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 coefficient 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?