Hadronic decays of mesons and baryons in the Dyson-Schwinger approach

We study hadronic decays of mesons and baryons in the context of the Dyson-Schwinger equations of QCD. Starting from a well-established effective interaction in rainbow-ladder truncation, we consistently calculate all ingredients of the appropriate decay diagrams. The resulting strong couplings are presented as functions of the quark mass from the chiral limit up to the respective decay thresholds. In particular, we investigate the $\rho \to \pi \pi$ and for the first time the $\Delta \to N\pi$ transitions. Both meson and baryon results compare well to available lattice QCD results as well as experimental data and present the first step towards a comprehensive covariant study of hadron resonances in the Dyson-Schwinger approach.

Figure 1

(Color online) Left panel: quark DSE (2) in rainbow truncation. Right panel: schematics of the meson BSE (3) and diquark BSE (5) in ladder truncation.

Figure 2

(Color online) The quark-diquark BSE, Eq. (7).

Figure 3

(Color online) The $\rho \pi \pi$ triangle diagram, Eq. (10).

Figure 4

(Color online) Decomposition of the $\Delta N\pi$ transition matrix element in the quark-diquark model, Eqs. (16, 17). Seagull terms are neglected. ’SC’ and ’AX’ denote the types of diquarks (i.e., scalar and axial-vector) that can appear in combination with the nucleon and $\Delta$ bound-state amplitudes.

Figure 5

(Color online) Left panel: Evolution of the $\rho \pi \pi$ coupling with the pion mass squared. The experimental point is indicated by the star and the symbols denote lattice data from Refs. 23, 24, 25, 26, 27, 28. For better readability the point of the Göckeler group is shifted slightly to the right. Right panel: The decay width of the $\rho$-meson versus the pion mass squared. The width of the bands illustrates the dependence on $\omega$ (see text).

Figure 6

(Color online) Left panel: the transition form factor $G_{\Delta N\pi }$ as a function of the squared pion momentum $Q^2$. The upper band corresponds to the full result and the lower band shows the impulse-approximation value where the pion couples to the quark line only. The width of the bands again shows the dependence on $\omega$. The solid lines are the results of the calculation in the kinematically allowed regions whereas the dashed lines are the respective dipole fits, cf. Equation (19). The pion mass shell $Q^2=-m_{\pi }^2$ is indicated by the vertical line and the star shows the experimental value $g_{\Delta N\pi }=G_{\Delta N\pi }(-m_{\pi }^2)$. Right panel: Current-mass evolution of $g_{\Delta N\pi }$ (upper band) and $G_{\Delta N\pi }(Q^2=0)$ (lower band). The vertical shaded area depicts the $\omega$-dependent location of the threshold $M_{\Delta }=M_N+m_{\pi }$.