**Principal Investigator:**

Sara Collins

**Affiliation:**

Institute for Theoretical Physics, Regensburg University (Germany)

**Local Project ID:**

hru29

**HPC Platform used:**

JUQUEEN of JSC

**Date published:**

*In this project we compute the decay constants of the D and Ds mesons using numerical simulation (Lattice QCD). The decay constants are required in order to extract from experiment the CKM matrix elements, the parameters of the Standard Model of particle physics associated with weak decays. High precision determinations of the CKM matrix elements from a variety of processes are sought in order to uncover hints of physics beyond the Standard Model. A significant systematic arising in Lattice QCD simulations is that due to the finite lattice spacing. We reduce this systematic by simulating at a very fine lattice spacing.*

**Introduction**

Our current understanding of the natural world is contained within the Standard Model of particle physics which describes three of the four fundamental forces --- the electromagnetic interaction between charged particles, the weak interaction which occurs during beta-decay and the strong interaction. The latter is responsible for binding quarks and gluons together to form protons and neutrons that exist within the nuclei of atoms. However, we know that the Standard Model is not a complete description as, for example, gravity (the fourth force) is not included and from astronomical evidence we know that there must be new particles which account for most of the mass of the universe. There are many on-going and planned experiments which are searching for hints of physics beyond the Standard Model. Some efforts are focused on detecting new particles, while others are looking for evidence of inconsistencies within the Standard Model itself. This project aims to contribute to the latter approach, which involves determining the parameters of the Standard Model to high precision using the many different processes which are detected in experiments such as the Large Hadron Collider at CERN.

At the high energies achieved in these experiments, particles (bound states of quarks and gluons) are produced which then subsequently decay. The parameters of the Standard Model associated with weak decay are known as the CKM matrix elements. Those CKM elements which are due to the weak decay of particles containing one of the heavy quarks (i.e. a charm or a bottom quark) are less well known. This project aims to enable the CKM elements associated with the charm quark to be better determined by computing the decay constants of D and Ds mesons. The decay constants describe how, for example, the charm and anti-down quark within a D meson annihilate during weak decay. These decay constants are needed in order to extract the CKM matrix elements from experimental data.

The decay constants are computed via numerical simulations of the D and Ds mesons. This involves simulating quantumchromodynamics (QCD), the theory which describes the interactions between the quarks and gluons. Monte Carlo simulations are performed on a finite four dimensional grid. Representative "configurations" of the quark and gluon interactions are generated. Correlation functions are computed on each configuration and the decay constants can be extracted once a statistical average is performed. In addition to the accompanying statistical error there are systematic uncertainties due, for example, to the finite grid or lattice spacing. This systematic can be significant for calculations of the properties of D and Ds mesons and the simulation has to be performed for different values of the lattice spacing so that a continuum limit can be performed. However, it becomes more difficult to generate a representative ensemble of configurations as the lattice spacing is reduced (and computationally more expensive).

In this project we generated ensembles at a very fine lattice spacing as part of the Coordinated Lattice Simulations (CLS) effort. This was possible through the use of open boundary conditions. The full set of ensembles available comprises spacings from *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. Due to the computational expense the simulations are performed for mostly unphysically large masses for the (degenerate) up and down (light) quarks and an extrapolation to the physical point is required. A unique feature of the CLS simulations is that ensembles have been generated which approach the physical point along two different trajectories.

Figure 1 displays the ensembles produced for which the average of the light and strange quarks is kept fixed. The masses of the light quarks for each of the ensembles is indicated by the value of the pion mass. The computer time granted for this project enabled us to extend the set of configurations labeled J500 and to generate a new ensemble with a lighter pion mass labeled J501. Not shown are another set of ensembles for which the strange quark mass is kept fixed to its physical value. The measurements necessary to compute the decay constants on the J500 and J501 ensembles and the rest of the CLS ensembles were carried out on the QPACE3 supercomputer of SFB/TR55.

As a by-product of the calculation we computed the mass spectrum of hadrons containing a single or two charm quarks. Preliminary results for the Ωcc and Ξcc doubly charmed baryons are given in Figure 3. The analysis is, again, on-going and we shift the results to agree with the mass of the newly discovered Ωcc at the physical point. The results indicate the mass of the yet-to-be-seen Ξcc is around 3710 MeV. Final results for the masses (and decay constants) will be obtained by performing a continuum and physical point extrapolation. Examples of such extrapolations are shown by the shaded regions in the figure.

**Results**

The analysis of the decay constants of the D and Ds mesons, fD and fDs, respectively, is on-going. Preliminary results for each ensemble are shown in Figure 2 as a function of the pion mass squared. The ratio of the decay constants is formed in order to cancel some of the systematics arising in the calculation. The fact that the results lie along two lines (corresponding to the two approaches to the physical point) indicates that the systematics are indeed not significant. The linear extrapolation to the physical point is shown only to guide the eye and demonstrate that the results are consistent with other lattice QCD calculations of this quantity as represented by the averages compiled by the Flavour Lattice Averaging Group (FLAG).

**Outlook**

The computer time granted was used to generate very fine lattices enabling a constrained continuum extrapolation of the results for the decay constants of the D and Ds mesons. This leads to a significant reduction in one of the main systematics arising in the calculation of these quantities and will enable us to produce very competitive final results. In addition, predictions will be made for a number of doubly charmed baryons, which have not been seen in experiment so far. The ensembles will also be used for a large number of other interesting projects.

**References**

Collins et al. (RQCD and ALPHA Collaborations), Leptonic decay constants for D-mesons from 3-flavour CLS ensembles, Proceedings of the 35th International Symposium on Lattice Field Theory (Lattice 2017): Granada, Spain, EPJ Web Conf. 175 (2018) 13019.

Collins et al. (RQCD and ALPHA Collaborations), Charmed pseudoscalar decay constants on three-flavour CLS ensembles with open boundaries, Proceedings of the 34th International Symposium on Lattice Field Theory (Lattice 2016): Southampton, UK, PoS LATTICE2016 (2017) 368.

S. Hofmann, PhD thesis (in preparation).

K. Eckert, PhD thesis (in preparation).

**Research Team:**

Gunnar Bali, Peter Bruns, Sara Collins (PI), Antonio Cox, Kevin Eckert, Benjamin Gläßle, Jochen Heitger, Stefan Hofmann, Stefano Piemonte, Andreas Schäfer, Jakob Simeth, Wolfgang Söldner, Philipp Wein.

**Scientific Contact**

Dr. Sara Collins

Institut für theoretische Physik

Universität Regensburg D-93040 Regensburg (Germany)

e-mail: sara.collins [@] ur.de

**Acknowledgements**

The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. for funding this project by providing computing time on the GCS Supercomputer JUQUEEN at Jülich Supercomputing Centre (JSC) as well as support from the Deutsche Forschungsgemeinschaft SFB/TR 55.

*JSC project ID: hru29*

*February 2019*