Paul Balduf#paperOfTheDay : "#Renormalization of a scalar field theory in strong coupling" from 1972.
Recall that phi^6 theory in 3 dimensions is a perturbatively renormalizable scalar #QuantumFieldTheory model, and in perturbation theory (using #FeynmanIntegral s), one expects there to be counterterms for the phi^6, phi^4, and phi^2 interactions to accommodate their anomalous scale dependence. In the present paper, Wilson uses a different approach, and introduces an approximation scheme for the quantum effective action, which is not inherently related to conventional perturbation theory. In this, he finds that the so-approximated model only acquires anomalous flow for phi^2, but not for phi^4 and phi^6. The approximation is relatively coarse, so one should not take this as a "solution" of phi^6 theory, but rather as a concrete example of what could in principle happen in a strongly coupled interacting field theory.
Note that with such results, there is no contradiction with perturbation theory: Low-order perturbation theory describes a behaviour very close to a free theory, but perturbation series are divergent and asymptotic. This means that the true functional form only emerges after resummation, and is in general very different from "inserting a large number for the coupling into the low-order perturbation series".
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.6.419
Recall that phi^6 theory in 3 dimensions is a perturbatively renormalizable scalar #QuantumFieldTheory model, and in perturbation theory (using #FeynmanIntegral s), one expects there to be counterterms for the phi^6, phi^4, and phi^2 interactions to accommodate their anomalous scale dependence. In the present paper, Wilson uses a different approach, and introduces an approximation scheme for the quantum effective action, which is not inherently related to conventional perturbation theory. In this, he finds that the so-approximated model only acquires anomalous flow for phi^2, but not for phi^4 and phi^6. The approximation is relatively coarse, so one should not take this as a "solution" of phi^6 theory, but rather as a concrete example of what could in principle happen in a strongly coupled interacting field theory.
Note that with such results, there is no contradiction with perturbation theory: Low-order perturbation theory describes a behaviour very close to a free theory, but perturbation series are divergent and asymptotic. This means that the true functional form only emerges after resummation, and is in general very different from "inserting a large number for the coupling into the low-order perturbation series".
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.6.419