gunton, M. S. Green, Temple University, 1973. Google Scholar

  • [74] J. Alexandre, V. Branchina, J. Polonyi, Phys. Lett. b 445, 351 (1999). http://dx.doi.org/10.1016/S0370-2693(98)01491-9CrossrefGoogle Scholar

  • [75] A. Ringwald, C. Wetterich, Nucl. Phys. b 334, 506 (1990); N. Tetradis, C. Wetterich, Nucl. Phys. b 383, 197 (1992). http://dx.doi.org/10.1016/0550-3213(90)90489-ZCrossrefGoogle Scholar

  • [76] R. B. Griffith, P. A. Pierce, Phys. Rev. Lett. 41, 917 (1978); J. Stat. Physics. 20, 499 (1979); O. Hudák, Phys. Lett. a 73, 273 (1979); C. Rebbi, R. H. Swendsen, Phys. Rev. b 21, 4094 (1980); R. B. Griffith, Physica a 106, 59 (1981); A. Stella, Physica a 108, 211 (1981). http://dx.doi.org/10.1103/PhysRevLett.41.917CrossrefGoogle Scholar

  • [77] A. Hasenfratz, P. Hasenfratz, Nucl. Phys. b 295, 1 (1988) http://dx.doi.org/10.1016/0550-3213(88)90224-6CrossrefGoogle Scholar

  • [78] H. W. Blöte, R. H. Swendsen, Phys. Rev. Lett. 43, 799 (1979); C. Lang, Nucl. Phys. b 280, 255 (1987); K. Decker, A. Hasenfratz, P. Hasenfratz, Nucl. Phys. b 295, 21 (1988); A. Gonzales-Arroyo, M. Okawa, Y. Shimizu, Phys. Rev. Lett. 60, 487 (1988); F. Igloi, C. Vanderzande, Physica a 135, 347 (1986). http://dx.doi.org/10.1103/PhysRevLett.43.799CrossrefGoogle Scholar

  • [79] R. B. Israel, in Random Fields (Esztergom, 1979), J. Fritz, J. L. Lebowitz, D. Szász eds. (North-Holland, Amsterdam, 1981); A. C. D. van Enter, R. Fernández, A. Sokal, J. Stat. Physics. 72, 879 (1993). Google Scholar

  • [80] J. Alexandre, J. Polonyi, Tree-level renormalization, hep-th/9906017. Google Scholar

  • [81] Sz. Borsanyi, A. Patkos, J. Polonyi, Zs. Szep, Phys. Rev. d 62, 085013 (2000). http://dx.doi.org/10.1103/PhysRevD.62.085013CrossrefGoogle Scholar

  • [82] S. Matinyan, G. Savvidy, Nucl. Phys. b 134, 539 (1978); G. Savvidy, Phys. Lett. b 71, 133 (1977); M. Duff, R. Ramón-Medrano, Phys. Rev. d 12, 3357 (1976); H. Pagels, E. Tomboulis, Nucl. Phys. b 143, 485 (1978). http://dx.doi.org/10.1016/0550-3213(78)90463-7CrossrefGoogle Scholar

  • [83] N. K. Nielsen, P. Olesen, Nucl. Phys. b 144, 376 (1978); Phys. Lett. b 79, 304 (1978); J. Amjorn, N. K. Nielsen, P. Olesen, Nucl. Phys. b 152, 75 (1979); H. B. Nielsen, P. Olesen, Nucl. Phys. b 160, 380 (1979); H. B. Nielsen, M. Ninomiya, Nucl. Phys. b 156, 1 (1979); J. Amjorn, P. Olesen, Nucl. Phys. b 60, 1980 (.) http://dx.doi.org/10.1016/0550-3213(78)90377-2CrossrefGoogle Scholar

  • [84] G. Dosch, Phys. Lett. b 190, 555 (1987); G. Dosch, Y. Simonov, Phys. Lett. b 205, 339 (1988); Y. Simonov, Yad. Fiz. 48, 1381 (1988); Yad. Fiz. 50, 213 (1988). Google Scholar

  • [85] G. ’tHooft, Phys. Rev. Lett. 37, 8 (1976). http://dx.doi.org/10.1103/PhysRevLett.37.8CrossrefGoogle Scholar

  • [86] J. Fingberg, J. Polonyi, Nucl. Phys. b 486, 315 (1997); V. Branchina and H. Mohrbach, J. Polonyi, Phys. Rev. d 60, 45006 (1999); Phys. Rev. d 60, 45007 (1999); M. Dufour Fournier, J. Polonyi, Phys. Rev. d 61, 065008 (2000). http://dx.doi.org/10.1016/S0550-3213(96)00649-9CrossrefGoogle Scholar

  • [87] K. Huang, J. Polonyi, Int. J. of Mod. Phys. 6, 409 (1991). http://dx.doi.org/10.1142/S0217751X91000253CrossrefGoogle Scholar

  • [88] I. Nándori, J. Polonyi, K. Sailer, Phys. Rev. d 63, 045022 (2001). http://dx.doi.org/10.1103/PhysRevD.63.045022CrossrefGoogle Scholar

  • [89] J. M. Kosterlitz, D. J. Thouless, Journal J. Phys. CC61181973; J. M. Kosterlitz, Journal J. Phys. CC710461974. Google Scholar

  • [90] J. V. Jose, L. P. Kadanoff, S. Kirkpatrick, D. R. Nelson, Phys. Rev. b 16, 1217 (1977). http://dx.doi.org/10.1103/PhysRevB.16.1217CrossrefGoogle Scholar

  • [91] S. Coleman, Phys. Rev. d 11, 3424 (1975). http://dx.doi.org/10.1103/PhysRevD.11.2088CrossrefGoogle Scholar

  • [92] S. Mandelstam, Phys. Rev. d 11, 3026 (1975). http://dx.doi.org/10.1103/PhysRevD.11.3026CrossrefGoogle Scholar

  • [93] S. Samuel, Phys. Rev. d 18, 1916 (1978). http://dx.doi.org/10.1103/PhysRevD.18.1916CrossrefGoogle Scholar

  • [94] J. Polonyi, The confinement-deconfinement mechanism, in Quark Gluon Plasma, World Scientific, 1990, R. Hwa, edt. Google Scholar

  • [95] N. D. Mermin, H. Wagner, Phys. Rev. Lett. 17, 1133 (1966); S. Coleman, Comm. Math. Phys. 31, 259 (1973). http://dx.doi.org/10.1103/PhysRevLett.17.1133CrossrefGoogle Scholar

  • [96] G. V. Gersdorff, C. Wetterich, Nonperturbative Renormalization Flow and Essential Scaling for the Kosterlitz-Thouless Transition, hep-th/0008114. Google Scholar

  • [97] C. Becchi, in Elementary Particles, Field Theory and Statistical Mechanics, Eds. M. Bonini, G. Marchesini, E. Onofri, Parma University 1993, hep-th/9607188; M. Bonini, M. D’Attanasio, G. Marchesini, Nucl. Phys. b 409, 1993 (441); Nucl. Phys. b 418, 1994 (81); Nucl. Phys. b 421, 1994 (429); Nucl. Phys. b 437, 1995 (163); M. Reuter, C. Wetterich, Nucl. Phys. b 417, 181 (1994); Nucl. Phys. b 427, 291 (1994); M. D’Attanasio, T. R. Morris, Phys. Lett. b 378, 213 (1996). Google Scholar

  • [98] D. F. Litim, J. M. Pawlowski, Phys. Lett. b 435, 181 (1998); F. Freire, D. F. Litim, J. M. Pawlowski, Phys. Lett. b 495, 256 (2000). http://dx.doi.org/10.1016/S0370-2693(98)00761-8CrossrefGoogle Scholar

  • [99] T. R. Morris, Nucl. Phys. b 573, 97 (2000), J. High of Energy Phys. 0012, 012 (2000). http://dx.doi.org/10.1016/S0550-3213(99)00821-4CrossrefGoogle Scholar

  • [100] M. Simionato, Int. J. of Mod. Phys. 15, 2121 (2000); Int. J. of Mod. Phys. 15, 2153 (2000). CrossrefGoogle Scholar

  • [101] J.Alexandre, J. Polonyi, K. Sailer, in preparation. Google Scholar

  • [102] K. Wilson, Rev. Mod. Phys. 55, 583 (1983). http://dx.doi.org/10.1103/RevModPhys.55.583CrossrefGoogle Scholar

  • [103] M. E. Fischer, in the Proceedings of Enrico Fermi International School of Physics, ed. M. S. Green (Academic Press, New York, 1971); M. P. Nightingale, Physica a 83, 561 (1976). Google Scholar

  • [104] K. Wilson, M. E. Fischer, Phys. Rev. Lett. 28, 248 (1972); K. Wilson, Phys. Rev. d 7, 2911 (1973). Google Scholar

  • [105] P. C. Hohenberg, B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977). http://dx.doi.org/10.1103/RevModPhys.49.435CrossrefGoogle Scholar

  • [106] G. ’tHooft, Proceedings of Colloquim on Renormalization of Yang-Mills Fields and Application to Particle Physics, ed. C. P. Korthes-Altes, (Marseilles, 1972); D. Gross, F. Wilcek, Phys. Rev. Lett. 30, 1343 (1973); H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973); S. Coleman, D. Gross, Phys. Rev. Lett. 31, 851 (1973). Google Scholar

  • [107] S. Weinberg, Physica a 96, 327 (1979); J. Gasser, H. Leutwyler, Ann. Phys. 158, 142 (1984); Nucl. Phys. b 250, 465 (1985); Phys. Rep. 87, 77 (1982); D. B. Kaplan, A. V. Manohar, Phys. Rev. Lett. 56, 2004 (1986). http://dx.doi.org/10.1016/0378-4371(79)90223-1CrossrefGoogle Scholar

  • [108] S. Weinberg, Phys. Lett. b 91, 51 (1980) I. J. R. Aitchison, Acta Phys. Polonica b 18, 191 (1987). http://dx.doi.org/10.1016/0370-2693(80)90660-7CrossrefGoogle Scholar

  • [109] N. Isgur, M. B. Wise, Phys. Lett. b 232, 113 (1989); Phys. Lett. b 237, 527 (1990). http://dx.doi.org/10.1016/0370-2693(89)90566-2CrossrefGoogle Scholar

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Published Online: 2003-03-01

Published in Print: 2003-03-01


Citation Information: Open Physics, ISSN (Online) 2391-5471, DOI: https://doi.org/10.2478/BF02475552.

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[93]
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[94]
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[95]
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[96]
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[97]
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[98]
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[99]
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[100]
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[101]
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[102]
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[103]
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[104]
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[105]
S. Diehl, H. Gies, J. M. Pawlowski, and C. Wetterich
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Journal of High Energy Physics, 2005, Volume 2005, Number 10, Page 050
[110]
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[111]
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[112]
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[113]
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[115]
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[116]
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Journal of Physics A: Mathematical and General, 2006, Volume 39, Number 25, Page 8119
[117]
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