Charm Sea Effects on Heavy Flavor Mesons Gauss Centre for Supercomputing e.V.


Charm Sea Effects on Heavy Flavor Mesons

Principal Investigator:
Francesco Knechtli

Bergische Universität Wuppertal

Local Project ID:

HPC Platform used:
SuperMUC and SuperMUC-NG of LRZ

Date published:


Quantum Chromodynamics (QCD) is the theory proposed in the early 1970’s to explain the properties of the strong interactions. The latter hold quarks together to form composite particles called hadrons. Quarks are elementary particles of spin 1/2 and represent, along with leptons, the smallest building blocks of matter, according to the Standard Model of Particle Physics. The strong interactions are mediated by spin 1 particles, the so-called gluons.

One of the main features of QCD is confinement. Quarks and gluons are never observed in isolation, but only within hadrons, that in turn are usually classified into mesons (composite particles made of a quark and an antiquark) and baryons (composite particles made of three quarks). The newly discovered states with a charmonium component could hint at the existence of exotic states such as tetra- or penta-quarks, but this is still under debate.

Quark confinement and hadron properties cannot be understood using perturbative methods and nowadays lattice QCD represents one of the most suitable tools to investigate QCD properties starting from first principles. In this approach QCD is discretized on a Euclidean four-dimensional space-time and the quantities of interest, like masses and decay constants of hadrons, can be computed numerically via Monte Carlo methods. This kind of study requires a huge computational effort, especially when considering a theory with dynamical quarks, and the use of supercomputers is necessary if we want to achieve results that can be compared with experiments.

QCD encompasses six flavors of quarks (up, down, strange, charm, bottom, top). However, since quark masses cover a large range of values that differ by several orders of magnitude, lattice QCD simulations often include the effects of only two, three or at most four flavors in the sea. In this project we estimate the effects including a sea charm quark. Its inclusion means a significant effort in tuning of simulations and we want to know at which level of precision it matters.

In order to do that we consider QCD with just a single species of quarks, the charm quarks, and we compare the results obtained with this simplified model to a theory without dynamical quarks (often called quenched QCD). This gives the possibility to use moderately large lattice volumes and perform reliable extrapolations to zero lattice spacing.

We study charmonium states, which are composite particles made of a charm quark and a charm antiquark. The charmonium system, frequently characterized as the “hydrogen atom” of meson spectroscopy owing to the fact that it is non-relativistic enough to be reasonably well described by certain potential models, is the perfect testing ground for a comparison of theory with experiment.

Over the last years, there has been a renewed interest in spectral calculations with charmonia because of the experimental discovery of many unexpected states, e.g. the so-called X, Y, Z states, which highlight the need for a more complete theoretical understanding. However, to accurately understand the charmonium spectrum, one must also investigate properties other than masses, such as decay rates. In the charmonium system, the lowest-lying states lie below the DD  threshold, resulting in relatively narrow widths due to the absence of OZI allowed strong decays. This means that radiative transitions, i.e. transitions from an initial state to a final state via the emission of a photon, can have significant experimentally accessible branching ratios. Therefore, a lattice calculation of the decay constants addressed here provides valuable theoretical insight for experiment at the fully non-perturbative level such that we consider them as natural and representative observables to quantify charm sea quark effects in hadron physics beyond the mass spectrum.

Results and methods

With Nf = 2 charm quarks, the calculation has been performed on a total of five ensembles that differ in the lattice spacing, while in the quenched (Nf = 0) case four ensembles were used [1]. The measurements of masses and decay constants over a wide range of lattice spacings, including very fine ones, enable a safe continuum extrapolation for both theories, which is shown in Fig. 1 for the decay constants of the ηand J/ψ mesons [2].

In the Nf = 2  theory we consider only ensembles with lattice spacings ≤ 0.05 fm, which is typically the smallest value which can be achieved in simulations including the light quarks. Our simplified setup allows to simulate lattice spacings smaller by a factor two. They are essential to control the continuum extrapolations which show the expected linear behavior in the lattice spacing squared (α2). The comparison of their values in the continuum limit allows to compute the impact of charm sea quarks on these observables.

For the decay constant fηc the relative difference between Nf = 2 and Nf = 0 is 0.48(34)%, for the decay constant fJ/ψ it is 0.12(77)%. These results mean that the decoupling of the charm quark at low energies works very well at an energy scale as large as about 500 MeV.

While the decay constant fηc has not been measured experimentally, the experimental value for the J/ψ meson is fJ/ψ ≅ 407(4) MeV (obtained from the partial decay width of J/ψ into an electron-positron pair [3]). The discrepancy with our value is due to several effects (light sea quarks, charm annihiliation, electromagnetism, number of charm quarks, meson mass).

We also study the effects of a dynamical charm quark on the hyperfine splitting of a Bc meson made of a bottom quark (anti-quark) and a charm anti-quark (quark) [2]. In particular we focus on its pseudoscalar (Bc) and vector channel (Bc*).

The properties of the charmed B meson system are of special interest in spectroscopy because they are the only heavy mesons consisting of heavy quarks with different flavors. Because they carry flavor, they cannot annihilate into gluons and so are more stable with widths less than 100 keV. At the LHC, with its higher luminosity, the spectroscopy and decay of Bc* mesons can now be experimentally measured such that on the theory side complementary precision studies of these meson states by means of lattice QCD become increasingly important.

Our simplified setup provides a first estimate of the charm sea effects on the ratio of the Bc* to Bc meson mass. For a “bottom” quark with a mass twice of the physical charm quark we find a relative difference for this mass ratio of 0.092(50)% between the Nf = 2 and Nf = 0 theories, which is comparable to what we found for the case of charmonium [1].

The computation of the meson masses and decay constants was performed with our measurement program [4], which invokes the openQCD [5] solvers for the inversions of the Dirac operator. The latter are state-of-the art Krylov space solvers that support even-odd preconditioning, an algebraic multi-grid method, mixed precision calculations and various further improvements. Optimized versions of the solvers for SuperMUC-NG (AVX instructions) exist.

Our package has been developed for a CPU environment and includes an MPI parallelization. For the inversion of the Dirac operator of the heavy quarks we use a modified version of the SAP GCR solver, which includes the distance preconditioning method.

Ongoing Research / Outlook

In this project, we have neglected the effects of charm annihilation in charmonium. They will be computed in a follow-up project, where also more channels (JPC quantum numbers) and mixing with glueballs will be investigated.

References and Links

[1] S. Calì, F. Knechtli, T. Korzec, Eur. Phys. J. C, 79 (7), 607, 2019.
[2] S. Calì, K. Eckert, J. Heitger, F. Knechtli, T. Korzec, Eur. Phys. J. C, 81 (8), 733, 2021.
[3] D. Hatton, C.T.H. Davies, B. Galloway, J. Koponen, G.P. Lepage, A.T. Lytle, Phys. Rev. D 102, 054511, 2020.

Research Team

Salvatore Calì2, Jochen Heitger3, Francesco Knechtli (PI)1,Tomasz Korzec1

1Bergische Universität Wuppertal
2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, USA
3Westfälische Wilhelms-Universität Münster

Scientific Contact

Prof. Dr. Francesco Knechtli
Bergische Universität Wuppertal
Fakultät für Mathematik und Naturwissenschaften
Theoretische Teilchenphysik
Gaußstraße 20, D-42119 Wuppertal (Germany)
e-mail: knechtli [at]

NOTE: This report was first published in the book "High Performance Computing in Science and Engineering – Garching/Munich 2020 (2021)" (ISBN 978-3-9816675-4-7)

Local project ID: pn56fo

September 2021

Tags: LRZ Bergische Universität Wuppertal EPP QCD