# M1: Accomplishments of the second funding period

• A. Dispersive formalism for hadronic light-by-light scattering
Besides the single pseudo-scalar channels, the production of two pseudo-scalar mesons are the next important contribution entering the hadronic light-by-light correction to $$(g-2)_\mu$$. To this end, we have performed a theoretical study of the $$\gamma\gamma^{*} \to \pi^+\pi^-, \pi^0\pi^0$$ processes from the threshold through the $$f_2(1270)$$ region in the $$\pi\pi$$ invariant mass. We have adopted the Omnès representation in order to account for rescattering effects in both s- and d-partial waves. For the description of the $$f_0(980)$$ resonance, we have implemented a coupled-channel unitarity. We achieved a reasonable description of the $$\gamma\gamma \to \pi^+\pi^-, \pi^0\pi^0$$ total cross sections in comparison with the recent empirical data from the Belle Collaboration. For the finite $$Q^2$$ we made a first dispersive prediction of the cross section including the $$f_2(1270)$$ region. The constructed amplitudes are serving as an essential framework to interpret the current experimental two-photon fusion program at BESIII. The corresponding data are being analysed within the experimental part of project M1. The obtained results will serve as one of the relevant inputs to constrain the hadronic piece of the light-by-light scattering contribution to the muon's anomalous magnetic moment. The results of this work have been submitted for publication.
Besides the $$2\pi$$ channels, we have also performed a theoretical study of the $$\gamma\gamma \to \pi\eta$$ process from the threshold up to 1.4 GeV in the $$\pi\eta$$ invariant mass. For the s-wave $$a_0(980)$$ resonance state a dispersive formalism was adopted using a coupled-channel Omnès representation, while the d-wave $$a_2(1320)$$ state was described as a Breit-Wigner resonance. An analytic continuation to the $$a_0(980)$$ pole position allows us to extract its two-photon decay width as $$\Gamma_{a_0\to\gamma\gamma}=0.27(4)$$ keV. The results have been published. We are currently extending this work to the virtual photon case $$\gamma\gamma^{*} \to \pi\eta$$.
• B. Light-by-light sum rules and its application to charmonium states
1. The light-quark meson contributions to three exact light-by-light scattering sum rules have been evaluated in light of new data by the Belle Collaboration for the transition form factors of the tensor meson $$f_2(1270)$$ and scalar meson $$f_0(980)$$. We confirm a previous finding that the $$\eta, \eta^\prime$$ and $$f_2(1270)$$ contributions saturate one of these sum rules up to photon virtualities around 1 GeV$$^2$$. At larger virtualities, our sum rule analysis shows an important contribution of the $$f_2(1565)$$ meson and provides a first empirical extraction of its helicity-2 transition form factor. Two further sum rules allow us to predict the helicity-0 and helicity-1 transition form factors of the $$f_2(1270)$$ meson. Furthermore, our analysis also provides an update for the scalar and tensor meson hadronic light-by-light contributions to $$(g-2)_\mu$$. The results have been published.
2. The hadronic contribution to the eight forward amplitudes of light-by-light scattering ($$\gamma^\ast \gamma^\ast \to \gamma^\ast \gamma^\ast$$) was computed in lattice QCD, in two-flavor QCD with pion masses extending down to 190 MeV at two different lattice spacings. Via dispersive sum rules, the amplitudes were compared to a model of the $$\gamma^\ast \gamma^\ast \to$$ hadrons cross sections in which the fusion process is described by hadronic resonances. The results provide an important test for the model estimates of hadronic light-by-light scattering in the anomalous magnetic moment of the muon. The results have been published.
3. The forward light-by-light scattering sum rules have also been applied to charmonium states. We have shown that these sum rules imply a cancellation between charmonium bound state contributions, which are mostly known from the $$\gamma \gamma$$ decay widths of these states, and continuum contributions above $$D \bar D$$ threshold, for which we provided a duality estimate. Two of these sum rules allow to predict the yet unmeasured $$\gamma^\ast \gamma$$ coupling, with one longitudinal and one transverse photon, of the $$\chi_{c1}(1P)$$ state as: $$\tilde \Gamma_{\gamma \gamma} \simeq \left( 1.6 \pm 0.30 \right) \; \mathrm{keV}$$, or equivalently $$\tilde \Gamma_{\gamma \gamma} / \Gamma_{tot} \simeq \left(1.9 \pm 0.4 \right) \times 10^{-3}$$. This prediction can be tested at present high-luminosity $$e^+ e^-$$ colliders. The results have been published.
4. In an attempt to express the hadronic contribution to $$(g-2)_\mu$$ directly in terms of measurable quantities, a sum rule, attributed to Schwinger, has been applied to relate $$(g - 2)_\mu$$ in a linear way to an integral of a photo-absorption cross sections on a muon. The linear property makes it suitable for a straightforward assessment of the hadronic contributions to $$(g-2)_\mu$$. It was tested on the leading order QED contribution, as well as on the hadronic vacuum-polarization contribution. The results have been published. A next application under stutdy is the single meson contribution to the light-by-light contribution.
• C. Spacelike meson TFF program at BESIII
The analysis of the two hadronic reactions $$\gamma^*\gamma \to \pi^0$$ and $$\gamma^*\gamma \to \pi^+\pi^-$$ are almost completed and it is planned to publish these channels in the course of the second funding period. In the case of the channel $$\gamma^*\gamma \to \pi^0$$ preliminary results have been shown at conferences. The $$Q^2$$ range covered in the analyses exceeds significantly towards smaller values as motivated by the hadronic light-by-light contribution to $$(g-2)_\mu$$. Furthermore, in the two-pion channel for the first time the $$M_{\pi\pi}$$ mass range down to threshold can be investigated. The analysis will shortly be entering the internal review of the collaboration; a memo of the analysis has been completed. The analysis of the neutral two-pion channel has started and is now carried out within a master thesis. A feasibility study for the reaction with two virtualities $$\gamma^*\gamma^* \to \pi^0$$ has been carried out.
• D.1 Timelike meson TFF program at BESIII
The timelike TFF measurement at high momentum transfer of the $$\eta$$ and $$\eta^\prime$$ are within internal review within the BESIII collaboration. The analysis of a measurement of the $$\pi^0$$ TFF has been carried out within a master thesis. We plan to publish all these analyses within the next year.
• D.2 Timelike meson TFF program at A2/MAMI
Although originally foreseen as a feasibility study, the analysis of the timelike TFF using the $$\pi^0$$ Dalitz decay, could already be published with competitive statistics and systematics. As a consequence, a new high statistics data sample of this reaction has successfully been collected in 2018. In the third funding period, the form factor will be analyzed and we expect to publish the world's best measurement of this form factor from a timelike measurement. Finally, also a high statistics measurement of the $$\omega$$ and $$\eta$$ TFFs have been published within a common paper.
• E. Hardware Project: Design study of a Zero Degree Detector (ZDD) for BESIII
Within a master thesis the readout of LYSO crystals with SiPM detectors has successfully been studied. In addition, investigations of the temperature behavior and the response to external light of LYSO crystals have been carried out. In November 2017 a successful test beam of a LYSO crystal together with the full readout scheme has been carried out. Data acquisition rates of 1 MHz and higher have been verified experimentally with a flash ADC and a FPGA board. The background environment of the ZDD detector has been studied with large scale Monte Carlo simulations.