In Polyakov's book about gauge fields & strings, in chapter devoted to non-linear sigma model he emphasizes problem with large $N$ expansion of this model. Lagrangian of 2D model is $$\frac{1}{2g^2}(\partial_{\mu}{\bf n})^2$$ with constraint ${\bf n}^2=1$. It is possible to add term into action which explicitly contains constraint by introducing additional field $\lambda(x)$. Then, one can integrate out fields ${\bf n}$ (path integral over these fields is gaussian) and obtain effective action in terms of field $\lambda$. Now it is time to use $N\rightarrow\infty$. It is possible to find saddle point of this effective action and see that it corresponds to $\lambda=m^2$, $m>0$ (up to sign or $i$). Then, we can ivestigate fluctuations near saddle point as $m^2+\alpha(x)$, where $\alpha$ is fluctuation.
After all calculations, we can compute all the correlation functions of initial model in terms of effective action. Near to the end of this chapter, he says that propagator of $\alpha(x)$ field in effective action has bad behavior, $$D(q^2)\rightarrow q^2/\ln(q^2/m^2), \quad q^2\rightarrow\infty,$$ and then says that if we do not impose constraint $n_in^i=1$ everything will be ok. Instead of constraint ${\bf n}^2=1$, it is possible to introduce quartic term into initial action and avoid "bad behavior" problem.
Why constraing $n_in^i=1$ creates problem with propagator behaviour for large momenta? Can somebody clarify this moment?
