Can the weak interaction be studied without bothering the electromagnetic one? It is well known that the electroweak interaction can be studied as a $SU(2)_L \otimes U(1)_Y$ gauge theory. It is also known that the electromagnetic interaction is a $U(1)$ gauge theory, and it works perfectly on its own (by "it works" I mean on the paper, obviously there are many scattering processes that can't be explained via QED only, but one can study QED as a standalone theory). My question is: can the same be done for the weak interaction, with all three gauge bosons? Can it be described as a standalone gauge theory?
To clarify, I'm aware of the Fermi theory, but that's not what I'm interested in, because it isn't a gauge field theory.
 A: There is no gauge theory describing the weak interaction after electroweak breaking. The meaning of symmetry breaking, after all, is that the resulting effective theory no longer has all the symmetries of the theory prior to breaking.
There is no problem at all in writing down a standalone $\mathrm{SU}(2)$ gauge theory, and you're free to choose the representations of the particles, so it can also only act on particles of a certain chirality if you want. Such a gauge theory will have three massless gauge bosons, whether you call them "W and Z bosons" or something else is entirely up to you.
The reason you don't see such a theory discussed in the context of the weak interaction is that that's not what the weak interaction is: After electroweak breaking, the W and Z bosons are massive and the Ws are electrically charged - this isn't the particle content of a SU(2) gauge theory, so studying SU(2) gauge theories won't tell you anything useful about the weak interaction. In fact, since the bosons have become massive, they cannot any longer be described as the force carriers of any Yang-Mills gauge theory.
In contrast to this, the photon remains massless and uncharged after electroweak breaking, and so is well-described by a $\mathrm{U}(1)$ gauge theory.
A: No; it's tricky. Recall, QED runs on the single generator of the four in the SM ($\tau_3+{\mathbb I}/2$) that escapes SSBreaking by leaving the Higgs doublet v.e.v. invariant upon exponentiation, even though it does not commute with all three of them.
So, singling out QED from the SM is the world before 1967 and is easy because the photon is massless, in contrast to the Ws and the Z, so massive that they damp themselves out at low energies; this is what makes the Weak interactions weak.
You could go to higher energies, around 85GeV and study the WI subtractively, by taking out QED, but this is a bit contrived, to the extent there is no surviving group for it which is unbroken, unlike in QED. All generators of the WI are SSBroken, and fail to commute with QED, so it's not a standalone theory.
In practice however, writing down the "after SSBreaking" Lagrangian is what one in fact does, focussing on the charged and neutral weak currents and their chiral peculiarities, and suitably renormalizing them, etc... The cynosure or signature in these studies is focussing on the three quasi-Goldstone-boson components of the Higgs multiplet, but I don't have a dramatic, cogent illustration of how to do this.
