The interactions of a massive particle fall off exponentially with distance (massless particles have long-range interactions), the exponent determined by the mass.
Mathematically, this dependence is governed by the quadratic term of the field in the action.

Now let's lump all fields together into a multi-index field. Then the vacuum state(s) corresponds to the minimum energy configuration(s), and the nearby shape of the landscape around that minimum (those minima) is determined by the massive fields. Adding more of these fields won't change the space of minima or the long-range behavior of interactions.

Or is that too nontechnical and hand-wavy?

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