  By applying a microscopic gauge-invariant kinetic theory in $d$-wave
superconductors, we analytically derive the energy spectra of the breathing
Higgs mode and, in particular, rotating Higgs mode that is unique for the
$d$-wave order parameter. Analytical investigation on their dynamic properties
is also revealed. We show that the breathing Higgs mode is optically visible in
the second-order regime. Whereas the rotating Higgs mode is optically inactive,
we show that this mode can be detected in the pseudogap phase by magnetic
resonance experiment. It is interesting to find that with a longitudinal
temperature gradient, the charge-neutral rotating Higgs mode generates a
thermal Hall current by magnetic field in the pseudogap phase, providing a
unique scheme for its detection.
