  We investigate the possibility of using the 4 dimensional $O(4)$ symmetric
$\phi^4$ model as an effective theory for the sigma-pion system. We carry out
lattice Monte Carlo simulations to establish the triviality bound in the case
of explicitly broken symmetry and to compare it with results from continuum
functional methods. In case of a physical parametrization we find that
triviality restricts the possible lattice spacings to a narrow range, therefore
cutoff independence in the effective theory sense is practically impossible for
thermal quantities. We match the critical line in the space of bare couplings
in the different approaches and compare vacuum physical quantities along the
line of constant physics (LCP).
