Quark-photon Vertex Research Articles

Quark and proton anomalous magnetic moments in confining models

We compute the one-loop quark-photon vertex and the quark magnetic moment in three different models with an infrared-modified gluon propagator, namely: the massive (Curci-Ferrari-like) model, the Gribov-Zwanziger model, and the refined Gribov-Zwanziger model. We show results for the F2 form factor and analyze the role played by the mass parameters associated with the confined gluon in these models. Using the framework of the constituent quark model, we further construct the observable proton magnetic moment including the effects of quantum chromodynamics (QCD) interactions generated by the different confining models. Our results show that there is no qualitative observable difference stemming from the presence of complex-conjugated poles. On the other hand, quantitative differences between the various confining models can be sizable, so that observable constraints would in principle be possible, provided one has sufficient information about the constituent quark mass and the QCD running coupling. Published by the American Physical Society 2024

Timelike behavior of the pion electromagnetic form factor in the functional formalism

The electromagnetic form factor of the pion is calculated within the use of functional formalism. We develop integral representation for the minimal set of Standard Model Green's functions and derive the dispersion relation for the form factor in the two-flavor QCD isospin limit ${m}_{u}={m}_{d}$. We use the dressed quark propagator as obtained from the gap equation in Minkowski space and within the Dyson-Schwinger equations formalism to derive the approximate dispersion relation for the form factor for the first time. We evaluate the form factor for the spacelike as well as for the timelike momentum in the presented formalism. A new Nakanishi-like form of integral representation is proved on the basis of the vector Bethe-Salpeter equation for the quark-photon vector with a ladder-rainbow kernel. The gauge technique turns out to be a part of the entire structure of the vertex. In the analytic approach presented here, it is shown that a large amount of the $\ensuremath{\rho}$-meson peak in the cross section ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ is governed by the gauge invariance of QED/QCD---i.e., by a gauge-technique-constructed quark-photon vertex. This approximation naturally explains the broad shape of the $\ensuremath{\rho}$-meson peak.

Elucidating the ρ-meson’s role as intermediate resonance in the time-like electromagnetic pion form factor

Motivated by the planned measurements of the time-like electromagnetic proton form factors an exploratory study of the time-like electromagnetic pion form factor is presented in a formalism which describes mesons as Poincaré-invariant bound states. In the respective quark interaction kernel, beyond the gluon-intermediated interactions for valence-type quarks, non-valence effects are included by allowing the pions to couple back in a self-consistent manner to the quarks. Consequently, the opening of the dominant ρ decay channel, ρ → ππ, and the presence of a multi-particle branch cut, setting in when the two-pion threshold is crossed, are included consistently, first in the correspondingly calculated quark-photon vertex, and then consequently in the time-like electromagnetic pion form factor. The obtained results are in agreement with the available experimental data and provide further evidence for the effcacy of a vector-meson dominance model. Effects of going beyond the here employed isospin-symmetric limit are discussed. Last but not least, an outlook is provided on how to include in Poincaré-covariant Faddeev approaches the effects of intermediate resonances in the calculation of baryon properties.

Quark anomalous magnetic moment and its effects on the ρ meson properties

A symmetry-preserving treatment of mesons, within a Dyson-Schwinger and Bethe-Salpeter equations approach, demands an interconnection between the kernels of the quark gap equation and meson Bethe-Salpeter equation. Appealing to those symmetries expressed by the vector and axial-vector Ward-Green-Takahashi identitiges (WGTI), we construct a two-body Bethe-Salpeter kernel and study its implications in the vector channel; particularly, we analyze the structure of the quark-photon vertex, which explicitly develops a vector meson pole in the timelike axis and the quark anomlaous magnetic moment term, as well as a variety of $\rho$ meson properties: mass and decay constants, electromagnetic form factors, and valence-quark distribution amplitudes.

Elucidating the effect of intermediate resonances in the quark interaction kernel on the timelike electromagnetic pion form factor

An exploratory study of the timelike pion electromagnetic form factor in a Poincar\'e-covariant bound state formalism in the isospin symmetric limit is presented. Starting from a quark interaction kernel representing gluon-intermediated interactions for valence-type quarks, nonvalence effects are included by introducing pions as explicit degrees of freedom. The two most important qualitative aspects are, in view of the presented study, the opening of the dominant $\ensuremath{\rho}$-meson decay channel and the presence of a multiparticle branch cut setting in which the two-pion threshold is crossed. Based on a recent respective computation of the quark-photon vertex, the pion electromagnetic form factor for spacelike and timelike kinematics is calculated. The obtained results for its absolute value and its phase compare favorably to the available experimental data, and they are analyzed in detail by comparing them to the expectations based on an isospin-symmetric version of a vector-meson dominance model.

Contribution of neutral pseudoscalar mesons to aμHLbL within a Schwinger-Dyson equations approach to QCD

A continuum approach to Quantum Chromodynamics (QCD), based upon Schwinger-Dyson (SD) and Bethe-Salpeter (BS) equations, is employed to provide a tightly constrained prediction for the $\gamma^{*} \gamma^{*} \rightarrow \{ \pi^0, \eta, \eta', \eta_c, \eta_b \}$ transition form factors (TFFs) and their corresponding pole contribution to the hadronic light-by-light (HLbL) piece of the anomalous magnetic moment of the muon ($a_\mu$). This work relies on a practical and well-tested quark-photon vertex Ansatz approach to evaluate the TFFs for arbitrary space-like photon virtualities, in the impulse approximation. The numerical results are parametrized meticulously, ensuring a reliable evaluation of the HLbL contributions to $a_\mu$. We obtain: $a_{\mu}^{\pi^0-\textrm{pole}} = (6.14 \pm 0.21) \times 10^{-10}$, $a_{\mu}^{\eta-\textrm{pole}} = (1.47 \pm 0.19) \times 10^{-10}$, $a_{\mu}^{\eta'-\textrm{pole}} = (1.36 \pm 0.08) \times 10^{-10}$, yielding a total value of $a_{\mu}^{\pi^0+\eta+\eta'-\textrm{pole}} = (8.97 \pm 0.48) \times 10^{-10}$, compatible with contemporary determinations. Notably, we find that $a_{\mu}^{\eta_c+\eta_b-\textrm{pole}} \approx a_{\mu}^{\eta_c-\textrm{pole}} = (0.09 \pm 0.01) \times 10^{-10}$, which might not be negligible once the percent precision in the computation of the light pseudoscalars is reached.

On the effect of resonances in the quark-photon vertex

.A calculation of hadronic timelike form factors in the Poincaré-covariant Bethe-Salpeter formalism necessitates knowing the analytic structure of the non-perturbative quark-photon vertex in the context of the Poincaré-covariant Bethe-Salpeter formalism. We include, in the interaction between quark and antiquark, the possibility of non-valence effects by introducing pions as explicit degrees of freedom. These encode the presence of intermediate resonances in the Bethe-Salpeter interaction kernel. We calculate the vertex for real as well as complex photon momentum. We show how the vertex reflects now the correct physical picture, with the rho resonance appearing as a pole in the complex momentum plane. A multiparticle branch cut for values of the photon momentum from -4m_{pi}^{2} to -infty develops. This calculation represents an essential step towards the calculation of timelike form factors in the Bethe-Salpeter approach.

Practical scheme from QCD to phenomena via Dyson-Schwinger equations

We deliver a new scheme to compute the quark propagator and the quark-gluon interaction vertex through the coupled Dyson-Schwinger equations (DSEs) of QCD. We take the three-gluon vertex into account in our calculations, and implement the gluon propagator and the running coupling function fitted by the solutions of their respective DSEs. We obtain the momentum and current mass dependence of the quark propagator and the quark-gluon vertex, and the chiral quark condensate which agrees with previous results excellently. We also compute the quark-photon vertex within this scheme and give the anomalous chromo- and electro-magnetic moment of quark. The obtained results also agree with previous ones very well. These applications manifest that the new scheme is realistic and then practical for explaining the QCD-related phenomena.

On the effect of resonances in the quark-photon vertex

On the effect of resonances in the quark-photon vertex

Electromagnetic transition form factors of baryons in the space-like momentum region

We present results from a calculation of the electromagnetic transition form factors between ground-state octet and decuplet baryons as well as the octet-only $\Sigma^0$ to $\Lambda$ transition. We work in the combined framework of Dyson-Schwinger equations and covariant Bethe-Salpeter equations with all elements, the baryon three body wave function, the quark propagators and the dressed quark-photon vertex determined from a well-established, momentum dependent approximation for the quark-gluon interaction. We discuss in particular the similarities among the different transitions as well as the differences induced by SU(3)-isospin symmetry breaking. We furthermore provide estimates for the slopes of the electric and magnetic $\Sigma^0$ to $\Lambda$ transitions at the zero photon momentum point.

Electromagnetic transition form factors of baryons in a relativistic Faddeev approach

The covariant Faddeev approach which describes baryons as relativistic three-quark bound states and is based on the Dyson-Schwinger and Bethe-Salpeter equations of QCD is briefly reviewed. All elements, including especially the baryons’ three-body-wave-functions, the quark propagators and the dressed quark-photon vertex, are calculated from a well-established approximation for the quark-gluon interaction. Selected previous results of this approach for the spectrum and elastic electromagnetic form factors of ground-state baryons and resonances are reported. The main focus of this talk is a presentation and discussion of results from a recent investigation of the electromagnetic transition form factors between ground-state octet and decuplet baryons as well as the octet-only Σ0 to Λ transition.

Effects of a dressed quark-gluon vertex in vector heavy-light mesons and theory average of theBc*meson mass

We extend earlier investigations of heavy-light pseudoscalar mesons to the vector case, using a simple model in the context of the Dyson-Schwinger-Bethe-Salpeter approach. We investigate the effects of a dressed quark-gluon vertex in a systematic fashion and illustrate and attempt to quantify corrections beyond the phenomenologically very useful and successful rainbow-ladder truncation. In particular we investigate the dressed quark-photon vertex in such a setup and make a prediction for the experimentally as yet unknown mass of the ${B}_{c}^{*}$, which we obtain at 6.334 GeV well in line with predictions from other approaches. Furthermore, we combine a comprehensive set of results from the theoretical literature. The theoretical average for the mass of the ${B}_{c}^{*}$ meson is $6.336\ifmmode\pm\else\textpm\fi{}0.002\text{ }\text{ }\mathrm{GeV}$.

Probing Nucleons with Photons at the Quark Level

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.

Role of momentum dependent dressing functions and vector meson dominance in hadronic light-by-light contributions to the muong−2

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.

Non-contact Interactions and the Hadronic Light-by-light Contribution to the Muon $g-2$

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.

A model of a transition neutral pion form factor measured in annihilation and scattering channels at high momentum transfer

We consider an alternative explanation of newly found growth of neutral pion transition form factor with high virtuality of one of photons. It is based on Sudakov suppression of quark-photon vertex. Some applications to scattering and annihilation channels are considered including the relevant experiments with lepton-proton scattering.

A fresh look at hadronic light-by-light scattering in the muon g - 2 with the Dyson-Schwinger approach

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}.

Pion generalized parton distributions with covariant and light-front constituent quark models

We investigate the model dependence of no-helicity flip generalized parton distribution of the pion upon different approaches for the quark-hadron and quark-photon vertices, in the spacelike region. In order to obtain information on contributions from both the valence and the nonvalence regions, we compare results for spacelike momentum transfers obtained from (i) an analytic covariant model with a bare quark-photon vertex, (ii) a light-front approach with a quark-photon vertex dressed through a microscopic vector-meson model, and (iii) a light-front approach based on the relativistic Hamiltonian dynamics. Our comparisons lead us to infer the same dynamical mechanism, the one-gluon-exchange dominance at short distances, as a source of both the electromagnetic form factor at large momentum transfer and the parton distribution close to the end points. The expected collinear behavior of the generalized parton distributions at high-momentum transfer, i.e. a maximum for x{approx}1, is also illustrated, independently of the different approaches. Finally, a comparison with recent lattice calculations of the gravitational form factors is presented.

Vector meson form factors and their quark-mass dependence

The electromagnetic form factors of vector mesons are calculated in an explicitly Poincar\'e covariant formulation, based on the Dyson-Schwinger equations of QCD, that respects electromagnetic current conservation and unambiguously incorporates effects from vector meson poles in the quark-photon vertex. This method incorporates a two-parameter effective interaction, where the parameters are constrained by the experimental values of chiral condensate and ${f}_{\ensuremath{\pi}}$. This approach has successfully described a large amount of light-quark meson experimental data, including ground-state pseudoscalar masses and their electromagnetic form factors and ground-state vector meson masses and strong and electroweak decays. Here we apply it to predict the electromagnetic properties of vector mesons. The results for the static properties of the \ensuremath{\rho} meson are as follows: charge radius $\ensuremath{\langle}{r}_{\ensuremath{\rho}}^{2}\ensuremath{\rangle}=0.54\phantom{\rule{0.3em}{0ex}}{\mathrm{fm}}^{2}$, magnetic moment $\ensuremath{\mu}=2.01$, and quadrupole moment $\mathcal{Q}=\ensuremath{-}0.41$. We investigate the quark-mass dependence of these static properties and find that our results at the charm quark mass are in agreement with recent lattice simulations. The charge radius decreases with increasing quark mass, but the magnetic moment is almost independent of the quark mass.

Pion form factor and quark mass evolution in a light-front Bethe-Salpeter model

We discuss the soft contribution to the elastic pion form factor with the mass evolution from current to constituent quark being taken into account using a light-front Bethe-Salpeter (LFBS) model, which is a light-front quark model (LFQM) with a running mass. It is shown that partial conservation of the axial-vector current (PCAC) is satisfied with a running quark mass. We examine the sensitivity of the pion form factor using two different functional forms of the quark propagator. The Ball-Chiu ansatz is used to maintain local gauge invariance of the quark-photon vertex. The extension of our model to the hard contribution is also discussed.