Do we have an analytic calculation to derive $\frac{F^2}{4}\,\text{Tr}\left\{\partial_\mu U\partial^\mu U^{\dagger}\right\}$ from the QCD Lagrangian?

Question

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.

Answer 1

However, I am not sure how the pion chiral lagrangian emerges. In my understanding, we just apply the effective field framework and write all possible terms symmetric under global 𝑆𝑈(2)_𝐿×𝑆𝑈(2)_𝑅 transformations

Correct: there is no analytic nonperturbative work producing such chiral effective lagrangians out of fundamental QCD, the part of the SM responsible for their emergence.

In the pre-QCD days of current algebra, people discovered the relevant symmetries and their realization, unbroken or dynamically SSBroken, and constructed QCD utilizing knowledge of these symmetries. With the advent of lattice simulations, they could "confirm" several such crude features (chiral symmetry SSBreaking, confinement, spectra), and could estimate basic constants like the condensate, pion decay constants, etc, crudely.

A systematic derivation of the full chiral model out of nonperturbative QCD on the lattice was "discussed" for decades, notably by Wilson and LePage, but it never really took off the ground.

Like many such grim facts, physicists just accepted the above and made sure they firmed up the trail map of how one should go from point A to point B, schematically, and provided analog lattice models of crucial processes. The full linkage of QCD and chiral effective theories continues to be elusive.