Toward unifying the description of meson and baryon properties

We present a Poincaré covariant Faddeev equation, which enables the simultaneous prediction of meson and baryon observables using the leading order in a truncation of the Dyson-Schwinger equations that can systematically be improved. The solution describes a nucleon's dressed-quark core. The evolution of the nucleon mass with current-quark mass is discussed. A nucleon-photon current, which can produce nucleon form factors with realistic $Q^2$ evolution, is described. Axial-vector diquark correlations lead to a neutron Dirac form factor that is negative, with $r_1^{\mathit{nu}}>r_1^{\mathit{nd}}$. The proton electric-magnetic form factor ratio falls with increasing $Q^2$.

Figure 1

Covariant Faddeev equation. ψ is the Faddeev amplitude for a nucleon of total momentum $P=p_q+p_d$. It expresses the relative momentum correlation between the dressed-quark and dressed-diquarks within the nucleon. The shaded region shows the equation's kernel: The single line is the dressed-quark propagator, $S(p)$ [see Eq. (4)]; $\gamma$ is the diquark correlation amplitude [see Eq. (5)]; and the double line is the diquark propagator (see Fig. 3).

Figure 2

Evolution with current-mass, $\hat{m}$, of scalar and axial-vector diquark masses: $m_{\mathrm{sc}}$ and $m_{\mathrm{av}}$ (thick bands). Bands demarcate sensitivity to variation $\omega =\overline{\omega }(\hat{m})\pm \delta \omega$, with the lower (upper) edge corresponding to largest (smallest) ω value. ($m_{\pi }$ is calculated from the rainbow-ladder meson Bethe-Salpeter equation: $\hat{m}=6.1\hspace{0.5em}\mathrm{MeV}\Rightarrow m_{\pi }=0.138$ GeV.) Evolution of ρ-meson mass [11] (solid curve). [The interaction of Eqs. (1, 2, 3) was deliberately constructed to yield a calculated overestimate of $m_{\rho }(m_{\pi }^2)$.] With $m_{\rho }$ we also depict results from simulations of lattice-regularized QCD [17] along with an analysis and chiral extrapolation [18] (short dashed curve). Evolution with $\hat{m}$ of the nucleon mass obtained from our Faddeev equation: $\hat{m}=6.1$ MeV, $M_N=1.26(2)$ GeV (thin band). For comparison we provide results from lattice-QCD [19, 20] and an analysis of such results [21] (dashed curve).

Figure 3

Sum of diagrams that determine the diquark propagators in Fig. 1. $\gamma$ represents the diquark correlation amplitudes, the single line is for the dressed-quark propagator, and ${\mathrm{K}}^{-1}$ is the inverse of the ladder-kernel in Eqs. (1, 2, 3).

Figure 4

Calculated result for ${\mu }_p{G}_E^p(Q^2)/G_M^p(Q^2)$ (shaded band). The bandwidth delimits the sensitivity to ω on the domain $\omega =\overline{\omega }(\hat{m})\pm \delta \omega$: larger (smaller) ω is denoted by lower (upper) border. Data are from Ref. [32] (open circles), Ref. [33] (filled squares), Ref. [34] (filled circles), and Ref. [35] (filled diamonds).