Abstract
We compute the hierarchicalφ 4- in terms of perturbation theory in a running coupling. In the three-dimensional case we resolve a singularity due to resonance of power counting factors in terms of logarithms of the running coupling. Numerical data are presented and the limits of validity explored. We also compute moving eigenvalues and eigenvectors on the trajectory as well as their fusion rules.
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References
G. A. Baker, Ising model with a scaling interaction,Phys. Rev. B 5:2622–2633 (1972).
P. M. Bleher and Ja. G. Sinai, Investigation of the critical point in models of the type of Dyson's hierarchical models,Commun. Math. Phys. 33:23–42 (1973).
P. Collet and J.-P. Eckmann,A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics (Springer-Verlag, Berlin, 1978).
F. J. Dyson, Existence of a phase-transition in a one-dimensional Ising ferromagnet,Commun. Math. Phys. 12:91–107 (1969).
J.-P. Eckmann and P. Wittwer, Multiplicative and additive renormalization, inLes Houches 1984, K. Osterwalder and R. Stora, eds. (1984), pp. 4554465.
G. Felder, Renormalization group in the local potential approximation,Commun. Math. Phys. 111:101–121 (1987).
J. S. Feldman, T. R. Hurd, L. Rosen, and J. D. Wright,QED: A Proof of Renormalizability, 2nd ed. (Springer-Verlag, Berlin, 1988).
G. Gallavotti, Some aspects of the renormalization problem in statistical mechanics,Mem. Accad. Lincei 15:23 (1978).
G. Gallavotti, Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods,Rev. Mod. Phys. 57(2):471 (1985).
K. Gawedzki and A. Kupiainen, Asymptotic freedom beyond perturbation theory, inLes Houches 1984, K. Osterwalder and R. Stora, eds. (1984), pp. 185–293.
G. Gallavotti and F. Nicolo, Renormalization theory in 4 dimensional scalar fields I,Commun. Math. Phys. 100:545–590 (1985).
G. Gallavotti and F. Nicolo, Renormalization theory in 4 dimensional scalar fields II,Commun. Math. Phys. 101:247–282 (1985).
J. Kogut and K. Wilson, The renormalization group and the ε-expansion,Phys. Rep. C 12(2):75–200 (1974).
H. Koch and P. Wittwer, On the renormalization group transformation for scalar hierarchical models,Commun. Math. Phys. 138:537–568 (1991).
J. Polchinski, Renormalization and effective Lagrangeans,Nucl. Phys. B 231:269–295 (1984).
A. Pordt, Renormalization theory for hierarchical models,Helv. Phys. Acta 66:105–154 (1993).
V. Rivasseau,From Perturbation Theory to Constructive Renormalization (Princeton University Press, Princeton, New Jersey, 1991).
C. Wieczerkowski, The renormalizedϕ 4-trajectory by perturbation theory in the running coupling, Münster Preprint MS-TPI-95-06, HEP-TH/9601142.
K. Wilson, Renormalization group and critical phenomena I and II,Phys. Rev. B 4:3174–3205 (1971).
C. Wieczerkowski and Y. Xylander, Improved actions, the perfect action, and scaling by perturbation theory in Wilson's renormalization group: The two dimensionalO(N)-invariant non linear σ-model in the hierarchical approximation,Nucl. Phys. B 440:393 (1994).
C. Wieczerkowski and Y. Xylander, Perfect observables for the hierarchical non linear (O(N)-invariant σ-model, DESY Preprint 95094 (1995).
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Rolf, J., Wieczerkowski, C. The hierarchicalφ 4-trajectory by perturbation theory in a running coupling and its logarithmby perturbation theory in a running coupling and its logarithm. J Stat Phys 84, 119–145 (1996). https://doi.org/10.1007/BF02179579
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DOI: https://doi.org/10.1007/BF02179579