  Following arXiv:1907.04737, we continue our investigation of the relation
between the renormalizability (with finitely many couplings) and integrability
in 2d $\sigma$-models. We focus on the "$\lambda$-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 $\lambda$, and the fields are
$g$, valued in a group $G$, and a 2d vector $A_\pm$ in the corresponding
algebra. We formulate the $\lambda$-model as a $\sigma$-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
$\lambda$ running. This is in contrast to the standard $\sigma$-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
$\beta$-function of the $\lambda$-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 $\beta$-function of
a "squashed" principal chiral model.
