Delta and omega electromagnetic form factors in a Dyson-Schwinger/Bethe-Salpeter approach

We investigate the electromagnetic form factors of the $\Delta$ and the $\Omega$ baryons within the Poincaré-covariant framework of Dyson-Schwinger and Bethe-Salpeter equations. The three-quark core contributions of the form factors are evaluated by employing a quark-diquark approximation. We use a consistent setup for the quark-gluon dressing, the quark-quark bound-state kernel and the quark-photon interaction. Our predictions for the multipole form factors are compatible with available experimental data and quark model estimates. The current-quark mass evolution of the static electromagnetic properties agrees with results provided by lattice calculations.

Figure 1

The quark-diquark BSE, Eq. (1)

Figure 2

Quark DSE (4) and diquark BSE (7) in rainbow-ladder truncation.

Figure 3

Effective coupling $\alpha (k^2)$ of Eq. (6), evaluated for $\Lambda =0.98\mathrm{GeV}$ corresponding to the $u/d$-quark mass and in the range $\eta \in [1.6,2.0]$.

Figure 4

Result for the quark-diquark $\Delta$ amplitude. The plot shows the zeroth Chebyshev moments of the dressing functions $f_k^{\Delta }$ in Eq. (9), where the labeling corresponds to Eq. 18 of Ref. 52. The remaining components $f_3$, $f_5$ and $f_8$ are small and not displayed.

Figure 5

A baryon’s electromagnetic current in the quark-diquark model.

Figure 6

Electromagnetic form factors of the $\Delta$. The bands represent the sensitivity to a variation of $\eta =1.8\pm 0.2$. The results are compared to unquenched lattice data of Ref. 40 at three different pion masses.

Figure 7

Static electromagnetic properties of the $\Delta$. Upper left panel: dimensionless form factors at $Q^2=0$. Upper right panel: dimensionless squared radii $R_i$, given in $(\mathrm{GeV}\mathrm{fm}{)}^2$. Lower left panel: $\Delta$ magnetic moment, expressed in static nuclear magnetons and compared to quenched [36, 38] and dynamical [37] lattice data, as well as dynamical (circles), quenched (triangle-up) and mixed-action results (triangle-down) from Ref. 40. Lower right panel: squared electric charge radius compared to lattice results of Refs. 38, 40. Stars denote experimental values [1]: for the $\Delta$ we plot ${\mu }_{{\Delta }^{++}}/2$; for the ${\Omega }^{-}$ we show $|{\mu }_{{\Omega }^{-}}|$. The dashed vertical lines indicate the positions of the $u/d$- and strange-quark mass. Note that there is no $s\overline{s}$ pseudoscalar meson in nature; the value $m_{s\overline{s}}=0.69\mathrm{GeV}$ corresponds to a meson-BSE solution at a strange-quark mass $m_s=150\mathrm{MeV}$ [61].