Do massive particles exchange virtual Higgs bosons between themselves?
If they do does the resulting Higgs force decay exponentially with distance?
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Yes, massive particles such as W-bosons, Z-bosons, quarks, and leptons couple to the Higgs field via the cubic (Yukawa) interaction, so they may also exchange the virtual Higgs. Yes, because the virtual particle is massive, one gets the Yukawa potential that includes the exponential dumping with distance.
This "Higgs force" is much less fundamental and important than the four fundamental interactions (strong and weak nuclear forces, electromagnetism, gravity) because
The third point is related to the fact that unlike electrostatic, magnetostatic, and gravitational static forces, the Higgs exchange doesn't become "more important" when some objects are at rest. Instead, the Feynman diagrams with the Higgs exchange are examples among many and they're usually important primarily for very quickly moving particles. When the speeds are close to the speed of light, one has to use the full quantum field theory and the concept of "force", relevant only in mechanics, becomes inadequate.
In my analysis, the complex Higgs doublet $(\phi^0, \phi^+)$ can be transformed by $SU(2)$ only to $(0, \phi^0+i\phi^+)$, which is not real. By coupling to gravity in the first order formalism $(e,w)$, a local conformal symmetry exists while gravity remains non-dynamic.
Therefore, the electroweak gauge can be spontaneously broken by fixing a conformal scale, and the complex Higgs doublet reduces to $(0, \mu+iH)$. The Yukawa couplings to fermions are now absent, and there is no Higgs exchange. The only remaining Higgs interaction other than to gravity is to the vector bosons as $(\mu^2+H^2)$, which contributes to their mass renormalisation.
Such a theory avoids the complications of adding an imaginary mass term to the Higgs Lagrangian, and the real Higgs mass emerges naturally from the $|\mu+iH|^4$ term. Higgs resonances can still be realized at high energies for the vector boson tadpole diagrams, but there is no Higgs force.