Light baryons and their excitations

We study ground states and excitations of light octet and decuplet baryons within the framework of Dyson-Schwinger and Faddeev equations. We improve upon similar approaches by explicitly taking into account the momentum-dependent dynamics of the quark-gluon interaction that leads to dynamical chiral symmetry breaking. We perform calculations in both the three-body Faddeev framework and the quark-diquark approximation in order to assess the impact of the latter on the spectrum. Our results indicate that both approaches agree well with each other. The resulting spectra furthermore agree one-to-one with experiment, provided well-known deficiencies of the rainbow-ladder approximation are compensated for. We also discuss the mass evolution of the Roper and the excited $\mathrm{\Delta }$ with varying pion mass and analyze the internal structure in terms of their partial wave decompositions.

Figure 1

Three-quark Faddeev equation.

Figure 2

Simplification of the Faddeev equation in Eq. (3) (top left) to the quark-diquark Bethe-Salpeter equation (5) (top right). The bottom panel shows the ingredients that enter in the equation and are calculated beforehand: the quark propagator, diquark Bethe-Salpeter amplitudes and diquark propagators.

Figure 3

Nucleon and $\mathrm{\Delta }$ baryon spectrum for $J^P=1/2^{\pm }$ and $3/2^{\pm }$ states determined within rainbow-ladder. The three-body results (open boxes) are compared to the quark-diquark spectrum with full diquark content for $\eta =1.7$ (filled boxes), together with the PDG values and their experimental uncertainties [4]. The widths of our results represent an error estimate based on the corresponding eigenvalue curves (see the Appendix for details).

Figure 4

Left: Vector, scalar and axialvector meson masses calculated in rainbow-ladder as functions of the $\eta$ parameter. For the scalars and axialvectors we employed $c=0.35$ in order to shift the $a_1$ and $b_1$ masses towards their experimental values (shown by the horizontal band). Right: Analogous plot for the diquark masses.

Figure 5

Nucleon and $\mathrm{\Delta }$ spectrum with reduced strength in the pseudoscalar and vector diquark channels; see text for a detailed discussion.

Figure 6

Mass evolution of the nucleon and Roper resonance (left) and the $\mathrm{\Delta }$ and its first excited state (right). The bands are results obtained in the three-body rainbow-ladder Faddeev framework with $1.6\le \eta \le 2.0$. In the left plot they are compared with lattice data for the nucleon and its first excitation [17, 19, 20, 21, 72, 73, 74]. In the right plot, only lattice data for the $\mathrm{\Delta }$ ground state are shown; data for the excited $\mathrm{\Delta }$ give values too high to be visible in the plot [75, 76, 77, 78].

Figure 7

Inverse eigenvalue spectra $1/{\lambda }_i(M)$ for the four nucleon (left) and four $\mathrm{\Delta }$ channels (right). In both cases the left panels show the results with all diquarks included and the right panels those with scalar and axialvector diquarks only.