Abstract
Renormalization group (RG) and resummation techniques have been used in $N$-component $\phi^4$ theories at fixed dimensions below four to determine the presence of non-trivial IR fixed points and to compute the associated critical properties. Since the coupling constant is relevant in $d<4$ dimensions, the RG is entirely governed by renormalization scheme-dependent terms. We show that the known proofs of the Borel summability of observables depend on the renormalization scheme and apply only in "minimal" ones, equivalent in $d=2$ to an operatorial normal ordering prescription, where the $\beta$-function is trivial to all orders in perturbation theory. The presence of a non-trivial fixed point can be unambiguously established by considering a physical observable, like the mass gap, with no need of RG techniques. Focusing on the $N=1$, $d=2$ $\phi^4$ theory, we define a one-parameter family of renormalization schemes where Borel summability is guaranteed and study the accuracy on the determination of the critical exponent $\nu$ as the scheme is varied. While the critical coupling shows a significant sensitivity on the scheme, the accuracy in $\nu$ is essentially constant. As by-product of our analysis, we improve the determination of $\nu$ obtained with RG methods by computing three more orders in perturbation theory.
Highlights
Perturbative renormalization group (RG) techniques have been extensively used in the analysis of critical phenomena that have a continuum limit description in terms of an effective Landau-Ginzburg Hamiltonian
We show that the known proofs of the Borel summability of observables depend on the renormalization scheme and apply only in “minimal” ones, equivalent in d 1⁄4 2 to an operatorial normal ordering prescription, where the β-function is trivial to all orders in perturbation theory
We have shown that in the renormalization scheme (RS) Sthe proof given in Ref. [7] no longer holds, and we are not aware of papers in constructive quantum field theory generalising the proofs in Refs. [4,5] to other RSs such as S
Summary
Perturbative renormalization group (RG) techniques have been extensively used in the analysis of critical phenomena that have a continuum limit description in terms of an effective Landau-Ginzburg Hamiltonian. Fixed points by looking for zeroes of a properly defined β-function These are necessarily strongly coupled and require a Borel resummation of the perturbative series to be established. The Borel summability of the fixed dimension perturbative series in the φ4 theory at parametrically small coupling and in the unbroken phase m2 > 0 has been established long ago [4,5] [3], based on a properly defined β-function, and recall in Sec. II B how the critical regime can be analyzed in minimal RSs We report in the Appendix an improvement in the determination of ν obtained by RG methods in the RS of Ref. [3] by adding three more orders to the known perturbative series and numerically resumming the resulting series
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