How can we calculate pion decay constant in Chiral Perturbation Theory?

Question

The above diagram is a one-loop contribution to the pion decay constant $f_\pi$. For example in this paper (Eq.7) they have written down the pion decay constant to one loop, but the calculation is not given. Usually the pion decay constant is defined in the following way: $$ \langle 0|J^a_\mu|\pi^b(p)\rangle = i p_\mu f_\pi \delta^{ab}$$ (link, Eq.4.19). I wanted to know how exactly are these two pictures related. That is, how do we extract the one-loop correction to $f_\pi$ by calculating the amplitude corresponding to the above diagram, in the context of Chiral Perturbation Theory or simply using the Lagrangian of the above mentioned paper Eq.3.

I have searched the internet extensively but could not find a place where such a calculation is done explicitly, so any help will be really appreciated.

Answer 1

In the end, you cannot calculate the pion decay constant in chiral perturbation theory. All you can do is adjust the parameters in your n-loop calculation so that the physical value is reproduced. The standard reference for the one-loop renormalization of ChPTh is Gasser and Leutwyler, Chiral Perturbation Theory to One Loop, Annals of Physics 158, 142 (1984).