Anomalous dimensions in the symmetric orbifold

Recently, the anomalous conformal dimensions of the symmetric orbifold under the $2$-cycle twisted sector deformation were calculated using the perturbed action of the supercharges. In particular, explicit and simple formulae for the dispersion relations of the torus magnons in the $w$-cycle twisted sector were derived for large $w$. In this paper we reproduce these results from a direct perturbed $2$-point function calculation. In the process we also develop techniques (and a Mathematica code) that allows one to do these calculations for arbitrary quarter BPS states at finite $w$.