When we have a doublet in electroweak gauge symmetry, we assign a weak hypercharge. The question is, if weak hypercharge is zero we could be talking about that there isn't electroweak interaction with another field?

All this before breaking the symmetry, at sufficiently high energies where the electromagnetic force and the weak force are unified.


I think you are asking: In a theory with gauged $SU(2)_L\times U(1)_Y$, suppose we have a doublet $D$ under $SU(2)_L$ that has hypercharge zero, $Y[D]=0$. What are the properties of this particle?

We assume that electroweak symmetry is broken to electromagnetism. We know that the electric charge is $Q = T^3 + Y$. Thus the two components of the $D$ have electric charge $\pm 1/2$ coming from their $T^3$ charges.

At high energies, the $D$ interacts with the $W^{1,2,3}$ gauge bosons with coupling $g$. We know that $W^{1,2}$ combine into $W^{\pm}$. The $W^3$ mixes with $B$ to form the $Z$ and $A$. Because the $D$ couples to $W^3$ (but not $B$), it picks up a charge under both $Z$ and the photon, $A$.

  • $\begingroup$ Ok i agree Henry. But my question is more fundamental: at very high energies (over the electroweak scale) the electroweak force really exist? So, if we interpret the hypercharge as the responsible for this force, means that Y=0 for a doublet, implies that there´s no electroweak interaction? in the sense of the electromagnetic charge is the responsible of the electromagnetic interaction. $\endgroup$ – Eduardo Castillo Ruiz Aug 22 '17 at 0:07
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    $\begingroup$ At very high energies the electroweak force absolutely, really exists, at least within the Standard Model, where this is tautological---that's what the theory assumes. You can ask whether or not "electromagnetism" exists at high scales, which is a more subtle question. Electromagnetism is just a subgroup of the full electroweak symmetry, so yes, this also exists. Though in the limit where electroweak symmetry is restored, it makes more sense to talk about the $W^{1,2,3}$ and $B$ bosons rather than the $W^\pm$, $Z$, and photon. $\endgroup$ – Henry Deith Aug 22 '17 at 3:41
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    $\begingroup$ I also do not follow your argument about the existence of a particle with certain charges having anything to do with the interpretation of the force. I must be misunderstanding you, because this is like saying "the neutron has no electric charge, thus does electricity really exist?" There may be a language confusion: the electroweak force refers to the combination $SU(2)_L\times U(1)_Y$. It's the combination of the weak force and hypercharge. Electromagnetism is a $U(1)$ subgroup of the electroweak group. $\endgroup$ – Henry Deith Aug 22 '17 at 3:46
  • $\begingroup$ Sorry about my english, maybe you've not understood the spirit of my question. I know that everything you posted is correct. That's what the SM books say. I'll ask you in another way: Let's think the model with electroweak symmetry $SU(3)_L\otimes U(1)_N$. In this model we can assign a hypercharge zero to lepton triplets $\Psi_L\sim (3,0)$. If we consider the term $\overline{\Psi_L}(\Psi_L)^c\Phi$, where $\Phi$ is a Higgs triplet (or sextet) with zero hypercharge to obtain Majorana lepton masses terms, Is there any electroweak interaction between the fields before the breakinkg? $\endgroup$ – Eduardo Castillo Ruiz Aug 24 '17 at 15:33
  • $\begingroup$ Hm, there may still be some confusing communication. Let's try to sort them out. First: you have referred to electroweak symmetry as $SU(3_L)\times U(1)_N$. As you know, this is no longer what we call electroweak symmetry in the Standard Model (it may contain the usual definition). For clarity, let me define this as EW' symmetry (with a prime) in case this may be a source of confusion. Second: can you clarify what you are asking when you say "is there any electroweak interaction between the fields"? Which fields? Do you mean EW or EW' ? $\endgroup$ – Henry Deith Aug 24 '17 at 17:20

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