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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+1doi:10.1088/1674-1137/38/6/063102 fatcat:bq3r6rbqxzcbhkyw3ybnrenwjm