The Lagrangian of chiral perturbation theory (with two quark flavors) is written using the following matrix U
U=eiσiϕi/f
where
σi are the Pauli matrices,
ϕi are three scalar fields and
f is a constant with mass dimension.
U is unitary, which makes the
ϕi fields real.
The Lagrangian at O(p2) order is
L2=f24tr(DμU†DμU)
the Lagrangian at next to leading order
O(p4) is
L4=l14tr(DμU†DμU)tr(DνU†DνU)+l24tr(DμU†DνU)tr(DμU†DνU)+l3+l416[tr(χU†+Uχ†)]2+l48tr(DμUDμU†)tr(χU†+Uχ†)+l5tr(U†FμνRUFLμν)+il62tr(FμνRDμUDνU†+FμνLDμU†DνU)−l716[tr(χU†−Uχ†)]2+h1+h3−l44tr(χ†χ)+h1−h3−l416([tr(χU†+Uχ†)]2+[tr(χU†−Uχ†)]2−2tr(χU†χU†+Uχ†Uχ†))−4h2+l52tr(FRμνFμνR+FLμνFμνL)
at next to leading order the coefficients li and renormalize with loops with vertices coming only from the O(p2) Lagrangian like
li=lri(μ)+γi32π2(−1ϵ−ln4π+γe−1)
where the gammas are
γ1=1/3γ2=2/3γ3=−1/2γ4=2γ5=−1/6γ6=−1/3γ7=0
Now comes the question. Let's consider the same theory with the same Lagrangian but allowing the ϕ fields in U to be complex.
My question is, would the γi−s change?