θ dependence of Tc in 4d SU(3) Yang-Mills theory with histogram method and the Lee-Yang zeros in the large N limit

The phase diagram on the θ-T plane in four dimensional SU(3) Yang-Mills theory is explored. We revisit the θ dependence of the deconfinement transition temperature, T_c(θ), on the lattice through the constraint effective potential for Polyakov loop. The θ term is introduced by the reweighting method, and the critical β is determined to θ ∼ 0.75, where the interpolation in β is carried out by the multipoint reweighting method. The θ dependence of T_c obtained here turns out to be consistent with the previous result by D'Elia and Negro [1, 2]. We also derive T_c(θ) by classifying configurations into the high and low temperature phases and applying the Clausius-Clapeyron equation. It is found that the potential barrier in the double well potential at T_c(θ) becomes higher with θ, which suggests that the first order transition continues robustly above θ ∼ 0.75. Using information obtained here, we try to depict the expected θ dependence of the free energy density at T ≲ T_c(0), which crosses the first order transition line at an intermediate value of θ. Finally, how the Lee-Yang zeros associated with the spontaneous CP violation appear is discussed formally in the large N limit, and the locations of them are found to be (θ_Rθ_I)= (2 m +1 )π 2/n +1 2 χ V₄) with m and n arbitrary integers.