Muons decay into other particles. Does the Higgs field know that muons will decay and does it interact with them differently compared to how it interacts with electrons?
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No, the Higgs field does not "know" or "care" what the muon will or will not do. All it knows, in the Standard Model, is the electroweak quantum numbers of the muon, which are identical to those of the electron, except it couples more strongly to the muon than it does to the electron by a factor of $m_\mu/m_e$. See this answer. The SM interaction is $$ -\frac{m_\mu\sqrt{2}}{v} \overline{ \begin{pmatrix} \nu_{\mu L} \\ \mu_L \end{pmatrix} } \cdot \begin{pmatrix} 0 \\ \frac{h+v}{\sqrt 2} \end{pmatrix} ~\mu_R +\hbox{h.c.}=-m_\mu \left ( 1+{h\over v}\right )\overline{\mu_L} \mu_R +\hbox{h.c.} ~. $$
Because of that, the renormalizations of these two couplings are slightly different, so the quantum corrections will be somewhat, but not significantly, different: estimate them.
If what you were asking is how the Higgs couplings contribute to muon decay, the answer is the decay goes through the charged current, the W, which involves the Goldstone boson partners of the physical Higgs. They "help" it decay, but the Yukawa coupling is not involved, because the "physical Higgs” is uncharged.
Cosmas Zachos
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Do you have a reference for an experimnetal confirmation of this? – my2cts Apr 08 '21 at 13:04
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This: 'it couples more strongly to the muon than it does to the electron by a factor of $m_\mu/m_e$'. – my2cts Apr 08 '21 at 13:19
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Well, the muon SM behavior has been confirmed by CMS, of course. Will dig for wispy e+e- decay, all indications being consistent with it, so far, but, of course, the ratio to τ+ τ- is unimpeachable. Surely you are not hoping to base departures from the SM on the lightest charged lepton? – Cosmas Zachos Apr 08 '21 at 13:22
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The ATLAS e+e- rabbit hole only affords limits, so far. – Cosmas Zachos Apr 08 '21 at 13:28