We study lattice cutoff effects on the confinement-deconfinement transition and the $Z_3$ symmetry in $SU(3)$-Higgs theory in $3+1$ dimensions. The Higgs in this study is a complex triplet with vanishing bare mass and quartic coupling. The lattice cutoff is regulated by varying the number of temporal lattice sites, $N_{\tau }$. Our results show that the nature of the confinement-deconfinement transition depends on $N_{\tau }$. For $N_{\tau }=2$ the transition is found to be the end point of a first-order transition and is first order for $N_{\tau }\ge 3$. The distributions of the Polyakov loop and other observables, sensitive to the $Z_3$ symmetry, show that the strength of $Z_3$ explicit breaking decreases with $N_{\tau }$. Up to $T\simeq 2T_c$, the free energy difference between $Z_3$ states decreases with $N_{\tau }$, suggesting the realization of $Z_3$ symmetry in the continuum limit.

• Received 24 November 2022
• Accepted 7 August 2023

DOI:https://doi.org/10.1103/PhysRevD.108.054503

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Published by the American Physical Society