What does off-diagonal term mean?

[physlad]

when we apply Higgs mechanism to the lagrangian of a complex scalar field... we get terms for a Glodstone boson, a massive scalar and a massive vector field in the new lagrangian..

we also get other terms.. one of them takes following form

$$A_\mu \partial^\mu \xi$$

this term is said to be off-diagonal in the fields... what does this mean? and what does it imply?

I mean how can we know just by looking at such a term that our lagrangian does not describe physical fields? and that we need to rewrite the lagranigan in terms of real fields... until we get the lagrangian which contains a massive vector and a massive scalar (Higgs particle)..

 

[azntoon]

The term you mentioned is off-diagonal because it involves two distinct fields, the vector field A_μ and the scalar field ξ. This implies that the Lagrangian does not contain physical fields describing a massive vector and massive scalar (Higgs particle), and must be rewritten in terms of real fields. This is done by introducing a unitary transformation on the scalar fields, which allows us to write the original complex scalar field as a combination of two real scalar fields. The same transformation can then be applied to the vector field, resulting in a massive vector and a massive scalar (the Higgs particle).