Difference between "C-violation without CP-violation" and "C-violation with CP -violation" Consider two possible decay channels of a massive particle as $X\to A+B$ and $X\to C+D$ with decay rates $r$ and $1-r$ respectively. Let the decay rates of its antiparticle into channels $\bar X\to \bar A+\bar B$ and $\bar X\to \bar C+\bar D$ are respectively $\bar r$ and $1-\bar r$.

For a theory with C-violation but CP-conservation, although the decay angular distribution for X and $\bar X$ would be different the decay rates integrated over all angles will be equal i.e., $\Gamma_X=\Gamma_{\bar X}$. But for a theory with both C and CP-violation would ensure different absolute rates in the two channels i.e., $\Gamma_X\neq\Gamma_{\bar X}$.

How can I understand/prove this statement? For the reference, see this.
 A: Let's look at the expression for the differential decay rate of $X\to f$, where $f$ contains an arbitrary number of particles with momenta $\mathbf p_i$ and spin $s_i$. This is given by: $$2m_X \text d\Gamma =(2\pi)^4\delta^4(P_X-\sum_i p_i)\vert \mathscr M (X\to f)\vert ^2 \text d \Phi,$$ where the phase space factor $$\text d \Phi = \prod _i \frac{\text {d}^3\mathbf p _i}{2E_i(2\pi)^3}$$ and $\mathscr M $ depends on the spin and the momenta of the final particle.
Now, if the theory is $CP$ conserving, the invariant amplitude $\mathscr M$, which is simply the $S$-matrix element with the $\delta$ function of momentum conservation omitted, must satisfy $$\mathscr M (\overline X \to \overline f)=\eta_f\mathscr M (X \to  f),$$where $\eta _f$ is a phase factor. Note that the $CP$ conjugate matrix element involves a reversal of momenta but leaves the polarizations unchanged. Even if $\text d\Phi$ is invariant under $\mathbf p _i \to -\mathbf p _i$, this does not imply that the differential decay rate must be invariant. I guess this is what your quotation means.
To give a concrete example, consider: $$n\to pe^-\overline \nu _e,$$ where the neutron's spin is completely polarized along the $z$-axis. The angular distribution of the electron turns out to be proportional to: $$ 1+A_e \cos \theta _{en},$$ where $\theta_{en}$ is the angle beetween the electron's momenta and the neutron's spin. $A_e\approx -0.12$, that is, the electron is preferentially emitted opposite to the neutron's spin.
Fermi's lagrangian which drives the beta decay is invariant under $CP$. However, the $CP$ mirror image of an electron flying opposite to the neutron's spin, is a positron flying in the direction of the antineutron's spin, which is the preferential configuration in the $CP$ conjugated process: $$\overline n \to \overline p e^+ \nu_e.$$ The angular distribution differs in the two processes, but the integrated rate is the same (the integral of $\cos \theta _en$ beetween $-1$ and $+1$ vanishes).
