Propagator for matrix fields (based on Srednicki ch80, p490)

[LAHLH]

Hi,

If I have a matrix valued field $B(x)_i^{..j}=B^a (x) (T^a)_i^{..j}$ and the relevant part of my Lagrangian is $L=Tr(-\tfrac{1}{2}\partial^{\mu}B\partial_{\mu}B+..)$ then how can I see that the propagator for the matrix field is $\Delta_{i..k}^{..j..l}(k^2)=\tfrac{(T^a)_i^j(T^a)_k^l}{k^2-i\epsilon}$ ?

I understand that if we expand the L in terms of the coefficent field we get $L=-\tfrac{1}{2}\partial^{\mu}B^a\partial_{\mu}B^{a}$ and this leads to the propagator for the coefficient field as $\Delta^{ab}(k^2)=\tfrac{\delta^{ab}}{k^2-i\epsilon}$, (just like usual for a massless scalar field) but not sure how to see the propagator of matrix field...

thanks for any help...

 

[LAHLH]

Also why exactly does each external propagator carry at $T^{a_i}$ factor?