# P1: Accomplishments of the second funding period

• A. Precision measurements of hadronic cross sections at BESIII
The cross section measurement of the exclusive hadronic channel $$e^+e^- \to \pi^+\pi^-$$ was published in 2016 (126 citations by October 2018). We have analysed the two-pion channel in the mass range between 600 and 900 MeV, which contributes about 50 % of the total hadronic vacuum polarization (HVP) contribution to $$(g-2)_\mu$$. In addition, the two-pion cross section has been studied at low ($$< 600$$ MeV) and at high masses ($$> 900$$ MeV) within one bachelor as well as two master thesis projects. A new PhD student has recently started and will continue with the publication of the two-pion channel at high masses. Within the second funding period the channels $$e^+e^- \to \pi^+\pi^-\pi^0$$ and $$e^+e^- \to \pi^+\pi^-2\pi^0$$ will be published. Preliminary results of these analyses have already been shown at conferences. We have also made important contributions to the measurement of the inclusive hadronic cross section using the scan data in the energy range above 2 GeV. This includes the estimate of the background from two-photon processes as well as detailed investigations of the model dependence of the underlying inclusive Monte-Carlo generators.
• B. Charmonium spectroscopy at BESIII
As a spin-off of the two-pion analysis mentioned above, the world's most precise measurement of the electronic width of the $$J/\psi$$ resonance has been achieved and published (exploiting the channel $$e^+e^- \to \mu^+\mu^-$$). Furthermore, two cross section measurements of the exclusive states $$e^+e^-\to \pi^+\pi^-J/\psi$$ and $$e^+ e^- \to \pi^+ \pi^- h_c$$ have been published in Phys. Rev. Lett. As a major result of the second funding period, a data taking (in total 3 weeks) has successfully been proposed by the Mainz ground and carried out in 2017. We search for the production of the $$\chi_{c1}$$ resonance ($$J^{P,C}=1^{+,+}$$) in $$e^+e^-$$ annihilation, which corresponds to the production of a non-vector state. The analysis of the reaction $$e^+e^-\to \chi_{c1} \to J/\psi \gamma \to \mu^+\mu^- \gamma$$ is essentially completed. We find a positive evidence for the production of the $$\chi_{c1}$$ resonance. This is the first time in the history of $$e^+e^-$$ experiments that the production of a non-vector resonance has been verified in an annihilation process. We plan to enter the BESIII internal review process shortly.
• C. Dark photon searches at BESIII
Recently, the search for a dark photon in the mass range between 1.5 and 3.4 GeV/$$c^2$$ has been published. This is a search in the visible dark photon decay model. The analysis of a search in the invisible decay model using single-photon events is ongoing.
• D. Preparation of the MAGIX experiment at MESA; development of GEM-based focal plane detectors
In several beam tests at MAMI, the high rate performance of GEM detectors with an area of 10x10 cm$$^2$$has successfully been tested. In order to reduce multiple scattering effects, it has been investigated within two master theses whether a GEM detector can be operated with a reduced material budget of the readout PCB (readout on kapton foil) and of the copper material (cromium GEM detector). While the kapton foil readout was found to be operational, no high-rate operation was possible with the cromium GEM detector. As a new baseline design we now foresee the construction of a time projection chamber, which will allow for reduced multiple scattering effects of the electrons passing the material of the focal plane detector. This work is carried out within a PhD and a master thesis.
• E. Lattice calculation of the hadronic vacuum polarization contribution to the muon $$g-2$$ in QCD with dynamical up, down and strange quarks
After last year's publication of our result for the leading hadronic vacuum polarization contribution, $$a_\mu^{\rm hvp}$$, in two-flavour QCD, our focus has shifted to a refined calculation in QCD with $$N_f=2+1$$ dynamical flavours. We have determined the individual contributions of the charm, strange and light quarks on a large set of gauge ensembles that cover four different lattice spacings and pion masses down to the physical one. The large region of lattice parameters and the use of fully O($$a$$)-improved versions of the lattice vector current allows for a reliable extrapolation to the physical point. Other systematic effects, such as finite-volume corrections, have been studied using a combination of data, effective field theories, as well as a dedicated calculation of the iso-vector correlator, which allows for a much more precise determination of the long-distance contributions to $$a_\mu^{\rm hvp}$$. Our preliminary result is obtained with an error of about 3 %, which is dominated by the uncertainty in the lattice scale. A status report has been presented at this year's lattice conference. The status of lattice calculations of the hadronic vacuum polarization and light-by-light scattering contributions to the muon $$g-2$$ were the subject of a large review article published in Prog. Part. Nucl. Phys.
• F. Reducing systematic effects in lattice calculations of the hadronic vacuum polarization
In order to reach the precision goal of lattice calculations for $$a_\mu^{\rm hvp}$$, several improvements must be made: the contributions from quark-disconnected diagrams must be included, as well as the effects arising from isospin breaking. For the former, we have used the technique of hierarchical probing to determine quark-disconnected diagrams on a subset of the available gauge ensembles. In addition, we have tested the Lorentz-covariant, Euclidean coordinate space method which indeed gives statistically more precise results for the disconnected contributions. We have started to apply the method to our gauge ensemble at the physical pion mass, where the contributions from this type of diagram is expected to be most important. We have also prepared for the determination of isospin-breaking corrections to $$a_\mu^{\rm hvp}$$, by extending the so-called “Rome formalism“ to the situation of QCD with open boundary conditions. As a test of the method we have computed the mass splittings among pseudoscalar mesons, and the calculation of the correlators that are relevant for the determination of isospin breaking in $$a_\mu^{\rm hvp}$$ are about to start. A third refinement is the improvement in the determination of the lattice scale, whose uncertainty dominates our estimate for the hadronic vacuum polarisation. In addition to reducing the statistical error we currently investigate a set of hadronic quantities that allow for the determination of the lattice scale with substantially reduced systematics, including those arising from isospin breaking.
• G. Studies of vector resonances
Our auxiliary calculation of the mass spectrum and scattering phase shifts in the iso-vector channel has been performed on ensembles with $$N_f=2$$ and $$2+1$$ flavours of dynamical quarks. In a comprehensive study of the two-flavour case, which formed the basis of a PhD thesis, it was established that the long-distance contribution to $$a_\mu^{\rm hvp}$$ can be constrained much more precisely. For QCD with $$N_f=2+1$$ flavours we currently investigate the most appropriate strategy to increase the precision of our calculations.
• H. Lattice calculation of the hadronic light-by-light scattering contribution to the muon $$\mathbf{g-2}$$
A conceptually new strategy for the determination of the hadronic light-by-light scattering contribution has been developed. Its key ingredient is an integral representation in position space, comprising the light-by-light scattering amplitude and a kernel function that represents the internal photon and muon lines. While the former can be evaluated in a lattice simulation, the kernel can be treated analytically in infinite volume. This is a major advantage, since the severe infrared problems associated with massless photon modes are entirely avoided. The validity of the approach has been tested by reproducing the known contributions of a lepton loop and of the neutral pion pole in the hadronic light-by-light contribution to $$(g-2)_\mu$$. In addition, the eight light-by-light forward scattering amplitudes computed on the lattice have been compared to phenomenological models of the two-photon fusion cross sections.
• I. Lattice calculation of the transition form factor for $$\mathbf{\pi^0\to\gamma^\ast\gamma^\ast}$$

The contribution from the pion pole is expected to dominate the hadronic light-by-light scattering contribution. Knowledge of the form factor for the transition between a $$\pi^0$$ and two virtual photons, $$\pi^0\to\gamma^\ast\gamma^\ast$$, allows for the evaluation of the pion pole contribution. We have performed a lattice calculation of the transition form factor, both in the singly and doubly virtual case. The dependence of the data on the photon virtuality was fitted using several model-inspired ansätze. In the singly virtual case they are in good agreement with experimental data and also with the Brodsky-Lepage limit. The pion pole contribution has been determined at the physical point with a total precision of 13 %, using the two-flavour ensembles. Currently, the study is being extended to the ensembles with $$2+1$$ dynamical flavours.