  We study collective modes near the quantum critical point of a
pair-density-wave (PDW) superconductor in 2+1 dimensions. The fate of gaps of
various collective modes is investigated by functional renormalization. For
incommensurate PDW superconductors, we show that the gapless Leggett mode,
protected by the emergent $U(1)$ symmetry, can induce an exponentially small
Higgs mass compared to the superconducting gap. Further, for commensurate PDW
superconductors, we find an emergent mass hierarchy in the collective modes,
i.e. the masses of Leggett boson, Higgs boson, and the superconducting gap can
differ by several magnitudes in the infrared. This may shed light to a
mechanism underlying the hierarchy problem in the Standard Model of particle
physics.
