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A massive Goldstone (MG) mode, often referred to as a Higgs amplitude mode,is a collective excitation that arises in a system involving spontaneousbreaking of a continuous symmetry, along with a gapless Nambu-Goldstone mode.It has been known in the previous studies that a pure amplitude MG mode emergesin superconductors if the dispersion of fermions exhibits the particle-hole(p-h) symmetry. However, clear understanding of the relation between thesymmetry of the Hamiltonian and the MG modes has not been reached. Here wereveal the fundamental connection between the discrete symmetry of theHamiltonian and the emergence of pure amplitude MG modes. To this end, weintroduce nontrivial charge-conjugation ($\mathcal C$), parity ($\mathcal P$),and time-reversal ($\mathcal T$) operations that involve the swapping of pairsof wave vectors symmetrical with respect to the Fermi surface. The product of$\mathcal{CPT}$ (or its permutations) represents an exact symmetry analogous tothe CPT theorem in the relativistic field theory. It is shown that a fermionicHamiltonian with a p-h symmetric dispersion exhibits the discrete symmetriesunder $\mathcal C$, $\mathcal P$, $\mathcal T$, and $\mathcal{CPT}$. We findthat in the superconducting ground state, $\mathcal T$ and $\mathcal P$ arespontaneously broken simultaneously with the U(1) symmetry. Moreover, werigorously show that amplitude and phase fluctuations of the gap function areuncoupled due to the unbroken $\mathcal C$. In the normal phase, the MG and NGmodes become degenerate, and they have opposite parity under $\mathcal T$.Therefore, we conclude that the lifting of the degeneracy in thesuperconducting phase and the resulting emergence of the pure amplitude MG modecan be identified as a consequence of the the spontaneous breaking of $\mathcalT$ symmetry but not of $\mathcal P$ or U(1).

Hidden charge-conjugation, parity, and time-reversal symmetries and massive Goldstone (Higgs) modes in superconductors