The nonanalyticity and the sign problem in the $Z_3$-symmetric heavy quark model at low temperature are studied phenomenologically. For the free heavy quarks, the nonanalyticity is analyzed in relation to the zeros of the grand canonical partition function. The $Z_3$-symmetric effective Polyakov-line model (EPLM) in the strong coupling limit is also considered as an phenomenological model of $Z_3$-symmetric QCD with large quark mass at low temperature. We examine how the $Z_3$-symmetric EPLM approaches the original one in the zero-temperature limit. The effects of the $Z_3$ symmetry affect the structure of zeros of the microscopic probability density function at the nonanalytic point. The average value of the Polyakov line can detect the structure, while the other thermodynamic quantities are not sensitive to the structure in the zero-temperature limit. The effect of the imaginary quark chemical potential is also discussed. The imaginary part of the quark number density is very sensitive to the symmetry structure at the nonanalytical point. For a particular value of the imaginary quark number chemical potential, large quark numbers may be induced in the vicinity of the nonanalytical point.

• Received 17 April 2021
• Accepted 21 June 2021

DOI:https://doi.org/10.1103/PhysRevD.104.014012

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Published by the American Physical Society