# M1: Accomplishments of ﬁrst funding period

• Sum rule studies of $$\mathbf{\gamma^\ast \gamma}$$ processes and light-by-light scattering
We derived two new exact super-convergence relations, which relate the forward light-by-light scattering to energy weighted integrals of the $$\gamma^\ast \gamma$$-fusion cross sections. They complement the known super-convergence relation based on the extension of the GDH sum rule to the light-light system. All of the new relations were veriﬁed exactly in perturbation theory at leading order in scalar and spinor QED, and proof to all orders in perturbation theory was given within the context of the $$\phi^4$$ quantum ﬁeld theory. We then applied the super-convergence relations to the $$\gamma^\ast \gamma$$-production of mesons, and showed quantitatively that they lead to relations between the $$\gamma^\ast \gamma$$-transition form factors for (pseudo-)scalar, axial-vector and tensor mesons.
• Hadronic light-by-light (HLbL) contributions to (g-2)$$\mathbf{_\mu}$$: Evaluation of axial-vector, scalar, and tensor meson pole contributions
We provided an improved estimate for the HLbL correction to the muon’s anomalous magnetic moment $$a_\mu\equiv (g-2)_\mu$$/2, by considering single meson contributions beyond the leading pseudo-scalar mesons. Based on available experimental input as well as constraints from light-by-light scattering sum rules, we provided new estimates for the effects of axial-vector, scalar, and tensor mesons to $$a_\mu$$.
• Phenomenological studies of $$\mathbf{\eta}$$ and $$\mathbf{\eta^\prime}$$ transition form factors (FFs) using rational approximants
The $$\eta$$ and $$\eta^\prime$$ transition form factors were analyzed in the space-like region in a model-independent way through the use of rational approximants. The slope and curvature parameters of the form factors as well as their values at zero and inﬁnity were extracted from experimental data. These results allow improved extractions of the mixing parameters of the $$\eta - \eta^\prime$$ system as well as improve on the pseudoscalar-exchange contributions to the HLbL contribution to $$a_\mu$$.
• HLbL contributions to $$\mathbf{a_\mu}$$: development of a dispersion relation framework
We started with the development of a new general dispersive formalism for evaluating the HLbL scattering contribution to the anomalous magnetic moment of the muon. In the suggested approach, this correction is related to the imaginary part of the muon’s electromagnetic vertex function. The latter may be directly related to measurable hadronic processes by means of unitarity and analyticity. As a test of the new formalism, we applied it to evaluate a 2-loop scalar three-point function, as well as to evaluate the meson pole exchange contribution to $$a_\mu$$. In both cases, an exact agreement was found with the direct two-loop calculation.
• Measurement of the spacelike $$\mathbf{\pi^0}$$ transition FF at BES-III
At BESIII a $$\gamma\gamma$$ program has been successfully launched by Mainz researchers. Before, no such analyses were carried out by the collaboration. We developed a single-tag analysis technique, which allows to access the spacelike transition FF of mesons over a wide Q$$^2$$ range. For the ﬁrst funding period of the CRC, the transition FFs of pseudoscalar mesons are in the focus of interest. The measurement of the $$\pi^0$$ transition FF is especially important as the pion exchange plays a dominant role in the hadronic light-by-light (HLbL) contribution. We currently analyze a data sample of 2.9 fb−1 taken at a cms energy of 3.77 GeV. By exploiting the single-tag technique, the transition FF between 0.3 GeV$$^2$$ and 3.1 GeV$$^2$$ is accessible. The BESIII measurement will yield a signiﬁcanly improved FF in that energy range compared to existing data from CLEO and CELLO. The BESIII measurement is complementary to relatively recent data from BABAR and BELLE above 4 GeV$$^2$$ and covers the most relevant energy range in view of the HLbL motivation. The analysis is in a very advanced stage, some systematic studies are however still ongoing. We are very conﬁdent that a publication within the ﬁrst half of 2015 will be possible.
• Measurement of the spacelike $$\mathbf{\eta}$$ and $$\mathbf{\eta^\prime}$$ transition FFs at BESIII
Also in the case of the spacelike $$\eta$$ and $$\eta^\prime$$ transition FFs, the single tag technique is employed. For both analyses, the $$\pi^+\pi^-\gamma\gamma$$ ﬁnal states is investigated with $$\eta\to\pi^+\pi^-\pi^0, \eta^\prime\to \pi^+\pi^-\eta$$ and the subsequent decay of $$\eta\to\gamma\gamma$$. Preliminary results are foreseen for the end of the ﬁrst funding period. Similar to the $$\pi^0$$ FF, the world’s best measurements below 3 GeV$$^2$$ will be achieved.
• Measurement of the spacelike $$\mathbf{\gamma^\ast\gamma\to\pi^+\pi^-}$$ process at BESIII
Although originally foreseen for the second funding period, we already started the analysis of the charged two-pion channed in the single-tag mode. The main reason for this is the fact that the channel came out to be of special importance for the HLbL motivation, but has never been measured before by any experiment so-far. It was shown that a special effort has to be devoted to the background suppression of $$\gamma^\ast\gamma\to\mu^+\mu^-$$ events as well as other QED background channels. We expect preliminary results to come out within the course of 2015, while the ﬁnal publication might be expected at the beginning of the second funding period.
• Measurement of the timelike $$\mathbf{\eta}$$ transition FF in the Dalitz decay $$\mathbf{\eta\to e^+ e^-\gamma}$$ with Crystal Ball/TAPS
A high-accuracy measurement of the Dalitz decay $$\eta\to e^+ e^-\gamma$$ was achieved with one order of magnitude greater statistics compared to our previous publication. This was achieved by analyzing a new data sample and by using more sophisticated analysis tools. Our new result is currently the most precise determination of the transition FF from $$\eta\to e^+ e^-\gamma$$ decays and yields the world’s best measurement of the timelike transition FF of the $$\eta$$ meson at low q$$^2$$, where q$$^2$$ is the invariant mass of the $$e^+ e^-$$ pair. Above the di-muon mass the results are comparable in statistics to measurements at CERN, where the decay to a photon and muon pairs has been analyzed. The FFs are found to be in good agreement in that q$$^2$$ range.
• Preparation of the timelike $$\mathbf{\eta^\prime}$$ transition FF measurement at Crystal Ball/TAPS
For the measurement of $$\eta^\prime$$ decays the end-point tagging spectrometer was installed in the beamline. To improve data taking rates upgrades of the DAQ- and trigger systems were successfully performed, leading to a three times higher η′ production rate than in 2012. It is planned to take in total 10 weeks of data for studying $$\eta^\prime$$ decays. This would mean a six times higher $$\eta^\prime$$ sample compared to 2012. The ﬁrst 4 weeks were already taken in summer 2014 and analyses are ongoing. A conservative estimate based on the ﬁrst 4 weeks of data gives roughly 600 analyzed $$\eta^\prime$$ Dalitz decays from 10 weeks of data taking.
• Access to timelike transition FFs through the processes $$\mathbf{e^+ e^- \to\eta\gamma}$$ and $$\mathbf{e^+ e^- \to\eta^\prime\gamma}$$ at BES-III
We are currenly analyzing the production of $$\eta$$ and $$\eta^\prime$$ mesons in conjuction with a photon exploiting recent BES-III data in the so-called XYZ region at a cms energy of 4.3 GeV with an integrated luminosity of 5.1 fb−1. Cross sections for the reactions and $$e^+e^-\to\eta^\prime\gamma$$ are determined using the decay modes $$\eta\to\pi^+\pi^-\pi^0$$ and $$\eta^\prime\to\pi^=\pi^-\eta$$ and subsequent decays of $$\pi^0, \eta$$ to two photons. From these cross sections timelike transition FFs for $$\eta$$ and $$\eta^\prime$$ are extracted at momentum transfers around q$$^2$$= 18.5 GeV$$^2$$. The statistical precision of the extracted timelike transition FFs is comparable to BABAR results in the timelike region at q$$^2$$= 112 GeV$$^2$$ (Phys.Rev. D74, 012002) as well as to BABAR measurements in the spacelike region for Q$$^2$$= 10 ... 30 GeV$$^2$$ (Phys.Rev. D84, 052001). Studies of systematic uncertainties are currently performed and analysis results will be prepared for internal review within the BES-III collaboration.
• Data driven approaches to constrain the hadronic contribution to $$\mathbf{a_\mu}$$
We organized three workshops to discuss strategies for improvements of the accuracy of the hadronic contributions to $$a_\mu$$. One workshop was held at the ECT* in Trento, and two workshops were held in Mainz. Each of the meetings had about 40 participants, and brought together the leading researchers working in this ﬁeld, both in theory and experiment. As a result of the lively discussions at these meetings, it was agreed to prepare a whitepapertowards a data driven approach to constrain the hadronic contribution to $$a_\mu$$. This will update and extend a previous whitepaper, which was prepared for the US Particle Physics ”Snowmass” Self Study in 2013.