Abstract
The formalism of projection Hamiltonians is applied to the N-component O(N)-invariant ϕ4 model in the Euclidean and p-adic spaces. We use two versions of the ε-expansion (with ε = 4 − d and with ε = α − 3d/2, where α is the renormalization group parameter) and evaluate the critical indices ν and η up to the second order of the perturbation theory. The results for the (4− d)-expansion then coincide with the known results obtained via the quantum-field renormalization-group methods. Our calculations give evidence that in dimension three, both expansions describe the same non-Gaussian fixed point of the renormalization group.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon, Oxford (1996).
K. Wilson and J. Kogut, The Renormalization Group and the ε Expasion, North-Holland, Amsterdam (1974).
P. M. Bleher and M. D. Missarov, Comm. Math. Phys., 74, 235–272 (1980).
M. D. Missarov, J. Stat. Phys., 38, 851–860 (1985).
P. M. Blekher and M. D. Missarov, Theor. Math. Phys., 74, 132–136 (1988).
E. Yu. Lerner and M. D. Missarov, J. Stat. Phys., 76, 805–817 (1994); M. D. Missarov, Theor. Math. Phys., 109, 1249–1259 (1996); 117, 1483–1498 (1998); “Renormalization group solution of fermionic Dyson model,” in: Asymptotic Combinatorics with Application to Mathematical Physics (NATO Sci. Ser. II, Math. Phys. Chem., Vol. 77, V. A. Malyshev and A. M. Vershik, eds.), Kluwer, Dordrecht (2002), pp. 151–166.
M. D. Missarov and R. G. Stepanov, Theor. Math. Phys., 139, 672–678 (2004).
É. Yu. Lerner and M. D. Missarov, Theor. Math. Phys., 78, 177–184 (1989).
V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Calculus and Mathematical Physics [in Russian], Nauka, Moscow (1994); English transl., World Scientific, Singapore (1994).
M. Moshe and J. Zinn-Justin, Phys. Rep., 385, 69–228 (2003); I. Ya. Aref’eva, Zap. Nauchn. Sem. LOMI, 77, 24–32 (1978).
I. Ya. Aref’eva, E. R. Nissimov, and S. J. Pacheva, Comm. Math. Phys., 71, 213–246 (1980); A. N. Vasil’ev and M. Yu. Nalimov, Theor. Math. Phys., 55, 423–431 (1983).
M. D. Missarov, “Renormalization group and renormalization theory in p-adic and adelic scalar models, ” in: Dynamical Systems and Statistical Mechanics (Adv. Sov. Math., Vol. 3, Ya. G. Sinai, ed.), Amer. Math. Soc., Providence, R. I. (1991), pp. 143–164; E. Yu. Lerner and M. D. Missarov, Comm. Math. Phys., 121, 35–48 (1989).
Author information
Authors and Affiliations
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 365–384, March, 2006.
Rights and permissions
About this article
Cite this article
Missarov, M.D., Stepanov, R.G. Epsilon-expansion in the N-component ϕ 4 model. Theor Math Phys 146, 304–320 (2006). https://doi.org/10.1007/s11232-006-0041-5
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11232-006-0041-5