Wow! What a comment!

> Thank you very much for this paragraph on the hypercharge interaction. So if I understand well we can consider that the photon and the Z boson are actually illusions, corresponding in a sense to a biased perception of reality. And, Gauge symmetries allowed us to correct that (all bosons should be massless).

No, it is the opposite. The photon and the Z are real, the two others are unphysical. Both the photon and the Z undergo hypercharge and weak interactions. The combination of these two interactions correspond exactly to electromagnetism for the photon.

More technically, after the breaking of the electroweak symmetry (my next post), the B and W<sub>3</sub> field got a non diagonal mass matrix. There are thus non physical and we need to mix them, to  diagonalize the mass matrix. This gives rise to the photon and the Z naturally, with the right properties. This is tough to explain without a blackboard. Maybe you should just ignore this paragraph (or ask more about it).

> I only teach about the charged weak bosons, so after reading what you wrote, I dug into what the Z boson actually does within the scope of the weak interaction. It is a neutral exchange particle that allows transfer of momentum between neutrinos and electrons, leaving both particles unchanged otherwise.
> Wow! (light bulb in my head, :-): it does a very similar job to the photon, except that because it has no electric charge, it could not be attributed to an electromagnetic interaction.

When one relies on gauge interactions to describe weak charged current, one ends up in finding this works very well. However, we got the neutral current for free out of the formalism, and the observed neutral current properties do not match the observed properties of the Z boson. 

The reason is simple: the interactions of the physical Z boson contain both a weak part and a hypercharge part. The combination of the two being different from those of the photon, the Z interactions are different from the photon ones.

Your diagram is actually wrong. At the level of the gauge symmetries (which should be on the top and not on the bottom), you have:
 - strong interactions, with a gluon mediator
 - weak interactions, with a W+, W-, W3 mediators
 - hypercharge interaction, with a B mediator

Those are the gauge fields, that may be different from the physical fields. The W3 and B fields will mix when the electroweak symmetry is broken, and they will give rise to the physical Z boson and the photon. Both will have interactions that are different combinations for the neutral weak and hypercharge interactions. The photon combination matches electromagnetism, and the Z boson ones are different.

In fact, the number of bosons does not change. The electroweak symmetry breaking mechanism only allows some of them to become massive.

And actually, your alpha and beta are just sine and cosine of a mixing angle (it is a simple rotation). The mixing angle is known as the electroweak mixing angle (sometimes called the Weinberg angle).

Is it clearer?

> In my understanding, the electroweak interaction is when both weak and EM (or weak and hypercharge) fuse into one single force at extremely high energy density (ultra high temperature) with its own electroweak boson. The symmetry is then of a much higher order. It breaks when the temperature become lower, and like a liquid becoming solid loses spacial symmetry, the electroweak force breaks into two (Weak and Hypercharge). Is my view correct?

You actually describe the electroweak phase transition :) We have on one side the strong times weak tims hypercharge gauge symmetry and on the other side the strong times electromagnetic gauge symmetry.