UA(1) breaking and scalar mesons in the Nambu and Jona-Lasinio model

We examine the role played by the ${\mathrm{U}}_{\mathit{A}}$(1) symmetry-breaking ’t Hooft interaction in scalar mesons, using the two- and three-flavor versions of the Nambu and Jona-Lasinio model. We first examine the two-flavor case within the Hartree + random phase approximation in order to (a) develop the formalism necessary for the treatment of the three-flavor case, and (b) to see if the scalar sector can be treated as ideally mixed, i.e., if the u,d quark mesons completely separate themselves from the ones with strange quarks. Then we calculate the scalar and pseudoscalar meson mass spectra in the three-flavor model within the mean-field approximation and establish relations between the scalar and pseudoscalar meson masses. This analysis leads to a new approximate sum rule that is a consequence of ${\mathrm{U}}_{\mathit{A}}$(1) breaking in the Nambu–Jona-Lasinio (NJL) model: ${\mathit{m}}_{{\mathrm{\eta }}^{\prime }}^2$+${\mathit{m}}_{\mathrm{\eta }}^2$-2${\mathit{m}}_{\mathit{K}}^2$≃ -${\mathit{m}}_{{\mathit{f}}_0}^{\prime 2}$-${\mathit{m}}_{{\mathit{f}}_0}^2$ +2${\mathit{m}}_{{\mathit{K}}_0^{\mathrm{*}}}^2$, where η′,η,K are the observed pseudoscalar mesons, ${\mathit{K}}_0^{\mathrm{*}}$ is the strange scalar meson at 1430 MeV, and ${\mathit{f}}_0$,${\mathit{f}}_0^{\prime }$ are ‘‘the eighth and the ninth’’ scalar mesons. It states that the mass splitting between the scalar singlet and the octet is of the same size as, but of opposite sign to, the corresponding splitting among the pseudoscalars. We discuss the large-${\mathit{N}}_{\mathit{C}}$ limit of our results. Two of our partial results leading to the sum rule are consistent with the large-${\mathit{N}}_{\mathit{C}}$ limit results of QCD, though one of them is not required by QCD.