The up quark $u$ and down quark $d$ form a doublet ($u$, $d$). I know that they have a SU(2) symmetry. However, they also have a U(1) symmetry because they are regarded as two states of the same type of particle. I am puzzled: What is the symmetry of this doublet? Do they have two symmetries, i.e., SU(2) and U(1), or U(2) = SU(2) $\otimes$ U(1)? How to understand their symmetry?

  • $\begingroup$ Related: physics.stackexchange.com/q/175667/2451 $\endgroup$ – Qmechanic Jul 28 '18 at 18:22
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    $\begingroup$ Yes, they have those two symmetries as direct product. Additionally, they have SU(3) color symmetry, as well. So they form $SU(2)_L$ doublet of $SU(3)_C$ triplets with $U(1)_Y$ hypercharge. Leptons, on the other hand, are color singlets. $\endgroup$ – Oktay Doğangün Jul 28 '18 at 18:47
  • $\begingroup$ In fact, $u$ and $d$ have different masses. So, is the $S(2)_{L}$ flavor symmetry broken or not exact? $\endgroup$ – Shen Aug 12 '18 at 18:23

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