  The lepton flavor violating decay of the Higgs boson $H\to l_Al_B$ is studied
within two qualitatively different extensions of the Yukawa sector: one
renormalizable and the other nonrenormalizable; both incorporating Lorentz
violation in a model-independent fashion. These extensions are characterized by
Yukawa-like matrices, the former by a constant Lorentz 2-tensor $Y^{AB}_{\mu
\nu}$, whereas the latter by a constant Lorentz vector $Y^{AB}_\mu$. It is
found that the experimental constraints on the decays $l_A\to l_B\gamma$
severely restrict lepton flavor violating Higgs signals in the renormalizable
scenario. In this context, it is found that $BR(H\to \mu^\pm e^\mp)$ and
$BR(H\to \tau^\pm \mu^\mp)$ cannot be larger than $10^{-18}$ and $10^{-11}$,
respectively. In the nonrenormalizable scenario, transitions mediated by the
Higgs or the $Z$ gauge boson are induced at tree level, and we find mild
restrictions on lepton flavor violation. Using the experimental limits on the
three-body decays $l_A \to l_B \bar{l}_Cl_C$ to constraint the vector
$Y^{AB}_\mu$, it is found that the branching ratio for the decays $H\to \mu^\pm
e^\mp$ is of about $4\times 10^{-9}$, more important, a branching ratio of
$7\times 10^{-4}$ is found for the $\tau^\pm \mu^\mp$ mode. Accordingly, the
decay $H \to \tau^\pm \mu^\mp$ could be at the reach of future measurements.
The lepton flavor violating decays of the $Z$ gauge boson were also studied. In
the renormalizable scenario, it was found the undetectable branching ratios
$BR(Z\to \mu^\pm e^\mp)<5.7\times 10^{-21}$ and $BR(Z\to \tau^\pm
\mu^\mp)<2.0\times 10^{-12}$. In the nonrenormalizable scenario, it was found
that $BR(Z\to \mu^\pm e^\mp)<0.67\times 10^{-12}$ and $BR(Z\to \tau^\pm
\mu^\mp)<1.12\times 10^{-7}$. Although the latter branching ratio is relatively
large, it still could not be within the range of future measurements.
