We performed a calculation of the \(\eta\)-\(\eta^\prime\) mixing in the framework of large-\(N_c\) chiral perturbation theory. A general expression for the \(\eta\)-\(\eta^\prime\) mixing at next-to-next-to-leading order (NNLO) was derived, including higher-derivative terms up to fourth order in the four momentum, kinetic and mass terms. In addition, the axial-vector decay constants of the \(\eta\)-\(\eta^\prime\) system were determined at NNLO. The numerical analysis of the results was performed successively at LO, NLO, and NNLO.

We calculated the anomalous decays \(\eta^{(\prime)}\to\gamma^{(\ast)}\gamma^{(\ast)}\) and \(\eta^{(\prime)}\to\pi^+\pi^-\gamma^{(\ast)}\) at the one-loop level up to next-to-next-to-leading order (NNLO) in large-\(N_c\) chiral perturbation theory. Both the decays to real photons and the decays involving virtual photons, providing access to the substructure of the mesons, have been discussed. The results were numerically evaluated successively at LO, NLO, and NNLO. The appearing low-energy constants were determined through fits to the available experimental data. In the case of \(\eta^{(\prime)}\to\gamma^{(\ast)}\gamma^{(\ast)}\) we investigated the decay widths to real photons, the widths of \(\eta^{(\prime)}\to \gamma l^+l^-\), where \(l=e, \mu\), and the single-virtual transition form factors. The considered observables of the decays \(\eta^{(\prime)}\to\pi^+\pi^-\gamma^{(\ast)}\) are the spectra of the decays involving a real photon, \(\eta^{(\prime)}\to\pi^+\pi^-\gamma\), as well as the spectra of \(\eta^{(\prime)}\to\pi^+\pi^-l^+l^-\), where \(l=e,\mu\), with respect to the invariant masses of the \(\pi^+\pi^-\) and \(l^+l^-\) systems.

*Photon-energy spectrum of \(\eta\to\pi^+\pi^-\gamma\) at LO (gray), NLO (blue)* *and NNLO (red). For the NLO and NNLO results the corresponding 1~\(\sigma\) error bands* *are shown. The experimental data are taken from P. Adlarson et al. [Wasa-at-COSY Collaboration],* *Phys.Lett.B 707, 243 (2012)*

**Dalitz plot analysis of the decay \(\mathbf{\eta^\prime \to\eta\pi^0\pi^0}\)**

We performed a Dalitz plot analysis of the decay \(\eta^\prime\to\eta\pi^0\pi^0\) using about 124,000 signal decays collected with the Crystal Ball/TAPS setup and the Endpoint Tagger (EPT) in 2014. The A2 dataset analysed in this paper is the largest sample of \(\eta^\prime\to\eta\pi^0\pi^0\) decays so far and results in the most precise to date determination of the parameters describing the matrix element for this decay.

The large dataset also allows, for the first time, the observation of a structure below the \(\pi^+\pi^-\) threshold, which is in good agreement with the cusp that was predicted based on the \(\pi\pi\)scattering length combination, \(a_0-a_2\), extracted from \(K\to3\pi\) decays.**Measurement of the Dalitz plot distribution of \(\mathbf{\eta^\prime \to\pi^0\pi^0\pi^0}\)**

[PhD project of Martin Wolfes.] Using the EPT data set of A2, we attempt to reconstruct the decay \(\eta^\prime\to\pi^0\pi^0\pi^0\), with the goal to perform a Dalitz plot analysis. Initial expectations based on the number of produced \(\eta^\prime\), the branching fraction, and a signal Monte Carlo study indicated that a signal yield comparable to the BESIII sample of 1900 reconstructed \(\eta^\prime\to\pi^0\pi^0\pi^0\) would be within reach.

However, performing the analysis we had to realise that a clean extraction of the signal decay is not possible due to the presence of a large, irreducible, background from \(3\pi^0\) production which does not proceed via a primary \(\eta^\prime\). This analysis therefore turns out to be not feasible within the scope of a PhD thesis.

Instead, a measurement of the cross section \(\gamma p \to \Sigma^+ K^0\) with \(\Sigma^+\to p\pi^0\) and \(K^0_{S}\to π^0π^0 \)is being performed. The analysis sees about 52000 reconstructed signal decays; a measurement of the differential cross section in several bins of \(E_\gamma\) is therefore possible. Results are expected for 2018.**Measurement of the branching fraction \( \mathbf{\eta^\prime\to\omega\gamma}\)**

[This measurement is the PhD project of Andreas Neiser.] The plan is to measure the branching fraction of \(\eta^\prime\to\omega\gamma\to\pi^0\gamma\) relative to the abundant reference channel \(\eta^\prime\to\gamma\gamma\). Signal yields of about 50,000 reference and 1300 signal decays are found in the full endpoint tagger data set. However, there is a systematic difference between data and simulation in the signal efficiency, so that the preliminary result for the branching fraction comes out significantly smaller than the PDG average. A publication using an alternative analysis chain is under preparation.**Measurement of the branching fraction\( \mathbf{\omega\to\eta\gamma}\)**

[PhD project of Oliver Steffen.] The decay \(\omega\to\eta\gamma\) was planned to be measured using the decay \(\omega\to\pi^0\gamma\) as a reference, using the approximately 60 × \(10^6\omega\) mesons produced in the EPT beam times. The reference channel is cleanly reconstructed with the expected rates, but identification of the signal decay is seriously hampered by the presence of a large background from the processes \(\gamma p\to\eta\pi^0 p\), in which one photon from the \(\pi^0\) is not detected. Additional backgrounds are also present, and the distribution of \(m(\eta\gamma)\) is not well described by simulation, making a determination of the signal yield very unreliable.

As an alternative result, the differential cross section for \(\omega\) photoproduction is measured in bins of the incident photon energy; this agrees within uncertainties with previous results.**Study of charmonium decays to three pions**We developed and applied a data selection for the channels \(J/\psi\to\pi^+\pi^-\pi^0\) and \(\psi^\prime\to\pi^+\pi^-\pi^0\) on the full BESIII data sets. Fits to these data using Veneziano amplitudes in close collaboration with A. Szczepaniak (Indiana University) were started and show a very good description of the \(J/\psi\) data; more work is needed for a thorough understanding of the \(\psi^\prime\) data.

**Dalitz plot analysis of \( \mathbf{D^0 \to K^0_S K^+ K^-}\)**

We finalised the Dalitz plot analysis and measurement of the branching fraction of the decay \(D^0 \to K^0_S K^+ K^-\) using the full 2.92 fb\(^{-1}\) data sample taken at \(\sqrt{s}=3.773\)~GeV. The branching fraction result is the most precise measurement to date. The uncertainty is dominated by systematic uncertainties and improves on the PDG average by a factor of 1.7. A journal publication is under internal review within BESIII; submission is expected in the first quarter of 2018.**Exotic charmonium states**Since the start of this funding period we performed the search for the two isospin singlet states \(\eta_c\eta\) in the reaction channel \(e^+e^-\to\eta_c\eta\pi\pi\) and \(J/\psi\eta\) in the reaction channel \(e^+e^-\to J/\psi~\eta\eta\) with data of the BESIII experiment. The first step was the optimization of the extraction of events with the correct signature in the detector.

To extract the \(\eta_c\eta\) events we had to sum over 16 different \(\eta_c\) decay channels in order to get a reasonable amount of \(\eta_c\)candidates. Concerning the background contributions we were not able to describe the remaining background distribution with the recent models. Therefore we decided to describe the remaining contribution in the data directly by fitting a combined model of a signal and a background function simultaneously to each of the 16 \(\eta_c\) decay channels. After the study of the systematic uncertainties the upper limit for the production of this reaction channel was calculated for the three energies at 4.23, 4.26 and 4.36 GeV. For this analysis the review process at BESIII already started two years ago and the draft of this publication has already been written. We decided to add two further data points at 4.42 and 4.60 GeV. The update of the memo and paper draft is expected within the next month.

For the \(J/\psi\eta\) case extensive studies were performed in order to subtract the background contributions. Measured cross sections do not exist for all contributions, therefore a rough analysis was done for the missing channels to determine at least the upper limit of those channels. Only very few events survive for the 5 energy points 4.23, 4.26, 4.36, 4.42 and 4.60 GeV. Therefore we calculated the upper limits for each energy and took into account all systematic uncertainties. The memo is almost ready and the review of this analysis is going to start soon.

In parallel we are working on the analyses \(e^+e^-\to(D\bar D^\ast)^+\pi^-\) and \(e^+e^-\to(D\bar D^\ast)^0\pi^0\) at 4.42 GeV. After finishing the event selection the two independent analyses were cross-checked in detail and differences understood. The reaction channel \(e^+e-\to(D\bar D^\ast)^+\pi^-\) turned out to still have a large background contribution after the selection. This contribution has either to be understood or removed before performing a partial wave analysis. In order to get a handle on how to describe the background contribution, the reaction channel of \(e^+e^-\to D\bar D^+\pi^-\) is being investigated at 4.18 GeV center of mass energy. The reaction channel \(e^+e^-\to(D\bar D^\ast)^0\pi^0\) has turned out to be almost background free. Therefore we are setting up the first fit by using the canonical formalism.