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Integrable sigma models and 2-loop RG flow

Following arXiv:1907.04737, we continue our investigation of the relation between the renormalizability (with finitely many couplings) and integrability in 2d $σ$-models. We focus on the "$λ$-model," an integrable model associated to a group or symmetric space and containing as special limits a (gauged) WZW model and an "interpolating model" for non-abelian duality. The parameters are the WZ level $k$ and the coupling $λ$, and the fields are $g$, valued in a group $G$, and a 2d vector $A_\pm$ in the corresponding algebra. We formulate the $λ$-model as a $σ$-model on an extended $G \times G \times G$ configuration space $(g, h, \bar{h})$, defining $h $ and $\bar{h}$ by $A_+ = h \partial_+ h^{-1}$, $A_- = \bar{h} \partial_- \bar{h}^{-1}$. Our central observation is that the model on this extended configuration space is renormalizable without any deformation, with only $λ$ running. This is in contrast to the standard $σ$-model found by integrating out $A_\pm$, whose 2-loop renormalizability is only obtained after the addition of specific finite local counterterms, resulting in a quantum deformation of the target space geometry. We compute the 2-loop $β$-function of the $λ$-model for general group and symmetric spaces, and illustrate our results on the examples of $SU(2)/U(1)$ and $SU(2)$. Similar conclusions apply in the non-abelian dual limit implying that non-abelian duality commutes with the RG flow. We also find the 2-loop $β$-function of a "squashed" principal chiral model.