Kobayashi-Kondo-Maskawa-’t Hooft interaction in pentaquarks

We review critically the predictions of pentaquarks in the quark model, in particular, those based on the flavor-spin-dependent (Glozman-Riska) hyperfine interaction and the color-spin (one-gluon-exchange Fermi-Breit) one. We include the antiquark interactions and find that: (1) the exotic SU(3) multiplets are not substantialy affected in the flavor-spin model, whereas some of the nonexotic multiplets are; and (2) the variational upper bound on the ${\Xi }^{--}-{\Theta }^{+}$ mass difference in the color-spin hyperfine interaction model is substantially reduced. This leads us to the $U_A(1)$ symmetry breaking Kobayashi-Kondo-Maskawa-’tHooft interaction. We discuss some of its phenomenological consequences for pentaquarks.

Figure 1

The weight diagram of the ${\overline{\mathbf{10}}}_{\mathrm{F}}$-plet in $\mathrm{SU}(3{)}_{\mathrm{F}}$.

Figure 2

Pauli allowed pentaquark states in the harmonic oscillator potential. The excitation energy $\hslash \omega$ depends on the color singlet state the pentaquark is in.

Figure 3

The $\mathrm{SU}(3{)}_{\mathrm{F}}$ weight diagrams of the ${\overline{\mathbf{10}}}_{\mathrm{F}}$- and the ${\mathbf{8}}_{\mathrm{F}}$-plet pentaquarks and their mixing members: the isodoublet $N$ and isovector $\Sigma$.

Figure 4

Pentaquark energy spectrum in the Glozman-Riska model of flavor-spin HFI with and without antiquark interactions, in units of $4C_{\chi }\simeq 120\mathrm{MeV}$. Spin $\frac{1}{2}$ is assumed for the pentaquark and spin zero for the four-quark system.

Figure 5

Pseudoscalar and vector meson spectrum in the Glozman-Riska model, without ($C_{\chi }=0$) and with ($C_{\chi }>0$) flavor-spin hyperfine interaction. The value of the GR coupling constant $C_{\chi }=30\mathrm{MeV}$, in accord with the standard baryon fit. Note that the predicted states are not properly ordered and that only one, the ${\eta }^{\prime }(958)$, of 18 predicted mesons (loosely) matches its experimental counterpart. The arrows are shown to guide the eye.

Figure 6

Feynman diagrammatic representation of the KKMT three-body interaction. $\mathcal{I}$ stands for instanton and $\overline{\mathcal{I}}$ stands for anti-instanton.

Figure 7

Feynman diagrammatic representation of the KKMT two-body interaction. $\mathcal{I}$ stands for instanton and $\overline{\mathcal{I}}$ stands for anti-instanton.

Figure 8

Meson mass spectrum in the nonrelativistic quark model with harmonic oscillator confinement without ($K=0$) and with ($K\ne 0$) Bonn version of the KKMT (’tHooft) interaction and without (${\alpha }_C=0$) and with ${\alpha }_C\ne 0$ color-spin (Breit) HFI. Note that the predicted states are properly ordered and that only two predicted states’ masses do not match precisely their experimental counterparts: the $\pi$ and the $\eta$.

Figure 9

Vector and pseudoscalar meson mass spectrum without ($K=0$) and with ($K\ne 0$) the Tokyo version of the KKMT interaction and without (${\alpha }_C=0$) and with ${\alpha }_C\ne 0$ color-spin (Breit) HFI. Note that the pseudoscalar states are properly ordered, whereas vector mesons are generally too low and out of order.

Figure 10

Pentaquark spectrum (mass splitting) with the spin-independent part of the KMT interaction. The spin-independent HFI energy $E_{\mathrm{S}.\mathrm{I}.}^{\mathrm{KMT}}$ is in units of $A=4C_{\chi }=95\pm 20\mathrm{MeV}$. Numbers at the bottom of columns indicate the $q^4$ states’ $\mathrm{SU}(3{)}_{\mathrm{F}}$ multiplets.

Figure 11

Pentaquark spectrum (mass splitting) of the $\mathbf{1134}$-plet with the complete (spin-independent and spin-dependent parts) of the two-body KMT interaction. Other SU(6) multiplets are vertically displaced according to their value of the quadratic Casimir operator. HFI energy $E_{\mathrm{HF}}^{\mathrm{KMT}}$ is in units of $A=4C_{\chi }=95\pm 20\mathrm{MeV}$. Numbers at the bottom of columns indicate the $q^4$ states’ $\mathrm{SU}(3{)}_{\mathrm{F}}$ multiplets, and the pentaquark $\mathrm{SU}(3{)}_{\mathrm{F}}$ multiplets (above). The spin of the pentaquark is denoted above the level.