Coefficient of effective chiral Lagrangian of $\pi\pi$ scattering I have been suffering from the coefficient in the expansion of chiral lagrangian.
Consider $$L=\frac{F^{2}}{4} \rm{Tr}(\partial_{\mu}U^{\dagger}\partial^{\mu}U),$$
where $$U=\exp(i\frac{\phi}{F}).$$ For $\pi\pi$ scattering,we need the 4 $\pi's$ terms $L^{\phi_{4}}$. To get $L^{\phi_{4}}$,we expand $$U \approx 1+i\frac{\phi}{F}-\frac{1}{2}\frac{\phi^{2}}{F^{2}}-\frac{i}{6}\frac{\phi^{3}}{F^{3}}+\frac{1}{24}\frac{\phi^{4}}{F^{4}},$$then I have seen lots of books which state the result of this calculation is $$\frac{1}{48} \rm{Tr}([\phi,\partial_{\mu}\phi][\phi,\partial^{\mu}\phi]),$$ however,I can not see why there is a factor of 3 in the denominator,since if we only consider four $\pi's$ term then only order less than $(\frac{\phi}{F})^{2}$ will survive. Can anyone explain how to do the calculation?
 A: Importantly, observe what you might know, namely the method in the madness of the Lie algebra elements involved,
$$L=\frac{F^{2}}{4} \rm{Tr}(\partial_{\mu}U^{\dagger}\partial^{\mu}U)= \frac{F^{2}}{4} \rm{Tr}(\partial_{\mu}U^{\dagger}U ~~ U^\dagger\partial^{\mu}U)
,$$
where the $\phi$s are Hermitean matrices!
But there is celebrated systematics in the  operator exponential derivatives, to $O(\phi^3)$,
$$
U^\dagger\partial U =    \frac{i}{F} \partial \phi  +  \frac{1}{2F^2} [\phi, \partial \phi]-  \frac{i}{6F^3}[\phi,   [\phi  ,\partial \phi ]]   +...   ~~~\implies \\
\partial U^\dagger ~ U =  - \frac{i}{F} \partial \phi  -  \frac{1}{2F^2} [\phi, \partial \phi]+  \frac{i}{6F^3}[\phi,   [\phi  ,\partial \phi ]]   +...   .
$$
The are each in the Lie Algebra!
Plug these inside the trace, and use its cyclicity to rearrange the commutators.
The quartic term you are seeking comes from the linear times cubic terms, and, of course, the squared quadratic ones,
$$
\frac{F^{2}}{4} \frac{1}{F^4}(1/3-1/4)\rm{Tr} ([\phi,\partial_{\mu}\phi][\phi,\partial^{\mu}\phi]) \\
=\frac{1}{48 F^2} \rm{Tr}([\phi,\partial_{\mu}\phi][\phi,\partial^{\mu}\phi]).$$
Note the surviving quadratic/kinetic term, and the vanishing of the cubic one.
