arXiv subjects
Publications and source records attributed to Gernot Eichmann.
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.
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.
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 $ππ$ 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 $σ/f_0(500)$ that emerged in the past two decades from dispersive approaches \cite{Pelaez:2015qba}.
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 $a_μ^{π^\pm-\text{box}} = -15.7 \,(2)(3) \times 10^{-11}$ and $a_μ^{K^\pm-\text{box}} = -0.48 \,(2)(4) \times 10^{-11}$ thus confirming the large suppression of box contributions beyond the leading pion box.
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.
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 $η$ and $η'$ pole contributions and discuss the dynamical effects in the $η-η'$ mixing due to the strange quarks. Our result for the total pseudoscalar pole contributions is $a_μ^{\text{PS-pole}} = 91.6 \,(1.9) \times 10^{-11}$ and for the pion-box contribution we obtain $a_μ^{π-\text{box}} = -16.3 \,(2)(4) \times 10^{-11}$.
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.
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-Padé 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.
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.
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.
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 $π^0\toγγ$ 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.
We summarize recent work in determining the transition form factor (TFF) of the neutral pion ($π^0 \to γ^*γ^*$), by solving the non-perturbative Dyson-Schwinger and Bethe-Salpeter equations. We first study the transition form factor, followed by the rare decay $π^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.
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.
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.
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.
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.
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.
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.