Abstract
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).
Highlights
The φ4 scalar model with an internal Oð4Þ symmetry in four spacetime dimensions has long been used as a model for spontaneous chiral symmetry breaking [1]
In the case of a physical parametrization we find that triviality restricts the possible lattice spacings to a narrow range, and cutoff independence in the effective theory sense is practically impossible for thermal quantities
Based on a calculation carried out by Lüscher and Weisz (LW) in the same model applied to the Higgs particle [4] one can estimate the lowest lattice spacing that can be reached in a parametrization adjusted to light mesons
Summary
The φ4 scalar model with an internal Oð4Þ symmetry in four spacetime dimensions has long been used as a model for spontaneous chiral symmetry breaking [1]. Based on a calculation carried out by Lüscher and Weisz (LW) in the same model applied to the Higgs particle [4] one can estimate the lowest lattice spacing that can be reached in a parametrization adjusted to light mesons This turns out to be aLmWin 1⁄4 0.40ð4Þ fm, which corresponds to a maximal cutoff in momentum representation to a few times 500 MeV. To compare the values of physical quantities, we need the relation between the lattice spacing a and the cutoff Λ This is determined by matching the critical line of the model at zero temperature with the one determined by Lüscher and Weisz in [2] using the hopping parameter expansion.
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