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I'm reading Schwartz's Quantum field theory and the standard model. I'm stuck at page 566. He says that when we take the decoupling limit, that's keeping $F_{\pi}$ fixed while sending $m$ and $\lambda$ to infinity, projects out the non-linear sigma model. Now, I don't understand what all this means. Doing this will take the coefficients of all terms in the potential to infinity while not affecting the kinetic terms.

So, How did he get:

$$L=(1/2) (\partial_{\mu}\pi)^2$$ after taking the decoupling limit?

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Substitute $F_\pi$ = $\frac{2m}{\sqrt{\lambda}}$ in (28.12 ) so the second term in between the parenthesis that's multiplied by $1/F^2_\pi$ becomes $\frac{1}{2} (\partial\pi)^2 + other \; terms$. Apply L'Hospital's rule and ignore all the $\infty$'s. Finally the Lagrangian becomes ($decoupled \: \pi$ & $\sigma$ fields): $\frac{1}{2} (\partial\pi)^2 + \frac{1}{2} (\partial\sigma)^2$

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