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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 outlattice Monte Carlo simulations to establish the triviality bound in the caseof explicitly broken symmetry and to compare it with results from continuumfunctional methods. In case of a physical parametrization we find thattriviality restricts the possible lattice spacings to a narrow range, thereforecutoff independence in the effective theory sense is practically impossible forthermal quantities. We match the critical line in the space of bare couplingsin the different approaches and compare vacuum physical quantities along theline of constant physics (LCP).

The $O(4)$ $\phi^4$ model as an effective light meson theory: A lattice-continuum comparison