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Construction of the Hierarchical φ4-Trajectory

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Abstract

We study the invariant unstable manifold of the trivial renormalization-group fixed point tangent to the φ4-vertex in the hierarchical approximation. We parametrize it by a running φ4-coupling with linear step β-function. The manifold is studied as a fixed point of the renormalization group composed with a flow of the running coupling. We present a rigorous construction of it beyond perturbation theory by means of a contraction mapping. Starting from a perturbative approximant of order seven, we obtain a convergent representation in dimensions 2 < D < 28/9 with certain restrictions. The perturbative approximant is logarithmically divergent in D = 3 dimensions.

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REFERENCES

  1. P. Albuquerque, La Liberté asymptotique du modèle φ 44 dans l'approximation hiérarchique et le thèoreme de la variété centrale, Diploma thesis (University of Geneva, 1991).

  2. P. M. Bleher, Usp. Nauk. 32:243 (1977).

    Google Scholar 

  3. T. Balaban, Ultraviolet stability in field theory, in Scaling and self-similarity in physics, J. Fröhlich, ed. (Birkhäuser, Boston, 1983).

    Google Scholar 

  4. D. Brydges, Functional integrals and their application, Troisième Cycle de la Physique en Suisse Romand (Lausanne, 1992).

  5. G. Benfatto, M. Cassandro, G. Gallavotti, N. Nicolò, E. Olivieri, E. Presutti, and E. Scacciatelli, Some probabilistic techniques in field theory, Commun. Math. Phys. 71:95–130 (1980); On the ultraviolet stability in the Euclidean scalar field theories, Commun. Math. Phys. 71:343 (1980).

    Google Scholar 

  6. D. Brydges, J. Dimock, and T. R. Hurd, The short distance behavior of φ 43 . Preprint 1993.

  7. G. Benfatto and G. Gallavotti, Renormalization group, Physics Notes No. 1 (Princeton University Press, 1995).

  8. P. M. Bleher and Ya. G. Sinai, Investigation of the critical point in models of the type of Dyson's hierarchical model, Commun. Math. Phys. 33:23 (1973).

    Google Scholar 

  9. P. M. Bleher and Ya. G. Sinai, Critical indices for Dyson's asymptotically hierarchical models, Commun. Math. Phys. 45:347 (1975).

    Google Scholar 

  10. P. Collet and J.-P. Eckmann, The ε-expansion for the hierarchical model, Commun. Math. Phys. 55:67–96 (1997).

    Google Scholar 

  11. P. Collet and J.-P. Eckmann, A renormalization group analysis of the hierarchical model in statistical physics, Lecture Notes in Physics 74 (Springer-Verlag, 1978).

  12. F. J. Dyson, Existence of a phase transition in a one-dimensional Ising ferromagnet, Commun. Math. Phys. 12:91–107 (1969); Nonexistence of spontaneous magnetization in a one-dimensional Ising ferromagnet, Commun. Math. Phys. 12:212–215 (1969).

    Google Scholar 

  13. J.-P. Eckmann and P. Wittwer, Multiplicative and additive renormalization, in Critical phenomena, random systems, gauge theories, K. Osterwalder and R. Stora, eds. (Les Houches Session XLIII 1984, North-Holland Physics Publishing, 1986).

  14. J. Feldman and K. Osterwalder, The Wightman axioms and the mass gap for weakly coupled φ 43 quantum field theories, Ann. Phys. 97:80–135 (1976).

    Google Scholar 

  15. R. Fernandez, J. Fröhlich, and A. Sokal, Random walks, critical phenomena, and triviality in quantum field theory (Springer Monographs in Physics, 1992).

  16. J. Feldman, J. Magnen, V. Rivasseau, and R. Sénéor, Construction and Borel summability of infrared φ 44 by a phase space expansion, Commun. Math. Phys. 109:137 (1987).

    Google Scholar 

  17. G. Gallavotti, Some aspects of the renormalization problems in statistical mechanics and field theory, Mem. Accad. Lincei 15:23 (1978).

    Google Scholar 

  18. G. Gallavotti, On the ultraviolet stability in statistical mechanics and field theory, Ann. Mat. Pura ed Applicata 120:1–23 (1979).

    Google Scholar 

  19. G. Gallavotti, Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods, Rev. Mod. Phys. 57(2):471–562 (1985).

    Google Scholar 

  20. J. Glimm and A. Jaffe, Positivity of the φ 43 -Hamiltonian, Fortschr. Phys. 21:327–376 (1973).

    Google Scholar 

  21. J. Glimm and A. Jaffe, Quantum Physics (Springer-Verlag, 1987).

  22. K. Gawedzki and A. Kupiainen, A rigorous block spin approach to massless lattice theories, Commun. Math. Phys. 77:31 (1980).

    Google Scholar 

  23. K. Gawedzki and A. Kupiainen, Triviality of φ 44 and all that in a hierarchical model, J. Stat. Phys. 29:683 (1982).

    Google Scholar 

  24. K. Gawedzki and A. Kupiainen, Rigorous renormalization group and asymptotic freedom, in Scaling and self-similarity in physics, renormalization in statistical mechanics and dynamics, J. Fröhlich, ed. Progress in Physics (Birkhäuser, 1983), Vol. 7.

  25. K. Gawedzki and A. Kupiainen, Nontrivial continuum limit of φ 44 -model with negative coupling constant, Nucl. Phys. B 257(FS 14):474–504 (1985).

    Google Scholar 

  26. K. Gawedzki and A. Kupiainen, Massless lattice φ 44 : Rigorous control of a renormalizable asymptotically free model, Commun. Math. Phys. 99:197 (1985).

    Google Scholar 

  27. K. Gawedzki and A. Kupiainen, Asymptotic freedom beyond perturbation theory, in Critical phenomena, random systems, gauge theories, K. Osterwalder and R. Stora, eds. (Les Houches Session XLIII 1984, North-Holland Physics Publishing, 1986).

  28. G. Gallavotti and F. Nicolo, Renormalization in four dimensional scalar fields I, Commun. Math. Phys. 100:545–590 (1985); Renormalization in four dimensional scalar fields II, Commun. Math. Phys. 101:247–282 (1985).

    Google Scholar 

  29. G. Gallavotti and V. Rivasseau, φ4 field theory in dimensions four, a modern introduction to its unsolved problems, Ann. Inst. Poincaré B40:185 (1984).

    Google Scholar 

  30. H. Koch and P. Wittwer, A non-Gaussian renormalization group fixed point for hierarchical scalar lattice field theories, Commun. Math. Phys. 106:495–532 (1986).

    Google Scholar 

  31. H. Koch and P. Wittwer, The unstable manifold of a nontrivial renormalization group fixed point. Manuscript, 1986.

  32. H. Koch and P. Wittwer, The unstable manifold of a nontrivial RG fixed point, Canadian Mathematical Society, Conference Proceedings 9:99–105 (1988).

    Google Scholar 

  33. H. Koch and P. Wittwer, Computing bounds on critical indices, in Proceedings of the NATO advanced study institute on non linear evolution and chaotic phenomena, Noto 1987, G. Gallavotti and P. Zweifel, eds., NATO ASI series, B 176 (Plenum Press, 1988).

  34. H. Koch and P. Wittwer, On the renormalization group transformation for scalar hierarchical models, Commun. Math. Phys. 138:537 (1991).

    Google Scholar 

  35. H. Koch and P. Wittwer, A nontrivial renormalization group fixed point for the Dyson-Baker hierarchical model, Commun. Math. Phys. 164:627–647 (1994).

    Google Scholar 

  36. J. Magnen and R. Seneor, Phase space cell expansion and bored summability for the Euclidean φ 43 theory, Commun. Math. Phys. 56:237–276 (1977).

    Google Scholar 

  37. A. Pordt, Convergent multigrid polymer expansions and renormalization for Euclidean field theory. DESY preprint 90–020.

  38. A. Pordt, Renormalization theory for hierarchical models, Helv. Phys. Acta 66:105–154 (1993).

    Google Scholar 

  39. V. Rivasseau, From perturbative to constructive renormalization (Princeton University Press, 1991).

  40. J. Rolf and C. Wieczerkowski, The hierarchical φ4-trajectory by perturbation theory in a running coupling and its logarithm, Jour. Stat. Phys. 84(1/2):119 (1996).

    Google Scholar 

  41. K. Wilson, Model of coupling constant renormalization, Phys. Rev. D 2(8):1438–1472 (1970).

    Google Scholar 

  42. K. Wilson, Renormalization group and strong interactions, Phys. Rev. D 3:1818 (1971).

    Google Scholar 

  43. K. Wilson, Renormalization group and critical phenomena, Phys. Rev. B 4:3174–3205 (1971).

    Google Scholar 

  44. K. Wilson, Renormalization of a scalar field in strong coupling, Phys. Rev. D 6:419 (1972).

    Google Scholar 

  45. K. Wilson, Quantum field theory models in less than four dimensions, Phys. Rev. D 7:2911 (1973).

    Google Scholar 

  46. K. Wilson, Confinement of quarks, Phys. Rev. D 10:2445–2459 (1974).

    Google Scholar 

  47. K. Wilson, The renormalization group and critical phenomena, Rev. Mod. Phys. 55:583–600 (1983).

    Google Scholar 

  48. K. Wilson and M. E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. B 28:240–243 (1972).

    Google Scholar 

  49. K. Wilson and J. Kogut, The renormalization group and the ε expansion, Phys. Rep. C 12(2):75–20 (1974).

    Google Scholar 

  50. C. Wieczerkowski, The renormalized φ 44 -trajectory by perturbation theory in the running coupling: (I) the discrete renormalization group, Nucl. Phys. B 488:441–465 (1997); The renormalized φ 44 -trajectory by perturbation theory in the running coupling: (II) the continuous renormalization group, Nucl. Phys. B 488:466–489 (1997); Renormalized g − log(g) double expansion for the invariant φ4-trajectory in three dimensions, MS–TP1–97–02.

    Google Scholar 

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  1. Institut für Theoretische Physik I, Universität Münster, D-48149, Münster, Germany;

    C. Wieczerkowski

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Wieczerkowski, C. Construction of the Hierarchical φ4-Trajectory. Journal of Statistical Physics 92, 377–430 (1998). https://doi.org/10.1023/A:1023028319101

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