Domain Wall Questions: Exploring L=kρσ Interaction

[DuckAmuck]

So I have learned that an interaction term in a Lagrangian like: L= kρσ
can result in "domain wall" problems between the fields ρ and σ.

I imagine this means that it can result in particles spontaneously changing mass, spin, or some other property, violating conservation laws. Is that why this is a problem, or is there some other issue?

Is this sort of thing akin to a B-L sphaleron?

Is this in any way related to neutrino oscillation, or theoretical possibility that a muon oscillates into an electron without producing any other decay products?

 

[MacRudi]

So I have learned that an interaction term in a Lagrangian like: L= kρσ
can result in "domain wall" problems between the fields ρ and σ.

I imagine this means that it can result in particles spontaneously changing mass, spin, or some other property, violating conservation laws. Is that why this is a problem, or is there some other issue?

Is this sort of thing akin to a B-L sphaleron?

Is this in any way related to neutrino oscillation, or theoretical possibility that a muon oscillates into an electron without producing any other decay products?

a very good question! or questions. Maybe @fzero has an answer.
I can only give a hint to this problem, because I'm not sure how to solve and if I'm right.

If we have a theory with a M background, then it must be covariant like in the superspace and then the Lagrangian does not contain non dynamical-fields nor Lagrange multipliers.
But I think @fzero is able to explain it much better.