Form factors of $Δ$(1232) and the electromagnetic $N-Δ$ transition

Abstract

In this work, the electromagnetic and gravitational form factors of $Δ$ isobars, as well as the electromagnetic $N-Δ$ transition form factors are studied systematically and continuously using a covariant quark-diquark approach with the pion cloud effect. In our model, the baryon is treated as the two-body system to simplify calculations, and the quarks are assumed to be surrounded by the pion cloud. The related physical properties, such as the charge radius and magnetic moment of $Δ$ and the helicity amplitudes of the $N-Δ$ transition are obtained and discussed. Our results for the form factors of both $Δ$(1232) and the electromagnetic $N-Δ$ transition are in reasonable agreement with the experimental or lattice results. Moreover, we found that the pion cloud plays an important role in the results through enlarging the magnetic transition form factor $G_M(t)$ and shifting the sign of the D-term of $Δ$ to negative.

Figures (9)

• Figure 1: Feynman diagrams for the transition current coupling with the quark (the first panel) and the diquark (the second and third panels). The grey and black points are the baryon-quark-diquark vertices for the nucleon and $\Delta$, respectively, and the green point stands for the dressed photon-quark vertex introduced in Sec. \ref{['PionCloud']}. The internal structure of the diquark is considered by coupling with the photon (the white points) and shown later in Fig. \ref{['f-dq']}.
• Figure 2: Feynman diagram for the internal structure of the diquark, where the black points are the quark-diquark coupling vertices with $\gamma^5$ for the scalar diquark and $\gamma^\alpha$ for the axialvector diquark. The kinematical variables have the relations $p_D=l-\frac{q}{2}$ and $p'_D=l+\frac{q}{2}$, where $l$ and $q$ are the momentums in the second or third panel of Fig. \ref{['f-emff']}.
• Figure 3: The photon-quark coupling process with the pion cloud correction. Diagram (a) shows the coupling process without the pion cloud. In diagrams (b) and (c), the electromagnetic current respectively couples with the quark and the pion, and the form factors $f_{i,0}^q$ and $f_{i,0}^{\pi}\, (i=1,2)$ are originated from these two diagrams. Specially, when calculating diagrams (b) and (c), the quark is assumed to be point-like to simplify the calculation. The blue points stand for the bare vertex $J^{\mu}_{PL}=\gamma^\mu$, and the white point represents the inner structure of the pion.
• Figure 4: EMFFs of $\Delta^+$. The results are compared with the lattice data in Ref. Alexandrou:2008bn, with pion mass $m_{\pi}=353$ MeV (blue triangles), $m_{\pi}=384$ MeV (red stars), and $m_{\pi}=410$ MeV (black points).
• Figure 5: EMFFs of $\Delta^0$.