Completing the Picture of the Roper Resonance

We employ a continuum approach to the three valence-quark bound-state problem in relativistic quantum field theory to predict a range of properties of the proton’s radial excitation and thereby unify them with those of numerous other hadrons. Our analysis indicates that the nucleon’s first radial excitation is the Roper resonance. It consists of a core of three dressed quarks, which expresses its valence-quark content and whose charge radius is 80% larger than the proton analogue. That core is complemented by a meson cloud, which reduces the observed Roper mass by roughly 20%. The meson cloud materially affects long-wavelength characteristics of the Roper electroproduction amplitudes but the quark core is revealed to probes with $Q^2\gtrsim 3m_N^2$.

Figure 1

Poincaré-covariant Faddeev equation. $\mathrm{\Psi }$ is the Faddeev amplitude for a baryon of total momentum, $P=p_q+p_d$. The shaded rectangle demarcates the kernel of the Faddeev equation: single line, dressed-quark propagator; $\mathrm{\Gamma }$, diquark correlation amplitude; and double line, diquark propagator.

Figure 2

Upper panel: Zeroth Chebyshev moment of all $S$-wave components in the nucleon’s Faddeev wave function, which is obtained from $\mathrm{\Psi }$ in Fig. 1, by reattaching the dressed-quark and -diquark legs. Lower panel: Kindred functions for the first excited state. Legend: $S_1$ is associated with the baryon’s scalar diquark, the other two curves are associated with the axial-vector diquark, and the normalization is chosen such that $S_1(0)=1$. Details are provided in Refs. [62, 63].

Figure 3

Upper panel: $F_1^{*}$ as a function of $x=Q^2/m_N^2$. Solid (black) curve, our prediction; dot-dashed (red) curve, contact-interaction result [28]; dotted (green) curve, inferred meson-cloud contribution; and dashed (blue) curve, anticipated complete result. Lower panel: $F_2^{*}(x)$ with same legend. Data in both panels: Circles (blue) [13], triangle (gold) [14], squares (purple) [16], and star (green) [9].