MatterAntimatter Asymmetry and the Search for fundamental Laws of Physics with JUGENE
The MatterAntimatter Asymmetry
Our visible universe today consists of matter with only some tiny trace amounts of antimatter, e.g. in cosmic rays. Although this statement seems to be rather trivial given our everyday experience, it has some farreaching consequences for the origin of our universe. Unless we are willing to accept that the universe started out in an asymmetric state, with more matter than antimatter, some mechanism has to exist that gives preference to the production of matter over antimatter in the course of evolution of our universe.
In 1967 Sakharov has listed the necessary ingredients for such a mechanism. Obviously one ingredient is a process that preferentially transforms matter into antimatter, and indeed such processes have been found in nature. The most prominent of these processes (see Fig. 1) is the socalled “kaon mixing”, where a neutral particle (the kaon or Kͦ) is transformed into its antiparticle (the antikaon or K̅ ͦ)
The reverse process, the transformation of an antikaon K̅ ͦ into a kaon K ͦ, does also exist, but it progresses at a slightly different rate, thus leaving a slight imbalance between matter and antimatter.
The obvious question now is whether this slight imbalance is able to explain the observed matter dominance of the universe. This question, however, is not so straightforward to answer because the particles involved, the kaons and antikaons, are not simple elementary particles. They are what we call mesons, bound states of one quark and one antiquark held together by the strong nuclear force. Because the effects of the strong nuclear force are very difficult to account for precisely, it is not straightforward to deduce from the properties of the (anti)kaons the fundamental properties of their constituent (anti)quarks that are ultimately relevant for explaining the observed materantimatter asymmetry.

Figure 1: An illustration of neutral kaon mixing. Both the kaon and its antiparticle contain one quark and one antiquark bound together by the strong nuclear force and they can transform into one another (grey).

Broadly speaking, there are two possible scenarios: either the experimentally observed neutral kaon mixing can be explained by the known properties of quarks within the current standard model of particle physics, the socalled CKM (CabibboKobayashiMaskawa) theory, or not. In the first case one would get a nontrivial confirmation of the standard model and a hint that the matterantimatter asymmetry has its origin at a much higher energy scale, because the standard model alone is known not to provide enough of it.
In the second case one would obtain clear evidence for the existence of “new physics”  new particles, forces, or mechanisms (e.g. extra dimensions) that go beyond the established standard model. In this scenario the “new physics” would provide some part of the experimentally observed kaon mixing and a potential answer for the materantimatter asymmetry in the universe.
QCD
In order to clearly distinguish between these two scenarios, the properties of kaons and their constituent quarks have to be related numerically in a precise way. This is not a straightforward task because the quarks that form a kaon are bound by the strong nuclear force that we can describe with a theory called quantum chromodynamics (QCD). QCD, however, does not allow for the existence of single, free quarks; quarks have to occur in bound states only. This essential feature of QCD, called quark confinement, is caused by the specific “color” interaction between quarks. In addition to the electrical charge, quarks do carry a “color” charge. This “color” has nothing to do with real color – it was labeled “color” because of the peculiar mixing behavior of the fundamental charges that is reminiscent of the mixing between the primary colors red, green, and blue.

Figure 2: Field lines of a pair of opposite electric charges.

While an electrical charge can only be positive or negative and have a different magnitude, color charge has three components that we may label red, green, and blue. This has surprising and far reaching consequences: let us first imagine placing two electrical
charges of the same magnitude but opposite sign into otherwise empty space (see Fig. 2). The electrical field, or its field quanta, the photons, can spread out freely into space. This is essentially due to the fact that the photons themselves are electrically neutral.
We now turn to the color field between two opposite color charges or quarks. The field quanta of the color interaction, the gluons, must relate the different color components and therefore cannot all be color neutral. As a consequence, the field lines themselves are color charged and do interact with each other. This results in a squeezing of the color flux tube between the two opposite charges (see Fig. 3a). Trying to pull the charges apart will result in the flux tube containing more and more energy. At a certain point there is enough energy in the flux tube to pairproduce a quarkantiquark pair out of the vacuum (see Fig. 3b). Instead of having split up the quarkantiquark bound state, or meson, into its constituents, we end up with two mesons eventually.


Figure 3: (a) The color flux lines between a pair of opposite color charges. (b) When separating the two charges, the flux tube gets elongated and ultimately breaks up by producing a quarkantiquark pair out of the vacuum.

Lattice QCD
It is now evident that QCD (in the low energy regime that we are interested in here) is very hard to treat numerically: the fundamental degrees of freedom
of the theory (quarks, gluons) are very different from the states that occur in nature (bound states like mesons or protons and neutrons). Usual perturbative techniques that deal with small corrections to the free fundamental degrees of freedom are evidently insufficient.
A more direct way of approaching this problem is lattice QCD. One discretizes the QCD equations on a spacetime lattice and then proceeds to perform
the quantum mechanical path integration stochastically. This approach is of course extremely demanding numerically. The integration has to be performed
on a space with a dimension given by the number of lattice points times the internal degrees of freedom. Since the lattice has to be both fine enough to reproduce all essential details and big enough to fit the entire system being investigated, the typical dimension is of order 108 1010.
In addition to the large dimensionality D of the stochastic integral that needs to be performed, it also needs to be performed over quarks, which are fermions. Fermionic degrees of freedom, however, are anticommuting numbers and their inclusion ultimately leads to the need of computing determinants of square matrices of dimension D2. Though these matrices are typically sparse, it is nonetheless a challenging task even for petaflopclass installations like JUGENE to perform this stochastic integration.
Extracting Physics
In order to extract final physics results from a stochastic integration of the QCD path integral, a number of further steps are necessary. The first one of these is reaching the “physical point”. This step is rather unique to lattice QCD, as in most other computer simulations, the fundamental parameters
of the theory are known beforehand, whereas the interesting observables are connected to the nontrivial behavior of large systems.
In lattice QCD we do not know beforehand the fundamental parameters of the theory – the masses of the quarks and the coupling constant. Free quarks
do not occur in nature and therefore their masses and the strength of their interactions cannot be measured directly. We therefore have to first solve the inverse problem and to determine which parameters of the theory allow us to reproduce the experimentally observed phenomena of the strong nuclear force.
In practice, we looked at the masses of three different bound states of quarks (two mesons, the kaon and the pion, and one heavier particle or baryon, that is a bound state of three quarks) and defined the physical point as the point where the ratio between the three masses assume their experimentally observed values. In our lattice simulations we had three parameters to tune: quark masses and the coupling constant. From the three bound state masses we use to identify the physical point, we obtain two independent mass ratios that constrain the space of three parameters to a one–parameter line of “physical points”.
We can further differentiate the physical points along the line by comparing any one mass, e.g. that of the pion, from experiment to the dimensionless mass of the same bound state as measured on the lattice. This procedure is called “scale setting” as it determines the fundamental length scale of the lattice, which is also not known beforehand. The second step which is then necessary to obtain physical results is to extrapolate along the line of physical points to the point where the lattice scale vanishes. This step is called “continuum extrapolation” as it removes the discretization mesh and therefore leads to continuum results.

Figure 4: Comparison of some experimentally observed masses of quark bound states to our QCD prediction.

Knowing how to obtain continuum results, we still have to deal with the fact that our lattice represents only a finite volume in space and a finite extent in time. The third step towards obtaining physical predictions is therefore the infinite volume extrapolation. Fortunately most of the finite volume corrections are exponential in the lattice extent so that a large lattice extent renders them negligibly small.
Controlled Errors
Once we have carried out this procedure of obtaining physical predictions, we might wonder what would happen if we used three other masses initially to construct our line of physical points. Would we have captured physics exactly with our discretized lattice, we would obtain the same line. In reality there are errors due to the truncation however, so one can trace out different lines of “physical points” depending on the input quantities used. If the theory and our methods of solving it are correct, however, all these lines must extrapolate
to the same continuum value. We can therefore obtain a strong crosscheck of QCD itself and our ability to solve it by checking that not only our input masses but also the masses of other bound states are correctly reproduced in the continuum limit at the physical point. As shown in Fig. 4 we have successfully performed such a crosscheck in 2008 using mainly the computing resources at the Jülich Supercomputing Centre [1].

Figure 5: Value of the parameter BK as measured in our lattice calculations. (a) One interpolation of the lattice values to the physical point and (b) one extrapolation of the lattice values to the continuum. 
This result exemplifies what is the most important aspect of our calculations: a full control over all sources of systematic error. When making a prediction that has the potential to disprove our standard physical theories about our world we need to make sure that we do not underestimate the inaccuracy of our prediction.
In order to achieve this goal, our first aim was trying to avoid any extrapolations. The supercomputing resources provided to us mainly by the Jülich Supercomputing Centre in combination with algorithmic and theoretical advances [2,3,4] in fact allowed us for the first time in this kind of calculations to entirely eliminate the extrapolation to the physical point and replace it by an interpolation (see Fig. 5a). The unavoidable continuum extrapolation
is very flat (see Fig. 5b) and was performed with different extrapolation functions. The spread between different results allowed us to reliably estimate the corresponding systematic uncertainty.

Figure 6: Our final value of the parameter BK compared to previous determinations (black) and a very conservative standard model prediction (blue). For more details see [4].

Consistency of the Standard Model
As a final result of our calculation, we obtain the socalled bag parameter
of the kaon BK. This number parameterizes the amount of matterantimatter asymmetry in neutral kaon mixing and previous calculations were placing it in the region of 0.720.75 with a typical uncertainty of about 5%. The standard model is favoring a larger number, albeit with a relatively large error, and some authors were seeing this as a possible indication of new physics [5].
Our new result (see Fig. 6) is 0.773(12) and has a precision of 1.5%, which is entirely consistent with the standard model expectations. It is also far more precise than the standard model prediction, which implies that an improved standard model prediction for BK would increase the accuracy of this test to the percent level.
In conclusion, our calculation provided strong evidence that the CKM mechanism of matterantimatter asymmetry in the standard model does correctly explain the observed neutral kaon mixing. The origin of the matterantimatter asymmetry in our universe is still a mystery of fundamental physics that needs to be explored in the future.
Acknowledgment
I would like to thank all the members of the BudapestMarseilleWuppertal collaboration with whom this work was performed. This project has been supported by DFG grant SFBTR 55, the necessary computer resources have been provided by NIC/Jülich Supercomputing Centre, GENCI, the University of Wuppertal, and the CPT Marseille.
References
[1] Durr, S. et al.
(BudapestMarseilleWuppertal collaboration), Abinitio Determination of Light Hadron Masses, Science 322 (2008) 1224
[2] Durr, S. et al.
(BudapestMarseilleWuppertal collaboration), Lattice QCD at the physical point: light quark masses, Phys. Lett. B701 (2011) 265
[3] Durr, S. et al.
(BudapestMarseilleWuppertal collaboration), Lattice QCD at the physical point: Simulation and analysis details, JHEP 1108 (2011) 148
[4] Durr, S. et al.
(BudapestMarseilleWuppertal collaboration), Precision computation of the kaon bag parameter, Phys. Lett. B705 (2011) 477
[5] Brod, J. and Gorbahn, M.
NexttoNexttoLeadingOrder CharmQuark Contribution to the CP Violation Parameter epsilon_K and Delta M_K, Phys. Rev. Lett. 108 (2012) 121801
• Christian Hoelbling Bergische Universität Wuppertal
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