Direct vs. indirect CP violation: theoretical foundations I know very little about the difference between direct and indirect CP violation. I've been studying QFT from Peskin and Schroeder's "An Introduction to QFT", and they don't seem to address this issue. So I would appreciate it if someone could explain what is the difference between direct and indirect CP violation. In relation to this, I understand that the phase of the CKM matrix can account only for direct CP violation. Why is it so?
 A: No, CKM matrix mediates undirect CP violation as well as direct. 
Direct and undirect CP-violation
Let's discuss the difference between direct and undirect CP violation for the phenomena of K mesons. Formally from the QCD spontaneous breaking of chiral symmetry we know that there exist the octet of mesons which become massless at zero quarks mass limit; for example, such mesons are $K^{0}, \bar{K}^{0}$. We may write down linear combinations of these mesons, which are CP-even and CP-odd:
$$
K_{1} \equiv \frac{1}{\sqrt{2}}\left(K^{0} + \bar{K}^{0}\right), \quad K_{2} \equiv \frac{1}{\sqrt{2}}\left( K^{0} - \bar{K}^{0}\right)
$$
If CP symmetry is exact, then $K_{1}$ may decay on pion pair $\pi \pi$ (which is CP even), while the decay of $K_{2}\to 2 \pi$ is forbidden. 
It was naturally to identify $K_{1}$ with short-lived specie $K_{S}$ of kaons and $K_{2}$ with long-lived $K_{L}$. This identification means that we state that the full hamiltonian which describes meson interaction is $CP$-invariant, so eigenstates of hamiltonian are CP-odd or CP-even. However, it was experimentally shown that process $K_{L} \to 2 \pi$ exist, which shows that there is CP violation mixing of $K_{1,2}$:
$$
K_{S} = \frac{K_{1}+\epsilon K_{2}}{\sqrt{1 + |\epsilon|^{2}}}, \quad K_{L} = \frac{K_{2} + \epsilon K_{1}}{\sqrt{1 + |\epsilon|^{2}}}
$$
The reason of such mixing is presence of nondiagonal elements in mass matrix for $K^{0}, \bar{K}^{0}$. These nondiagonal elements violate CP symmetry, but undirectly: we only have that physical states aren't eigenstates of CP violation. Undirect CP violation predicts nonzero relation
$$
\tag 1 \frac{A(K_{L} \to \pi^{+} \pi^{-})}{A(K_{S} \to \pi^{+} \pi^{-})} = \frac{A(K_{L} \to \pi^{0} \pi^{0})}{A(K_{S} \to \pi^{0} \pi^{0})} = \epsilon
$$
There exist, however, "another" CP violation which states that there is possible process $K_{2} \to 2 \pi$. It leads to inequality $\frac{A(K_{L} \to \pi^{+} \pi^{-})}{A(K_{S} \to \pi^{+} \pi^{-})}$ and $\frac{A(K_{L} \to \pi^{0} \pi^{0})}{A(K_{S} \to \pi^{0} \pi^{0})}$ in $(1)$:
$$
\frac{A(K_{L} \to \pi^{+} \pi^{-})}{A(K_{S} \to \pi^{+} \pi^{-})} = \epsilon + \epsilon{'}, \quad \frac{A(K_{L} \to \pi^{0} \pi^{0})}{A(K_{S} \to \pi^{0} \pi^{0})} = \epsilon - 2\epsilon{'}
$$
CP violation source in SM
As you know, there exist one CP violation source in SM - CKM matrix $V_{CKM}$: in canonical mass basis of quarks this matrix present in the sector of charged current interaction:
$$
L_{CC} = -c \begin{pmatrix}\bar{u} & \bar{c} & \bar{t} \end{pmatrix}_{L}\gamma^{\mu}V_{CKM}\begin{pmatrix} d \\ s \\ b\end{pmatrix}_{L}W_{\mu}^{+} + h.c. 
$$
In fact, you know that we may rotate 5 quark fields to eliminate 5 phases of $V_{CKM}$, so that in convenient parametrization it has one arbitrary phase and 3 angles. Such phase is source of CP violation. 
Role of CKM phase in direct and undirect CP violations
CKM phase is the only CP violating source in the Standard model (another one is QCD theta term, but it is bounded from above by very small value), and it causes undirect CP violation as well as direct. 
Undirect CP-violation arise because of box diagrams of type
$$
d +\bar{s} \to W^{*}(u,c,t)^{*}W^{*}(u,c,t)^{*} \to s \bar{d}
$$
while the direct CP violation happenings because of penguin diagrams 
