Distribution Amplitudes of the η and η' Mesons

Principal Investigator:
Andreas Schäfer

Institute for Theoretical Physics, Regensburg University (Germany)

Local Project ID:

HPC Platform used:

Date published:

In this project, properties of two mesons, i.e. particles made up of a quark and an anti-quark, namely the η and the η', are studied by numerically simulating the underlying theory, QCD, on a four-dimensional spacetime lattice. The focus of previous studies was on establishing the masses of these particles, which are intricately related to the so-called axial anomaly. In this project, for the first time, the internal structure of these mesons was simulated too. This can be characterized in lightcone kinematics, which is relevant for collider experiments, by so-called distribution amplitudes (DAs). The normalization of a DA is also known as a decay constant. Four such previously unknown constants have been determined with full assessment of all systematics.


Quantum chromodynamics (QCD) is the theory describing fundamental interactions between quarks and gluons. Photons bind electrically charged nuclei and electrons into atoms while the gluons of QCD bind strongly charged quarks (and anti-quarks) into hadrons, e.g., the protons and neutrons that can be found within a nucleus. Such bound states of three quarks are called baryons while states made up of a quark and an anti-quark are called mesons, e.g., the pion. While the photon itself is electrically neutral, the gluon carries strong (also called colour) charges. The resulting new interactions give rise to intricate collective phenomena such as the non-existence of isolated colour charges (confinement), i.e. in nature, as long as the temperature is lower than 155 MeV, i.e. 1.8×1012K quarks and gluons can only be found within baryons and mesons. Solving this theory in the low energy limit, where the interactions are strong, requires numerical simulation (Lattice QCD).

Another such feature is that even in the limit, in which all quarks are massless, most hadrons are massive. This is known as the spontaneous breaking of the chiral symmetry and this mechanism is responsible for over 95% of the non-dark mass of the universe (baryonic mass). The remainder is due to the Higgs mechanism. Another consequence of the spontaneous breaking of this symmetry is the existence of massless Nambu-Goldstone mesons: the pions, the kaons and the octet η meson, in analogy to massless magnons in condensed matter systems where spontaneous magnetization breaks the rotational symmetry of space. In QCD with three light quarks one may naively expect 32=9 light pseudo-Goldstone mesons, however, one only finds eight. The reason is the explicit breaking of the axial U(1) global symmetry by quantum corrections, known as the axial anomaly. The meson associated with the anomalous current is the singlet η meson. In nature, where the strange quark is heavier than the up and down quarks, the observed η and η' mesons are mixtures of singlet and octet ηs, where the heavier η' has dominantly singlet properties and the lighter η is dominated by the octet. This system is particularly interesting and complicated since the axial anomaly intimately links the meson properties to topological features of gluon field configurations. At the same time also the extent of the mixing itself is an object of scientific debate.

The computer time on HPC system JUQUEEN of JSC was mostly used to generate gauge ensembles, including one at the physical mass point, while the measurements and the post processing were carried out on the QPACE3 supercomputer of SFB/TR55 and on the JURECA-Booster partition of JSC. New numerical and analysis methods have allowed us to extract not only the masses of these particles for a number of different quark mass combinations and lattice spacings, but also the normalization of their distribution amplitudes, the decay constants.


A large number of Coordinated Lattice Simulations (CLS) gauge ensembles was analysed ranging from coarse lattices with a spacing a = 0.086 fm (β = 3.4) down to a = 0.050 fm (β = 3.7), where the continuum limit is reached as a quadratic function of the lattice spacing a. In Figure 1 we show the positions of these ensembles within the plane spanned by the light quark (ordinate) and the strange quark (abscissa) masses. The by far most expensive, left-most physical point ensemble was generated within this project.

In Figure 2 the resulting η and η' masses are shown, as a function of a particular combination of meson masses that is suggested by large N chiral perturbation theory. The vertical line marks the physical point and the two black stars the experimental values. The points to the left of the line correspond to the diagonal trajectory of Figure 1, along which the sum of the quark masses is constant, while the points to the right correspond to the horizontal trajectory of Figure 1, where the strange quark mass is approximately constant. Indeed, the results agree with experiment. For the first time the whole quark mass dependence of these states has been mapped out. The two points slightly to the left of the vertical line correspond to the (slightly too light) physical point ensemble.

The pioneering result of this simulation are the singlet and octet decay constants for both mesons,

where the state is destroyed by flavour singlet (0) and octet (8) local axial currents, respectively. The former constants depend on the renormalization scheme and scale, which has so far mostly been neglected in the literature. Formulating the η mixing questions in terms of decay constants allows for defining and determining mixing angles, which we did too. An interesting outcome of this is study is that the η' meson indeed has a strong gluonic component. In Figure 3 we show the decay constants, again as a function of a combination of meson masses, normalized with respect to the pion decay constant. For the left-most points the quark masses are equal and the octet decay constant of the η' vanishes while that of the η agrees with that of the pion, which is expected from flavour symmetry. Also in this case the singlet decay constant of the η must vanish and it does. We see that at the physical point (vertical line) the singlet decay constant of the η remains close to zero, however, the η' acquires a sizeable octet component. Again, the results from the different lattice spacings are plotted on top of each other.


Now that lattice QCD simulations have become clearly quantitative, it is very important to have full control over all systematic sources of uncertainty, in particular the continuum limit needs to be carried out to obtain definite results. This means that the time to publication can be long. At present the results are being finalised and phenomenological implications, e.g., regarding η' formfactors studied. For the octet η meson, we also studied the second Gegenbauer moment of the distribution amplitude. Moreover, the coupling to the gluonic topological charge is still being investigated. The ensembles generated, in particular the physical point ensemble, will be used for a multitude of other interesting projects.


J. Simeth et al. (RQCD Collaboration), η and η' masses and decay constants, Proceedings of the 35th International Symposium on Lattice Field Theory (Lattice 2017): Granada, Spain, EPJ Web Conf. 175 (2018) 05028.

J. Simeth, PhD thesis (in preparation).

Research Team:

Gunnar S. Bali, Vladimir M. Braun, Sara Collins, Benjamin Gläßle, Meinulf Göckeler, Fabian Hutzler,Piotr Korcyl, Rudolf Rödl, Andreas Schäfer (PI), Enno E. Scholz, Jakob Simeth, Wolfgang Söldner, André Sternbeck, Philipp Wein, Thomas Wurm.

Scientific Contacts:

Prof. Dr. Gunnar Bali, Prof. Dr. Andreas Schäfer
Institut für Theoretische Physik, Universitaet Regensburg
D-93040 Regensburg/Germany
e-mail: gunnar.bali [@] ur.de, andreas.schaefer [@] physik.uni-r.de

JSC project ID: hru28

February 2019

Tags: EPP Universität Regensburg JSC Large-Scale Project