Is is correct to say that, elementary particles have different masses, because they have different coupling strengths to Higgs field? And if yes. Does it make sense to question, why they have different coupling strengths?
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It is correct to say that fermions acquire their mass via their coupling to the Higgs field so that when a VEV is acquired, it combines with the Yukawa coupling to form the coefficient of the one-particle-in-one-particle-out term.
The gauge bosons acquire their mass via the Higgs field in a different way, namely via the SSB of the Electroweak by the acquisition of a VEV, and the gauge invariance that "eats" the goldstone modes.
Both of these are poorly understood, that is we don't know why the Yukawa couplings are what they are, and we don't know a lot about the electroweak SSB. We don't understand the Higgs potential in the first place.
The Higgs mechanism is how the gauge bosons get their mass. Your question is indeed correct in the form that the Yukawa couplings determine the coefficients of the 1-in-1-out term, also known as mass, but this has nothing to do with the Higgs mechanism, in fact, the Higgs contribution to these couplings is the same for all couplings as per the VEV.
MrDBrane
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