Recently the series for two RG functions (corresponding to the anomalous dimensions of the fields φ and φ 2 ) of the 3D φ 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

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ABSTRACT:We prove that the linear delta expansion for energy eigenvalues of the quantum mechanical anharmonic oscillator converges to the exact answer if the order dependent trial frequency Ω is chosen to scale with the order as Ω = CN γ ; 1/3 < γ < 1/2, C > 0 as N → ∞. It converges also for γ = 1/3, if C ≥ α c g 1/3 , α c ≃ 0.570875, where g is the coupling constant in front of the operator q 4 /4. The extreme case with γ = 1/3, C = α c g 1/3 corresponds to the choice discussed earlier by Seznec and Zinn-Justin and, more recently, by Duncan and Jones.

We consider here renormalizable theories without relevant couplings and present an I.R. consistent technique to study corrections to short distance behavior (Wilson O.P.E. coefficients) due to a relevant perturbation. Our method is the result of a complete reformulation of recent works on the field, and is characterized by a more orthodox treatment of U.V. divergences that allows for simpler formulae and consequently an explicit all order (regularization invariant) I.R. finitess proof. Underlying hypotheses are discussed in detail and found to be satisfied in conformal theories that constitute a natural field of application of this approach.

We improve and generalize in several accounts the recent rigorous proof of convergence of delta expansion order dependent mappings (variational perturbation expansion) for the energy eigenvalues of quartic anharmonic oscillator. For the single-well oscillator the uniformity of convergence in g # [0, ] is proven. The convergence proof is extended also to complex values of g lying on a wide domain of the Riemann surface of E( g). Via the scaling relation aÁ la Symanzik, this proves the convergence of delta expansion for the double well in the strong coupling regime (where the standard perturbation series is non Borel summable), as well as for the complex``energy eigenvalues'' in certain metastable systems. Difficulties in extending the convergence proof to the cases of higher anharmonic oscillators are pointed out. Sufficient conditions for the convergence of delta expansion are summarized in the form of three general theorems, which should apply to a wide class of quantum mechanical and higher dimensional field theoretic systems.