Generalized Nonlinear Lagrangian for Pseudoscalar and Scalar Mesons

Abstract

We construct a nonlinear Lagrangian to describe the scalar and pseudoscalar mesons such that the chiral SU(3) \ifmmode\times\else\texttimes\fi{} SU(3) symmetry is realized by an octet of Goldstone pseudoscalar mesons while the scalar particles behave neither as parity partners of the pseudoscalar mesons nor as Goldstone bosons in the SU(3) \ifmmode\times\else\texttimes\fi{} SU(3)-symmetry limit. The symmetry-breaking Lagrangian is assumed to transform as the $(3,\overline{3})+(\overline{3},3)$ representation of the SU(3) \ifmmode\times\else\texttimes\fi{} SU(3) group and contain explicit SU(3) - and SU(2)-violating terms. The transformation properties of the scalar fields together with these breaking terms in the Lagrangian enable our model to have an SU(3) -broken vacuum. We exhibit the masses of the scalar and pseudoscalar particles as well as the decay constants defining PCAC (partial conservation of axial-vector current) and PCVC (partial conservation of vector current) relations in terms of the parameters of the model, and obtain various well-known relationships (including the Glashow-Weinberg sum rules) between the physical quantities. The smoothness assumption is shown to imply approximate SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) symmetry of the Lagrangian with a small SU(3) violation in the vacuum (Gell-Mann-Oakes-Renner model), while an appropriate change in the smoothness assumption leads to approximate SU(3) symmetry of the Lagrangian with an almost SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) -invariant vacuum (Brandt-Preparata model). We calculate the symmetry-breaking parameters of the Lagrangian and vacuum and predict the mass and decay constant of the $\ensuremath{\kappa}$ meson. Furthermore, from the width of the decay ${\ensuremath{\pi}}_{N}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\pi}$, we obtain the decay widths of all the scalar mesons. Finally, we investigate the nonelectromagnetic SU(2) breaking and find that it gives a major contribution to the kaon mass difference but only a 5% correction to the pion mass difference. The fact that the scalar mesons are needed to get ${f}_{K}\ensuremath{\ne}{f}_{\ensuremath{\pi}}$ and different wave-function renormalization constants for the fields, which enables us to have solutions other than that of Gell-Mann, Oakes, and Renner, demonstrates the importance of the scalar mesons in SU(3)-symmetry-breaking effects.