Operator analysis of effective spin-flavor interactions for $L=1$ excited baryons

We match the nonrelativistic quark model, with both flavor-dependent and flavor-independent effective quark-quark interactions, to the spin-flavor operator basis of the $1/N_c$ expansion for the $L=1$ nonstrange baryons. We obtain analytic expressions for the coefficients of the $1/N_c$ operators in terms of radial integrals that depend on the shape and relative strength of the spin-spin, spin-orbit and tensor interactions of the model, which are left unspecified. We obtain several new, parameter-free relations between the seven masses and the two mixing angles that can discriminate between different spin-flavor structures of the effective quark-quark interaction. We discuss in detail how a general parametrization of the mass matrix depends on the mixing angles and is constrained by the assumptions on the effective quark-quark interaction. We find that, within the present experimental uncertainties, consistency with the best values of the mixing angles as determined by a recent global fit to masses and decays does not exclude any of the two most extreme possibilities of flavor-dependent (-independent) quark-quark interactions, as generated by meson (gluon) exchange interactions.

Figure 1

The angle correlation implied by mass relation R1, Eq. (34), is shown as a full line. The dashed line corresponds to R1’, Eq. (35), which differs from R1 by $1/N_c^2$ corrections.

Figure 2

Angle correlations from the relations R1 and R2, given by Eq. (34) and Eq. (37). Left panel: The curves giving the ${\theta }_1-{\theta }_3$ angle correlations intersect at $\text{sol}–\mathrm{A}=(0.80,0.01)$ (black dot) and $\text{sol}–\mathrm{B}=(-0.04,-0.23)$ (circle), obtained using the central values of the masses. The smaller point in between corresponds to the best fit of the angles of Ref. [34]. Right panel: Scatter plot obtained including the experimental errors on the masses. The rectangles correspond to sol–A, sol–B and the best fit (dashed) with errors as in Table 4.

Figure 3

OGE mass-angle relation R3 [Eq. (39)] (dashed green) and general correlations R1 [Eq. (34)] (red) and R2 [Eq. (37)] (blue) Left: Using the central values of the masses. Right: Scatter plot obtained including the experimental errors on the masses and OGE angles with errors (dashed rectangle) as in Table 4.

Figure 4

OME mass relations R4 [Eq. (42)] (orange) and R5 [Eq. (43)] (green), and general correlations R1 [Eq. (34)] (red) and R2 [Eq. (37)] (blue). Left: Using the central values of the masses R4 is not satisfied. Right: Scatter plot obtained including the experimental errors on the masses and OME angles with errors (dashed rectangle) as in Table 4.

Figure 5

Mixing angles for sol–A, sol–B, the OME and OGE model interactions as in Table 4 and the GGS best global fit values of Ref. [34].

Figure 6

Matrix elements as a function of the mixing angles $({\theta }_1,{\theta }_3)$. We also show the mixing angles for sol–A (black dot) and sol–B (circle). The smaller point in between corresponds to the GGS best-fit angles. The red curve shows the correlation between mixing angles given by relation R1 [Eq. (34)].

Figure 7

The constraint $D_1+D_2$ (dashed contour in left panel) and the magnitude of the tensor interaction $\sqrt{D_1^2+D_2^2}$ (right panel) as a function of the mixing angles. R1, sol–A, sol–B and best fit value are as in Fig. 6.

Figure 8

The magnitude of the spin-orbit interaction $\sqrt{P_1^2+P_2^2+P_3^2+P_4^2}$ as a function of the mixing angles. R1, sol–A, sol–B and best fit value are as in Fig. 6.

Figure 9

Operator coefficients as a function of the mixing angles $({\theta }_1,{\theta }_3)$. We also show the mixing angles for sol–A (black dot) and sol–B (circle). The smaller point in between corresponds to the best-fit angles. The red curve shows the correlation between mixing angles given by relation R1 [Eq. (34)].