Abstract
Recently it was shown that the scaling dimension of the operator $\phi^n$ in $\lambda(\bar\phi\phi)^2$ theory may be computed semiclassically 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 subleading $n$. This result was subsequently generalised to operators of fixed charge $Q$ in $O(N)$ theory and verified up to four loops in perturbation theory at leading and subleading $Q$. More recently, similar semiclassical calculations have been performed for the classically scale-invariant $U(N)\times U(N)$ theory in four dimensions, and verified up to two loops, once again at leading and subleading $Q$. Here we extend this verification to four loops. We also consider the corresponding classically scale-invariant theory in three dimensions, similarly verifying the leading and subleading semiclassical results up to four loops in perturbation theory.
Highlights
The investigation of renormalizable scalar field theories with scale-invariant self-interactions has attracted renewed attention in recent years
We have extended the investigations of Ref. [11] into large charge operators in scalar UðNÞ × UðNÞ theory
We have proceeded in a slightly different direction; the authors of Ref. [11] used their leading order (LO) and next-to-leading order (NLO) semiclassical result, combined with a known two-loop result for a particular charge (Q 1⁄4 2) to deduce the next-to-next-to-leading order (NNLO) two-loop terms for general Q, obtaining the full two-loop result for general Q
Summary
The investigation of renormalizable scalar field theories with scale-invariant self-interactions has attracted renewed attention in recent years. [9], the Uð1Þ result was compared with perturbation theory up to two loops, and in Refs. The classically scale-invariant UðNÞ × UðNÞ model has been investigated in Ref. Once again the leading and nonleading terms in a large charge expansion have been derived by a semi-classical calculation, and compared with perturbative results up to two loop order. In the current work we extend this check to three and four loops in perturbation theory, and further perform a similar analysis and check for the classically scale-invariant UðNÞ × UðNÞ theory in three dimensions. As usual we work within dimensional regularization with d 1⁄4 4 − ε and with the divergences for this theory canceled by replacing the couplings by bare counterparts given to leading (one-loop) order by
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