One-loop approximation in chiral sigma models

Consider a principal chiral model $$S[g] = frac{1}{4pilambda^2}int(|g^{-1}dg|^2) = frac{1}{4pilambda^2}int(g^{-1}dgwedgestar g^{-1}dg) ,tag{2.1}$$ where $g:Sigma rightarrow G $ is the so called principal chiral field (this theory is invariant under a $G_Rtimes G_L$ transformation). In this paper, equation (2.6) is the action with a 1-loop correction, but I thought I could use $$Gamma[g] = S[g] + trleft[ln S^{(2)}[g] right] + cdots ,$$ where $$S^{(2)}[g] = frac{delta^2 S[h]}{delta h(x) delta h(y)}|_{h=g}.$$ But apparently it is not the case, it is a correction related to the cut-off $Lambda$. I remember doing it in $phi^4$ theory, considering Feynman diagrams, but I don’t remember a systematic way of doing it, so I’m having troubles on this model. Can you help me to obtain the equation 2.6?

This correction is related to a correction on the G-invariant metric. If ${t_a}$ are the generator of the lie algebra $mathfrak{g} = Lie(G)$, write $g(x)=e^{alpha^a(x)t_a}$, the action becomes $$tr(g^{-1}dgwedgestar g^{-1}dg) = g_{ab} dalpha^awedge star dalpha^b$$ where $g_{ab}=tr(t_at_b)$ is the G-invariant metric. But the correction is: $$ mu|tr(g^{-1}dg)|^2 = mu tr(t_a)tr(t_b) dalpha^awedge star dalpha^b$$. Therefore the correction is just a correction on the metric:

$$ tr(t_at_b) longmapsto tr(t_at_b) + mu tr(t_a)tr(t_b) $$