I have studied the quark condensate and chiral perturbation theory. However, I am not sure that where the kinetic of the pion $$\frac{F^2}{4}\operatorname{Tr}\left\{\partial_\mu U\partial^\mu U^{\dagger}\right\},$$ come from. In my understanding, we just apply the effective field framework and write all possible term to construct the pion Lagrangian that respects the global $SU(2)_L\times SU(2)_R$ symmetry.

If my understanding is not correct, do we have analytic framework to derive the $\frac{F^2}{4}\operatorname{Tr}\left\{\partial_\mu U\partial^\mu U^{\dagger}\right\}$ from the standard model Lagrangian? I have no idea that how this term would arise.

I have studied the quark condensate and chiral perturbation theory. However, I am not sure where the "kinetic term" of the pion $$\frac{F^2}{4}\operatorname{Tr}\left\{\partial_\mu U\partial^\mu U^{\dagger}\right\},$$ comes from. In my understanding, we just apply the effective field framework and write all possible term to construct the pion Lagrangian that respects the global $SU(2)_L\times SU(2)_R$ symmetry.

If my understanding is not correct, do we have an analytic framework to derive the $\frac{F^2}{4}\operatorname{Tr}\left\{\partial_\mu U\partial^\mu U^{\dagger}\right\}$ from the standard model Lagrangian? I have no idea how this term could arise.

I have studied the quark condensate and the chiral perturbation theory. However, I am not sure that how the kinetic of the pion \begin{align} \frac{F^2}{4}Tr[\partial_\mu U\partial^\mu U^{\dagger}], \end{align} come from? In my understanding, we just apply the effective field framework and write all possibly term to construct the pion Lagrangian with respecting to the global $SU(2)_L\times SU(2)_R$ symmetry.

If my understanding is not true, does we have analytic framework to provide $F^2Tr[\partial_\mu U\partial^\mu U^{\dagger}]/4$ from the SM Lagrangian? I have no idea that how this term come from.

I have studied the quark condensate and chiral perturbation theory. However, I am not sure that where the kinetic of the pion $$\frac{F^2}{4}\operatorname{Tr}\left\{\partial_\mu U\partial^\mu U^{\dagger}\right\},$$ come from. In my understanding, we just apply the effective field framework and write all possible term to construct the pion Lagrangian that respects the global $SU(2)_L\times SU(2)_R$ symmetry.

If my understanding is not correct, do we have analytic framework to derive the $\frac{F^2}{4}\operatorname{Tr}\left\{\partial_\mu U\partial^\mu U^{\dagger}\right\}$ from the standard model Lagrangian? I have no idea that how this term would arise.

I have studied the quark condensate and the chiral pertubation theory. However, I am not sure that how the kinetic of the pion \begin{align} \frac{F^2}{4}Tr[\partial_\mu U\partial^\mu U], \end{align} come from?. In my understanding, we just apply the effective field framework and write all possibly term to construct the pion Lagrangian with respecting to the global $SU(2)_L\times SU(2)_R$ symmetry.

If my understanding is not true, does we have analytic framework to provide $F^2Tr[\partial_\mu U\partial^\mu U]/4$ from the SM Lagrangian? I have no idea that how this term come from.

I have studied the quark condensate and the chiral perturbation theory. However, I am not sure that how the kinetic of the pion \begin{align} \frac{F^2}{4}Tr[\partial_\mu U\partial^\mu U^{\dagger}], \end{align} come from? In my understanding, we just apply the effective field framework and write all possibly term to construct the pion Lagrangian with respecting to the global $SU(2)_L\times SU(2)_R$ symmetry.

If my understanding is not true, does we have analytic framework to provide $F^2Tr[\partial_\mu U\partial^\mu U^{\dagger}]/4$ from the SM Lagrangian? I have no idea that how this term come from.