  In this work, we compare the tetraquark mixing model and meson molecules in
describing the two physical nonets in the $J^P=0^+$ channel, the light nonet
[$a_0 (980)$, $K_0^* (700)$, $f_0 (500)$, $f_0 (980)$] and the heavy nonet
[$a_0 (1450)$, $K_0^* (1430)$, $f_0 (1370)$, $f_0 (1500)$]. In particular, we
focus on whether successful aspects of the tetraquark mixing model that apply
to all members of each nonet can be reproduced from a model of meson molecules.
By combining two mesons in the lowest-lying pseudoscalar nonet, we construct
SU$_f$(3) molecular nonets that can be tested for the two physical nonets. This
molecular approach can make two flavor nonets just as the tetraquark mixing
model but this model has some difficulties in describing the universal features
of the two nonets %Because of this, this molecular model cannot reproduce
successful aspects of the tetraquark mixing model, such as mass splitting
between the two nonets, and enhancement or suppression of the coupling
strengths of the two nonets into two pseudoscalar mesons. We also compare the
fall-apart modes of the tetraquark mixing model and the two-meson modes from
the molecular model. A clear distinction can be seen by the two-pion modes in
the isovector resonances. The two-pion modes appear in the molecular model, but
not in the tetraquark mixing model. The absence of the two-pion modes is
supported by the experimental decay modes of the isovector resonances.
