Abstract
Recently it was shown that the scaling dimension of the operator $\phi^n$ in $\lambda(\phi^*\phi)^2$ theory may be computed semi-classically at the Wilson-Fisher fixed point in $d=4-\epsilon$, for generic values of $\lambda n$ and this was verified to two loop order in perturbation theory at leading and sub-leading $n$. In subsequent work, this result was generalised to operators of fixed charge $Q$ in $O(N)$ theory and verified up to three loops in perturbation theory at leading and sub-leading order. Here we extend this verification to four loops in $O(N)$ theory, once again at leading and sub-leading order. We also investigate the strong-coupling regime.
Highlights
Renormalizable theories with scale invariant scalar selfinteractions have been subjects of enduring interest
The theory with a single scalar field exhibits a Wilson-Fisher fixed point (FP) where the coupling constant λ is OðεÞ, and this infrared (IR) attractive FP is associated with a second order phase transition
We extend the comparison with perturbation theory up to four loops, and discuss the large ðgQ Þ case, generalizing the large λn analysis of Ref. [9]
Summary
Renormalizable theories with scale invariant scalar selfinteractions have been subjects of enduring interest. The majority of work in renormalizable quantum field theories has involved the weak coupling expansion, in other words the Feynman diagram loop expansion This expansion fails or becomes ponderous at either strong coupling or (less obviously) for φn amplitudes at large n. [9] the scaling dimension of the same operator in the Uð1Þ-invariant λðφφÞ2 theory (corresponding to the special case N 1⁄4 2) was computed at the Wilson-Fisher fixed point λà as a semiclassical expansion in λÃ, for fixed λÃn.
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