Structure of the nucleon’s low-lying excitations

A continuum approach to the three valence-quark bound-state problem in quantum field theory is used to perform a comparative study of the four lightest $(I=1/2,J^P=1/2^{\pm })$ baryon isospin doublets in order to elucidate their structural similarities and differences. Such analyses predict the presence of nonpointlike, electromagnetically active quark-quark (diquark) correlations within all baryons; and in these doublets, isoscalar-scalar, isovector-pseudovector, isoscalar-pseudoscalar, and vector diquarks can all play a role. In the two lightest $(1/2,1/2^{+})$ doublets, however, scalar and pseudovector diquarks are overwhelmingly dominant. The associated rest-frame wave functions are largely $S$-wave in nature; and the first excited state in this $1/2^{+}$ channel has the appearance of a radial excitation of the ground state. The two lightest $(1/2,1/2^{-})$ doublets fit a different picture: accurate estimates of their masses are obtained by retaining only pseudovector diquarks; in their rest frames, the amplitudes describing their dressed-quark cores contain roughly equal fractions of even- and odd-parity diquarks; and the associated wave functions are predominantly $P$-wave in nature, but possess measurable $S$-wave components. Moreover, the first excited state in each negative-parity channel has little of the appearance of a radial excitation. In quantum field theory, all differences between positive- and negative-parity channels must owe to chiral symmetry breaking, which is overwhelmingly dynamical in the light-quark sector. Consequently, experiments that can validate the contrasts drawn herein between the structure of the four lightest $(1/2,1/2^{\pm })$ doublets will prove valuable in testing links between emergent mass generation and observable phenomena and, plausibly, thereby revealing dynamical features of confinement.

Figure 1

Poincaré covariant Faddeev equation: a homogeneous linear integral equation for the matrix-valued function $\mathrm{\Psi }$, being the Faddeev amplitude for a baryon of total momentum $P=p_q+p_d$, which expresses the relative momentum correlation between the dressed quarks and diquarks within the baryon. 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: (a) Baryon rest-frame quark-diquark orbital angular momentum fractions, as defined in Eq. (17). Lower panel: (b) Relative contribution of various quark-diquark orbital angular momentum components to the mass of a given baryon. In both panels, the results were computed with $g_{\mathrm{DB}}=0.43$, except for the identified bar triplets with lighter shading, for which $g_{\mathrm{DB}}=1$. Legend: $N_0^{+}$ is the ground-state nucleon, $N_1^{+}=N(1440)1/2^{+}$, $N_0^{-}=N(1535)1/2^{-}$, $N_1^{-}=N(1650)1/2^{-}$.

Figure 3

Upper panel: (a) Relative strengths of various diquark components within the indicated baryon’s Faddeev amplitude, as defined in Eq. (20). Lower panel: (b) Relative contribution to a baryon’s mass from a given diquark correlation in that baryon’s Faddeev amplitude. In both panels, the results were computed with $g_{\mathrm{DB}}=0.43$, except for the identified bar quadruplets with lighter shading, for which $g_{\mathrm{DB}}=1$. Legend: $N_0^{+}$ is the ground-state nucleon, $N_1^{+}=N(1440)1/2^{+}$, $N_0^{-}=N(1535)1/2^{-}$, $N_1^{-}=N(1650)1/2^{-}$.

Figure 4

Order-zero Chebyshev projections, Eq. (21), of the nucleon (left) and Roper (right) quark-core Faddeev wave functions, with $S$-wave in the top row and $P$-wave in the bottom. For a given baryon, all functions are rescaled by the associated ${\ell }^2=0$ value of the zeroth moment of ${\tilde{\mathcal{s}}}_1^{+}$.

Figure 5

Order-zero Chebyshev projections, Eq. (21), of the $N(1535)1/2^{-}$ (left) and $N(1650)1/2^{-}$ (right) quark-core Faddeev amplitudes, with $S$-wave in the top row and $P$-wave in the bottom. For a given baryon, all functions are rescaled by the associated ${\ell }^2=0$ value of the zeroth moment of ${\tilde{\mathcal{a}}}_2^{-}$.

Figure 6

Solid curve (blue)—quark mass function generated by the parametrization of the dressed-quark propagator specified by Eqs. (a3) and (a4) (A5); and band (green)—exemplary range of numerical results obtained by solving the gap equation with the modern DCSB-improved kernels described and used in Refs. [16, 81, 82, 83].