The quantity which was measured is a simple one to describe: by smashing together protons and antiprotons at the Tevatron 2-TeV collider, you sometimes (well, rarely in fact: once every ten billion collisions!) produce top-antitop quark pairs. So one thing you can do is to count the number of positively-charged top quarks emitted in one direction -say the one of the proton beam- and subtract the number of the same in the opposite direction: if you divide by the total, you get a fraction which you may call "asymmetry". You fully expect that to be consistent with zero.
In fact, there are subtle effects -fully understood within the standard model- which make the asymmetry very slightly different from zero; the effect is also dependent on the total energy of the reactions, so it makes sense to measure the asymmetry as a function of the total invariant mass of the top-antitop system (which you can easily measure). In the first determinations by CDF and DZERO, this asymmetry was seen to be growing with the top-antitop mass: exactly the kind of effect one would expect if some new physics at very high energy were "driving" this asymmetry to differ from zero. The effect was significant at the 3-sigma level in the first determinations.
The story of the 3-sigma effect "ended" recently, when the latest determinations by the Tevatron experiments showed reduced asymmetries, now compatible with standard model predictions. Or better, the story has not ended yet: only, if there is a departure from current model predictions, it is not business that the Tevatron can investigate - more statistical accuracy and higher energies are needed. For a review please see the excellent posting by Jester on the matter. (Previously I also wrote about the matter, see e.g. this 2011 post or this more recent summary).
Unfortunately, the asymmetry that new physics may cause is much better studied at a proton-antiproton collider than at a proton-proton collider (such as the one we happen to have at work now, the LHC): the production processes of top-antitop quark pairs at the LHC are different, and new physics might successfully hide its presence there for a while longer.
So the situation is less intriguing today than it was a few years ago, but it is still quite interesting. In particular, it looks like a situation where new ideas may make a difference, while we wait for more useful data (a higher-energy proton-antiproton collider, or a very high luminosity LHC). One such interesting idea came from CDF recently. If new physics may cause an asymmetry of top quark production, it might do the same to bottom-antibottom production: the bottom quark is the partner of the top in the third generation, and it would be quite naturally sensitive to the same kind of effects.
Very nice. But how to study bottom-antibottom asymmetries ? While top quarks have a very distinctive way of decaying to a b-quark and a W boson, and the W boson charge betrays whether it was a top or an antitop (so that you can clearly identify whether it was a top or an antitop which went forwards or backwards), for bottom quarks this is not as easy. The bottom quark also decays via W boson emission, but the W is virtual, and it is not as easy to "tag" its charge as in the case of the real, energetic top-produced Ws.
Of course, one could decide to select jet pairs coming from bottom-antibottom hadronization by "tagging" their b-quark content via the identification of a soft electron or muon in the jet, as a b-quark emits a negative W and this will yield a negative electron or muon, while a anti-bottom quark emits a positive W and thus a positive electron or muon. Unfortunately, this is a background-ridden technique, and one of scarce efficiency (as it is not easy to identify leptons within jet cores).
The technique used by Dante Amidei, Tom Wright, and Jon Wilson in CDF was to select jet pairs with secondary vertex b-tagging, and then look at the observable called "jet charge": by performing a momentum-weighted sum of the observed positive and negative electric charge of charged particles within the jets, one obtains some discrimination between b- and anti-b-originated jets.
The analysis is actually not at all easy to perform, as not only is the jet-charge observable only slightly discriminating the two charges; there is significant background from non-b-quark jets to consider, which may also exhibit some spurious asymmetry which needs to be accounted for. As usual, the more precisely you design an analysis, the more subtleties you have to keep under control. My ex colleagues have done a very good job, but this is no place to describe those subtleties. Some details of their technique can be found in the public page of their analysis, or in a recent conference note.
In the end, the measurement is not as precise as the one obtained with top quark pairs, but it is still a nice proof of principle of the viability of the method, and it also excludes some new physics models alright. The asymmetry measurement is summarized in the graph below, where the horizontal axis shows the invariant mass of the two b-jets, and the vertical axis describes the observed charge asymmetry.
As you can see, the error bars are large enough to prevent a conclusion on whether there is a better agreement with standard model predictions or with the new physics models considered as an example by the authors. A particle called "axigluon" could mess up the SM predictions (in pink), but the asymmetry it would cause strongly depends on that particle's mass, such that e.g. one can observe that the measured asymmetry disagrees with the one (in blue) which would be produced by a light (200 GeV) axigluon, while it is consistent with what a heavier one would produce (in purple for a mass of 345 GeV). So a 200 GeV axigluon is indeed excluded by the CDF data.
Finally, let me say I am happy to congratulate with my ex CDF colleagues for their will to exploit to the fullest the wonderful dataset that CDF produced in Run 2 in the last decade... I am sure there are other interesting physics results that can be extracted from the data, using new ideas and inventiveness!