Publication : t97/040

Abstract:
Recently the series for two RG functions (corresponding to the anomalous dimensions of the fields $\phi$ and $\phi^2$) of the 3D $\phi^4$ field theory have been extended to next order (seven loops) by Murray and Nickel. We examine here the influence of these additional terms on the estimates of critical exponents of the $N$-vector model, using some new ideas in the context of the Borel summation techniques. The estimates have slightly changed, but remain within errors of the previous evaluation. Exponents like $\eta$ (related to the field anomalous dimension), which were poorly determined in the previous evaluation of Le Guillou--Zinn-Justin, have seen their apparent errors significantly decrease. More importantly, perhaps, summation errors are better determined.\par The change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values.\par Finally, because an error has been discovered in the last order of the published $\varepsilon=4-d$ expansions (order $\varepsilon^5$), we have also reanalyzed the determination of exponents from the $\varepsilon$-expansion. \par The conclusion is that the general agreement between $\varepsilon$-expansion and 3D series has improved with respect to Le Guillou--Zinn-Justin.

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