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It is widely known that the CP violation provided by the Standard Model ( SM )is insufficient to explain the baryon asymmetry that we observe. However, I never came across any quantitative argumentation about this fact. By that, I mean that we know that

$\bullet$ The baryon asymetry is measured to be $\eta \simeq 6 \cdot 10^{-10}$

$\bullet$ The CP violation coming from the CKM matrix is "predicted" ( we still need experiments to find the values of the CKM parameters ) in the SM to be $J\simeq 3\cdot 10^{-5}$, where $J$ is the Jarkslog invariant. We could also take, as quantitative measurement of the CP violation coming from the CKM matrix, the value of the mixing parameter $\epsilon \simeq 2\cdot 10^{-3}$ in the $K-\bar{K}$ system.

My question is the following. Is there any prediction of $\eta$ assuming that the CP violation is only coming from the CKM matrix or, conversely, is there any estimation of what should be the value of $J$ or $\epsilon$ to get the right value of $\eta$. I understand that it should heavily depends other parameters due to the 2 other Sakharov conditions but I wonder if any privilegied model has been chosen to make this statement that the SM do not provide enough CP violation.

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