Anomalous dimensions at large charge in $d=4$ $O(N)$ theory

Recently it was shown that the scaling dimension of the operator ${\varphi }^n$ in $\lambda (\overline{\varphi }\varphi {)}^2$ 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 $n$. In subsequent work, this result was generalized to operators of fixed charge $\overline{Q}$ in $O(N)$ theory and verified up to three loops in perturbation theory at leading and subleading $\overline{Q}$. Here we extend this verification to four loops in $O(N)$ theory, once again at leading and subleading $\overline{Q}$. We also investigate the strong-coupling regime.

Figure 1

One- and two-loop diagrams for ${\gamma }_{T_{\overline{Q}}}$ contributing at leading $n$. Here and elsewhere, the lozenge represents the $T_{\overline{Q}}$ vertex.

Figure 2

Three-loop diagrams for ${\gamma }_{T_{\overline{Q}}}$ contributing at leading $n$.

Figure 3

Three-loop diagrams for ${\gamma }_{T_{\overline{Q}}}$ contributing at next-to-leading $n$.

Figure 4

Four-loop diagrams for ${\gamma }_{T_{\overline{Q}}}$ contributing at leading $n$.

Figure 5

Four-loop diagrams for ${\gamma }_{T_{\overline{Q}}}$ contributing at next-to-leading $n$.

Figure 6

Four-loop diagrams for ${\gamma }_{T_{\overline{Q}}}$ contributing at next-to-leading $n$ (continued).