Independent field-theory derivation of the logarithmic anomaly coefficient

Derive independently, from boundary field-theory data, the anomaly coefficient \(\mathsf{A}=\sqrt{21}/70\) governing the logarithmic running of the condensate of the dimension-four operator \(\mathcal{O}_{-}\), either through the momentum-space triple-K anomaly of its three-point function or through the Osborn/Callan–Symanzik metric anomaly in the trace sector, thereby fixing the overall normalization.

Background

The paper computes a universal coefficient $\mathsf{A}=\sqrt{21}/70$ for the logarithmic running induced at second order by the irrelevant operator $\mathcal{O}_{-}$ with dimension $\Delta=4$ in a three-dimensional boundary theory. In the bulk, this coefficient is tied to the cubic coupling $V_{,---}$ of the second scalar and appears in the logarithmic term of the induced condensate.

The authors identify the logarithm with a boundary three-point-function anomaly and state that an independent field-theory calculation—using either momentum-space triple-K methods or the Osborn/Callan–Symanzik anomaly—would complete the holographic identification by verifying the normalization.