KyleSingh said:
I figured that this was the case but couldn't articulate why they could not just posit three different particles.
There is another issue as well: interactions between the particles don't just appear out of nowhere. They have to be in the Lagrangian. The Lagrangian in the paper has no interaction terms. An example of a Lagrangian for a single scalar particle with an interaction term is this:
$$
L = - \frac{1}{2} \partial^\mu \partial_\mu \phi - m^2 \phi^2 - \lambda \phi^4
$$
The ##\lambda \phi^4## term is the interaction term: basically, it says that a pair of these ##\phi## particles can scatter off of each other.
A theory of three different scalar particles would have three different fields in it; you could call them ##\phi_\alpha##, ##\phi_\beta##, and ##\phi_\gamma##. The total Lagrangian would include a kinetic ##\partial^\mu \partial_\mu## term and a mass ##m^2## term for each field, plus interaction terms that could be between the same field, such as a ##\lambda## term for each field, or could be between different fields, such as a term like ##g \phi_\alpha^2 \phi_\beta^2##, which would describe scattering of an ##\alpha## and ##\beta## particle off of each other. Other interactions are also possible. But they all have to appear in the Lagrangian; you can't just conjure them up out of nowhere.
(Btw, the paper also seems to think that there can be separate derivative operators for each of the fictitious particles, as ##\partial_\alpha##, ##\partial_\beta##, etc. This is another error. The derivative operator has a spacetime index; it describes variations of fields in spacetime. It is the same operator regardless of which field, which kind of "particle", it is applied to.)
