How do we measure meson decay constants? 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 \big[  i \int \,d^4x {\cal H} \big] | \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. In particular,


*

*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 $ ?

 A: To toot my own horn a little1 may I suggest the series of papers published by the $f_\pi$ collaboration from Hall C at Jefferson Lab:


*

*Determination of the pion charge form factor for $Q^2=0.60-1.60 \,\mathrm{GeV^2}$, 2007
DOI:10.1103/PhysRevC.75.055205 and arXiv:nucl-ex/0607007

*Charged pion form factor between $Q^2=0.60$ and $2.45 \,\mathrm{GeV^2}$. I. Measurements of the cross section for the $^1H(e,e′\pi^+)n$ reaction, 2008  
DOI:10.1103/PhysRevC.78.045202 and arXiv:0809.3161

*Separated Response Function Ratios in Exclusive, Forward $\pi^{+/-}$ Electroproduction, 2014
DOI:10.1103/PhysRevLett.112.182501 and arXiv:1404.3985
This study achieved access to the decay constant by separating the structure functions for highly forward scattering to examine the pion-mass pole in the t-channel (where the decay constant is linearly present in the dominate term). The result is then used to tune the best available theoretical calculations by varying the input $f_\pi$ to get a good match to the experimental results.
You might also find some useful information about the experimental arrangments in the original paper


*

*Measurement of the Charged Pion Electromagnetic Form Factor, 2001  
DOI:10.1103/PhysRevLett.86.1713 and arXiv:nucl-ex/0010009
The very long period between the taking of the data and the extraction of the decay constant should serve as a channel marker for the considerable difficulty of work in the transition energy regime.

1 The real tooting of horms belongs to my collaborators as I was only a beginning grad student and concentrated on one systematic correction, not the whole project.
A: *

*How can we measure meson decay constants?
I am not an experimental physicst, but I think that the best way to obtain the decay constant is to study processes like $\pi^+\to \mu^+ \nu$ and extract them from the branching ratio:
$$\rm{Br}(\pi^+\to \mu^+\nu)=\dfrac{G_F^2 m_{\pi^+} m_\mu^2}{8 \pi}\left(1-\dfrac{m_\mu^2}{m_{\pi^+}^2} \right)^2 f_{\pi^+}^2 |V_{ud}|^2 \tau_{\pi^+} ,$$
which is measured nowadays with great precision. 
@dmckee answer's suggests that we can also extract the decay constant from the pion form factor, but this method seems less precise, because it is more difficult to measure form factors than decay constants (but maybe I'm wrong...). If you take a look at PDG, you'll see that the process $\pi^+\to \mu^+\nu$ is measured with an incredible precision.
One last comment about decay constants: actually, these quantities can be computed for pions using Lattice QCD methods and the theoretical error bars are comparable to the experimental ones! You can even find very precise computations for more exotic mesons, like $D$, $B$ and $B_s$.


*

*For your theoretical question:
It depends on the process you are considering! For example, if you have $\pi^+\to \mu^+\nu$, then you must take a second order term. In this term, you need a current $J^\mu_{q}$ related to the annihilation $u \bar{d}\to W^+$ and a leptonic current $J^\mu_\ell$ related to the creation $W^+\to\mu^+\nu$. Then, the time ordered product will only apply to the $W^+W^-$ term and it will give you simply the $W^+$ boson propagator.
From my experience, I would suggest you to integrate-out the vector bosons, because the corrections to the fermi theory are negligible. In this case, you can write an effective Hamiltonian:
    $$\mathcal{H}_{\text{eff}}=-\sqrt{2} G_F V_{ub}  [\overline{u}\gamma_\mu (1-\gamma_5)d][\bar{\mu}_L \gamma^\mu {\nu_\mu}_L] +\text{h.c.},  $$
and it is much simpler to read the amplitude and to relate it with the decay constant, because the hadronic part factorizes:
$$\mathcal{A}=-i\langle \mu^+,\nu | {H}_{\text{eff}} |\pi^+\rangle =i\sqrt{2} G_F V_{ub}  \langle 0 |\overline{u}\gamma_\mu \gamma^5 d|\pi^+\rangle\cdot \bar{u}(p_\nu)[\gamma^\mu(1-\gamma_5)/2 ]v(p_\mu),$$
where $$\langle 0 |\overline{u}\gamma_\mu \gamma^5 d|\pi^+\rangle=-i p_\mu f_{\pi^+}.$$
