Understanding the effective Lagrangian of minimal composite Higgs model In the original paper of minimal composite Higgs model based on $SO(5)/SO(4)$, the authors say that, after integrating out the heavy fields of strong dynamics, the general effective Lagrangian at the quadratic level is Eq. (9), i.e.
$$\frac{1}{2}\left(g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2}\right)\big[\Pi_0^B(p)B^\mu B^\nu+\Pi_0(p)\text{tr}[A^\mu A^\nu]+\Pi_1(p)\text{tr}[\Sigma A^\mu A^\nu\Sigma^T]\big],\tag{9}$$
where $\Sigma=(0,0,0,0,1)e^{i\sqrt{2}\Pi^{\hat a}T^{\hat a}/f}$, and $\Pi^{\hat a}$ is the Goldstone.
But I can't understand how to obtain above equation, especially the $(g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2})$ factor. Can it be derived from the CCWZ construction? I tried the Maurer-Cartan form
$$U^\dagger(A_\mu+i\partial_\mu)U=\bar d_\mu^{\hat a}T^{\hat a}+\bar e_\mu^aT^a,$$
but from $\bar d_\mu^{\hat a}\bar d^{\mu\hat a}$ I can only get some terms like $g_{\mu\nu}\Pi_0(p)\text{tr}[A^\mu A^\nu]$, not
$$(g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2})\Pi_0(p)\text{tr}[A^\mu A^\nu],$$
as classified in Eq. (9). So I get confused. I've searched and read lots of papers, but they just take Eq.(9) as granted and don't explain the details. Could someone tell me how to get Eq. (9), or lead me to some reference papers?
 A: The factor that you've brought up is a projection operator onto the transverse degrees of freedom:
$$P_T(p) = g_{\mu\nu} - \frac{p_\mu p_\nu}{p^2}$$
This should look familiar from the propagator of a massive gauge boson in Landau gauge. Recall that Landau gauge ($\xi = 0$) is the gauge choice where we've separated out the Goldstone bosons. This is a nice gauge to have in the back of our minds for composite Higgs theories because we're specifically treating the Goldstone sector as coming from the SO(5)/SO(4) breaking.
You might find Contino's lectures a useful pedagogical reference. He introduces the projector around equation (19). The general parameterization of the effective Lagrangian that you quote is Contino's equation (42). What's going on here is a big trick, the culmination of which is in the paragraph above Contino's equation (48). 
Here's how it works:


*

*Start by pretending that the SO(5) is gauged. Then we can think of the $P_T$ as a projection operator for Landau gauge. (I suppose one could have picked any gauge.)

*Then turn off the gauging except for the subgroup that is actually identified with electroweak symmetry. This boils down to throwing out all of the "fake" gauge fields. This gives the Goldstone interactions with the actual gauge bosons in terms of the form factors, $\Pi(p)$.
Hope that helps!
