At scales below $\Lambda_{QCD}$ global QCD symmetry $SU_{L}(3)\times SU_{R}(3)$ is spontaneously broken down to $SU_{f}(3)$, and pseudogoldstone degrees of freedom called mesons arise. It is possible to obtain explicit for of lagrangian for them: we substract goldstone degrees of freedom from quarks field, $q = U \tilde{q}$, where $U = e^{i\gamma_{5}t^{a}\epsilon_{a}}$ and $\epsilon_{a}$ parametrise goldstone degrees of freedom; then we replace $\bar{\tilde{q}}\tilde{q}, \bar{\tilde{q}}\gamma_{5}\tilde{q}$ by their VEVs $v, 0$. In the result chiral effective theory arises.

The question: I understand how (by using method described above) to derive term $\text{Tr}[\partial_{\mu}U^{\dagger}\partial^{\mu}U]$, but I don't understand how terms like

$$

\tag 1 \text{Tr}[\partial_{\mu}U^{\dagger}\partial^{\mu}U\partial_{\nu}U^{\dagger}\partial^{\nu}U]

$$ arise, since in QCD lagrangian there are only bilinear combinations of quark fields (if I understand correctly, term $(1)$ arise without taking into account gluon field), so product of only two $U$ matrix can arise.

I would be grateful for explanation how terms like $(1)$ arise.