Search

In this letter, we discuss a novel method to search for inhomogeneous chiral symmetry breaking in theories with fermions. The prime application we have in mind is QCD, but the method is also applicable for other theories including solid state applications. It is based on an extension of the chiral susceptibility to inhomogeneous phases and it works as a stability analysis. Our method allows us to determine when homogeneous solutions have the tendency to create spatially modulated condensates. As proof of principle, we apply this technique to a rainbow-ladder QCD model and confirm the conjectured shape of the instability region published in Ref. [1].

In this work we continue our efforts to study the existence of a phase with an inhomogeneous, i.e., spatially varying, chiral condensate in QCD. To this end we employ a previously established method of stability analysis of the two-particle irreducible effective action in a truncation that corresponds to a rainbow-ladder approximation of the quark-gluon interaction of QCD. If the analysis is restricted to homogeneous phases, the phase diagram features a first-order chiral transition in the lower-temperature regime. Performing the stability analysis along the lower-chemical-potential border of the corresponding spinodal region, we find that below a certain temperature the homogeneous chirally symmetric solution is unstable against inhomogeneous condensation. We argue that this instability may persist to chemical potentials above the homogeneous first-order phase boundary, in which case it signals the existence of an inhomogeneous ground state. Our methodology is also applicable for more sophisticated truncations of the QCD effective action.

A symmetry-preserving continuum approach to the calculation of baryon properties in relativistic quantum field theory is used to predict all form factors associated with nucleon-to-$\Delta$ axial and pseudoscalar transition currents, thereby unifying them with many additional properties of these and other baryons. The new parameter-free predictions can serve as credible benchmarks for use in analysing existing and anticipated data from worldwide efforts focused on elucidation of $\nu$ properties., Comment: 8 pages, 6 figures, 3 tables

An important goal of ongoing and future experiments is to explore spectra and transition form factors of baryons with non-zero strangeness. Of particular interest is the transition form factor$\gamma^{(*)} \Sigma^0 \rightarrow \Lambda$ in the time-like momentum region that can be extracted from Dalitz decays. On the road towards a theoretical description of these form factors we extend a covariant dynamical quark-diquark model for the baryon Faddeev equation to the strange-quark sector. Based on an excellent description of the mass spectrum of selected baryon octet and decuplet states and reasonable results for the nucleon form factors we determine the elastic electromagnetic form factors of $\Lambda$ and $\Sigma^+, \Sigma^0, \Sigma^-$ hyperons in the space-like region as well as the ones for the octet transition $\gamma^{(*)} \Sigma^0 \rightarrow \Lambda$. We discuss qualitative and quantitative features of the diquark-quark picture and compare systematically with previous results from a three-body Faddeev approach and lattice data where available., Comment: version accepted by EPJA

The possible occurrence of crystalline or inhomogeneous phases in the QCD phase diagram at large chemical potential has been under investigation for over thirty years. Such phases are present in models of QCD such as the Gross-Neveu model in 1+1 dimensions, Nambu-Jona-Lasinio (NJL) and quark meson models. Yet, no unambiguous confirmation exists from actual QCD. In this work, we propose a new approach for a stability analysis that is based on the two-particle irreducible effective action and compatible with full QCD calculations within the framework of functional methods. As a first test, we reproduce a known NJL model result within this framework. We then discuss the additional difficulties which arise in QCD due to the non-locality of the quark self-energy and suggest a method to overcome them. As a proof of principle and as an illustration of the analysis, we consider the Wigner-Weyl solution of the quark Dyson-Schwinger equation (DSE) within a simple truncation of QCD in the chiral limit and analyse its stability against homogeneous chiral-symmetry breaking fluctuations. For temperatures above and below the tricritical point we find that the boundary of the instability region coincides well with the second-order phase boundary or the left spinodal, respectively, obtained from the direct solutions of the DSEs. Finally, we outline how this method can be generalized to study inhomogeneous fluctuations., Comment: refs added; data files organised

A quark+diquark Faddeev equation treatment of the baryon bound state problem in Poincar\'e-invariant quantum field theory is used to deliver parameter-free predictions for all six $\Delta$-baryon elastic weak form factors. Amongst the results, it is worth highlighting that there are two distinct classes of such $\Delta$-baryon form factors, $(g_1, g_3, G_{\pi\Delta\Delta})$, $(h_1, h_3, H_{\pi\Delta\Delta})$, the functions within each of which are separately linked via partial conservation of axial current (PCAC) and Goldberger-Treiman (GT) relations. Respectively within each class, the listed form factors possess qualitatively the same structural features as the nucleon axial, induced pseudoscalar, and pion-nucleon coupling form factors. For instance, the $\Delta$-baryon $g_1$ axial form factor can reliably be approximated by a dipole function, characterised by an axial charge $g_A^{\Delta^+}=0.71(9)$ and mass-scale $m_A^\Delta=0.95(2)m_\Delta$. Moreover, the two distinct $\Delta$-baryon PCAC form factor relations are satisfied to a high degree of accuracy on a large range of $Q^2$; the associated GT relations present good approximations only on $Q^2/m_\Delta^2 \simeq 0$; and pion pole dominance approximations are reliable within both classes. There are two $\pi\Delta\Delta$ couplings: $g_{\pi\Delta\Delta} = 10.46(1.88)$; $h_{\pi\Delta\Delta}= 35.73(3.75)$; and the associated form factors are soft. Such couplings commonly arise in phenomenology, which may therefore benefit from our analyses. A flavour decomposition of the axial charges reveals that quarks carry $71$\% of the $\Delta$-baryon spin. The analogous result for the proton is $\approx 65$\%., Comment: 19 pages, 9 figures, 2 tables

We study electromagnetic as well as strong isospin breaking effects in the isospin mass splittings of light pseudoscalar and vector mesons. To this end we employ a coupled system of quark Dyson-Schwinger and meson Bethe-Salpeter equations whose interaction kernels contain gluon, pion and photon exchange interactions. In bound states, QCD-induced isospin breaking is manifest on different levels. On the one hand, a different explicit up- and down-quark mass directly affects the propagators of the constituent quarks. On the other hand, it leads to different interaction kernels within the isospin multiplets. In addition, electromagnetic isospin breaking is induced via a photon exchange diagram. Using the kaon iso-doublet and the charged pion masses as input to determine the up, down and strange quark masses we find for the pion, kaon and rho meson mass splittings different patterns each. In particular, our results provide evidence that the effects from two sources of mass splittings, the different quark masses and the different quark charges, do not add up linearly., Comment: 9 pages, 2 figures

A symmetry-preserving Poincar\'e-covariant quark+diquark Faddeev equation treatment of the nucleon is used to deliver parameter-free predictions for the nucleon's axial and induced pseudoscalar form factors, $G_A$ and $G_P$, respectively. The result for $G_A$ can reliably be represented by a dipole form factor characterised by an axial charge $g_A=G_A(0)=1.25(3)$ and a mass-scale $M_A = 1.23(3) m_N$, where $m_N$ is the nucleon mass; and regarding $G_P$, the induced pseudoscalar charge $g_p^\ast = 8.80(23)$, the ratio $g_p^\ast/g_A = 7.04(22)$, and the pion pole dominance Ansatz is found to provide a reliable estimate of the directly computed result. The ratio of flavour-separated quark axial charges is also calculated: $g_A^d/g_A^u=-0.16(2)$. This value expresses a marked suppression of the size of the $d$-quark component relative to that found in nonrelativistic quark models and owes to the presence of strong diquark correlations in the nucleon Faddeev wave function -- both scalar and axial-vector, with the scalar diquark being dominant. The predicted form for $G_A$ should provide a sound foundation for analyses of the neutrino-nucleus and antineutrino-nucleus cross-sections that are relevant to modern accelerator neutrino experiments., Comment: 8 pages, 6 figures, 2 tables. Accepted for publication in Phys. Lett. B

We review results for the phase diagram of QCD, the properties of quarks and gluons and the resulting properties of strongly interacting matter at finite temperature and chemical potential. The interplay of two different but related transitions in QCD, chiral symmetry restoration and deconfinement, leads to a rich phenomenology when external parameters such as quark masses, volume, temperature and chemical potential are varied. We discuss the progress in this field from a theoretical perspective, focusing on non-perturbative QCD as encoded in the functional approach via Dyson-Schwinger and Bethe-Salpeter equations. We aim at a pedagogical overview on the physics associated with the structure of this framework and explain connections to other approaches, in particular with the functional renormalization group and lattice QCD. We discuss various aspects associated with the variation of the quark masses, assess recent results for the QCD phase diagram including the location of a putative critical end-point for $N_f=2+1$ and $N_f=2+1+1$, discuss results for quark spectral functions and summarise aspects of QCD thermodynamics and fluctuations., Comment: 80 pages; invited review article, v2: typos corrected, figures updated, comments added; version accepted by PPNP

We report results on the quark spectral function in the Landau gauge at finite temperature determined from its Dyson-Schwinger equation. Compared to earlier quenched results this study encompasses unquenched $N_f=2+1$ fermion flavors in the medium. For the reconstruction of real-time spectra we deploy a recent Bayesian approach (BR method) and develop a new prior in order to better assess the inherent systematic uncertainties. We identify the quark quasi-particle spectrum and analyze the (non-)appearance of zero modes at or around the pseudo-critical temperature. In both, the fully unquenched system and a simpler truncation using a model for the gluon propagator we observe a characteristic three-peak structure at zero three-momentum. The temperature dependence of these structures in case of the gluon propagator model is different than observed in previous studies. For the back-coupled and unquenched case we find interesting modifications at and around the pseudo-critical transition temperature., Comment: 18 pages, 21 figures

We complement studies of the neutral pion transition form factor pi^0 --> gamma^(*) gamma^(*) with calculations for the electromagnetic decay widths of the processes pi^0 --> e^+ e^-, pi^0 --> e^+ e^- gamma and pi^0 --> e^+ e^- e^+ e^-. Their common feature is that the singly- or doubly-virtual transition form factor serves as a vital input that is tested in the non-perturbative low-momentum region of QCD. We determine this form factor from a well-established and symmetry-preserving truncation of the Dyson-Schwinger equations. Our results for the three- and four-body decays match results of previous theoretical calculations and experimental measurements. For the rare decay we employ a numerical method to calculate the process directly by deforming integration contours, which in principle can be generalized to arbitrary integrals as long as the analytic structure of the integrands are known. Our result for the rare decay is in agreement with dispersive calculations but still leaves a 2 sigma discrepancy between theory and experiment., Comment: 13 pages, 9 figures, 3 tables, v2: typos corrected, Refs. added

We study the transition of non-perturbative to perturbative QCD in situations with possible violations of scaling limits. To this end we consider the singly- and doubly-virtual pion transition form factor $\pi^0\to\gamma\gamma$ at all momentum scales of symmetric and asymmetric photon momenta within the Dyson-Schwinger/Bethe-Salpeter approach. For the doubly virtual form factor we find good agreement with perturbative asymptotic scaling laws. For the singly-virtual form factor our results agree with the Belle data. At very large off-shell photon momenta we identify a mechanism that introduces quantitative modifications to Efremov-Radyushkin-Brodsky-Lepage scaling., Comment: 5 pages, 7 figures, v3:contents revised, version published in PLB

We study ground states and excitations of light octet and decuplet baryons within the framework of Dyson-Schwinger and Faddeev equations. We improve upon similar approaches by explicitly taking into account the momentum-dependent dynamics of the quark-gluon interaction that leads to dynamical chiral symmetry breaking. We perform calculations in both the three-body Faddeev framework and the quark-diquark approximation in order to assess the impact of the latter on the spectrum. Our results indicate that both approaches agree well with each other. The resulting spectra furthermore agree one-to-one with experiment, provided well-known deficiencies of the rainbow-ladder approximation are compensated for. We also discuss the mass evolution of the Roper and the excited Delta with varying pion mass and analyse the internal structure in terms of their partial wave decompositions., Comment: 12 pages, 7 figures

We review the spectrum and electromagnetic properties of baryons described as relativistic three-quark bound states within QCD. The composite nature of baryons results in a rich excitation spectrum, whilst leading to highly non-trivial structural properties explored by the coupling to external (electromagnetic and other) currents. Both present many unsolved problems despite decades of experimental and theoretical research. We discuss the progress in these fields from a theoretical perspective, focusing on nonperturbative QCD as encoded in the functional approach via Dyson-Schwinger and Bethe-Salpeter equations. We give a systematic overview as to how results are obtained in this framework and explain technical connections to lattice QCD. We also discuss the mutual relations to the quark model, which still serves as a reference to distinguish 'expected' from 'unexpected' physics. We confront recent results on the spectrum of non-strange and strange baryons, their form factors and the issues of two-photon processes and Compton scattering determined in the Dyson-Schwinger framework with those of lattice QCD and the available experimental data. The general aim is to identify the underlying physical mechanisms behind the plethora of observable phenomena in terms of the underlying quark and gluon degrees of freedom., Comment: 140 pages, 52 figures, 3 tables. Invited review for Progress in Particle and Nuclear Physics. Changes from v1: minor additions, corrected typos and added references; brief discussion on light-front and holographic QCD in Sec. 3.2; updated Figs. 3.22-3.23 and corresponding discussion

We present a calculation of the electric and magnetic form factors of ground-state octet and decuplet baryons including strange quarks. We work with a combination of Dyson-Schwinger equations for the quark propagator and covariant Bethe-Salpeter equations describing baryons as bound states of three (non-perturbative) quarks. Our form factors for the octet baryons are in good agreement with corresponding lattice data at finite $Q^2$; deviations in some isospin channels for the magnetic moments can be explained by missing meson cloud effects. At larger $Q^2$ our quark core calculation has predictive power for both, the octet and decuplet baryons., Comment: 14 pages, 6 figures, 7 tables. Version 2: layout improved. Version 3: Matches the published version

We investigate the impact of unquenching effects on QCD Green's functions, in the form of quark-loop contributions to both the gluon propagator and three-gluon vertex, in a three-loop inspired truncation of the three-particle irreducible (3PI) effective action. The fully coupled system of Dyson-Schwinger equations for the quark-gluon-, ghost-gluon- and three-gluon vertices, together with the quark propagator, are solved self-consistently; our only input are the ghost and gluon propagators themselves that are constrained by calculations within Lattice QCD. We find that the two different unquenching effects have roughly equal, but opposite, impact on the quark-gluon vertex and quark propagator, with an overall negative impact on the latter. By taking further derivatives of the 3PI effective action, we construct the corresponding quark-antiquark kernel of the Bethe-Salpeter equation for mesons. The leading component is gluon exchange between two fully-dressed quark-gluon vertices, thus introducing for the first time an obvious scalar-scalar component to the binding. We gain access to time-like properties of bound-states by analytically continuing the coupled system of Dyson--Schwinger equations to the complex plane. We observe that the vector axial-vector splitting is in accord with experiment and that the lightest quark-antiquark scalar meson is above 1 GeV in mass., Comment: 13 pages, 14 figures, 1 table, v2: minor changes, version accepted by PRD

The location of the critical end point of QCD has been determined in previous studies of $N_f=2+1$ and $N_f=2+1+1$ dynamical quark flavors using a (truncated) set of Dyson-Schwinger equations for the quark and gluon propagators of Landau-gauge QCD. A source for systematic errors in these calculations has been the omission of terms in the quark-gluon interaction that can be parametrized in terms of baryonic degrees of freedom. These have a potentially large dependence on chemical potential and therefore may affect the location of the critical end point. In this exploratory study we estimate the effects of these contributions, both in the vacuum and at finite temperature and chemical potential. We find only a small influence of baryonic contributions on the location of the critical end point. We estimate the robustness of this result by parameterizing further dependencies on chemical potential., Comment: 12 pages, 8 figures, v2: minor changes, version accepted by PRD

We present a numerical solution of the four-quark Bethe-Salpeter equation for ground-state scalar tetraquarks with J^PC = 0^++. We find that the four-body equation dynamically generates pseudoscalar-meson poles in the Bethe-Salpeter amplitude. The resulting tetraquarks are genuine four-quark states that are dominated by pseudoscalar meson-meson correlations. Diquark-antidiquark contributions are subleading because of their larger mass scale. In the light quark sector, the sensitivity of the tetraquark wave function to the pion poles leads to an isoscalar tetraquark mass M_sigma ~ 350 MeV which is comparable to that of the sigma/f0(500). The masses of its multiplet partners kappa and a0/f0 follow a similar pattern. This provides support for a tetraquark interpretation of the light scalar meson nonet in terms of 'meson molecules'., Comment: 7 pages, 4 figures; version accepted by PLB

We present a numerical solution of the four-quark Bethe-Salpeter equation for a scalar tetraquark. We find that the four-body equation dynamically generates pseudoscalar poles in the Bethe-Salpeter amplitude. The sensitivity to the pion poles leads to a light isoscalar tetraquark mass M ~ 400 MeV, which is comparable to that of the sigma/f0(500). The masses of its multiplet partners kappa and a0/f0 follow a similar pattern, thereby providing support for the tetraquark interpretation of the light scalar nonet., Comment: 7 pages, 4 figures. Proceedings of the workshop "Excited QCD 2015", Tatranska Lomnica, Slovakia, March 8-14, 2015. To be published in Acta Physica Polonica B Proc. Suppl

Four-point functions are at the heart of many interesting physical processes. A prime example is the light-by-light scattering amplitude, which plays an important role in the calculation of hadronic contributions to the anomalous magnetic moment of the muon. In the calculation of such quantities one faces the challenge of finding a suitable and well-behaved basis of tensor structures in coordinate and/or momentum space. Provided all (or many) of the external legs represent similar particle content, a powerful tool to construct and organize such bases is the permutation group S4. We introduce an efficient notation for dealing with the irreducible multiplets of S4, and we highlight the merits of this treatment by exemplifying four-point functions with gauge-boson legs such as the four-gluon vertex and the light-by-light scattering amplitude. The multiplet analysis is also useful for isolating the important kinematic regions and the dynamical singularity content of such amplitudes. Our analysis serves as a basis for future efficient calculations of these and similar objects., Comment: 23 pages, 9 figures, 5 tables

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., Comment: 11 pages, 7 figures, 2 tables. Proceedings of the 'XIth Quark Confinement and the Hadron Spectrum', September 8-12, 2014, St. Petersburg, Russia. Added references & fixed typos

We present a calculation of the spectrum of charmonia, bottomonia and B_c-meson states with `ordinary' and exotic quantum numbers. We discuss the merits and limitations of a rainbow-ladder truncation of Dyson-Schwinger and Bethe-Salpeter equations and explore the effects of different shapes of the effective running coupling on ground and excited states in channels with quantum numbers J <= 3. We furthermore discuss the status of the X(3872) as a potential (excited) quark-antiquark state and give predictions for the masses of charmonia and bottomonia in the tensor channels with J=2,3., Comment: 9 pages, 8 figures, 1 table. Version to appear in Eur. Phys J. A

We present the mini-proceedings of the workshops Hadronic contributions to the muon anomalous magnetic moment: strategies for improvements of the accuracy of the theoretical prediction and $(g-2)_{\mu}$: Quo vadis?, both held in Mainz from April 1$^{\rm rst}$ to 5$^{\rm th}$ and from April 7$^{\rm th}$ to 10$^{\rm th}$, 2014, respectively., Comment: 84 pages, 23+18 contributions. Editors: Tom Blum, Pere Masjuan, and Marc Vanderhaeghen

We extend the calculation of relativistic bound-states of a fermion anti-fermion pair in the Bethe-Salpeter formalism to the case of total angular momentum $J=3$. Together with results for $J \le 2$ this allows for the investigation of Regge trajectories in this approach. We exemplify such a study for ground and excited states of light unflavored mesons as well as strange mesons within the rainbow-ladder approximation. For the $\rho$- and $\phi$-meson we find a linear Regge trajectory within numerical accuracy. Discrepancies with experiment in other channels highlight the need to go beyond rainbow-ladder and to consider effects such as state mixing and more sophisticated quark-antiquark interaction kernels., Comment: 11 pages, 8 figures, 2 tables. Minor typos corrected; accepted in EPJA

We investigate the phase structure of QCD at finite temperature and light-quark chemical potential. We improve upon earlier results for N_f=2+1 dynamical quark flavors and investigate the effects of charm quarks in an extension to N_f=2+1+1. We determine the quark condensate and the Polyakov loop potential using solutions of a coupled set of (truncated) Dyson-Schwinger equations for the quark and gluon propagators of Landau gauge QCD. At zero chemical potential we find excellent agreement with results from lattice-QCD. With input fixed from physical observables we find only a very small influence of the charm quark on the resulting phase diagram at finite chemical potential. We discuss the location of the emerging critical end-point and compare with expectations from lattice gauge theory., Comment: 10 pages, 9 figures, v2: typos corrected, minor changes, version accepted by PRD

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., Comment: 13 pages, 6 figures; v2: typos corrected, colors improved

We report on recent results of a calculation of the nucleon and delta masses in a covariant bound-state approach, where to the simple rainbow-ladder gluon-exchange interaction kernel we add a pion-exchange contribution to account for pion cloud effects. We observe good agreement with lattice data at large pion masses. At the physical point our masses are too large by about five percent, signaling the need for more structure in the gluon part of the interaction., Comment: 4 pages, 3 figures, Proceedings of The 13th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon (MENU 2013), Rome

In this work we explore the effect of pion cloud contributions to the mass of the nucleon and the delta baryon. To this end we solve a coupled system of Dyson-Schwinger equations for the quark propagator, a Bethe-Salpeter equation for the pion and a three-body Faddeev equation for the baryons. In the quark-gluon interaction we explicitly resolve the term responsible for the back-coupling of the pion onto the quark, representing rainbow-ladder like pion cloud effects in bound states. We study the dependence of the resulting baryon masses on the current quark mass and discuss the internal structure of the baryons in terms of a partial wave decomposition. We furthermore determine values for the nucleon and delta sigma-terms., Comment: 9 pages, 4 figures, 2 tables. v2: Numerics corrected; results updated; discussion extended. Version accepted for publication in Phys.Lett.B

We investigate the effect of a strong magnetic field on dynamical chiral symmetry breaking in quenched and unquenched QCD. To this end we apply the Ritus formalism to the coupled set of (truncated) Dyson-Schwinger equations for the quark and gluon propagator under the presence of an external constant Abelian magnetic field. We work with an approximation that is trustworthy for large fields eH > \Lambda_{QCD}^2 but is not restricted to the lowest Landau level. We confirm the linear rise of the quark condensate with large external field previously found in other studies and observe the transition to the asymptotic power law at extremely large fields. We furthermore quantify the validity of the lowest Landau level approximation and find substantial quantitative differences to the full calculation even at very large fields. We discuss unquenching effects in the strong field propagators, condensate and the magnetic polarization of the vacuum. We find a significant weakening of magnetic catalysis caused by the back reaction of quarks on the Yang-Mills sector. Our results support explanations of the inverse magnetic catalysis found in recent lattice studies due to unquenching effects., Comment: 21 pages, 9 figures; v2: references added, typos and numerics corrected; results qualitatively unchanged

We employ the Dyson-Schwinger equations for quark and gluon propagators in order to study QCD with 2+1 flavours at finite temperature and density. In a suitable truncation for these equations, we determine the position of the critical end-point as well as the deconfinement temperature at all chemical potentials. For the latter, the Polyakov-loop potential is obtained from the QCD propagators. This is possible for the first time at finite chemical potential, with implications for effective models., Comment: Proceedings for the 8th International Workshop on Critical Point and Onset of Deconfinement (CPOD 2013). 5 pages, 5 figures

The Polyakov loop potential serves to distinguish between the confined hadronic and the deconfined quark-gluon plasma phases of QCD. For Nf=2+1 quark flavors with physical masses we determine the Polyakov loop potential at finite temperature and density and extract the location of the deconfinement transition. We find a cross-over at small values of the chemical potential running into a critical end-point at mu/T > 1., Comment: 5 pages, 9 figures

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., Comment: Proceedings of Excited QCD 2013, Bjelasnica Mountain, Sarajevo, February 3rd-9th. 6 pages

We summarize results from our recent paper* on nucleon Compton scattering in the Dyson-Schwinger approach. We study the Compton scattering phase space and its various kinematic limits, and we introduce a covariant, transverse tensor basis for the scattering amplitude that is free of kinematic singularities. We illustrate the decomposition of the Compton amplitude at the quark-gluon level and calculate its nonperturbative handbag part, which also includes all t-channel meson resonances. The calculation requires numerical solutions for the dressed quark propagator, the nucleon's Faddeev amplitude and the quark Compton vertex as its input; they are consistently obtained from Dyson-Schwinger, Bethe-Salpeter and Faddeev equations. As a first application, we verify that the result for the Compton scattering amplitude reproduces the pi-gamma-gamma transition form factor when the pion pole in the t channel is approached. *G. Eichmann and C. S. Fischer, arXiv:1212.1761 [hep-ph]., Comment: 8 pages, 4 figures, 1 table. Proceedings of the Xth Quark Confinement and the Hadron Spectrum, October 7-12, Munich, Germany

We analyze the nucleon's Compton scattering amplitude in the Dyson-Schwinger/Faddeev approach. We calculate a subset of diagrams that implements the nonperturbative handbag contribution as well as all t-channel resonances. At the quark level, these ingredients are represented by the quark Compton vertex whose analytic properties we study in detail. We derive a general form for a fermion two-photon vertex that is consistent with its Ward-Takahashi identities and free of kinematic singularities, and we relate its transverse part to the onshell nucleon Compton amplitude. We solve an inhomogeneous Bethe-Salpeter equation for the quark Compton vertex in rainbow-ladder truncation and implement it in the nucleon Compton scattering amplitude. The remaining ingredients are the dressed quark propagator and the nucleon's bound-state amplitude which are consistently solved from Dyson-Schwinger and covariant Faddeev equations. We verify numerically that the resulting quark Compton vertex and nucleon Compton amplitude both reproduce the pion-photon transition form factor when the pion pole in the t-channel is approached., Comment: 36 pages, 10 figures, 5 tables

In this talk we present results for the propagators and the phase diagram of QCD, obtained from a truncated set of Dyson-Schwinger equations at finite temperature T and quark chemical potential mu. We include back-coupling effects of the dressed quark and gluon propagators, which also allows us to study the influence of strange quarks. For the phase diagram we find a critical end-point at mu/T = 1.9 and coinciding chiral and deconfinement phase transitions., Comment: 5 pages, 3 figures. FAIRNESS 2012 proceedings

We present a refined calculation of the quark-loop contribution to hadronic light-by-light scatter- ing that focuses upon the impact of the transverse components of the quark-photon vertex. These structures are compared and contrasted with those found within the extended NJL-models. We discuss similarities and differences between the two approaches and further clarify the important role of momentum dependent dressing functions. As expected we find that the transverse structures of the quark-photon vertex lead to a suppression of the quark-loop contribution to the anomalous magnetic moment of the muon. However, we find evidence that this suppression is overestimated within models with simple approximations for the quark-photon interaction., Comment: 13 pages, 5 figures, v2: typos corrected, references added, version submitted to PRD

We investigate the phase structure of QCD at finite temperature and chemical potential by solving a coupled set of truncated Dyson-Schwinger equations for the quark and gluon propagator. In contrast to previous calculations we take into account the full back-reaction of the quarks onto the Yang-Mills sector and we include the effects of strange quarks. We discuss the resulting thermal mass of the unquenched gluon propagator and extract order parameters for the chiral and deconfinement transition from the quarks. Our result for the temperature dependence of the quark condensate at zero chemical potential agrees well with corresponding lattice calculations. We determine the phase diagram at finite chemical potential and find a potential critical endpoint at (mu_q,T) = (190,100) MeV., Comment: 11 pages, 9 figures, 1 table

We compute the axial and pseudoscalar form factors of the nucleon in the Dyson-Schwinger approach. To this end, we solve a covariant three-body Faddeev equation for the nucleon wave function and determine the matrix elements of the axialvector and pseudoscalar isotriplet currents. Our only input is a well-established and phenomenologically successful ansatz for the nonperturbative quark-gluon interaction. As a consequence of the axial Ward-Takahashi identity that is respected at the quark level, the Goldberger-Treiman relation is reproduced for all current-quark masses. We discuss the timelike pole structure of the quark-antiquark vertices that enters the nucleon matrix elements and determines the momentum dependence of the form factors. Our result for the axial charge underestimates the experimental value by 20-25% which might be a signal of missing pion-cloud contributions. The axial and pseudoscalar form factors agree with phenomenological and lattice data in the momentum range above Q^2 ~ 1...2 GeV^2., Comment: 17 pages, 7 figures, 1 table

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 $a_\mu^\mathrm{LBL}=(217 \pm 91) \times 10^{-11}$., Comment: Contribution to the proceedings of 'International school of nuclear physics, 33rd course', Erice-Sicily: 16 - 24 September 2011

We derive the expression for the nonperturbative coupling of a hadron to two external currents from its underlying structure in QCD. Microscopically, the action of each current is resolved to a coupling with dressed quarks. The Lorentz structure of the currents is arbitrary and thereby allows to describe the hadron's interaction with photons as well as mesons in the same framework. We analyze the ingredients of the resulting four-point functions and explore their potential to describe a variety of processes relevant in experiments. Possible applications include Compton scattering, the study of two-photon effects in hadron form factors, pion photo- and electroproduction on a nucleon, nucleon-pion or pion-pion scattering., Comment: 15 pages, 6 figures

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 $a_\mu$., Comment: 9 pages, 8 figures

We investigate the critical scaling of the quark propagator of N_f=2 QCD close to the chiral phase transition at finite temperature. We argue that it is mandatory to take into account the back-reaction effects of pions and the sigma onto the quark to observe critical behavior beyond mean field. On condition of self-consistency of the quark Dyson-Schwinger equation we extract the scaling behavior for the quark propagator analytically. Crucial in this respect is the correct pion dispersion relation when the critical temperature is approached from below. Our results are consistent with the known relations for the quark condensate and the pion decay constant from universality. We verify the analytical findings also numerically assuming the critical dispersion relation for the Goldstone bosons., Comment: 9 pages, 6 figures

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 $a_\mu=116\,591\,891.0(105.0)\times 10^{-11}$., Comment: 14 pages, 16 figures, 2 tables. Version to appear in Phys. Rev. D

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., Comment: 3 pages, 5 figures; Contribution to the proceedings of `Quark Confinement And The Hadron Spectrum IX', Madrid, Sept. 2010

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}., Comment: 5 pages, 7 figures, v2: title slightly changed, minor corrections, version accepted by EPJ

We report on an analysis of the quark spectral representation at finite temperatures based on the quark propagator determined from its Dyson-Schwinger equation in Landau gauge. In Euclidean space we achieve nice agreement with recent results from quenched lattice QCD. We find different analytical properties of the quark propagator below and above the deconfinement transition. Using a variety of ansaetze for the spectral function we then analyze the possible quasiparticle spectrum, in particular its quark mass and momentum dependence in the high temperature phase. This analysis is completed by an application of the Maximum Entropy Method, in principle allowing for any positive semi-definite spectral function. Our results motivate a more direct determination of the spectral function in the framework of Dyson-Schwinger equations.

We study a gauge invariant order parameter for deconfinement and the chiral condensate in SU(2) and SU(3) Yang-Mills theory in the vicinity of the deconfinement phase transition using the Landau gauge quark and gluon propagators. We determine the gluon propagator from lattice calculations and the quark propagator from its Dyson-Schwinger equation, using the gluon propagator as input. The critical temperature and a deconfinement order parameter are extracted from the gluon propagator and from the dependency of the quark propagator on the temporal boundary conditions. The chiral transition is determined using the quark condensate as order parameter. We investigate whether and how a difference in the chiral and deconfinement transition between SU(2) and SU(3) is manifest., Comment: 15 pages, 9 figures. For clarification one paragraph and two references added in the introduction and two sentences at the end of the first and last paragraph of the summary. Appeared in EPJC

We investigate masses of light mesons from a coupled system of Dyson--Schwinger (DSE) and Bethe--Salpeter equations (BSE), taking into account dominant non-Abelian, sub-leading Abelian, and dominant pion cloud contributions to the dressed quark-gluon vertex. The axial-vector Ward-Takahashi identity preserving Bethe-Salpeter kernel is constructed and the spectrum of light mesons calculated. Our model goes significantly beyond the rainbow-ladder. We find that sub-leading Abelian corrections are further dynamically suppressed, and that our results supersede early qualitative predictions from simple truncation schemes., Comment: 4 pages, 3 figures, 1 table. Talk given at the 5-th Int. Conf. on Quarks and Nuclear Physics, Beijing, September 21-26,2009

We determine the quark condensate and the dressed Polyakov loop from the finite temperature Landau gauge quark propagator evaluated with U(1)-valued boundary conditions in an approximation to quenched QCD. These gauge invariant quantities allow for an investigation of the chiral and deconfinement transition. We compare results from Dyson-Schwinger equations on a lattice with infinite volume continuum results and study the temperature and quark mass dependence of both quantities. In particular we investigate the chiral condensate and the dressed Polyakov loop in the chiral limit. We also consider an alternative order parameter for the deconfinement transition, the dual scalar quark dressing, and compare with the dressed Polyakov loop. As a result we find only slightly different transition temperatures for the chiral and the deconfinement transition at finite quark masses; in the chiral limit both transitions coincide., Comment: 15 pages, 7 figures; v2: minor corrections and clarifications, version accepted by PRD

We investigate masses and decay constants of light mesons from a coupled system of Dyson--Schwinger and Bethe--Salpeter equations. We explicitly take into account dominant non-Abelian contributions to the dressed quark-gluon vertex stemming from the gluon self-interaction. We construct the corresponding Bethe-Salpeter kernel that satisfies the axial-vector Ward-Takahashi identity. Our numerical treatment fully includes all momentum dependencies with all equations solved completely in the complex plane. This approach goes well beyond the rainbow-ladder approximation and permits us to investigate the influence of the gluon self-interaction on the properties of mesons. As a first result we find indications of a nonperturbative cancellation of the gluon self-interaction contributions and pion cloud effects in the mass of the rho-meson., Comment: 4 pages, 5 figures. Matches published version in PRL