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

Above diagram is an 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: $$<0|J^a_\mu|\pi^b(p)> = i p_\mu f_\pi \delta^{ab}$$ (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.

Thanks

• At the risk of this being tagged as not on topic, I have to ask, have you reached a conclusion on this subject? Check page 36, (5.2) of arxiv.org/abs/hep-ph/9509427, and let me know if it helps, cause I'm curious as well. – user103310 Jan 8 '16 at 23:48