We performed a calculation of the \(\eta\)-\(\eta'\) mixing in the framework of large-\(N_c\) chiral perturbation theory. A general expression for the \(\eta\)-\(\eta'\) 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 \(\eta\)-\(\eta'\) system were determined at NNLO. The numerical analysis of the results was performed successively at LO, NLO, and NNLO.

**Quantum corrections to the chiral anomaly in large-\(N_c\) chiral perturbation theory**We calculated the anomalous decays \(\eta^{(')}\to\gamma^{(\ast)}\gamma^{(\ast)}\)and \(\eta^{(')}\to\pi^+\pi^-\gamma^{(\ast)}\) 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 \(\eta^{(')}\to\gamma^{(\ast)}\gamma^{(\ast)}\) we investigated the decay widths to real photons, the widths of \(\eta^{(')}\to \gamma l^+l^-\), where \(l=e, \mu\), and the single-virtual transition form factors. The considered observables of the decays \(\eta^{(')}\to\pi^+\pi^-\gamma^{(\ast)}\) are the spectra of the decays involving a real photon, \(\eta\prime\to\pi^+\pi^-\gamma\), as well as the spectra of \(\eta^{(')}\to\pi^+\pi^-l^+l^-\), where \(l=e,\mu\), with respect to the invariant masses of the \(\pi^+\pi^-\) and \(l^+l^-\) systems.

Left: Photon-energy spectrum of \(\eta\to\pi^+\pi^-\gamma\) at LO (gray) and NLO (blue). The blue band is the \(1\sigma\) error band. The experimental data are taken from P. Adlarson {\it et al.} [WASA-at-COSY Collaboration],Phys. Lett. B {\bf 707}, 243 (2012). Right: Invariant-mass spectrum of the \(\pi^+\pi^-\) system in \(\eta'\to\pi^+\pi^-\gamma\) at LO (gray) and NLO (blue) fitted up to \(0.59\) GeV (dash-dotted),\(0.64\) GeV (dashed), \(0.72\) GeV (solid). The experimental data are taken from A. Abele {\it et al.} [Crystal Barrel Collaboration], Phys.\ Lett.\ B {\bf 402}, 195 (1997).}

**Quark-mass and \(1/N_c\) corrections to the \(VP\gamma\) interaction**We analyzed quark-mass and \(1/N_c\) corrections to the \(VP\gamma\) interaction within a chiral effective Lagrangian approach. In the chiral limit and at leading order in \(1/N_c\), all processes are driven by a single Lagrangian,

where \(F\) denotes the pion-decay constant in the three-flavor chiral limit, \(F_{\mu\nu}\) is the electromagnetic field-strength tensor, \(Q\) the quark-charge matrix, and \(V\) and \(\Phi\) refer to the vector-meson and pseudoscalar-meson nonets, respectively. The NLO \(1/N_c\) corrections provide three new structures, the quark-mass corrections two additional structures. We included singlet-octet mixing in both multiplets and analyzed the experimental data of the 12 physical processes. Below, the results for the decay widths of the leading-order (cyan) and full (blue) analyses are shown in comparison with data (black).

As to be expected, the inclusion of \(1/N_c\) corrections and SU(3)-symmetry-breaking quark-mass corrections leads to an improved description.**Maintenance and improvements of the A2 detector setup**Personnel supported by project M2 have played an instrumental role in the maintenance and further development of the A2 detector facility at MAMI. These detector-related projects include improvements to the overall control system, a speed-up of the data acquisition scheme, a new and improved high-voltage distribution to the photomultipliers of the Crystal Ball detector (CB), and the replacement of about 30 broken PMTs in CB. Together with the upgrade of the focal plane detector of the main tagger performed within projects S2 and S3, the work outlined above greatly improves the reliability and efficiency of data taking and operation of the A2 setup.

**\(\eta\) and \(\eta\prime\) photoproduction at MAMI**We performed high-statistics measurements of the \(\gamma p \to \eta p\) and \(\gamma p \to \eta\prime p\) differential cross sections from reaction thresholds up to \(E_\gamma = 1577\) MeV (\(W = 1960\) MeV).Within its kinematic range, our measurement is the most precise determination of the differential \(\gamma p \to \eta\prime p\) cross section. It clearly demonstrates that the detector setup is suitable to measure \(\eta\prime\) decays and validates the used analysis chain that will be used for other \(\eta\prime\) related analyses.

**Dalitz plot analyses of \(\eta\prime\) decays to \(\eta \pi^0 \pi^0\) and \(\pi^0\pi^0\pi^0\)**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 shows, for the first time, clear evidence of a cusp at the \(\pi^+\pi^-\) threshold. The cusp is in good agreement with predictions based on the combination of isospin \(I=0\) and \(I=2 \pi\pi\) scattering lengths, \(a_0 - a_2\), extracted from \(K \to 3\pi\) decays.BESIII published an analysis of the same decay almost simultaneously. The BESIII analysis is based on the analysis of the reaction \(J/\psi \to \gamma \eta\prime\) using \(1.3\times 10^9\) \(J/\psi\) events, yielding roughly half the \(\eta\prime\) statistics compared to the A2 dataset. In contrast to the analysis of A2 data, the statistics of the then-current BESIII data did not require the cusp to achieve a satisfactory fit.**Measurement of the branching fraction \(\eta\prime \to \omega \gamma\)**The goal of this analysis is to measure the branching fraction of \(\eta\prime \to \omega \gamma \to \pi^{0} \gamma \gamma\) relative to the well-measured reference channel \(\eta\prime \to \gamma \gamma\). This is one of the decays of the type \(VP\gamma\) studied in the theoretical part of M2, and its branching fraction is currently known only with 5% accuracy. For the analysis of this decay channel, we used two independent analysis chains. One analysis is performed using the toolchain that was also used for the measurement of the \(\gamma p \to \eta\prime p\) differential cross section. The second analysis, used the new analysis framework, ANT, that is being developed within the group supported by project M2. A publication of the branching fraction \(\eta\prime \to \omega \gamma\) is under preparation.**Measurement of the \(\omega \to \pi^0 e^{+}e^{-}\) Dalitz decay**The Dalitz decay \(\omega \to \pi^{0} e^{+}e^{-}\) gives access to the time-like singly-virtual \(\omega\) transition form factor (TFF). This does not only play an important contribution to hadronic light-by-light scattering (see project M1), but plays an important role in understanding the properties of the \(\omega\) and will be studied by the theoretical side of project M2 in the context of the \(VP\gamma^*\) studies.Using data collected with the A2 setup at electron beam energies of \(1508\)MeV and \(1557\)MeV, a total of \(1.1 \times 10^3\) \(\omega \to \pi^{0} e^{+}e^{-}\) events could be reconstructed and the timelike TFF is determined from \(e^+e^-\) threshold up to \(m_{ee} = 0.6\) GeV. Parametrising \omegaTFF with a simple pole model, as suggested by VMD, results in a slope parameter of \(\Lambda^{-2}_{\omega\pi^{0}} = (1.99 \pm 0.21_\text{tot}) \text{GeV}^{-2}\). This is compatible with, but somewhat lower than, the NA60 result for the slope parameter, \(\Lambda_{\omega\pi^{0}}^{-2} = (2.223 \pm 0.026_\text{stat} \pm 0.037 _\text{syst})\;\text{GeV}^{-2}\), which is much larger than the expectation from a simple VMD ansatz, and in disagreement with predictions from dispersion theory. The tension between the NA60 data on the one hand and A2 data and dispersion theory calculations on the other hand call for an improved measurement of \omegaTFF.

**Study of the \(\eta^\prime\rightarrow\gamma\pi^+\pi^-\) decay dynamics**The decay \(\eta\prime \to \gamma \pi^{+} \pi^{-}\) is dominated by \(\eta\prime \to \rho^0 \gamma\) that is relevant for the \(VP\gamma^{(\ast)}\) system. Other possible contributions are from \(\omega \gamma\) with \(\omega \to \pi^{+} \pi^{-}\) and the box anomaly or excited \(\rho\) states, but neither of those has been firmly established so far.Using a sample of \(9.7\times 10^5\)\(J/\psi \to \gamma \eta\prime, \eta\prime \to \gamma \pi^{+}\pi^{-}\) decays, for the first time the contribution of \(\omega\) to the decay \(\eta\prime \to \gamma \pi^{+} \pi^{-}\) could be established. A model-dependent analysis of the \(\pi^{+}\pi^{-}\) mass spectrum revealed that, in addition to contributions from \(\rho^0\) and \(\omega\), another amplitude either for the box anomaly or for \(\rho^\prime\) is needed to obtain a satisfactory fit.With this model-dependent analysis, the branching fraction for \(\eta\prime \to \rho^0\gamma\) is measured to be \(B(\eta\prime \to \rho^0\gamma) = (33.34 \pm 0.06 \pm 1.60)\%\), somewhat higher than the PDG average that is dominated by a CLEO result from 2009. This analysis provides additional constraints for the theoretical study of \(VP\gamma^{(\ast)}\) in the theoretical part of M2.

**Study of charmonium decays to three pseudoscalars**The decays of charmonia to three pions allow to study pion dynamics in great detail and make inroads at resolving the \(\rho\pi\)-puzzle. Advanced models involving rescattering and including complete Regge-trajectories (Veneziano amplitudes) are available for these studies. We have extended the existing analysis to larger data samples taken in 2012 and reproduced the result of the branching fraction measurements. Using this data set, a preliminary partial wave analysis using the Veneziano amplitudes was performed with promising results, in particular in the case of the \(J/\psi\) decay. We also started to develop the toolkit necessary for a simultaneous analysis of the \(J/\psi\) and \psip data. In the meantime, BESIII collected an even larger data set on the \(J/\psi\) resonance, which we intend to analyze in the third funding period.

**Hadronic \(D\) decays**We finalised the Dalitz plot analysis and measurement of the branching fraction of the decay \(D^{0}K^{0}_{S}K^{+}K^{-}\) using the full \(2.93 fb^{-1}\) data sample taken at \(\sqrt{s}=3.773\) GeV. The Dalitz plot analysis was performed with a double-tag sample to allow the distinction between signal \(D^{0}\) and \(Dzb\) decays. The selection of contributing resonances was performed using the Least Absolute Shrinkage and Selection Operator (LASSO) method that balances model complexity and fit quality with an objective function. This analysis is one of the first within BESIII to use such a method from machine learning.Using the amplitude model measured above, the branching fraction was measured with an untagged method,

*i.e.*reconstructing only the signal decay, in order increase the available statistics and to avoid complications due to quantum correlations between signal and tag side. The measured branching fraction is \(\mathcal{B}(D^{0}\to K^{0}_{S} K^{+}K^{-}) = (4.51\pm 0.05 \pm 0.16) \times 10^{-3}\), dominated by systematic uncertainties and with a significantly improved total uncertainty compared to the current world average.This analysis was performed within the newly-developed PWA framework ComPWA. In addition, many analysis techniques not yet established within BESIII were explored and applied. These include the use of the LASSO method to select the contributing resonances, the \(Q\)-factor method to identify and remove background, and a goodness-of-fit test suitable for unbinned maximum-likelihood tests.The journal draft has entered the final stage of internal review and will be submitted to Phys. Rev. in fall 2019.**Search for charmonium isospin singlet states**Two triplets of states, the \(Z_c(3900)\) and the \(Z_c(4020)\) isospin triplet, were discovered in the last years which decay to \(J/\psi \pi\) or \(h_c \pi\), respectively. Since these triplets were found it is also possible that the corresponding isospin singlet partners exist. The existence or non-existence of these states gives complementary information about the nature of these states as they narrow the possibilities of theoretical interpretations.Since the beginning of the second funding period we performed a search for isospin singlet states decaying to \(\eta_c \eta\) in the reaction channel \(e^+e^{-} \rightarrow \eta_c\eta \pi\pi\) and \(J/\psi \eta\) in the reaction channel \(e^+e^{-} \rightarrow J/\psi \eta\eta\) with data of the BESIII experiment. The first step was the optimization of the selection of events with the correct signature in the detector. After a look into the data it was clear that only a few events can be found for the complete reaction channel, therefore we determine the upper limits at the 90% confidence level for both reaction channels in addition to the observed cross section with statistical and systematic error estimation.

**Interpretation of Exotic Charmonium states**The charged and neutral \(Z_c\) states were found in the decay of the \(Y(4260)\) at BESIII. For these established charmonium-like states, detailed studies of the production and decay mechanisms will help to elucidate the interrelationships between the \(Y\) and \(Z\) states and shed light on their internal structure. The goal of this sub-project was to prepare for a global analysis of the final states \(J/\psi \pi\pi\) and \(D\bar{D}^*\pi\) whilst making the best possible use of the available BESIII data. A simultaneous fit to all the BESIII data above the open charm threshold will then show at which center of mass energies there are \(Y\) states decaying to \(Z\) and whether additional states are present in the decay.The first step of the preparation for the simultaneous fit, the optimization of the selection of the reaction channels \(e^+e^{-} \rightarrow (D \bar{D}^*)^+ \pi^{-}\) and \(e^+e^{-} \rightarrow (D \bar{D}^*)^0 \pi^{0}\) is finished. As an example, the distributions of the invariant masses of the charged mode are shown in the following figure. In these plots, a large change in the distribution within the kinematic limits in dependence on the center of mass energy can be observed.

Distribution of the invariant masses of the \(\pi^- D^*\) subsystem versus the \(\pi^- D\) subsystem for the six different center of mass energies as indicated in the plots.

**Analysis of \(e^+e^{-} \rightarrow (D \bar{D}^*)^+ \pi^{-}\) and \(e^+e^{-} \rightarrow D^0 D^+ \pi^{-}\):**A first attempt to perform a partial wave analysis was made by using the canonical formalism. Overall, the fit describes the data already quite well. Nevertheless, there are still deviations between data and the fit visible, which are related to the remaining background contributions. Detailed background studies showed that the background events are mainly from channels that also contain two \(D\) mesons in the final state. Unfortunately, neither the cross section nor the angular distribution of these channels were measured before in detail. Therefore, we are not able to properly model these channels and include them in the fit. A large fraction of time was spent in order to get these background channels under control.Since it was not successful we changed the reaction channel to the reaction channel \(e^+e^{-} \rightarrow D^0 D^+ \pi^{-}\). This is a simpler system with only three particles in the final state, in which only very few intermediate states are expected. This is ideal to identify the background contributions and to learn how to handle this contribution in preparation for the partial wave analysis. The reaction channel is interesting by itself and has not yet been measured. Also in this channel, a large fraction of background events is present with two \(D\) mesons in the final state. Several techniques were investigated to suppress or subtract the background contribution. The so-called Q-factor method was found to perform best to statistically subtract all background events. After this reweighing the data set, a first partial wave analysis was performed which indicates only small deviations between data and the fit result.

**\(\bar{D}^*\):**For the selection of the reaction channel \(e^+e^{-} \rightarrow (D \bar{D}^*)^0 \pi^{0}\) four final states were taken into account: \(D^0 \bar{D}^{*0} \pi^{0}\) and \(D^- D^{*+} \pi^{0}\) and their charge conjugate states. There was a large amount of cross feed found between these final states. To keep as many final states as possible the solution was to investigate the four-body final state \(D^0 \bar{D}^{0} \pi^0 \pi^{0}\). A comparison between data and Monte Carlo simulations indicates the two contributions \(\pi^0 D^0 \bar{D}^{*0}\) and c.c. and \(D^{*0} \bar{D}^{*0}\) are clearly dominating the distribution in data.A first attempt to perform a partial wave analysis failed because of the difficulties in describing the resolution function in the region of the \(D^*\) resonances. Since the \(D^{*0} \bar{D}^{*0}\) final state was not measured in detail yet we narrowed the data sample by selecting only events in the \(D^{*0} \bar{D}^{*0}\) region, where only four partial waves are contributing to the final state. The steps of setting up the fit procedure and performing checks for quality assurance are done by using simulated data sets. The result of the fit to data looks promising.