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We study a perturbative expansion of the squashed 3-sphere ($S^3_b$)partition function of 3d $\mathcal{N}=2$ gauge theories around the squashingparameter $b= 1$. Our proposal gives the coefficients of the perturbativeexpansion as a finite sum over the saddle points of thesupersymmetric-localization integral in the limit $b \rightarrow 0$ (theso-called Bethe vacua), and the contribution from each Bethe vacua can besystematically computed using saddle-point methods. Our expansion provides anefficient and practical method for computing basic CFT data ($F,C_T,C_{JJ}$ andhigher-point correlation functions of the stress-energy tensor) of the IRsuperconformal field theory without performing the localization integrals.

Expanding 3d $\mathcal{N}=2$ Theories around the Round Sphere