The forward-backward asymmetry $A_{\mathrm{FB}}$ in top quark pair production, measured at the Tevatron, is probably related to the contribution of new particles. The Tevatron result is more than a $2\sigma$ deviation from the standard model prediction and motivates the application of alternative models introducing new states. However, as the standard model predictions for the total cross section ${\sigma }_{tt}$ and invariant mass distribution $M_{tt}$ for this process are in good agreement with experiments, any alternative model must reproduce these predictions. These models can be placed into two categories: One introduces the $s$-channel exchange of new vector bosons with chiral couplings to the light quarks and to the top quark, and another relies on the $t$-channel exchange of particles with large flavor-violating couplings in the quark sector. In this work, we employ a model which introduces both $s$- and $t$-channel nonstandard contributions for the top quark pair production in proton-antiproton collisions. We use the minimal version of the $SU(3{)}_C\otimes SU(3{)}_L\otimes U(1{)}_X$ model (3-3-1 model) that predicts the existence of a new neutral gauge boson, called $Z^{\prime }$. This gauge boson has both flavor-changing couplings to up and top quarks and chiral coupling to the light quarks and to the top quark. This very peculiar model coupling can correct the $A_{\mathrm{FB}}$ for top quark pair production for two ranges of $Z^{\prime }$ mass while leading to a cross section and invariant mass distribution quite similar to the standard model ones. This result reinforces the role of the 3-3-1 model for any new physics effect.

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