It's said in Chapter VI.4 of A. Zee's book *Quantum Field Theory in a Nutshell*, a theory defined as $L(U(x))=\frac{f^2}{4}Tr(\partial_{\mu}U^{\dagger}\cdot\partial^{\mu}U)$, can be write in the form of a non-linear $\sigma$ model (up to some order)

$L=\frac{1}{2}(\partial\vec{\pi})^2+\frac{1}{2f^2}(\vec{\pi}\cdot\partial\vec{\pi})^2+...$,

where $U(x)=e^{\frac{i}{f}\vec{\pi}\cdot\vec{\tau}}$ is a matrix-valued field belonging to $SU(2)$, $\vec{\pi}$ is a three components vector, $\vec{\tau}$ are Pauli matrices. Maybe it's not hard but I meet some problems to derive it.

I suppose the first step is the Taylor expansion of $U$, $U=1+\frac{i}{f}\vec{\pi}\cdot\vec{\tau}-\frac{1}{2f^2}(\vec{\pi}\cdot\vec{\tau})^2+...$, and then $\partial^{\mu}U=\frac{i}{f}\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})-\frac{1}{f^2}(\vec{\pi}\cdot\vec{\tau})\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})$, then

$(\partial_{\mu}U^{\dagger})(\partial^{\mu}U)=\frac{1}{f^2}[\partial(\vec{\pi}\cdot\vec{\tau})]^2+\frac{1}{f^4}.[(\vec{\pi}\cdot\vec{\tau})\partial(\vec{\pi}\cdot\vec{\tau})]^2$.

Now there are my questions,

(1) Can I write $\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})=\partial^{\mu}\vec{\pi}\cdot\vec{\tau}$? Then by $\vec{\tau}^2=1$, I get

$L=\frac{1}{4}(\partial\vec{\pi})^2+\frac{1}{4f^2}(\vec{\pi}\cdot\partial\vec{\pi})^2$,

which is almost correct but differ to the wished answer by a pre-factor $\frac{1}{2}$.

(2) Suppose $\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})=\partial^{\mu}\vec{\pi}\cdot\vec{\tau}$ is correct, however, if I do $\partial^{\mu}U=\frac{i}{f}U\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})=\frac{i}{f}U\partial^{\mu}\vec{\pi}\cdot\vec{\tau}$ first, it seems $\partial^{\mu}U^{\dagger}\cdot\partial^{\mu}U=|\frac{i}{f}U\partial^{\mu}\vec{\pi}\cdot\vec{\tau}|^2=\frac{1}{f^2}(\partial\vec{\pi})^2$, say, only the first term of the wished answer.

I probably made something wrong somewhere, can anyone hit me?

This post imported from StackExchange Physics at 2014-03-17 05:58 (UCT), posted by SE-user hongchaniyi