Bound states from the spectral Bethe-Salpeter equation

Eichmann, Gernot ; Gómez, Andrés ; Horak, Jan ; Pawlowski, Jan M. ; Wessely, Jonas ; et al - 2024-05-17

We compute the bound state properties of three-dimensional scalar ${\varphi }^4$ 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.

Hidden-flavor four-quark states in the charm and bottom region

Hoffer, Joshua ; Eichmann, Gernot ; Fischer, Christian S. - 2024-04-22

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 ...

Going to the light front with contour deformations

Eichmann, Gernot ; Ferreira, Eduardo ; Stadler, Alfred - 2022-02-10

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 exten ...

Mass generation in Landau-gauge Yang-Mills theory

Eichmann, Gernot ; Pawlowski, Jan M. ; Silva, João M. - 2021-12-13

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 corroborat ...

Disentangling different structures in heavy-light four-quark states

Wallbott, Paul C. ; Eichmann, Gernot ; Fischer, Christian S. - 2020-09-25

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 $cq\overline{q}\stackrel{}{\mathrm{c<}}$ ...

$\sigma$-meson: Four-quark versus two-quark components and decay width in a Bethe-Salpeter approach

Santowsky, Nico ; Eichmann, Gernot ; Fischer, Christian S. ; Wallbott, Paul C. ; Williams, Richard ; et al - 2020-09-11

We study the dynamical generation of resonances in isospin singlet channels with mixing between two- and four-quark states. To this end, we generalize a Bethe-Salpeter approach to four-quark states employed previously to accommodate for mixing diagrams. The $q\overline{q}q\overline{q}$ and $q\overline{q}$ components of the Bethe-Salpeter wave function (with light quarks $q\in \{u$ ...

Kaon-box contribution to the anomalous magnetic moment of the muon

Eichmann, Gernot ; Fischer, Christian S. ; Williams, Richard - 2020-03-12

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 reevaluate the charged pion-box contribution taking effects due to isospin breaking into account. Our results are $a_{\mu }^{}$ ...

Scattering amplitudes and contour deformations

Eichmann, Gernot ; Duarte, Pedro ; Peña, M. T. ; Stadler, Alfred - 2019-11-01

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 $2\to 2$ 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 ...

Corrigendum to: “Single pseudoscalar meson pole and pion box contributions to the anomalous magnetic moment of the muon” [Phys. Lett. B 797 (2019) 134855]

Eichmann, Gernot ; Fischer, Christian S. ; Weil, Esther ; Williams, Richard - 2019-10-23

Single pseudoscalar meson pole and pion box contributions to the anomalous magnetic moment of the muon

Eichmann, Gernot ; Fischer, Christian S. ; Weil, Esther ; Williams, Richard - 2019-08-20

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 system ...

$X\left(3872\right)$ as a four-quark state in a Dyson-Schwinger/Bethe-Salpeter approach

Wallbott, Paul C. ; Eichmann, Gernot ; Fischer, Christian S. - 2019-07-30

We generalize 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 $I\left(J^{PC}\right)=0\left(1^{++}\right)$ of the $X\left(3872\right)$ and study the four-quark states with quark contents $cq\overline{q}\stackrel{}{}$ ...

Nucleon resonances in Compton scattering

Eichmann, Gernot ; Ramalho, G. - 2018-11-28

We calculate the nucleon resonance contributions to nucleon Compton scattering, including all states with $J^P=1/2^{\pm }$ and $J^P=3/2^{\pm }$ 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 fa ...

On the large- Q 2 behavior of the pion transition form factor

Eichmann, Gernot ; Fischer, Christian S. ; Weil, Esther ; Williams, Richard - 2017-10-04

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 π0→γγ 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 modi ...

The light scalar mesons as tetraquarks

Eichmann, Gernot ; Fischer, Christian S. ; Heupel, Walter - 2015-12-18

We present a numerical solution of the four-quark Bethe–Salpeter equation for ground-state scalar tetraquarks with JPC=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 Mσ∼350 MeV whi ...