Perturbation theory at large order. I. Theφ2Ninteraction

A new method for calculating the large orders of perturbation theory in quantum field theories has been discussed recently by Lipatov. We show that the same method applied to anharmonic oscillators in quantum mechanics allows one to rederive and generalize results previously obtained by Bender and Wu. We have also verified and generalized Lipatov's results to the case of an internal $O\left(n\right)\mathrm{}$ symmetry. These results show the divergence of the Wilson-Fisher $\epsilon$ expansion and indicate its Borel summability which is used for critical exponents. Similarly, the Callan-Symanzik functions for the ${\phi }^4$ theory in three dimensions are characterized.