Examples of the vacuum $| 0 \rangle$ break C, P, T symmetry? Can the vacuum $| 0 \rangle$ of some QFT break C, P, T symmetry?

*

*It could be that the QFT break C, P, T symmetry explicitly. Are there good examples?

p.s. The weak interaction breaks P explicitly and maximally, I believe.


*It could be that the QFT does break C, P, T symmetry explicitly. But the vacuum $| 0 \rangle$ of some QFT break C, P, T symmetry spontaneously. Are there good examples?

p.s. We can also change the parity P to the reflection symmetry. Follow people's advice.
 A: I am writing the answer based on my conversations with Prof. Michael Peskin.
We both agree that there are QFTs that do not break C, P, T symmetry explicitly. But the vacuum |0⟩ of some QFT break C, P, T symmetry spontaneously.
There are many theories in which the weak interactions are fundamentally P and C symmetric, but P and C are spontaneously broken.  One of the original papers is
G. Senjanovic and R. N. Mohapatra
Phys. Rev. D 12, 1502 (1975)
Consider an  $SU(2) \times  SU(2)$  left-right symmetric theory of weak interactions, in which  $P$ exchanges the two $SU(2)$'s.   Postulate a set of Higgs fields in the representation  $(1,0) + (0,1)$  where the numbers are the isospin under the corresponding $SU(2)$.   It is easy to write a potential energy in which one Higgs field acquires a vacuum expectation value but both cannot.    The the parity symmetry is spontaneously broken.    The vacuum in which the first $SU(2)$ is broken is equivalent to the vacuum in which the second $SU(2)$ is broken.   In both cases, the unbroken symmetry is $SU(2) \times   U(1)$, which is just right.
Prof. Michael Peskin: "Some people writing in the stackexchange post   seem to be going around in mathematical circles.   They do not seem to have picked up a textbook to find out what spontaneous symmetry breaking is or what the actual properties of the weak interaction are."
Prof. Michael Peskin generously suggested: "So how can I help them?" Look at Prof. Michael Peskin's textbook Concepts of Elementary Particle Physics.  Spontaneous symmetry breaking is discussed in a physical way.
