Search

We present results for single axial-vector and scalar meson pole contributions to the hadronic light-by-light scattering (HLbL) part of the muon's anomalous magnetic moment. In the dispersive approach to these quantities (in narrow width approximation) the central inputs are the corresponding space-like electromagnetic transition form factors. We determine these directly using a functional approach to QCD by Dyson-Schwinger and Bethe-Salpeter equations in the very same setup we used previously to determine pseudo-scalar meson exchange ($\pi$, $\eta$ and $\eta'$) as well as meson ($\pi$ and $K$) box contributions. Particular care is taken to preserve gauge invariance and to comply with short distance constraints in both the form factors and the HLbL tensor. Our result for the contributions from a tower of axial-vector states including short distance constraints is $a_\mu^{\text{HLbL}}[\text{AV-tower+SDC}] = 27.5 \,(3.2) \times 10^{-11}$. For the combined contributions from $f_0(980), a_0(980), f_0(1370)$ and $a_0(1450)$ we find $a_\mu^{\text{HLbL}}[\text{scalar}] = -1.6 \,(5) \times 10^{-11}$., Comment: 14 pages, 10 figures

We present quantitative results for masses and the internal structure of four-quark states with two heavy quarks, i.e. $QQ'\bar{q}\bar{q}'$ with $Q,Q'\in \{c,b\}$ and $q,q' \in \{u,d,s\}$, and $J^{P} \in {1^+}$. The composition of these states in terms of meson-meson and diquark-antidiquark pairs, extracted from a relativistic four-body Faddeev-Yakubowski equation, is dynamically determined from underlying QCD forces. We find states at energy levels in very good agreement with lattice QCD and, where available, with experimental states. Their internal structure, most notably between the $T^+_{cc}$, $T^+_{bc}$ and $T^+_{bb}$, show significant and sizeable variations., Comment: 4 pages, 6 figures, 4 appendices

We discuss the spectrum and the internal composition of ground and excited four-quark states in the charm and bottom energy region. To this end we extend previous calculations within the framework of the relativistic four-body Faddeev-Yakubovsky equation to include quantum numbers with $J^{P C} = 0^{++} , 0^{-+} , 1^{--} , 1^{+-}$ and $1^{+ +}$ and study their internal composition in terms of heavy-light meson pairs, hadroquarkonia and diquark-antidiquark clusters. We observe similar patterns in the charm and bottom energy region with different compositions of the four-quark states depending on $J^{P C}$ quantum numbers. Most notably, we find that all states with $C \cdot P = +1$ are dominated by heavy-light meson contributions, whereas for axialvector states with $J^{P C} = 1^{+-}$ including the $Z_c (3900)$ we find a much more complicated picture depending on the flavour content. We systematically compare our results for the spectrum with existing experimental results and provide predictions for future analyses., Comment: 14 pages, 11 figures

We compute the bound state properties of three-dimensional scalar $\phi^4$ theory in the broken phase. To this end, we extend the recently developed technique of spectral Dyson-Schwinger equations to solve the Bethe-Salpeter equation and determine the bound state spectrum. We employ consistent truncations for the two-, three- and four-point functions of the theory that recover the scaling properties in the infinite coupling limit. Our result for the mass of the lowest-lying bound state in this limit agrees very well with lattice determinations., Comment: 15 pages, 16 figures

We outline the physics opportunities provided by the Electron Ion Collider (EIC). These include the study of the parton structure of the nucleon and nuclei, the onset of gluon saturation, the production of jets and heavy flavor, hadron spectroscopy and tests of fundamental symmetries. We review the present status and future challenges in EIC theory that have to be addressed in order to realize this ambitious and impactful physics program, including how to engage a diverse and inclusive workforce. In order to address these many-fold challenges, we propose a coordinated effort involving theory groups with differing expertise is needed. We discuss the scientific goals and scope of such an EIC Theory Alliance., Comment: 44 pages, ReVTeX, White Paper on EIC Theory Alliance

In these introductory notes I give a brief overview on some theoretical aspects of light baryon spectroscopy. The first part contains a discussion of the symmetries of the baryon spectrum, the construction of flavor wave functions and some basic features of quark models. The second part examines the need for relativistic quantum field theory and focusses on spectrum calculations with functional methods., Comment: 20 pages, 11 figures, 1 table. Contribution to the 2021 School on the Physics of Baryons, Oct 18-22 2021, Sevilla

In covariant gauges, the gluonic mass gap in Yang-Mills theory manifests itself in the basic observation that the massless pole in the perturbative gluon propagator disappears in nonperturbative calculations, but the origin of this behavior is not yet fully understood. We summarize a recent study of the respective dynamics with Dyson-Schwinger equations in Landau-gauge Yang-Mills theory. We identify the parameter that distinguishes the massive Yang-Mills regime from the massless decoupling solutions, whose endpoint is the scaling solution. Similar to the PT-BFM scheme, we find evidence that mass generation in the transverse sector is triggered by longitudinal massless poles., Comment: 10 pages, 6 figures. Proceedings contribution to the XXXIII International (Online) Workshop on High Energy Physics "Hard Problems of Hadron Physics: Non-Perturbative QCD & Related Quests", November 8-12, 2021

We explore a new method to calculate the valence light-front wave function of a system of two interacting particles, which is based on contour deformations combined with analytic continuation methods to project the Bethe-Salpeter wave function onto the light front. In this proof-of-concept study, we solve the Bethe-Salpeter equation for a scalar model and find excellent agreement between the light-front wave functions obtained with contour deformations and those obtained with the Nakanishi method frequently employed in the literature. The contour-deformation method is also able to handle extensions of the scalar model that mimic certain features of QCD such as unequal masses and complex singularities. In principle the method is suitable for computing parton distributions on the light front such as PDFs, TMDs and GPDs in the future., Comment: 20 pages, 24 figures. Coincides with the version published in Phys. Rev. D

A longstanding question in QCD is the origin of the mass gap in the Yang-Mills sector of QCD, i.e., QCD without quarks. In Landau gauge QCD this mass gap, and hence confinement, is encoded in a mass gap of the gluon propagator, which is found both in lattice simulations and with functional approaches. While functional methods are well suited to unravel the mechanism behind the generation of the mass gap, a fully satisfactory answer has not yet been found. In this work we solve the coupled Dyson-Schwinger equations for the ghost propagator, gluon propagator and three-gluon vertex. We corroborate the findings of earlier works, namely that the mass gap generation is tied to the longitudinal projection of the gluon self-energy, which acts as an effective mass term in the equations. Because an explicit mass term is in conflict with gauge invariance, this leaves two possible scenarios: If it is viewed as an artifact, only the scaling solution survives; if it is dynamical, gauge invariance can only be preserved if there are longitudinal massless poles in either of the vertices. We find that there is indeed a massless pole in the ghost-gluon vertex, however in our approximation with the assumption of complete infrared dominance of the ghost this pole is only present for the scaling solution. We also put forward a possible mechanism that may reconcile the scaling solution, with an infrared dominance of the ghost, with the decoupling solutions based on longitudinal poles in the three-gluon vertex as seen in the PT-BFM scheme., Comment: 33 pages, 24 figures. Identical to the version published in Phys. Rev. D

In a recent comment [1] Blankleider and Kvinikhidze claim that our work [2] is based on a set of inconsistent Bethe-Salpeter equations for the coupling of four-quark to two-quark states. Here we demonstrate that their argument is insubstantial., Comment: 3 pages, 1 figure

In this feature article we summarise and highlight aspects of the treatment of four-quark states with functional methods. Model approaches to those exotic mesons almost inevitably have to assume certain internal structures, e.g. by grouping quarks and antiquarks into (anti-)diquark clusters or heavy-light $q\bar{q}$ pairs. Functional methods using Dyson-Schwinger and Bethe-Salpeter equations can be formulated without such prejudice and therefore have the potential to put these assumptions to test and discriminate between such models. So far, functional methods have been used to study the light scalar-meson sector and the heavy-light sector with a pair of charmed and a pair of light quarks in different quantum number channels. For all these states, the dominant components in terms of internal two-body clustering have been identified. It turns out that chiral symmetry breaking plays an important role for the dominant clusters in the light meson sector (in particular for the scalar mesons) and that this property is carried over to the heavy-light sector. Diquark-antidiquark components, on the other hand, turn out to be almost negligible for most states with the exception of open-charm heavy-light exotics., Comment: 19 pages, 9 figures, v2: typos corrected, version accepted by Few Body Syst

We study the dynamical generation of resonances in isospin singlet channels with mixing between two- and four-quark states. To this end we generalise a Bethe-Salpeter approach to four-quark states employed previously \cite{Heupel:2012ua} to accommodate for mixing diagrams. The $q\bar{q}q\bar{q}$ and $q\bar{q}$ components of the Bethe-Salpeter wave function (with light quarks $q\in\{u,d\}$) are determined consistently in a symmetry-preserving truncation of the underlying Dyson-Schwinger equations. As a prominent example we deal with the isospin-singlet $0^{++}$ meson with light quark content. We find that the $\pi\pi$ contribution of the four-quark component is mainly responsible for the low (real part of the) mass of the resulting state. We also study the analytic structure in the complex momentum plane and find a branch cut at the two-pion threshold and a singularity in the second Riemann sheet indicating a considerable decay width. Our findings are in excellent qualitative agreement with the general picture for the $\sigma/f_0(500)$ that emerged in the past two decades from dispersive approaches \cite{Pelaez:2015qba}., Comment: 10 pages, 7 figures, v2: minor changes, version accepted by PRD

Models proposed to explain recently discovered heavy-light four-quark states already assume certain internal structures, i.e. the (anti)quark constituents are grouped into diquark/antidiquark clusters, heavy-meson/light-meson clusters (hadrocharmonium) or heavy-light meson molecules. We propose and use an approach to four-quark states based on Dyson-Schwinger and Bethe-Salpeter equations that has the potential to discriminate between these models. We study the masses of heavy-light $cq\bar{q}\bar{c}$ and $cc\bar{q}\bar{q}$ four-quark states with $q=u,d,s$ and quantum numbers $I(J^{PC})=0(1^{++}),1(1^{+-}),0(0^{++})$ and $1(0^+),0(1^+),1(1^+)$. We identify the dominant components of the ground states with these quantum numbers and suggest candidates for corresponding experimental states. Most notably, we find strong heavy-light meson-meson and negligible diquark-antidiquark components in all $cq\bar{q}\bar{c}$ states, whereas for $cc\bar{q}\bar{q}$ states diquarks are present. A potential caveat in the $I=0$ channels is the necessary but costly inclusion of $c\bar{c}$ components which is relegated to future work., Comment: 10 pages, 4 figures, v2: minor corrections, version published in PRD rapid communications

We give a brief summary of the Dyson-Schwinger and Bethe-Salpeter approach to hadron spectroscopy and report on recent progress in determining resonance properties in this framework. We exemplify the extraction of resonances using a scalar model, where we solve the scattering equation for the four-point scattering amplitude and extract the pole locations on the second Riemann sheet as well as the phase shifts., Comment: 7 pages, 5 figures. Proceedings for Light Cone 2019, September 16-20, 2019, Ecole Polytechnique, Palaiseau, France

We present results for the charged kaon-box contributions to the hadronic light-by-light (HLBL) correction of the muon's anomalous magnetic moment. To this end we determine the kaon electromagnetic form factor within the functional approach to QCD using Dyson-Schwinger and Bethe-Salpeter equations and evaluate the kaon-box contribution as defined in the dispersive approach to HLBL. As an update to previous work we also re-evaluate the charged pion-box contribution taking effects due to isospin breaking into account. Our results are $a_\mu^{\pi^\pm-\text{box}} = -15.7 \,(2)(3) \times 10^{-11}$ and $a_\mu^{K^\pm-\text{box}} = -0.48 \,(2)(4) \times 10^{-11}$ thus confirming the large suppression of box contributions beyond the leading pion box., Comment: 6 pages, 4 figures; v3: improved numerics, updated number for kaon-box, conclusions unchanged. arXiv admin note: text overlap with arXiv:1903.10844

We employ a scalar model to exemplify the use of contour deformations when solving Lorentz-invariant integral equations for scattering amplitudes. In particular, we calculate the onshell 2 -> 2 scattering amplitude for the scalar system. The integrals produce branch cuts in the complex plane of the integrand which prohibit a naive Euclidean integration path. By employing contour deformations, we can also access the kinematical regions associated with the scattering amplitude in Minkowski space. We show that in principle a homogeneous Bethe-Salpeter equation, together with analytic continuation methods such as the Resonances-via-Pad\'e method, is sufficient to determine the resonance pole locations on the second Riemann sheet. However, the scalar model investigated here does not produce resonance poles above threshold but instead virtual states on the real axis of the second sheet, which pose difficulties for analytic continuation methods. To address this, we calculate the scattering amplitude on the second sheet directly using the two-body unitarity relation which follows from the scattering equation., Comment: 21 pages, 25 figures

We generalise the framework of Dyson-Schwinger and Bethe-Salpeter equations for four-quark states to accommodate the case of unequal quark masses. As a first application, we consider the quantum numbers $I(J^{PC})=0(1^{++})$ of the $X(3872)$ and study the four-quark states with quark contents $cq\bar{q}\bar{c}$ and $cs\bar{s}\bar{c}$. Their Bethe-Salpeter amplitudes are represented by a basis of heavy-light meson-meson, hadro-charmonium and diquark-antidiquark operators, which allows for a dynamical distinction between different internal configurations. In both cases we find the heavy-light meson-meson component to be dominant. For the putative $X(3872)$ we obtain a mass of $3916(74)$ MeV; the corresponding $cs\bar{s}\bar{c}$ state is predicted at $4068(61)$ MeV., Comment: 8 pages, 2 figures; v2: minor typos corrected

We present preliminary lattice results for the nonperturbative tensor structure of the vector and axial-vector quark-antiquark vertices in QCD. Our lattice calculations are for $N_f=2$ mass-degenerate Wilson fermion flavors whose quark mass values include an almost physical one. We compare our lattice results with the corresponding continuum solutions of the inhomogeneous Bethe-Salpeter equations in a rainbow-ladder truncation. We find similarities in the momentum dependencies of the form factors but also clear deviations at low momentum., Comment: 7 pages, 3 figures, Proceedings of the 36th Annual International Symposium on Lattice Field Theory (Lattice 2018), 22-28 July 2018, Michigan State University, East Lansing, Michigan USA

We present results for single pseudoscalar meson pole contributions and pion box contributions to the hadronic light-by-light (LBL) correction of the muon's anomalous magnetic moment. We follow the recently developed dispersive approach to LBL, where these contributions are evaluated with intermediate mesons on-shell. However, the space-like electromagnetic and transition form factors are not determined from analytic continuation of time-like data, but directly calculated within the functional approach to QCD using Dyson-Schwinger and Bethe-Salpeter equations. This strategy allows for a systematic comparison with a strictly dispersive treatment and also with recent results from lattice QCD. Within error bars, we obtain excellent agreement for the pion electromagnetic and transition form factor and the resulting contributions to LBL. In addition, we present results for the $\eta$ and $\eta'$ pole contributions and discuss the dynamical effects in the $\eta-\eta'$ mixing due to the strange quarks. Our result for the total pseudoscalar pole contributions is $a_\mu^{\text{PS-pole}} = 91.6 \,(1.9) \times 10^{-11}$ and for the pion-box contribution we obtain $a_\mu^{\pi-\text{box}} = -16.3 \,(2)(4) \times 10^{-11}$., Comment: 11 pages, 9 figures; v2: minor changes, diagram added, improved value for pion-box; version published in PLB

We calculate the nucleon resonance contributions to nucleon Compton scattering, including all states with J^P = 1/2^{+-} and J^P = 3/2^{+-} where experimental data for their electromagnetic transition form factors exist. To this end, we construct a tensor basis for the Compton scattering amplitude based on electromagnetic gauge invariance, crossing symmetry and analyticity. The corresponding Compton form factors provide a Lorentz-invariant description of the process in general kinematics, which reduces to the static and generalized polarizabilities in the appropriate kinematic limits. We derive the general forms of the offshell nucleon-to-resonance transition vertices that implement electromagnetic and spin-3/2 gauge invariance, which automatically also defines onshell transition form factors that are free of kinematic constraints. We provide simple fits for those form factors, which we use to analyze the resulting Compton form factors and extract their contributions to the nucleon's polarizabilities. Apart from the Delta(1232), the resonance contributions to the scalar and spin polarizabilites are very small, although the N(1520) could play a role for the proton's magnetic polarizability., Comment: 43 pages, 20 figures, 21 tables

We summarize recent work in determining the transition form factor (TFF) of the neutral pion ($\pi^0 \to \gamma^*\gamma^*$), by solving the non-perturbative Dyson-Schwinger and Bethe-Salpeter equations. We first study the transition form factor, followed by the rare decay $\pi^0 \to e^+ e^- $, which requires the TFF as input. In addition to the aspects of truncation required to compute the solution, we discuss unexpected behavior in the large $Q^2$ regime (with $Q^2$ the photon virtuality), and also touch on a path deformation in the complex plane to access the total decay rate for the rare decay., Comment: Contribution to the Proceedings of the Workshop FAIRNESS 2017. 5 pages, 3 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 discuss the light baryon spectrum obtained from a recent quark-diquark calculation, implementing non-pointlike diquarks that are self-consistently calculated from their Bethe-Salpeter equations. We examine the orbital angular momentum content in the baryons' rest frame and highlight the fact that baryons carry all possible values of L compatible with their spin, without the restriction P=(-1)^L which is only valid nonrelativistically. We furthermore investigate the meaning of complex conjugate eigenvalues of Bethe-Salpeter equations, their possible connection with 'anomalous' states, and we propose a method to eliminate them from the spectrum., Comment: 9 pages, 4 figures. Presented at Light Cone 2016, September 5-8 2015, IST, Universidade de Lisboa, Lisbon, Portugal

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 give a brief account of the Dyson-Schwinger and Faddeev-equation approach and its application to nucleon resonances and their transition form factors. We compare the three-body with the quark-diquark approach and present a quark-diquark calculation for the low-lying nucleon resonances including scalar, axialvector, pseudoscalar and vector diquarks. We also discuss the timelike structure of transition form factors and highlight the advantages of form factors over helicity amplitudes., Comment: 8 pages, 8 figures, 2 tables. Proceedings of the ECT* Workshop "Nucleon Resonances: From Photoproduction to High Photon Virtualities", October 12-16, 2015, Trento, Italy

We outline a microscopic framework to calculate nucleon Compton scattering from the level of quarks and gluons within the covariant Faddeev approach. We explain the connection with hadronic expansions of the Compton scattering amplitude and discuss the obstacles in maintaining electromagnetic gauge invariance. Finally we give preliminary results for the nucleon polarizabilities., Comment: 6 pages, 5 figures. Proceedings of Light Cone 2015, INFN Frascati, September 21-25, 2015

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

Baryons are treated as three-quark systems using QCD degrees of freedom in Poincare-covariant bound-state equations. The quark self-energy as well as the interaction between quarks are approximated by a vector-vector interaction via a single dressed-gluon exchange (rainbow-ladder truncation), thereby allowing a unified study of quark, meson and baryon properties. Here we will focus on the calculation of electromagnetic properties of spin-1/2 and spin-3/2 ground state baryons., Comment: 6 pages, 5 figures; to be published in the Proceedings of the International Conference on Exotic Atoms and Related Topics - EXA2014

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

Recent progress in our studies of propagators and three-point functions in Landau gauge for QCD and QCD-like theories is presented. Special emphasis is put on the properties of the three-gluon vertex and the quark-gluon vertex. The effect of unquenching is investigated. Furthermore, an exploratory study for a large number of light flavours is described, from where clear evidence for the qualitative behaviour of propagators in the so-called conformal window can be extracted., Comment: 13 pages, 9 figures; Proceedings of the workshop QCD-TNT-III, ECT* Trento, Sept. 2-6, 2013; references updated

We present a calculation of the three-gluon vertex from its Dyson-Schwinger equation in Landau-gauge Yang-Mills theory. All tensor structures are considered and back-coupled self-consistently. Within the chosen truncation, two-loop diagrams as well as diagrams containing Green's functions beyond the primitively divergent ones are neglected. Only the three-gluon vertex is chosen to be dynamical; the other propagators and vertex functions are provided as separate solutions of their Dyson-Schwinger equations or by Ansatz. For both scaling- and decoupling-type solutions we observe, in agreement with other studies, a sign change in the tree-level tensor dressing at a non-perturbative scale., Comment: 6 pages, 2 figures, proceedings of Excited QCD 2014 conference

The description of electromagnetic interactions with hadrons from the quark level requires knowledge of the underlying quark-gluon ingredients. I discuss some properties of the quark-photon vertex and quark Compton vertex, along with the role of electromagnetic gauge invariance and vector-meson dominance. A simple parametrization for the quark-photon vertex is given., Comment: 7 pages, 4 figures, 1 table. Presented at the Workshop 'Excited QCD 2014', Bjelasnica Mountain, Sarajevo, Bosnia-Herzegovina, February 2-8, 2014

We present the first Dyson-Schwinger calculation of the three-gluon vertex in Landau-gauge QCD in which its full covariant structure is back-coupled self-consistently. We truncate a Bose-symmetrized version of the DSE at the level of one-loop diagrams, model the four-gluon vertex, and neglect terms that contain non-primitively divergent n-point functions; the ghost-gluon vertex is taken bare to good approximation. Fit functions for the ghost and gluon propagators that interpolate between scaling and decoupling are presented. In all aspects of our study Bose symmetry is manifest, from the truncation to the basis decomposition and to the momentum invariants. We explore the uniform and soft-collinear infrared limits and obtain the expected infrared exponents. The presence of a zero crossing in the tree-level component of the vertex is confirmed for both scaling- and decoupling-type scenarios. The zero crossing appears at a scale ~1 GeV; however, its location might be sensitive to the four-gluon vertex and missing components in the DSE., Comment: 20 pages, 13 figures

We summarize recent results for the spectrum and properties of light pseudoscalar and scalar hadrons in a functional approach to QCD using Dyson-Schwinger and Bethe-Salpeter equations. We focus in particular on the role of pions as virtual constituents in light mesons ('pion cloud effects') as well as their role as constituents of tetraquark bound states. An extension of our framework into the charm quark sector reveals an all-charm tetraquark in the mass region around 5.5 GeV., Comment: 10 pages, 7 figures, contribution to the 51st International Winter Meeting on Nuclear Physics, 21-25 January 2013, Bormio (Italy)

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

We determine the mass of tetraquark bound states from a coupled system of covariant Bethe-Salpeter equations. Similar in spirit to the quark-diquark model of the nucleon, we approximate the full four-body equation for the tetraquark by a coupled set of two-body equations with meson and diquark constituents. These are calculated from their quark and gluon substructure using a phenomenologically well-established quark-gluon interaction. For the lightest scalar tetraquark we find a mass of the order of 400 MeV and a wave function dominated by the pion-pion constituents. Both results are in agreement with a meson molecule picture for the f_0(600). Our results furthermore suggest the presence of a potentially narrow all-charm tetraquark in the mass region 5-6 GeV., Comment: 4 pages, 2 figures, Refs added, Typos corrected, version accepted by PLB

I briefly summarize the application of the Dyson-Schwinger/Faddeev approach to baryon form factors. Recent results for nucleon electromagnetic and axial form factors as well as N-Delta-Gamma electromagnetic transition form factors are discussed. The calculation of the current diagrams from the quark-gluon level enables an analysis of common features, such as the implications of dynamical chiral symmetry breaking and quark orbital angular momentum, the timelike structure of the form factors, and their interpretation in terms of missing pion-cloud effects., Comment: 12 pages, 7 figures. Talk given at the International Workshop on QCD Green's Functions, Confinement, and Phenomenology (QCD-TNT-II), September 5-9, 2011, ECT*, Trento, Italy

We compute the vector-meson, nucleon and delta/omega-baryon masses and their evolution with the current-quark mass using a covariant generalized Bethe-Salpeter equation approach. The interaction kernel is truncated to a dressed gluon exchange. We study the model dependence of our results with the quark-gluon dressing to assess the validity of the truncation., Comment: 8 pages, 1 figure, 1 table. Talk presented by H. Sanchis-Alepuz at the International Workshop on QCD Green's Functions, Confinement, and Phenomenology - QCD-TNT II, September 05-09 2011, ECT* Trento, Italy

We study the N-Delta-gamma transition in the Dyson-Schwinger approach. The nucleon and Delta baryons are treated as quark-diquark bound states, where the ingredients of the electromagnetic transition current are computed self-consistently from the underlying dynamics in QCD. Although our approach does not include pion-cloud effects, we find that the electric and Coulomb quadrupole form-factor ratios R_EM and R_SM show good agreement with experimental data. This implies that the deformation from a spherical charge distribution inside both baryons can be traced back to the appearance of p waves in the nucleon and Delta bound-state amplitudes which are a consequence of Poincare covariance. On the other hand, the dominant transition amplitude, i.e. the magnetic dipole transition form factor, underestimates the data by ~25% in the static limit whereas agreement is achieved at larger momentum transfer, which is consistent with missing pion-cloud contributions. We furthermore find that the static properties of the form factors are not very sensitive to a variation of the current-quark mass., Comment: 21 pages, 9 figures, 4 tables. Added references

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 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 the solution of the Poincare-covariant Faddeev equation for the Delta(1232) and Omega(1672) baryons. The covariant structure of the corresponding baryon amplitudes and their decomposition in terms of internal spin and orbital angular momentum is explicitly derived. The interaction kernel is truncated to a rainbow-ladder dressed-gluon exchange such that chiral symmetry and its dynamical breaking are correctly implemented. The resulting physical masses agree reasonably with experiment and their evolution with the pion mass compares favorably with lattice calculations. Evidence for the non-sphericity of the Delta-resonance is discussed as well., Comment: 9 pages, 4 figures, 3 tables