Global symmetry and particle multiplets

Question

In chapter 20, of Peskin and Schroeder's quantum field theory book, they start with a comment that a global symmetry that is manifest lead to particle multiplets with restricted interactions. Can someone please explain how does a global symmetry does so with an example. I cannot understand this point.

Answer 1

An example would be strong isospin, which is an approximate global symmetry of the Standard Model.

The isospin doublet in this case is $$ \begin{pmatrix} p \\ n \end{pmatrix}.$$ But there also is an isospin triplett $$ \begin{pmatrix} \pi^+ \\ \pi^0 \\ \pi^- \end{pmatrix}.$$

The symmetry restricts the ways in which these particles can decay. Since this is only an approximate symmetry, there are decays violating stron isospin, but their branching ratios are strongly supresed w.r.t. isospin conserving decays.

Answer 2

Maybe this is the interpretation: every global (and continuos) symmetry of the lagrangian implies the existence of a conserved charge, using Noether's theorem. Charge commutes with the Hamiltonian, so there are some particle multiplets (same mass) labelled by the charge. The simplest example is probably the global symmetry of a complex scalar field for a phase, that is $\phi \rightarrow e^{i \alpha} \phi$. In this case the associated charge can be seen as the electric charge of the field. Restricted interactions may be interpreted as the fact that under electromagnetic interactions only charged fields interact.

Similarly, a phase symmetry can give you conservation of barionic/leptonic number.