At scales below ΛQCD global QCD symmetry SUL(3)×SUR(3) is spontaneously broken down to SUf(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˜q, where U=eiγ5taϵa and ϵa parametrise goldstone degrees of freedom; then we replace ˉ˜q˜q,ˉ˜qγ5˜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 Tr[∂μU†∂μU], but I don't understand how terms like
Tr[∂μU†∂μU∂νU†∂νU]
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.