Symmetries in particle physics

[Shen712]

We often use SO(N) and SU(N) to describe symmetries in particle physics. I am not clear which one to choose when I try to discuss a symmetry. For example, why do we use SU(3) but not SO(3) to describe the symmetry of the three colors of quarks? Similarly, why do we use SU(2) but not SO(2) to describe a quark doublet, for instance, the (u d) doublet?

 

[Orodruin]

The obvious answer would be ”because the orthogonal groups would not describe nature”.

On a more technical note, the quark fields allow complex rephasing and the kinetic terms in the Lagrangian are invariant under the global SU(N) transformations. Promoting the global symmetry to a local gauge symmetry forces you to introduce the minimal coupling to the corresponding gauge field.