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Does an anti-Higgs boson have some role in mediating the mass of these particles? If not, how does that work? Or does it not matter (so to speak)? 
 @cf713a4e - most things work exactly the same if you stick antiparticles in where you had particles.  So yes: the anti-Higgs boson mediates the mass of antiparticles just like the Higgs mediates the mass of particles.   

BUT: the Higgs boson is its own antiparticle!   In this respect it's like the photon.

As far as we know, the only thing that changes when you replace particles by antiparticles are a few *numbers* describing how *strongly* the Higgs interacts with them.   This makes antimatter behave in a very subtly different way than matter. 
 @b4c50e1b @cf713a4e why exactly would you say that the anti-Higgs field mediates the mass of antiparticles? As far as I understand it, Dirac fermions only arise as soon as the nonzero Higgs expectation value introduces a heterochiral coupling between the symmetry-broken left-handed and the right-handed Weyl fermions. Antiparticles then are distinguished from particles by the phase angle between the coupled left-handed and right-handed part of their wavefunction. So basically, the Higgs mass terms "happen" at an earlier conceptual stage than Dirac antiparticles. 
 @f15ef1f3 @cf713a4e - I said the anti-Higgs mediates the mass of antiparticles and then said the anti-Higgs is the Higgs.  I was not trying to lecture this person on quantum field theory, vacuum condensates, Weyl and Dirac spinors, etc. I was just pointing out that the Higgs field is just as responsible for the mass of antiparticles as it is for partices - which is unsurprising, since the Higgs particle we've detected in the lab is its own antiparticle. 
 @b4c50e1b Good point 😅