I'm trying to understand how people actually measure decay constants that are discussed in meson decays. As a concrete example lets consider the pion decay constant. The amplitude for $\pi ^-$ decay is given by,

\begin{equation}

\big\langle 0 | T \exp \left[ i \int \,d^4x {\cal H} \right] | \pi ^- ( p _\pi ) \big\rangle

\end{equation}

To lowest order this is given by,

\begin{equation}

i \int \,d^4x \left\langle 0 | T W _\mu J ^\mu | \pi ^- ( p _\pi ) \right\rangle

\end{equation}

If we square this quantity and integrate over phase space then we will get the decay rate.

On the other hand, the pion decay constant is defined through,

\begin{equation}

\left\langle 0 | J ^\mu | \pi ^- \right\rangle = - i f _\pi p _\pi ^\mu

\end{equation}

This is clearly related to the above, but it seems to me there are a couple of subtleties.

- How do we get rid of the time-ordering symbol?
- Since we don't have a value for $ W _\mu $ how can we go ahead and extract $f _\pi $ ?