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There is no need of starting with $SU(2)$ symmetry and then extending it to $SU(2)_L\otimes SU(2)_R$. In fact, the moment you write down the Lagrangian, the symmetry is by default $U(2)_L\otimes U(2)_R$ with $U(2)_L$ acting on the left handed quark doublet $(u , d)_L$ and $U(2)_R$ acting on the right handed quark doublet $(u , d)_R$, which can be extended ...

2

Question 1 First of all, when you discuss chiral symmetry spontaneous breaking, you need to assume pure QCD theory. QCD lagrangian with $u-,d-,s-$quarks (they have relatively small masses in compare with $b, t, c$-quarks) has the form  \tag 1 L_{QCD} = \bar{q}_{i}i\gamma_{\mu}D^{\mu}_{ij}q_{j} - \frac{1}{4}G_{\mu \nu}^{a}G^{\mu \nu}_{a} - ...

3

A more practical answer is that in many cases it is more useful to consider them separately. You could compare with electromagnetism. If I want to design a motor, it is much easier to work with the magnetic field generated by the coils than to invoke the whole glory of Maxwell's equations. Similarly, if I want to explain the propagation of light waves, ...

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Well, the so called "electroweak unification" is really more of an "electroweak mixing". I want to show you how the mixing is done, so that you yourself can decide whether you prefer to call it unification or mixing. You do not need to completely understand the equations, I will try to highlight the important points. The standard model is written in the ...

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