Abstract:
We investigate the impact of finite volume effects on the critical number of flavours, N_f^c, for chiral symmetry restoration in QED3. To this end we solve a set of coupled Dyson-Schwinger equations on a torus. For order parameters such as the anomalous dimension of the fermion wave function or the chiral condensate we find substantial evidence for a large dependence on the volume. We observe a shift in N_f^c from values in the range of 3.61 \le N_f^c \le 3.84 in the infinite volume/continuum limit down to values below N_f \le 1.5 at finite volumes in agreement with earlier results of Gusynin and Reenders in a simpler truncation scheme. These findings explain discrepancies in N_f^c between continuum and lattice studies.

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 summarise recent results for the quark loop part of the light-by-light scattering contribution to the muons anomalous magnetic moment. In particular we focus on the impact of a momentum dependent quark and quark-photon vertex. We compare the Dyson-Schwinger description with that of the extended Nambu--Jona-Lasinio model (ENJL) and find important quantitative differences. In particular 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 as found in the ENJL model.

Abstract:
We investigate the impact of finite volume effects on the critical number of flavours, N_f^c, for chiral symmetry restoration in QED3. To this end we solve a set of coupled Dyson-Schwinger equations on a torus. For order parameters such as the anomalous dimension of the fermion wave function or the chiral condensate we find substantial evidence for a large dependence on the volume. We observe a shift in N_f^c from values in the range of 3.61 \le N_f^c \le 3.84 in the infinite volume/continuum limit down to values below N_f \le 1.5 at finite volumes in agreement with earlier results of Gusynin and Reenders in a simpler truncation scheme. These findings explain discrepancies in N_f^c between continuum and lattice studies.

Abstract:
We present a calculation of the hadronic vacuum polarization (HVP) tensor within the framework of Dyson--Schwinger equations. To this end we use a well-established phenomenological model for the quark-gluon interaction with parameters fixed to reproduce hadronic observables. From the HVP tensor we compute both the Adler function and the HVP contribution to the anomalous magnetic moment of the muon, $a_\mu$. We find $a_\mu^{HVP}= 6760\times 10^{-11}$ which deviates about two percent from the value extracted from experiment. Additionally, we make comparison with a recent lattice determination of $a_\mu^{HVP}$ and find good agreement within our approach. We also discuss the implications of our result for a corresponding calculation of the hadronic light-by-light scattering contribution to $a_\mu$.

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:
In this work we devise a new method to study quark anti-quark interactions beyond simple ladder-exchange that yield massless pions in the chiral limit. The method is based on the requirement to have a representation of the quark-gluon vertex that is explicitly given in terms of quark dressings functions. We outline a general procedure to generate the Bethe-Salpeter kernel for a given vertex representation. Our method allows not only the identification of the mesons' masses but also the extraction of their Bethe-Salpeter wave functions exposing their internal structure. We exemplify our method with vertex models that are of phenomenological interest.

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 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:
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.