  A massive Goldstone (MG) mode, often referred to as a Higgs amplitude mode,
is a collective excitation that arises in a system involving spontaneous
breaking 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 emerges
in superconductors if the dispersion of fermions exhibits the particle-hole
(p-h) symmetry. However, clear understanding of the relation between the
symmetry of the Hamiltonian and the MG modes has not been reached. Here we
reveal the fundamental connection between the discrete symmetry of the
Hamiltonian and the emergence of pure amplitude MG modes. To this end, we
introduce nontrivial charge-conjugation ($\mathcal C$), parity ($\mathcal P$),
and time-reversal ($\mathcal T$) operations that involve the swapping of pairs
of wave vectors symmetrical with respect to the Fermi surface. The product of
$\mathcal{CPT}$ (or its permutations) represents an exact symmetry analogous to
the CPT theorem in the relativistic field theory. It is shown that a fermionic
Hamiltonian with a p-h symmetric dispersion exhibits the discrete symmetries
under $\mathcal C$, $\mathcal P$, $\mathcal T$, and $\mathcal{CPT}$. We find
that in the superconducting ground state, $\mathcal T$ and $\mathcal P$ are
spontaneously broken simultaneously with the U(1) symmetry. Moreover, we
rigorously show that amplitude and phase fluctuations of the gap function are
uncoupled due to the unbroken $\mathcal C$. In the normal phase, the MG and NG
modes become degenerate, and they have opposite parity under $\mathcal T$.
Therefore, we conclude that the lifting of the degeneracy in the
superconducting phase and the resulting emergence of the pure amplitude MG mode
can be identified as a consequence of the the spontaneous breaking of $\mathcal
T$ symmetry but not of $\mathcal P$ or U(1).
