  The $f_{1}(1420)$ productions via pion and kaon induced reactions on a proton
target are investigated in an effective Lagrangian approach. Two treatments of
the $t$-channel Born term, the Feynman model and the Regge model, are
introduced to calculate the total and differential cross sections of the
$\pi^{-}p\rightarrow f_{1}(1420)n$ and $K^{-}p\rightarrow f_{1}(1420)\Lambda $
reactions. The numerical results indicate that the experimental data of the
total cross section of the $\pi ^{-}p\rightarrow f_{1}(1420)n$ reaction can be
reproduced in both the Feynman and the Regge models. Based on the parameters
determined in the pion induced reaction, the cross sections of the
$K^{-}p\rightarrow f_{1}(1420)\Lambda $ reaction, about which there is little
data found in the literature, are predicted in the same beam momentum range. It
is found that the line shapes of the total cross section obtained in a kaon
induced reaction with two treatments are quite different. The cross sections
for both reactions are at an order of magnitude of $\mu$b, or larger, at a beam
momenta up to 10 GeV/c. The differential cross sections for both pion and kaon
induced reactions are also present. It is found that in the Regge model, the
$t$ channel provides a sharp increase at extreme forward angles. The results
suggest that the experimental study of the $f_1(1420)$ in the kaon induced
reaction on a proton target is as promising as in the pion induced reaction.
Such an experimental measurement is also very helpful to clarify the production
mechanism of the $f_1(1420)$.
