Abstract:
We summarize our results for hadronic contributions to the anomalous magnetic moment of the muon ($a_\mu$), the one from hadronic vacuum-polarisation (HVP) and the light-by-light scattering contribution (LBL), obtained from the Dyson-Schwinger equations (DSE's) of QCD. In the case of HVP we find good agreement with model independent determinations from dispersion relations for $a_\mu^\mathrm{HVP}$ as well as for the Adler function with deviations well below the ten percent level. From this we conclude that the DSE approach should be capable of describing $a_\mu^\mathrm{LBL}$ with similar accuracy. We also present results for LBL using a resonance expansion of the quark anti-quark T-matrix. Our preliminary value is $a_\mu^\mathrm{LBL}=(217 \pm 91) \times 10^{-11}$.

Abstract:
A novel approach towards the hadronic contributions to the anomalous magnetic moment of the muon $a_{\mu}$ is presented, namely the Dyson-Schwinger equations of QCD. It has the advantage of being valid for all momentum scales and has the potential to address off-shell amplitudes. We present our first results for the pseudoscalar (PS) meson exchange and the quark loop contributions. The meson exchange ($\pi^0, \eta, \eta'$), $a_\mu^{\textrm{LBL;PS}}=(84 \pm 13)\times 10^{-11}$, is commensurate with previous calculations, while the quark loop contribution $a_\mu^{\textrm{LBL;quarkloop}} = (107 \pm 48)\times 10^{-11}$, is strongly enhanced by vertex dressing effects in the quark photon vertex. Taken seriously this leads to the estimate of $a_\mu=116\,591\,865.0(96.6)\times 10^{-11}$, giving a 1.9 $\sigma$ deviation between theory and experiment.

Abstract:
We give a status report on the hadronic light-by-light scattering contribution to the muon's anomalous magnetic moment from the Dyson-Schwinger approach. We discuss novel, model-independent properties of the light-by-light amplitude: we give its covariant decomposition in view of electromagnetic gauge invariance and Bose symmetry, and we identify the relevant kinematic regions that are probed under the integral. The decomposition of the amplitude at the quark level and the importance of its various diagrams are discussed and related to model approaches.

Abstract:
We determine the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon using the framework of Dyson-Schwinger and Bethe-Salpeter equations of QCD. Our result for the pseudoscalar ($\pi^0, \eta, \eta'$) meson exchange diagram is commensurate with previous calculations. In our calculation of the quark loop contribution we improve previous approaches by implementing constraints due to gauge invariance. As a consequence, our value $a_\mu^{\textrm{LBL;quarkloop}} = (136 \pm 59)\times 10^{-11}$ is significantly larger. Taken at face value, this then leads to a revised estimate of the total $a_\mu=116\,591\,891.0(105.0)\times 10^{-11}$.

Abstract:
We present first results for the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon a_{\mu} in the framework of Dyson-Schwinger and Bethe-Salpeter equations. We determine the quark loop and pseudoscalar ({\pi}^0, {\eta}, {\eta}') meson exchange diagram using a phenomenological model for the combined strength of the gluon propagator and the quark-gluon interaction as the only input. Our result for meson exchange, a_{\mu}^{LBL;PS}=(84 \pm 13) x 10^{-11}, is commensurate with previous calculations. However, our number for the quark loop contribution, a_{\mu}^{LBL;quarkloop} = (107 \pm 2 \pm 46) x 10^{-11}, is significantly larger due to dressing effects in the quark propagator and the quark-photon vertex. Taken at face value, this then leads to a revised estimate of the total a_{\mu}=116 591 865.0(96.6) x 10^{-11}, which reduces the difference between theory and experiment to about 1.9 {\sigma}.

Abstract:
We summarize our recent results for the quark loop part of the light-by-light scattering contribution as well as the hadronic vacuum polarisation contributions to the anomalous magnetic moment of the muon. In particular we focus on the role played by the momentum dependence of the quark- and quark-photon vertex dressing functions. We give a detailed comparison of the Dyson-Schwinger description of this contribution to the corresponding picture emerging from hadronic models in particular the extended Nambu--Jona-Lasonio model (ENJL). We find that the details of the momentum dependence are important on a quantitative level. Especially the transverse parts of the quark-photon-vertex, which serve as a dynamical extension of simple vector meson dominance models, do not yield the large suppression of the light-by-light contribution found in the ENJL model if realistic dressings are taken into account.

Abstract:
We review the current status of nonperturbative studies of gauge field theory using the Dyson-Schwinger equation formalism and its application to hadronic physics. We begin with an introduction to the formalism and a discussion of renormalisation in this approach. We then review the current status of studies of Abelian gauge theories [e.g., strong coupling quantum electrodynamics] before turning our attention to the non-Abelian gauge theory of the strong interaction, quantum chromodynamics. We discuss confinement, dynamical chiral symmetry breaking and the application and contribution of these techniques to our understanding of the strong interactions.

Abstract:
We study hadronic decays of mesons and baryons in the context of the Dyson-Schwinger equations of QCD. Starting from a well-established effective interaction in rainbow-ladder truncation, we consistently calculate all ingredients of the appropriate decay diagrams. The resulting strong couplings are presented as functions of the quark mass from the chiral limit up to the respective decay thresholds. In particular, we investigate the \rho \pi \pi and for the first time the \Delta N \pi transitions. Both meson and baryon results compare well to available lattice QCD results as well as experimental data and present the first step towards a comprehensive covariant study of hadron resonances in the Dyson-Schwinger approach.

Abstract:
In these proceedings we present a mini-review on the topic of the Dyson-Schwinger/Bethe-Salpeter approach to the study of relativistic bound-states in physics. In particular, we present a self-contained discussion of their derivation, as well as their truncation such that important symmetries are maintained.

Abstract:
A model for verifying a consistency of the next-to-leading order hadronic contributions to the muon anomalous magnetic moment with those of the leading order is proposed. A part of the next-to-leading order hadronic contributions related to the vacuum polarization is rather accurately reproduced in the model. I find a new numerical value for the light-by-light hadronic contribution that leads to agreement with recent experimental result for the muon anomalous magnetic moment.