The Standard Model of Particle Physics includes a three by three matrix called the CKM matrix, that determines the probability that an up quark that emits a W+ boson will transform into each of the three possible down type quarks, and the probability that a down type quark that emits a W- boson will transform into each of the three possible up type quarks (the probability is equal to the square of the relevant entry in the matrix).
It is true by definition that this matrix can be fully explained with four parameters and that there are an infinite number of ways to do this, although only two or three ways of doing so are commonly used. One common approach is to use three real valued mixing angles and one imaginary valued CP violation phase.
The CP violation phase makes the probability of a CKM matrix with a term containing larger in the case of a particular starting point quark to a particular finishing point quark than the same transition in the reverse direction (i.e. it is equivalent to time asymmetry). This is the only source of CP violation in the Standard Model. Any other kind of observed CP violation would constitute "New Physics" beyond the Standard Model.
When applied to actual weak force decays of hadrons, the Standard Model predicts that baryons, vector mesons and charged mesons will have generally quite small CP violation which current experiments will struggle to detect except in the case of the heaviest B mesons, but that there will be significant observable CP violation in the decay of neutral pseudoscalar mesons, i.e., neutral kaons, neutral D mesons, and neutral B mesons (with and without strange quark components respectively).
In fact, five sigma evidence of CP violation at Standard Model predicted rates (the particle physics gold standard for calling a phenomena a "discovery" rather than a mere potential statistical fluke) has been observed in neutral kaons, neutral B mesons (with and without strange quark components), and even in charged B mesons.
The Latest Experimental Results For Neutral D Mesons
But, CP violation has not yet been observed in neutral D mesons, according to the most LHC results announced in a preprint today. The new results continue to show CP violations that are consistent to within experimental error with zero in neutral D meson decays (consistent with past attempts to measure the same thing that have been ongoing since the charm quark was first discovered).
This non-observation of CP violation, however, turns out to confirm the Standard Model, rather than contradicting it. This is because the expected amount of CP violation in D meson decays is on the order of 0.1% of the decay rates or less (although this rate hasn't successfully been calculated very accurately because QCD calculations is very difficult). But, current experimental measurements of these decay rates are insufficiently precise to make a statistically significant detection of this subtle amount of CP violation.
Why Is CP Violation Hard To See In D Mesons?
Why is CP violation so uncommon in the decays of neutral D mesons, compared to other neutral mesons?
Among the four mixing systems (K0, D0, Bd and Bs) the D0 system is in a sense unique. The mixing mechanism relies on internal d-type quarks; due to the smaller mass of the b quark compared to the t quark, the kinematics of the dispersive part of the mixing amplitude are not completely dominated by the heavy third generation quark. Furthermore, due to the specific structure of the Cabibbo-Kobayashi-Maskawa couplings, the absorptive part Γ12 will feature an extremely efficient Glashow-Iliopoulos-Maiani (GIM) mechanism. We will discuss this in detail later on and show that it leads to a suppression of the leading contribution by several orders of magnitude.
Put another way, CP violation is generally greater in heavier mesons, and the fact that B mesons are much heavier than D mesons makes the CP violation in D mesons easier to see.
Neutral kaons are an exception to this rule because they are really a linear combination of two different kinds of mesons (the K long and the K short), which decay at very different rates (the decay rates differ by a factor of 500) and we are seeing the CP violation in the disparity between K long and K short decays (rather than within the subgroup of K long, or K short decays), whereas in D and B mesons, we are measuring CP disparities in the decays of a single kind of meson with a single decay rate.
Prospects For Future Discoveries Of CP Violation In D Mesons
Experimental measurements need to be on the order of 100 times more precise than existing ones to be confidently expected to directly or indirectly observe CP violation in D meson decays (and perhaps the decays of some other kinds of hadrons) at the levels that the Standard Model predicts theoretically when the Standard Model's CP violating phase is fitted to the observed levels of CP violation in B meson and neutral kaon decays.
Even the full LHC data set, the LHC is predicted to be just barely be able to discovery CP violation in D meson decays if it can see them at all, given its expected sensitivity of about 10 times more precision than the results reported in preprint form today. Direct observation of this phenomena predicted by the Standard Model would require a new collider that is much more powerful than the LHC (which the most ambitious fundamental physics experiment in human history).
Thus, while further LHC measurements are expected to put even tighter bounds on beyond the Standard Model physics models, they are not expected to directly observe CP violation in neutral D meson decays.
Of course, any beyond the Standard Model physics that enhances CP violation in neutral D mesons by a factor of 10 to 100 or so, will be detected at the LHC.
Implications for BSM physics
While direct or indirect confirmation that there is CP violation in neutral D meson decays does not yet exist, the experimental measurements to date can rule out any beyond the Standard Model theory that predicts a significant enhancement of CP violation in neutral D mesons relative to the Standard Model expectation greater than about 0.3%, or that otherwise deviates materially from the Standard Model expectation for CP violation in the instances where CP violation has been observed.
CP violation is particularly sensitive to the existence of undiscovered heavy quarks and these measurements provided some of the earliest predictions of the charm and top quarks. The non-detection of elevated levels of CP violation, therefore, tends to suggest particularly strongly that there really are just three generations of quarks, rather than four or more.