Large order asymptotics and convergent perturbation theory for critical indices of the ${\varphi }^4$ model in $4-\epsilon$ expansion

Abstract

Large order asymptotic behaviour of renormalization constants in the minimal subtraction scheme for the ${\varphi }^4$ $(4-\epsilon )$ theory is discussed. Well-known results of the asymptotic $4-\epsilon$ expansion of critical indices are shown to be far from the large order asymptotic value. A {\em convergent} series for the model ${\varphi }^4$ $(4-\epsilon )$ is then considered. Radius of convergence of the series for Green functions and for renormalisation group functions is studied. The results of the convergent expansion of critical indices in the $4-\epsilon$ scheme are revalued using the knowledge of large order asymptotics. Specific features of this procedure are discussed.

8 pages, 2 figures. Invited talk at the 5th International Conference Renormalization Group 2002 (High Tatra Mountains, Slovakia, 10-16 March 2002)

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