One of the most intriguing effects in subatomic physics is the phenomenon of violation of the discrete symmetry called "CP". It is intriguing at various levels.

First of all, CP violation is intriguing because of the depth of the concept: proof of that be that it is not at all easy to explain it to outsiders (I will make an attempt below, but I am likely to fail!).

Second, its elusive nature makes it even more mysterious and difficult to study: only a few subatomic physical systems exhibit it, and the effect is visible only as a modification of measurable quantities at the level of a few parts in a thousand.

Third, and a subtle point: the smallness of the explored CP-violating phenomena make them sensitive to the contribution of "loop diagrams", reactions which proceed through an intermediate state composed of two or more virtual particles. Indeed, it is much easier to find a small perturbation in a physical quantity which is very close to zero, than to a bigger number! These quantum loops would allow yet undiscovered particles to contribute to the CP asymmetric phenomena, while remaining hidden in the intermediate loop. This important fact creates the opportunity to find new physics without the need to directly produce these new particles!

And fourth, CP violation is an intrinsically interesting and exciting field of investigation, because it might be the key to understand the origin of the observed matter-antimatter asymmetry in our observable Universe.

Now, if I have not managed to convince you yet that you need to read the rest of this article, I will add that the present experimental situation of CP violation measurements is quite confusing and contradictory! So if you already know CP violation inside out you might want to skip the next section, which provides some introduction to the topic conceived mainly for the benefit of outsiders, and go directly to the following one, where some experimental results worth discussing are illustrated.

A Crash Course in CP Violation

Ever since the discovery of CP violation in 1964 in a landmark experiment by Christenson, Cronin, Fitch and Turlay, high-energy physicists have attempted to study in detail the phenomenon, wherever the effect might be hiding; unfortunately, one may count these different places on one's fingertips.

To observe CP violation, you must measure a property of a physical system and the same property in its CP-conjugate state. Only special physical systems can be used: ones which constitute a so-called "eigenstate" of the CP transformation, i.e. ones that have a well-defined, and experimentally measurable, CP state. To understand this, imagine I asked you to compare Mozilla and Internet Explorer using as a metric their smell of coffee: this does not make much sense, does it ? The browsers are not "eigenstates of smell" - they do not have a well-defined value of that property, so asking whether Mozilla has a more toasted aroma does look silly.   

But what is the property we are talking about ? Well, C in CP stands for "charge conjugation", and P stands for "parity". Saying that a system is invariant under a CP transformation means that all of its properties and its behaviour -all the physics that there is to study about it, that is- are exactly the same if you take every particle in the system and replace with its charge conjugate (what is called a "C" transformation), and then you study the mirror image of the resulting configuration (a "P" transformation).

A few years before 1964 physicists had already discovered that weak interaction processes -the ones mediated by weak vector bosons, a prime example being nuclear beta decay- were modified by the C and P operations separately: there were observable differences between the weak interactions of a system and its C, or its P conjugate. Yet it was believed that the product of the two symmetry transformations would be a perfect symmetry even for weak interactions. Big was their surprise when a pure state of CP-negative neutral kaons were observed to decay into a pair of charged pions -a CP-positive state.

The explanation of how it is possible to incorporate CP violation in the theory describing weak interactions came only in the early '70, when Kobayashi and Maskawa realized that one needed a complex phase in the matrix describing the amplitude of reactions transmuting a quark flavor into another. With a complex phase one could accommodate CP violation (a simple explanation of this is in the nice Wikipedia entry on this topic), but a complex phase in the matrix was only possible if there were at least three families of a total of six quarks! This was a remarkable find: quarks were then just mathematical, abstract entities, and just three of them had been hypothesized yet - the November 1974 revolution was still to come.

In order to make use of the complex phase hypothesized by Kobayashi and Maskawa, CP-violating transitions are realized through the interference of two different amplitudes. Ohh, now interference is something I should also explain - but this post would diverge too much. Let me say that if two wave trains overlap their influence in a region of space, the net result may be a strengthening of the oscillatory motion, or a cancellation, depending on how the wave crests combine. A similar thing happens in subnuclear reactions, although I wouldn't push the analogy further.

Nowadays we study three different physical systems to evidence the smallish difference of the two (+1 and -1) autostates they can form: neutral kaons, neutral B mesons, and neutral D mesons. Notice how all of these are electrically neutral particles, and all contain a heavy (-ish, in the case of kaons) quark: a strange quark in the kaons, a charm quark in D mesons, and a bottom quark in B mesons. The reason of this choice is simple: heavy quark decays by weak interactions, which are the only ones which have so far been observed to violate the CP symmetry.

What physicists do with these particles, which are readily produced in copious amounts in proton-proton collisions at the LHC, but can of course also be produced even more effectively with electron-positron collisions of the right energy, is not too difficult: they study the decays of a particle and its antiparticle to the same final state, of well-defined CP quantum number (what we called above a "eigenstate"). They can then measure the difference between the rate of the two processes, which should be exactly zero if CP is conserved for the particle interactions involved in the decays. This is usually cast in the form of a "CP asymmetry", such as
A = (X-Y)/(X+Y).

Let us take for example a D0 meson, a particle composed of a charm quark and a anti-up antiquark. The D0 meson can decay to a K+ K- final state, where K+ is the charged kaon, a particle made up by a up and anti-strange quark pair. But of course also the anti-D0, the particle made up by a up-quark and a anti-charm antiquark, can decay to the same K+ K- final state. The difference in these decay amplitudes, normalized by dividing it by the sum of the same amplitudes, is the CP asymmetry which is measured - and found slightly but significantly different from zero in this case.

Of course there are huge complications to this simple scheme. Most of them stem from the difficulty in answering the simple question: how can one decide whether it was a D0 or a anti-D0 particle which was present in the first place ?

Remember, physicists usually determine the identity of a particle by looking at its decay products - these D mesons live lifetimes smaller than a trillionth of a second, and usually travel just a hundred microns or so in the detector before decaying. In this case, though, the decay cannot be used to distinguish the two -indeed, we want to use to determine a difference!

What is used is a "tag" of the identity of the meson, which can be constructed by studying the rest of the particles produced in the collision which created the D meson. Indeed, D mesons are not created alone: in the simplest case, we create a charm-anticharm quark pair in the hard collision; each of these then "dresses up" as a charmed meson. The D0 will contain the charm, and a anti-D0 will contain the anti-charm. Now suppose we observe, together with a D meson decay to a K+ K- final state, another D meson decaying to K+ pi-. The fact that this other D has decayed to a positive kaon means that it was a anti-D, because the K+ contains a anti-strange quark, which results from the decay of a anti-charm. If we had observed instead the K- pi+ combination, the conclusion would be opposite.

The above is sketchy and a bit imprecise, but I cannot really explain more detail about the subtleties of these physical systems and phenomena. However, you should well realize that experimental issues concern the perfect knowledge of the experimental efficiency to reconstruct mesons of different charge and angular distribution: we are talking about per-mille asymmetries here, which therefore require systematical uncertainties to be beaten down to less than that.

The Experimental Situation in the D0 system, Circa 2013

I believe I have written too much already, and I still need to fill you up with the controversial situation we are facing, given a bunch of different measurements by competing experiments that use different apparata, collisions, experimental techniques, decay modes to evidence a CP asymmetry in one of the three systems I mentioned above.

So perhaps we can live with a summary picture: the graph below shows several measurements by LHCb, Babar, Belle, CDF. These points should line up on a vertical line within their error bars, but they appear to scatter a bit more than statistics would want them to.

All these measurements are asymmetries determined in the D-meson system. Note how different experiments measure things in slight disagreement with one another. This is intriguing, because these experiments rely on different techniques, so one could read the discrepancies as an indication of some underlying similarity or dissimilarity of the methods rather than on the physical quantity being derived.

Also, one should always be aware, when such complex physics measurements are produced, that one often neglected source of uncertainty is the ill-defined nature of the quantity being estimated. This could play a role in the case of these asymmetri measurements.

So, theorists are at work trying to put forward different possible explanations of yet another three-sigma experimental effect. And the number and quality of different possible subtle effects that might play a role is large: we are talking about asymmetries of the order of a few parts in a thousand, so even really small effects could be involved.

And if there were new physics at play, capable of modifying some of these measurements but leave others unaffected ? Well, that's of course also possible a priori; however, note how this hypothesis is killed by applying Occam's Razor: if standard physics can explain the discrepancies, there's no need to invoke new entities. The court is still out on this one, so it is a bit premature to get excited yet !

Let us give now a look at two of these estimates. Of all the results, the LHCb measurement with the smallest error bar is the fourth from the top, which relies on a dataset of 1 inverse femtobarn of collisions. This determination is obtained by "pion tagging" the nature of the D0 mesons: D0 mesons accompanied by a soft positive pion are most likely to be true D0, while negative pions accompany most frequently anti-D0 particles; the reason is simple: these D0 particles come from the decay of a D*+ excited states (which are made up of a charm and anti-bottom quark), while anti-D0 mesons come from the decay of a D*- (made by a anti-charm and a bottom quark).

Note that LHCb disagrees with a different estimate the collaboration performs using "muon-tagged" D0 mesons, where the D0 meson is associated with a muon when it is produced in the decay of a B meson: since B mesons decay with a b->c quark transition, emitting a W+, or with a anti-b -> anti-c transition emitting a W-, then by identifying the charge of the muon produced by the decay of the virtual charged current boson, one can tell whether one is studying a D0 or a anti-D0. This technique produces a result which disagrees at almost three standard deviations with the "soft-pion" tagged result. The error bar in this case is larger because the method suffers from statistics - you need to collect semileptonic decays of B hadrons to perform the measurement.

The two LHCb measurements I have mentioned above are:

Delta A_CP (pion tag) = -0.34 +-0.15 +-0.10 %
Delta A_CP(mu tag) = 0.49 +-0.30 +- 0.14 %

and they are at 2.2 standard deviations apart.

And since I can't leave you without my own (steep) hypothesis...

It is quite likely that the above discrepancies will go away when more precise determinations will be produced. However, one could also speculate that some details of QCD are not well understood. While discussing briefly this issue during a meeting with some colleagues, some days ago, some of us joked that the observed difference of CP asymmetries measured by LHCb could maybe be due to Bose-Einstein correlations not accounted for by the analysis!

What are Bose-Einstein correlations ? They are an effect due to the bosonic nature of mesons such as pions or kaons. Identical bosons tend to occupy the same quantum state, so if I have a process whereby a soft positive pion is emitted together with a D0 meson which then decays into a pi+ pi- pair, the positive pion will be "attracted" toward the same quantum state of the soft pion emitted just before. In other words, there will be a slight, but definite, preference for the two identical bosons to have similar momenta (which means also to travel in similar spatial directions!).

Okay, you might be saying - but what does this have to do with the CP asymmetry ? Well, LHCb measures a "double ratio" asymmetry as follows:

ΔA = A_CP(K) - A_CP(π)

where A_CP(K) = [Γ(D0->KK) - Γ(anti-D0->KK)] / [Γ(D0->KK) + Γ(anti-D0->KK)], and similarly for the pions.

Here, you note that what is measured is the difference between two asymmetries. By doing that, LHCb does benefit from some cancellation of systematic uncertainties which might be favouring one charge configuration with respect to the other. However, you also notice that when you tag the D0 flavour with a soft pion, and then observe the D0 to decay to pions, this process must be contributed by Bose-Einstein correlations; while when you observe the D0 to decay into kaons, there is no Bose-Einstein correlation to speak of, because a positive kaon and a positive pion are two very different things!

Maybe I have confused you above. What I said is that since the flavour of the D0 is tagged by the sign of a pion traveling nearby, and produced by the decay D*->π + D0, then when the D0 decays to pions there may be correlations with the soft pion emitted by the D*, while when the D0 decays to kaons no such effect is present.

Note also that Bose-Einstein correlations are usually not present in Monte Carlo simulations available on the market....

The different physics playing in the two different asymmetries that LHCb subtracts one off the other may create a discrepancy with expectations if Bose-Einstein correlations are not properly taken into account. However, bear in mind that this is just a unproven assumption - it basically hypothesizes that there are some unaccounted for systematic uncertainties that affect more the Bose-Einstein-rich subset of D0 decays than the one free from those effects.

In any case, the physics of these CP violation measurements is really rich and complex. I will admit to knowing little of it, but I am glad I can offer at least this explanation here!

More information on the LHCb measurements is available in this recent paper.