Spectrum and structure of octet and decuplet baryons and their positive-parity excitations

A continuum approach to the three-valence-quark bound-state problem in quantum field theory, employing parametrizations of the necessary kernel elements, is used to compute the spectrum and Poincaré-covariant wave functions for all $\mathrm{flavor}\text{−}SU(3)$ octet and decuplet baryons and their first positive-parity excitations. Such analyses predict the existence of nonpointlike, dynamical quark-quark (diquark) correlations within all baryons; and a uniformly sound description of the systems studied is obtained by retaining flavor-antitriplet–scalar and flavor-sextet–pseudovector diquarks. Thus constituted, the rest-frame wave function of every system studied is primarily $S$ wave in character; and the first positive-parity excitation of each octet or decuplet baryon exhibits the characteristics of a radial excitation. Importantly, every ground-state octet and decuplet baryon possesses a radial excitation. Hence, the analysis predicts the existence and masses of positive-parity excitations of the $\mathrm{\Xi }$, ${\mathrm{\Xi }}^{*}$, and $\mathrm{\Omega }$ baryons, states which have not yet been empirically identified. This body of analysis suggests that the expression of emergent mass generation is the same in all $u$, $d$, and $s$ baryons and, notably, that dynamical quark-quark correlations play an essential role in the structure of each one. It also provides the basis for developing an array of predictions that can be tested in new generation experiments.

Figure 1

Poincaré covariant Faddeev equation: a 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 nonpointlike diquarks within the baryon. The shaded rectangle demarcates the kernel of the Faddeev equation: single line, dressed-quark propagator (Sec. 2b); $\mathrm{\Gamma }$, diquark correlation amplitude (Sec. 2c); and double line, diquark propagator (Sec. 2d).

Figure 2

Comparison between the masses listed in Table 1 (black circles), computed herein using Faddeev equation kernels built with dressed quarks and diquarks described by QCD-like momentum-dependent propagators and amplitudes and those obtained using a symmetry-preserving treatment of a vector $\times$ vector contact interaction (blue stars) [45]. Upper panel, octet states. Lower panel, decuplet states. The vertical riser indicates the response of our results to a coherent $\pm 5\%$ change in the mass scales associated with the diquarks and dressed quarks, Eqs. (6) and (a5), respectively. The horizontal axis lists a particle name with a subscript that indicates whether it is a ground-state ($n=0$) or first positive-parity excitation ($n=1$).

Figure 3

Upper panel, pictorial representation of octet masses in Table 1. Circles (black), computed masses. The vertical riser indicates the response of our predictions to a coherent $\pm 5\%$ change in the mass scales associated with the diquarks and dressed quarks, Eqs. (6) and (a5), respectively. Diamonds (green), empirical Breit-Wigner masses [10]. The horizontal axis lists a particle name with a subscript that indicates whether it is a ground-state ($n=0$) or first positive-parity excitation ($n=1$). Lower panel, analogous plot for the decuplet masses in Table 1. Where noticeable, the estimated uncertainty in the location of a resonance’s Breit-Wigner mass is indicated by an error bar.

Figure 4

Upper panel, relative strengths of various diquark components within the indicated baryon’s Faddeev amplitude, defined via Eqs. (22) and (23). Lower panel, relative contribution to a baryon’s mass from a given diquark correlation in that baryon’s Faddeev amplitude, defined in association with Eqs. (24).

Figure 5

Octet baryons and their first positive-parity excitations. Upper panel, baryon rest-frame quark-diquark orbital angular momentum fractions, as defined in Eqs. (27). Lower panel, relative contribution of various quark-diquark orbital angular momentum components to the mass of a given baryon.

Figure 6

Decuplet baryons and their first positive-parity excitations. Upper panel, baryon rest-frame quark-diquark orbital angular momentum fractions, as defined in Eqs. (27). Lower panel, relative contribution of various quark-diquark orbital angular momentum components to the mass of a given baryon.

Figure 7

$\mathrm{\Lambda }$-baryon Faddeev wave functions, Chebyshev-moment projections, Eq. (28). $S$ wave: ${\mathrm{\Lambda }}_{n=0}$ (upper left) and ${\mathrm{\Lambda }}_{n=1}$ (upper right). Legend: $\mathrm{A}\to {\tilde{\mathfrak{s}}}_1^1$; $\mathrm{B}\to {\tilde{\mathfrak{s}}}_1^{[2,3]}$; $\mathrm{C}\to ({\tilde{\mathfrak{a}}}_3^{[6,8]}+2{\tilde{\mathfrak{a}}}_5^{[6,8]})/3$; and $\mathrm{D}\to {\tilde{\mathfrak{a}}}_2^{[6,8]}$. $P$ wave: ${\mathrm{\Lambda }}_{n=0}$ (lower left) and ${\mathrm{\Lambda }}_{n=1}$ (lower right). Legend: $\mathrm{A}\to {\tilde{\mathfrak{s}}}_2^1$; $\mathrm{B}\to {\tilde{\mathfrak{s}}}_2^{[2,3]}$; $\mathrm{C}\to ({\tilde{\mathfrak{a}}}_4^{[6,8]}+2{\tilde{\mathfrak{a}}}_6^{[6,8]})/3$; $\mathrm{D}\to {\tilde{\mathfrak{a}}}_1^{[6,8]}$; and $\mathrm{E}\to ({\tilde{\mathfrak{a}}}_4^{[6,8]}-{\tilde{\mathfrak{a}}}_6^{[6,8]})$. $D$ waves are negligible for all octet baryons considered herein.

Figure 8

${\mathrm{\Xi }}^{*}$-baryon Faddeev wave functions, Chebyshev-moment projections, Eq. (28). $S$ wave: ${\mathrm{\Xi }}_{n=0}^{*}$ (top left) and ${\mathrm{\Xi }}_{n=1}^{*}$ (top right). Legend: $\mathrm{A}\to {\tilde{\mathfrak{a}}}_1^6+(-{\tilde{\mathfrak{a}}}_6^6+{\tilde{\mathfrak{a}}}_8^6)/3$; $\mathrm{B}\to {\tilde{\mathfrak{a}}}_1^9+(-{\tilde{\mathfrak{a}}}_6^9+{\tilde{\mathfrak{a}}}_8^9)/3$. $P$ wave: ${\mathrm{\Xi }}_{n=0}^{*}$ (middle left) and ${\mathrm{\Xi }}_{n=1}^{*}$ (middle right). Legend: $\mathrm{A}\to {\tilde{\mathfrak{a}}}_4^6$; $\mathrm{B}\to {\tilde{\mathfrak{a}}}_4^9$; $\mathrm{C}\to (2{\tilde{\mathfrak{a}}}_2^6-{\tilde{\mathfrak{a}}}_5^6-2{\tilde{\mathfrak{a}}}_7^6)/3$; $\mathrm{D}\to (2{\tilde{\mathfrak{a}}}_2^9-{\tilde{\mathfrak{a}}}_5^9-2{\tilde{\mathfrak{a}}}_7^9)/3$; $\mathrm{E}\to {\tilde{\mathfrak{a}}}_2^6-({\tilde{\mathfrak{a}}}_5^6-{\tilde{\mathfrak{a}}}_7^6)/5$; and $\mathrm{F}\to {\tilde{\mathfrak{a}}}_2^9-({\tilde{\mathfrak{a}}}_5^9-{\tilde{\mathfrak{a}}}_7^9)/5$; $D$ wave: ${\mathrm{\Xi }}_{n=0}^{*}$ (bottom left) and ${\mathrm{\Xi }}_{n=1}^{*}$ (bottom right). Legend: $\mathrm{A}\to {\tilde{\mathfrak{a}}}_3^6$; $\mathrm{B}\to {\tilde{\mathfrak{a}}}_3^9$; $\mathrm{C}\to -({\tilde{\mathfrak{a}}}_6^6+2{\tilde{\mathfrak{a}}}_8^6)/3$; $\mathrm{D}\to -({\tilde{\mathfrak{a}}}_6^9+2{\tilde{\mathfrak{a}}}_8^9)/3$; $\mathrm{E}\to -{\tilde{\mathfrak{a}}}_6^6+{\tilde{\mathfrak{a}}}_8^6$; and $\mathrm{F}\to -{\tilde{\mathfrak{a}}}_6^9+{\tilde{\mathfrak{a}}}_8^9$. $F$-wave components are negligible for all decuplet baryons considered herein.