Madame Wu and collegues showed us that the current mediated by what nowadays is called "W boson" is chiral, meaning that only left chiral fermions (and right anti-fermions) take part into the interaction. From the other hand, in the theoretical description, only matters in (SU(2)_L) doublets interacts with the W boson. So, if you want to describe what Wu teached us, you need to let left chiral fermions (and right anti-f.) to act as SU(2)_L doublets in your description of the world, while right handed fermions (and left a.f.) as SU(2)_L singlets (to avoid their interaction with the W boson). Otherwise you would have a theory not describing what you see, so not a physical theory. Of course both chiralities need to be charged under U(1)_Y, because you want to have a theory in which both fermion "species" interact with same strength to the photon, such as Q_em = Y + I^3_L (in appropriate conventions for Y and I^3_L, respectively the Y charge and the usual third generator of SU(2)_L). In the last equation SU(2) singlets will have I^3_L = 0 (by definition of singlet), so you need them to be charged under U(1)_Y if you want them to be electrically charged. The story could go on, but too far from your question ;)
I think your question should be more specific, like doublet under what? The SM SU(2) ? One of the problems you might face is that you will have trouble to write a gauge invariant Yukawa Lagrangian with a Higgs doublet. In fact then (I suppose), you can put gauge invariant quark mass terms by hand ... The quarks will not couple to the Standard Higgs which will be phenomenologically inconsistent ... You will find many such problems ..
Now if you mean the right handed quarks form doublet under a different SU(2) .... Indeed, there are models entertaining this possibility ... I suppose, a good example is the Left-Right-Symmetric models, where the right handed quarks form a doublet under SU(2)_R [Broken at a high scale] which is different from the SM group SU(2)_L.
Hope this helps.
Because by construction the theory is SU(2)_L x U(1)_Y symmetric.
On can also construct a SU(2)_L x SU(2)_R x U(1)_{B-L} theory which
is called a left right symmetric model. Here left handed and right handed
quarks are treated similarly. Right handed quarks also form doublets;
but not under SU(2)_L. They form doublets under SU(2)_R instead.
The right handed SU(2)_R is broken at a high scale in this type of theories.
As a result at the low energy we observe only the left handed SU(2) symmetry.
That is we can experimentally observe W+/- and the Z boson. The right
handed gauge bosons are much heavier, thus they cannot be observed
by Collider experiments.
The SM only postulates what we have observed. To answer a why question you would need a more fundamental theory. And this more fundamental theory would have to have sufficient explanatory power to recover the characteristic properties of the SM. String theory fails here.
There is a theory which is able to do this, see http://arxiv.org/abs/0908.0591 which predicts the gauge group and the fermions of the SM from simpe first principles. But it has been ignored up to now, despite publication in Foundations of Physics, probably because the basic assumptions are too far away from the mainstream. In particular, this model requires a preferred frame, which is a no-go for most physicists.
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