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I want to get something clear that I do not seem to understand. I used to read that the photon $A$ and the $Z$ boson are (different) linear combinations between the $\mathrm{W}^0$ (neutral weak boson before $SU(2)$ breaking) and the $B$ ("photon precursor" before $SU(2)$ symmetry breaking).

Nowadays it is more common to say that the $Z$ gets its mass from the Higgs through Yukawa coupling.

Which is the right way to look at things? What is wrong or incomplete in the previous description? Where exactly does the $Z$ mass come from - and is it due to the Higgs mass or not?

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    $\begingroup$ no, the Z mass is not coming from Yukawa coupling (Yukawa coupling means a coupling between a scalar and 2 fermions, while the Z boson is a spin 1 boson). It comes from the gauge interaction, Higgs boson carrying both weak hypercharge and weak isospin quantum numbers. $\endgroup$
    – Paganini
    Jul 28, 2015 at 19:39
  • $\begingroup$ Paganini, thank you. So what is the exact origin of the Z mass? And does the Higgs play a role in the mass of the Z or not? $\endgroup$
    – Hans973
    Jul 29, 2015 at 4:06

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To summarise:

  • It is correct that the photon and the $Z$ boson are linear combinations of $W^0$ and $B$.

  • The $Z^0$ (and the $W\pm$) get their mass via the Higgs mechanism, i.e. the vacuum expactation value of the Higgs to which both $B$ and $W$ bosons couple.

  • The $Z$ mass is related to the Higgs vacuum expectation value (VEV), not the Higgs mass. (These two are related, but not the same. We knew the value of the Higgs VEV before we knew the Higgs mass.)

  • This has nothing to do with Yukawa couplings -- those are a separate facet of the Higgs mechanism and provide masses to the charged leptons and quarks. (Generally, a Yukawa coupling couples two fermions and a scalar. It does not involve gauge bosons.)

  • As a side remark, the Higgs mechnism is not responsible for all the observed masses. In particular, the masses of hadrons (such as protons and neutrons) are overwhelmingly due to the strong interaction.

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The Z boson, like the W, acquires mass via the Higgs mechanism. The mixing between the SU(2) and U(1) gauge fields is the reason why the Z mass and the W mass are not identical:

$$m_\mathrm{Z} = \frac{m_\mathrm{W}}{\cos \theta_\mathrm{W}}$$

Where $\theta_\mathrm{W}$ is the weak mixing angle.

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It seems to me that there is nothing fundamentally wrong about your statements, thus I also don't see any contradictions. The electroweak unification states that, as you said, the $Z$ and the $\gamma$ are different linear combinations of $B^0$ and $W^0$. This all works very nicely, there is just the problem of the masses which is then fixed my the Higgs mechanism.

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    $\begingroup$ .... but it is wrong to attribute the Z’s mass to Yukawa couplings! $\endgroup$ Feb 12, 2020 at 1:15

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