Gauge singlet under SM What does it mean when they say something is a gauge singlet under the Standard Model group? I would like to understand this concept of Singlets and Doublets. Thanks in advance.
 A: Singlet under the Standard Model (SM) group stands for a particle without quantum numbers from the Standard Model gauge group, so its representation doesnt transform under the SM gauge group.  One example can be a hypothetical complex scalar field which has no SM quantum numbers.    These particles appear in the literature often in the context of Dark Matter.
Regarding doublets.  The Higgs boson for instance arises from a complex scalar field that is a doublet under the $\mathrm{SU(2)}_L$ gauge group of the Standard Model.  The Higgs field transforms then in the fundamental representation of $\mathrm{SU(2)}$.  You can write the Higgs doublet as
$$\phi=\binom{\phi^{+}}{\phi^{0}}=\frac{1}{\sqrt{2}} \binom{\phi_{3}+i\phi_{4}}{\phi_{1}+i\phi_{2}}$$
and it transforms under a $\mathrm{SU(2)}_L$ gauge transformation as
$$\phi \rightarrow e^{i   T^{i}  \beta_{i}(x)} \phi $$
where $T^i = \tau^i/2$ ($\tau^i$ are the three Pauli matrices) denote the generators of the fundamental representation of $\mathrm{SU(2)}_L$.   I used Einstein index summation notation. 
