# P1: Accomplishments of the second funding period

• Precision measurements of hadronic cross sections at BES III
The cross section measurement of the exclusive hadronic channel $$e^+e^- \to \pi^+\pi^-$$ was published in 2016. This channel happens to be the by far most relevant contribution to the hadronic vacuum polarization (HVP) contribution to $$(g-2)_\mu$$. The new BES III result, which confirms a large deviation between the Standard Model (SM) prediction for $$(g-2)_\mu$$ and the direct BNL measurement, has received a major interest in the community. Furthermore, in various master theses the analysis of the two-pion cross section at very low and very high masses has been initiated. 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. Paper drafts, which are in the internal review process within the collaboration, are furthermore existing for the exclusive channels $$e^+e^- \to \pi^+\pi^-\pi^0$$, $$e^+e^- \to \pi^+\pi^-2\pi^0$$, and $$e^+e^- \to \pi^+\pi^-3\pi^0$$. Preliminary results have already been shown at conferences.
• Charmonium spectroscopy at BES III
As a spin-off of the two-pion analysis mentioned above, the world’s most precise measure- ment of the electronic width of the $$J/\psi$$ resonance was 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. In these analyses new hadronic states have been discovered, indicating the existence of charmonium-like resonances. As a major result of the second funding period, a data taking (in total 3 weeks) has successfully been proposed and carried out, in which we search for the production of a non-vector state in $$e^+e^-$$ annihilation. The analysis of the reaction $$e^+e^-\to \chi_{c1} \to J/\psi \gamma \to \mu^+\mu^- \gamma$$ is currently ongoing. A strong cooperation with the group of H. Czyż is ongoing, especially regarding the interpretation of the recent $$\chi_{c1}$$ scan.
• Dark photon searches at BES III
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 almost completed. The parameter range tested in this new BES III analysis exceeds upon existing limits from BABAR and NA64 by a large factor.
• 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 been successfully tested. In two master theses the readout of the GEM detectors on the kapton foil itself as well as the stable operation with thinner detectors is investigated. If successful, this will lead to a reduction of the material budget of the detector by almost a factor of five, yielding much lower multiple scattering effects in the focal plane of the MAGIX spectrometers.
• Lattice calculation of the hadronic vacuum polarization contribution to the muon $$g−2$$ in two-flavour QCD
The calculation of the leading hadronic vacuum polarization contribution $$a_\mu^{\rm hvp}$$ in QCD with two dynamical flavours has been completed and is now published in JHEP. The paper contains a comprehensive investigation of the overall methodology of the lattice approach. This includes discussing different methods for controlling the low-energy regime (based on integral representations either in position or momentum space), the calculation of quark-disconnected diagrams, the reliable estimation of finite-volume corrections and the impact of the scale setting uncertainty on the overall precision in $$a_\mu^{\rm hvp}$$. The calculation reports a total error of 6%, of which 4.8% are due to statistics. A crucial ingredient for the evaluation of quark-disconnected diagrams is the cancellation of stochastic noise between the light and strange quark contributions, which has by now become a standard technique which is applied by most other lattice collaborations. The project formed the basis for two PhD thesis. The results have also been used as input for a phenomenological determination which combines perturbation theory, QCD sum rules and lattice QCD, and which has been published in Phys. Rev. D.
• Reducing systematic effects in lattice calculations of the hadronic vacuum polarization
We have started the calculation of $$a_\mu^{\rm hvp}$$ using gauge ensembles with dynamical light and strange quarks. The study include several significant improvements compared to the earlier two-flavour study: (1) we compute correlators for the O($$a$$) improved vector currents, which allows for an accelerated convergence towards the continuum limit, (2) a gauge ensemble at the physical pion mass has been generated which will significantly reduce the uncertainty arising from the chiral extrapolation, (3) the infrared regime of the isovector correlator will be constrained by a dedicated spectrum calculation which eliminates any residual model dependence, (4) a new technique (“hierarchical probing”) will be used to compute quark-disconnected diagrams more efficiently, and (5) we have started to determine the effects of isospin breaking which are thought to be relevant at the percent level. Our preliminary results for $$a_\mu^{\rm hvp}$$ are already of superior quality than those in the published two-flavour study. In addition, we have published a new conceptual approach for comput- ing $$a_\mu^{\rm hvp}$$, which is based on a Lorentz-covariant, Euclidean coordinate-space expression that offers more flexibility for an implementation in lattice QCD.
• Studies of vector resonances
We have begun a comprehensive study of the $$\rho$$-resonance as an auxiliary project for the determination of $$a_\mu^{\rm hvp}$$. It is based on the Lüscher formalism and requires the determination of energy levels in the iso-vector channel in a finite volume, which allows for the determination of the scattering phase shift. The formalism has been extended to the calculation of matrix elements, which can be linked to the pion form factor in the time-like regime. In a pilot study based on a subset of two-flavour gauge configurations, we have obtained results for the scattering phase shift and the timelike pion form factor. We have also demonstrated that the long-distance contribution of the vector correlator to $$a_\mu^{\rm hvp}$$ can be obtained much more precisely, regarding both the statistical error and the uncer- tainty arising from the usual modelling of the infrared part. First results for QCD with $$2+1$$ flavours have clearly isolated the contribution from the two-pion state which starts to dominate the correlator at distances of 3 fm.
• Lattice calculation of the hadronic light-by-light scattering contribution to the muon $$g−2$$
A new conceptual strategy for the determination of the hadronic light-by-light scattering contribution has been developed. Its key ingredient is an integral representation in po- sition space, comprising the light-by-light scattering amplitude and a kernel function that absorbs the effects of 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 can be entirely avoided. The validity of the approach has been tested by inserting the expression for the muon loop for the scattering amplitude which reproduced the known result for muonic light-by-light scattering. The QCD part of the integral representation has been tested by linking the light-by-light forward scattering amplitude to two-photon fusion cross sections that can be compared with phenomenological models.
• Lattice calculation of the transition form factor for $$\pi^0\to\gamma^\ast\gamma^\ast$$
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.