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We investigate whether the six-loop beta function of the $\lambda \phi^4_4$theory exhibits evidence for an ultraviolet zero. As part of our analysis, wecalculate and analyze Pad\'e approximants to this beta function. Extending ourearlier results at the five-loop level, we find that in the range of $\lambda$where the perturbative calculation of the six-loop beta function is reliable,the theory does not exhibit robust evidence for an ultraviolet zero.

Study of the Six-Loop Beta Function of the $\lambda\phi^4_4$ Theory