PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2003 | 1 | 1 | 1-71
Tytuł artykułu

Lectures on the functional renormalization group method

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca

Czasopismo
Rocznik
Tom
1
Numer
1
Strony
1-71
Opis fizyczny
Daty
wydano
2003-03-01
online
2003-03-01
Bibliografia
  • [1] N.N. Bogoliubov, D.V. Shirkov, Introduction to the Theory of Quantized Fields, 3rd ed., John Wiley and Sons, New York, 1973.
  • [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)90023-4[Crossref]
  • [3] P.W. Anderson, G.J. Yuval, J. Phys. c 4, 607 (1971). http://dx.doi.org/10.1088/0022-3719/4/5/011[Crossref]
  • [4] M. Gellman, F. Low, Phys. Rev. 95 1300 (1954). http://dx.doi.org/10.1103/PhysRev.95.1300[Crossref]
  • [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.64.1361[Crossref]
  • [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.2.1541[Crossref]
  • [7] J. Alexandre, V. Branchina, J. Polonyi, Phys. Rev. d 58, 16002 (1988). http://dx.doi.org/10.1103/PhysRevD.58.016002[Crossref]
  • [8] F.J. Wegner, A. Houghton, Phys. Rev. a 8, 401 (1973). http://dx.doi.org/10.1103/PhysRevA.8.401[Crossref]
  • [9] J. Polchinski, Nucl. Phys. b 231, 269 (1984). http://dx.doi.org/10.1016/0550-3213(84)90287-6[Crossref]
  • [10] T. Morris, Int. J. Mod. Phys. A9 (1994) 2411. http://dx.doi.org/10.1142/S0217751X94000972[Crossref]
  • [11] J. Collins, Renormalization, Cambridge University Press, 1984.
  • [12] J. Polonyi, K. Sailer, Phys. Rev. d 63, 105006 (2001). http://dx.doi.org/10.1103/PhysRevD.63.105006[Crossref]
  • [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)90528-7[Crossref]
  • [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/S0217751X95001170[Crossref]
  • [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)90384-4[Crossref]
  • [16] C. Wetterich, Phys. Lett. b 301, 90, (1993). http://dx.doi.org/10.1016/0370-2693(93)90726-X[Crossref]
  • [17] T. Morris, Phys. Lett. b 329, 241 (1944).
  • [18] D. Litim, Optimized renormalization group flow, hep-th/0103195.
  • [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-F[Crossref]
  • [20] U. Ellwanger, Phys. Lett. b 335, 364 (1994). http://dx.doi.org/10.1016/0370-2693(94)90365-4[Crossref]
  • [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/S0217732395002520[Crossref]
  • [22] J. Alexandre, J. Polonyi, Ann. Phys. 288, 37 (2001). http://dx.doi.org/10.1006/aphy.2000.6109[Crossref]
  • [23] S. Coleman, E. Weinberg, Phys. Rev. d 7, 1888 (1973). http://dx.doi.org/10.1103/PhysRevD.7.1888[Crossref]
  • [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.53.2417[Crossref]
  • [25] K. I. Aoki et al, Progr. of Theor. Phys. 95, 409 (1996). http://dx.doi.org/10.1143/PTP.95.409[Crossref]
  • [26] S. B. Liao, M. Strickland, Nucl. Phys. b 497, 611 (1997). http://dx.doi.org/10.1016/S0550-3213(97)00212-5[Crossref]
  • [27] A. Margaritis, G. Ódor, A. Patkos, Z. Phys. c 39, 109 (1998) http://dx.doi.org/10.1007/BF01560398[Crossref]
  • [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)91228-9[Crossref]
  • [29] T. Morris, Phys. Lett. b 334, 355 (1994). http://dx.doi.org/10.1016/0370-2693(94)90700-5[Crossref]
  • [30] N. Tetradis, C. Wetterich, Nucl. Phys. b 422, 541 (1994). http://dx.doi.org/10.1016/0550-3213(94)90446-4[Crossref]
  • [31] M. Alford, Phys. Lett. b 336, 237 (1994). http://dx.doi.org/10.1016/0370-2693(94)01005-6[Crossref]
  • [32] S. Liao, J. Polonyi, M. Strickland, Nucl. Phys. b 567, 493 (2000). http://dx.doi.org/10.1016/S0550-3213(99)00496-4[Crossref]
  • [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)00748-6[Crossref]
  • [34] S. Seide, C. Wetterich, Nucl. Phys. b 562, 524 (1999). http://dx.doi.org/10.1016/S0550-3213(99)00545-3[Crossref]
  • [35] J. Berges, C. Wetterich, Nucl. Phys. b 487, 675 (1997). http://dx.doi.org/10.1016/S0550-3213(96)00670-0[Crossref]
  • [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/BF01564716[Crossref]
  • [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.54.9509[Crossref]
  • [38] J. Polonyi, Current-density functional for disordered systems, cond-mat/0203090.
  • [39] A. Bonanno, M. Reuter, Phys. Rev. d 62, 043008 (2000). http://dx.doi.org/10.1103/PhysRevD.62.043008[Crossref]
  • [40] H. Gies, C. Wetterich, Renormalization group flow of bound states, hep-th/0107221.
  • [41] J. Polonyi, K. Sailer, Effective actions for the density functional theory, cond-mat/0108179.
  • [42] C. Bagnuls, C. Bervillier, hep-th/0002034; Berges, N. Tetradis, C. Wetterich, hep-th/0005122.
  • [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).
  • [44] C. Harvey-Fros, PhD Thesis, university of Southampton, hep-th/0108018; M. D. Turner, PhD Thesis, University of Southampton, hep-th/0108031.
  • [45] F. Wegner, J. Phys. c 7, 2098 (1974). http://dx.doi.org/10.1088/0022-3719/7/12/004[Crossref]
  • [46] J. I. Latorre, T. Morris, Int. J. of Mod. Phys. 16, 2071 (2001). http://dx.doi.org/10.1142/S0217751X01004724[Crossref]
  • [47] L. Schulmann, Techniques and Applications of Path Integrations, John Wiley and Sons, New York, 1981.
  • [48] D. McLaughlin, L. Schulman, J. Math. Phys. 12, 2520 (1971). http://dx.doi.org/10.1063/1.1665567[Crossref]
  • [49] J. Polonyi, Ann. Phys. 252, 300 (1996). http://dx.doi.org/10.1006/aphy.1996.0133[Crossref]
  • [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-G[Crossref]
  • [51] J. Comellas, Phys. Lett. b 509, 662 (1998).
  • [52] T. Morris, Nucl. Phys. b 495, 477 (1997); Phys. Rev. Lett. 77, 1658 (1996). http://dx.doi.org/10.1016/S0550-3213(97)00233-2[Crossref]
  • [53] A. Hasenfratz, P. Hasenfratz, Nucl. Phys. b 270, 687 (1986). http://dx.doi.org/10.1016/0550-3213(86)90573-0[Crossref]
  • [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.53.3252[Crossref]
  • [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/BF01239018[Crossref]
  • [56] T. Morris, Phys. Lett. b 329, 241 (1994). http://dx.doi.org/10.1016/0370-2693(94)90767-6[Crossref]
  • [57] C. Bagnulus, C. Bervillier, Phys. Rev. b 41, 402 (1990). http://dx.doi.org/10.1103/PhysRevB.41.402[Crossref]
  • [58] T. Morris, Phys. Lett. b 345, 139 (1995); http://dx.doi.org/10.1016/0370-2693(94)01603-A[Crossref]
  • [59] R. Jackiw, S. Templeton, Phys. Rev. d 23, 2291 (1981). http://dx.doi.org/10.1103/PhysRevD.23.2291[Crossref]
  • [60] J. Polchinski, in Proceedings of the 1992 TASI Elementary Particle Physics, Eds. J. Polchinski and J. Harvey, World Sci., 1992.
  • [61] S. Weinberg, Nucl. Phys. b 413, 567 (1994). http://dx.doi.org/10.1016/0550-3213(94)90001-9[Crossref]
  • [62] R. Shankar, Rev. Mod. Phys. 66, 129 (1994). http://dx.doi.org/10.1103/RevModPhys.66.129[Crossref]
  • [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-G[Crossref]
  • [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/BF02724229[Crossref]
  • [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)90041-5[Crossref]
  • [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)91494-9[Crossref]
  • [67] V. A. Miransky, Int. J. of Mod. Phys. 6, 1641 (1991). http://dx.doi.org/10.1142/S0217751X91000861[Crossref]
  • [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).
  • [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.10.3906[Crossref]
  • [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/BF02823296[Crossref]
  • [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.15.1712[Crossref]
  • [72] V. Branchina, J. Polonyi, Nucl. Phys. b 433, 99 (1995). http://dx.doi.org/10.1016/0550-3213(94)00367-N[Crossref]
  • [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.
  • [74] J. Alexandre, V. Branchina, J. Polonyi, Phys. Lett. b 445, 351 (1999). http://dx.doi.org/10.1016/S0370-2693(98)01491-9[Crossref]
  • [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-Z[Crossref]
  • [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.41.917[Crossref]
  • [77] A. Hasenfratz, P. Hasenfratz, Nucl. Phys. b 295, 1 (1988) http://dx.doi.org/10.1016/0550-3213(88)90224-6[Crossref]
  • [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.43.799[Crossref]
  • [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).
  • [80] J. Alexandre, J. Polonyi, Tree-level renormalization, hep-th/9906017.
  • [81] Sz. Borsanyi, A. Patkos, J. Polonyi, Zs. Szep, Phys. Rev. d 62, 085013 (2000). http://dx.doi.org/10.1103/PhysRevD.62.085013[Crossref]
  • [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)90463-7[Crossref]
  • [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)90377-2[Crossref]
  • [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).
  • [85] G. ’tHooft, Phys. Rev. Lett. 37, 8 (1976). http://dx.doi.org/10.1103/PhysRevLett.37.8[Crossref]
  • [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)00649-9[Crossref]
  • [87] K. Huang, J. Polonyi, Int. J. of Mod. Phys. 6, 409 (1991). http://dx.doi.org/10.1142/S0217751X91000253[Crossref]
  • [88] I. Nándori, J. Polonyi, K. Sailer, Phys. Rev. d 63, 045022 (2001). http://dx.doi.org/10.1103/PhysRevD.63.045022[Crossref]
  • [89] J. M. Kosterlitz, D. J. Thouless, Journal J. Phys. CC61181973; J. M. Kosterlitz, Journal J. Phys. CC710461974.
  • [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.16.1217[Crossref]
  • [91] S. Coleman, Phys. Rev. d 11, 3424 (1975). http://dx.doi.org/10.1103/PhysRevD.11.2088[Crossref]
  • [92] S. Mandelstam, Phys. Rev. d 11, 3026 (1975). http://dx.doi.org/10.1103/PhysRevD.11.3026[Crossref]
  • [93] S. Samuel, Phys. Rev. d 18, 1916 (1978). http://dx.doi.org/10.1103/PhysRevD.18.1916[Crossref]
  • [94] J. Polonyi, The confinement-deconfinement mechanism, in Quark Gluon Plasma, World Scientific, 1990, R. Hwa, edt.
  • [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.17.1133[Crossref]
  • [96] G. V. Gersdorff, C. Wetterich, Nonperturbative Renormalization Flow and Essential Scaling for the Kosterlitz-Thouless Transition, hep-th/0008114.
  • [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).
  • [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)00761-8[Crossref]
  • [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)00821-4[Crossref]
  • [100] M. Simionato, Int. J. of Mod. Phys. 15, 2121 (2000); Int. J. of Mod. Phys. 15, 2153 (2000). [Crossref]
  • [101] J.Alexandre, J. Polonyi, K. Sailer, in preparation.
  • [102] K. Wilson, Rev. Mod. Phys. 55, 583 (1983). http://dx.doi.org/10.1103/RevModPhys.55.583[Crossref]
  • [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).
  • [104] K. Wilson, M. E. Fischer, Phys. Rev. Lett. 28, 248 (1972); K. Wilson, Phys. Rev. d 7, 2911 (1973).
  • [105] P. C. Hohenberg, B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977). http://dx.doi.org/10.1103/RevModPhys.49.435[Crossref]
  • [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).
  • [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)90223-1[Crossref]
  • [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)90660-7[Crossref]
  • [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)90566-2[Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.-psjd-doi-10_2478_BF02475552
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.