这是一份liverpool利物浦大学PHYS377的成功案例

Therefore, even though $C$ violation implies that
$$
\Gamma\left(X \rightarrow q_{L} q_{L}\right) \neq \Gamma\left(\bar{X} \rightarrow \bar{q}{L} \bar{q}{L}\right)
$$
CP conservation would imply
$$
\Gamma\left(X \rightarrow q_{L} q_{L}\right)=\Gamma\left(\bar{X} \rightarrow \bar{q}{R} \bar{q}{R}\right)
$$
and also
$$
\Gamma\left(X \rightarrow q_{R} q_{R}\right)=\Gamma\left(\bar{X} \rightarrow \bar{q}{L} \bar{q}{L}\right)
$$
Then we would have
$$
\Gamma\left(X \rightarrow q L q_{L}\right)+\Gamma\left(X \rightarrow q_{R} q_{R}\right)=\Gamma\left(\bar{X} \rightarrow \bar{q}{R} \bar{q}{R}\right)+\Gamma\left(\bar{X} \rightarrow \bar{q}{L} \bar{q}{L}\right)
$$

PHYS377 COURSE NOTES ：

To get an unremovable phase, we need to consider a more complicated theory. For example, with two Dirac fermions we can have
$$
\mathcal{L}=\sum_{i=1}^{2} \bar{\psi}{i}\left(i ;-m{i} e^{i \theta_{i} \gamma_{s}}\right) \psi_{i}+\mu\left(\bar{\psi}{1} e^{i \alpha \gamma{s}} \psi_{2}+\text { h.c. }\right)
$$
After field redefinitions we find that
$$
\mathcal{L} \rightarrow \sum_{i=1}^{2} \bar{\psi}{i}\left(i \not-m{i}\right) \psi_{i}+\mu\left(\bar{\psi}{1} e^{i\left(\alpha-\theta{1} / 2-\theta_{2} / 2\right) \gamma_{\mathrm{s}}} \psi_{2}+\text { h.c. }\right)
$$
so the invariant phase is $\alpha-\frac{1}{2}\left(\theta_{1}+\theta_{2}\right)$.
It is also useful to think about this example in terms of 2 -component Weyl spinors, where the mass term takes the form
$$
\sum_{i=1}^{2} \psi_{L, i}^{\dagger} e^{i \theta_{i}} \psi_{R, i}+\mu \psi_{L, 1}^{\dagger} e^{i \alpha} \psi_{R, 2}+\text { h.c. }
$$