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With every interaction between elementary particles, there is a particle associated with the transmission of the force at hand (example: the photon transmitting the e.m force).

What's the particle conveying the interaction between the Higgs field and elementary particles fields?

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There is no such mediating particle or field. The interaction between the Higgs field and the fermion fields is a Yukawa interaction where a fermion bilinear directly couples to the scalar field as a term $\bar\psi\psi h$ in the Lagrangian.

In fact, you might view the Higgs itself as the mediating particle for an associated Yukawa force with potential $V(r) \propto \mathrm{e}^{-\mu r} / r$, where $\mu$ is the mass of the scalar, but since the Higgs is rather massive this "force" is so short-ranged as to be non-existent at the classical scales where it would make sense to speak of a force. The significance of this interaction in the standard model is mainly that it gives rise to the mass term of the fermions when the Higgs field acquires its VEV/mass.

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  • $\begingroup$ So that makes it a point interaction? $\endgroup$ – descheleschilder Dec 28 '16 at 14:08
  • $\begingroup$ @descheleschilder I don't really know what that's supposed to mean - what's a "point interaction"? It is the same kind of interaction as that between electrons and photons - if you want to call that a "point interaction" then it's a point interaction, if not then not. $\endgroup$ – ACuriousMind Dec 28 '16 at 14:13
  • $\begingroup$ Yes, that's what I mean. The Higgs field directly couples to a fermion field, so if we assume the Higgs point-like, it hits a point-like fermion at the point where they both coïncide. Which by the way seems to me almost impossible, just as impossible like when a point-like photon meets a point-like electron. Which in turn makes you wonder if elementary particles are really point-like. $\endgroup$ – descheleschilder Dec 28 '16 at 14:26
  • $\begingroup$ @descheleschilder A Feynman diagram is not a representation of what happens in a scattering process, but it basically represents an equation. Don't take them too literally. What we know is $m$ things goes in and $n$ things comes out and those diagrams are closely related to the probability of that happening. $\endgroup$ – FrodCube Dec 28 '16 at 14:35
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    $\begingroup$ @descheleschilder "Point-like" in quantum field theory means particles that have no substructure, i.e. are elementary, not objects that are literally localized at a point. The particles of QFT are difficult to even get a notion of position/localization for, and in no way is thinking about one point "hitting" another point a useful way to think about QFT interactions. $\endgroup$ – ACuriousMind Dec 28 '16 at 14:41

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