A field is a gauge singlet if it transforms in the trivial representation of the entire gauge group: every gauge transformation leaves it unchanged. A singlet of only one factor need not be a gauge singlet of the product group. For example a right-handed neutrino with representation is a Standard Model gauge singlet, whereas a color-singlet charged lepton is still acted on by the electroweak group. A singlet can have ordinary derivatives in its kinetic term and gauge-invariant masses, subject to Lorentz invariance and any additional global symmetries.
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