  WWithin mixed-action chiral perturbation theory (MA$\chi$PT), Sasa's
derivation of the bubble contribution to scalar $a_0$ meson is extended to
those of scalar $\kappa$ and $\sigma$ mesons. We revealed that $\kappa$ bubble
has two double poles and $\sigma$ bubble contains a quadratic-in-$t^2$ growth
factor stemming from the multiplication of two double poles for a general mass
tuning of valence quarks and sea quarks. The corresponding preliminary
analytical expressions in MA$\chi$PT with 2+1 chiral valence quarks and 2+1
staggered sea quarks will be helpful for lattice studies of scalar mesons.
