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Finite size scaling study of $N_{\text{f}}=4$ finite density QCD on the lattice

We explore the phase space spanned by the temperature and the chemical potential for 4-flavor lattice QCD using the Wilson-clover quark action. In order to determine the order of the phase transition, we apply finite size scaling analyses to gluonic and quark observables including plaquette, Polyakov loop and quark number density, and examine their susceptibility, skewness, kurtosis and Challa-Landau-Binder cumulant. Simulations were carried out on lattices of a temporal size fixed at $N_{\text{t}}=4$ and spatial sizes chosen from $6^3$ up to $10^3$. Configurations were generated using the phase reweighting approach, while the value of the phase of the quark determinant were carefully monitored. The $μ$-parameter reweighting technique is employed to precisely locate the point of the phase transition. Among various approximation schemes for calculating the ratio of quark determinants needed for $μ$-reweighting, we found the Taylor expansion of the logarithm of the quark determinant to be the most reliable. Our finite-size analyses show that the transition is first order at $(β, κ, μ/T)=(1.58, 0.1385, 0.584\pm 0.008)$ where $(m_π/m_ρ, T/m_ρ)=(0.822, 0.154)$. It weakens considerably at $(β, κ, μ/T)=(1.60, 0.1371, 0.821\pm 0.008)$ where $(m_π/m_ρ, T/m_ρ)=(0.839, 0.150)$, and a crossover rather than a first order phase transition cannot be ruled out.