Gauge Boson Propagators in Spontaneously Broken Gauge Theories

[TriTertButoxy]

The propagator for gauge bosons in a spontaneously broken (non-abelian) gauge theory in the $R_\xi$ gauge is (see Peskin and Schroeder eqn. 21.53)

$$\tilde{D}^{\mu\nu}_F(k)^{ab}=\frac{-i}{k^2-M^{ab}}\left[g^{\mu\nu}-(1-\xi)\frac{k^\mu k^\nu}{k^2-\xi M^{ab}}\right]\,,$$​

where $M^{ab}$ is the gauge boson mass matrix, and $\xi$ is the gauge fixing parameter. The matrices in the denominator should be interpreted as matrix inverses. To make perturbative calculations, I am supposed to diagonalize the mass matrix $M^{ab}$, and write the propagator in terms of the eigenvalues.

I would like to make my calculations as general as possible, and avoid having to go to a particular model to diagonalize the mass matrix. Is there a way to rationalize the propagator above so that the matrices are in the numerator?

 

[azntoon]

I would like to write the propagator as\tilde{D}^{\mu\nu}_F(k)^{ab}=\frac{-i}{k^2}\left[g^{\mu\nu}+(1-\xi)\frac{M^{ab}}{k^2-\xi M^{ab}}k^\mu k^\nu\right]\,.If this is not possible, what is the best way to proceed? Is it just necessary to choose a particular model and diagonalize the mass matrix?