  We investigate the impact of a $\lambda_6 \varphi^6$ term included in a
chirally invariant lattice Higgs-Yukawa model. Such a term could emerge from
BSM physics at some larger energy scale. We map out the phase structure of the
Higgs-Yukawa model with positive $\lambda_6$ and negative quartic self coupling
of the scalar fields. To this end, we evaluate the constraint effective
potential in lattice perturbation theory and also determine the magnetization
of the model via numerical simulations which allow us to reach also
non-perturbative values of the couplings. As a result, we find a complex phase
structure with first and second order phase transitions identified through the
magnetization. Further we analyze the effect of such a $\varphi^6$ term on the
lower Higgs boson mass bound to see, whether the standard model lower mass
bound can be altered.
