The $K_{l3\mathrm{}}$ form factors are investigated in the one-loop approximation within the context of a linear renormalizable SU(3) $\sigma$ model. The model incorporates SU(3) nonets of pseudoscalar ($\pi$,$K$,$\eta$,${\eta }^{\prime }$) and scalar ($\epsilon$,$\kappa$,$\sigma$,${\sigma }^{\prime }$) mesons. The ${\eta }^{\prime }$, $\epsilon$, $\kappa$, $\sigma$, and ${\sigma }^{\prime }$ mesons are associated with the $X^0\mathrm{}\left(957\right)\mathrm{}$, $\delta \mathrm{}\left(980\right)\mathrm{}$, $\kappa \mathrm{}\left(1400\right)\mathrm{}$, $S^{*}\mathrm{}\left(980\right)\mathrm{}$, and $\epsilon \mathrm{}\left(1300\right)\mathrm{}$, respectively. The Lagrangian contains the most general renormalizable chiral-SU(3) × SU(3)-invariant couplings as well as explicit linear symmetry-breaking terms belonging to the (3,3*) ⊕ (3*,3) representation of SU(3) × SU(3). All calculations are carried out in the one-loop approximation. With this model we first obtain a reasonable approximation to the scalar and pseudoscalar mass spectrum and the known leptonic decay constants. Most of the masses and decay constants are reproduced within 10%. In addition, the second-order corrections were usually in the neighborhood of 15-20% or less, supporting the conjecture that higher-order strong-interaction effects may, in most cases, be rather small. Employing these solutions, we calculate the $K_{l3\mathrm{}}$ form factors and compare these with recent experimental studies. The predictions for ${\lambda }_{+}$ are too small probably due to the fact that, at this level of approximation, the model contains no spin-one poles (vector mesons). However, the model predictions for ${\lambda }_0$ and $\xi \mathrm{}\left(0\right)\mathrm{}$ are fairly good. A number of theoretical predictions for the $K_{l3\mathrm{}}$ form factors based on current algebra and chiral perturbation theory are then investigated. In particular, we were interested in the magnitude of the corrections to various predictions derived using specific symmetry assumptions. As the model reproduces quite closely the model-independent calculation of $f_{+}\mathrm{}\left(0\right)\mathrm{}$ from chiral perturbation theory, we feel that our conclusions in this area may have a more general significance. For example, in the tree approximation the Callan-Treiman relation is an identity. In the one-loop approximation, the magnitude of the strong-interaction effects is much larger than the symmetry-breaking effects and the relation is still obeyed quite well (within ∼ 2%), reflecting the influence of the underlying SU(2) × SU(2) symmetry. Overall, our results support calculations based on chiral perturbation theory.

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