Determine the anomalous dimensions of the higher-dimensional non-BPS double-particle operator

Determine the anomalous dimensions of the primary and descendant components of the non-BPS double-particle operator $(L_{-1}\widetilde{L}_{-1}O_f^{++})^2$, whose connected correlator is obtained from the supergravity HHLL computation, and compare them with anomalous dimensions calculated by the independent method based on single-particle correlators.

Background

The authors initially compute the connected correlator associated with the double-particle operator (L−1L~−1Of++)2(L_{-1}\widetilde{L}_{-1}O_f^{++})^2. Unlike the simpler double-particle operator studied in detail, this operator has a relatively large bare dimension and mixes with other operators of the same bare dimension.

Because of this mixing, extracting and independently checking its anomalous dimension is technically more difficult. The paper explicitly postpones that calculation, while noting in the conclusion that the same correlator could be analyzed using the method developed for the simpler example.

References

However, $(\mathcal{O})2$ has a quite large bare dimension, equal to 6, and computing its anomalous dimension\footnote{More precisely, $(\mathcal{O})2$ is not a primary, but the superposition of some primaries and some descendants; each of these have their own anomalous dimensions.}, so as to perform the test described above, is complicated by the mixing with other operators with the same bare dimension. Thus, we leave this task for the future.

— Holographic correlators with non-supersymmetric multi-particle states  (2609.20534 - Giorgi et al., 17 Sep 2026) in Section 3, paragraph following Eq. (3.1), and concluding remarks