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Register a new userImplications of imaginary chemical potential for model building of QCD
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Authors
Kouji Kashiwa
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Properties of QCD at finite imaginary chemical potential are revisited to utilize for the model building of QCD in low energy regimes. For example, the electric holonomy which is closely related to the Polyakov-loop drastically affects thermodynamic quantities beside the Roberge-Weiss transition line. To incorporate several properties at finite imaginary chemical potential, it is important to introduce the holonomy effects to the coupling constant of effective models. This extension is possible by considering the entanglement vertex. We show justifications of the entanglement vertex based on the derivation of the effective four-fermi interaction in the Nambu--Jona-Lasinio model and present its general form with the local approximation. To discuss how to remove model ambiguities in the entanglement vertex, we calculate the chiral condensate with different $mathbb{Z}_3$ sectors and the dual quark condensate.
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The Polyakov loop extended Nambu--Jona-Lasinio (PNJL) model with imaginary chemical potential is studied. The model possesses the extended ${mathbb Z}_{3}$ symmetry that QCD does. Quantities invariant under the extended ${mathbb Z}_{3}$ symmetry, such as the partition function, the chiral condensate and the modified Polyakov loop, have the Roberge-Weiss (RW) periodicity. The phase diagram of confinement/deconfinement transition derived with the PNJL model is consistent with the RW prediction on it and the results of lattice QCD. The phase diagram of chiral transition is also presented by the PNJL model.
We draw the three-flavor phase diagram as a function of light- and strange-quark masses for both zero and imaginary quark-number chemical potential, using the Polyakov-loop extended Nambu-Jona-Lasinio model with an effective four-quark vertex depending on the Polyakov loop. The model prediction is qualitatively consistent with 2+1 flavor lattice QCD prediction at zero chemical potential and with degenerate three-flavor lattice QCD prediction at imaginary chemical potential.
We first extend our formulation for the calculation of $pi$- and $sigma$-meson screening masses to the case of finite chemical potential $mu$. We then consider the imaginary-$mu$ approach, which is an extrapolation method from imaginary chemical potential ($mu=i mu_{rm I}$) to real one ($mu=mu_{rm R}$). The feasibility of the method is discussed based on the entanglement Polyakov-loop extended Nambu--Jona-Lasinio (EPNJL) model in 2-flavor system. As an example, we investigate how reliable the imaginary-$mu$ approach is for $pi$- and $sigma$-meson screening masses, comparing screening masses at $mu_{rm R}$ in the method with those calculated directly at $mu_{rm R}$. We finally propose the new extrapolation method and confirm its efficiency.
Phase transitions in the imaginary chemical potential region are studied by the Polyakov loop extended Nambu-Jona-Lasinio (PNJL) model that possesses the extended Z3 symmetry. The extended Z3 invariant quantities such as the partition function, the chiral condensate and the modifed Polyakov loop have the Roberge-Weiss (RW) periodicity. There appear four types of phase transitions; deconfinement, chiral, Polykov-loop RW and chiral RW transitions. The orders of the chiral and deconfinement transitions depend on the presence or absence of current quark mass, but those of the Polykov-loop RW and chiral RW transitions do not. The scalar-type eightquark interaction newly added in the model makes the chiral transition line shift to the vicinity of the deconfiment transition line.
We study properties of 2+1-flavor QCD in the imaginary chemical potential region by using two approaches. One is a theoretical approach based on QCD partition function, and the other is a qualitative one based on the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model. In the theoretical approach, we clarify conditions imposed on the imaginary chemical potentials $mu_{f}=itheta_{f}T$ to realize the Roberge-Weiss (RW) periodicity. We also show that the RW periodicity is broken if anyone of $theta_{f}$ is fixed to a constant value. In order to visualize the condition, we use the PNJL model as a model possessing the RW periodicity, and draw the phase diagram as a function of $theta_{u}=theta_{d}equiv theta_{l}$ for two conditions of $theta_{s}=theta_{l}$ and $theta_{s}=0$. We also consider two cases, $(mu_{u},mu_{d},mu_{s}) =(itheta_{u}T,iC_{1}T,0)$ and $(mu_{u},mu_{d},mu_{s})=(iC_{2}T,iC_{2}T,itheta_{s}T)$; here $C_{1}$ and $C_{2}$ are dimensionless constants, whereas $theta_{u}$ and $theta_{s}$ are treated as variables. For some choice of $C_{1}$ ($C_{2}$), the number density of up (strange) quark becomes smooth in the entire region of $theta_{u}$ ($theta_{s}$) even in high $T$ region.
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