Propagator for W boson I've found in different literature that some write the propagator for the W boson as $\frac{g_{\mu\nu}-\frac{k_\mu k_\nu}{M^2_W}}{k^2-M^2_W+iM_W\Gamma_W}$ and others like $\frac{g_{\mu\nu}-\frac{k_\mu k_\nu}{M^2_W}}{k^2-M^2_W}$, without the $iM_W\Gamma_W$ term in the denominator.
Is there a reason for that or am I missing something?
 A: Putting an $iM\Gamma$ in the denominator of a propagator for bosonic particle with mass $M$ makes the particle unstable, decaying with rate $\Gamma$.  However, it also makes the propagator non-unitary, since the decaying particle doesn't turn into anything else; it just disappears, and probability is lost.  So the propagator with the $\Gamma_{W}$ cannot be the true propagator.  Nevertheless, if you want to do a tree-level calculation for a certain process that contains a $W^{\pm}$ intermediate, including the instability of the boson will improve the accuracy of your calculation.
In fact, neither of the propagators you cite is actually the "true" $W$ propagator.  I put "true" in quotation marks, because the kind of propagator you need to use to calculate radiative corrections is actually dependent on the gauge.  Moreover, in addition to the vector bosons, you typically also need to include scalar (Fadeev-Popov) ghosts in the representation of the gauge field.  These complications arise in both non-Abelian gauge theories and spontaneously broken gauge theories; and the electro-weak theory describing the $W^{\pm}$ is both!
So the propagators listed in the question will not work if they are used for anything beyond tree-level calculations.  If you try to use such expressions in radiative corrections, you will get gauge-dependent and nonunitary results.  Thus, if you are already limited to tree level, you might as well include the boson decay term, which is why many people do.
