The QCD equation of state at finite density from analytical continuation

We determine the equation of state of QCD at finite chemical potential, to order $(\mu_B/T)^6$, for a system of 2+1 quark flavors. The simulations are performed at the physical mass for the light and strange quarks on several lattice spacings; the results are continuum extrapolated using lattices of up to $N_t=16$ temporal resolution. The QCD pressure and interaction measure are calculated along the isentropic trajectories in the $(T,~\mu_B)$ plane corresponding to the RHIC Beam Energy Scan collision energies. Their behavior is determined through analytic continuation from imaginary chemical potentials of the baryonic density. We also determine the Taylor expansion coefficients around $\mu_B=0$ from the simulations at imaginary chemical potentials. Strangeness neutrality and charge conservation are imposed, to match the experimental conditions.


Introduction
To analyse the quark gluon plasma that is created in heavy ion collision experiments at the LHC or RHIC a theoretical understanding of the quark gluon plasma in QCD is needed. In the region of the deconfinement transition lattice QCD is a good tool to study QCD since this area can not be accessed perturbatively. There have been several ideas in lattice QCD on how to obtain results at real finite chemical potential. However at the moment direct simulations that are continuum extrapolated and at physical quark masses are restricted to vanishing or imaginary chemical potential. On the other hand the collisions especially at RHIC take place away form the axis of zero chemical potential [1]. Therefore information in that region are needed. Even though it is not possible to do direct lattice simulations, it is possible to extrapolate observables form zero or imaginary chemical potential. This method is called analytical continuation.
In terms of the baryon chemical potential the pressure can be written as In this proceedings we will present the continuum extrapolated Taylor coefficients of the pressure up to c 6 based on [2]. As well as the extrapolation for the transition temperature published in [3]. Our analysis is done with n S = 0 and n Q = 0.4 n B . (2)

Analysis
Our lattice calculation uses the 4stout improved staggered action introduced in [4]. We use simulations on three different lattice size: 40 3 × 10, 48 3 × 12 and 64 3 × 16 and up to six different values for μ B /T . As the simulations on the different lattice sizes correspond to slightly different temperatures and we want to know the derivative with respect to T , a first step in the analysis is the interpolation of ψ ψ , χψ ψ and χ S S for the transition temperature and n μBT 2 , nB μBT 2 and dμQ dμB for the equation of state for every μ ( j) B . The next step in the analysis is to fit the data in theμ 2 B direction. For this we use three different fit functions to estimate the systematic error. In for imaginary μ B all three fit functions can describe the data well. However their extrapolation to real μ B can be different. The next step is the continuum extrapolation. We use a linear fit though the data from N t = 10, 12 and 16. As an alternative analysis we also combine the the fit in the μ B direction and the continuum extrapolation.
From the different fit functions we can determine the Taylor coefficients of the pressure. The results are shown in figure 1. The statistical error is determined via the bootstrap method with 1000 bootstrap samples. For the systematic error we use a histogram of all different analysises and take the central 68% thus employing the histogram method introduced in [5].
The fits can also be used to extrapolate to real chemical potential. The results for the transition temperature are shown in figure 2.
When extrapolating to real chemical potential the higher order coefficients become more important the further we extrapolate as can be seen from figure 3. From the Taylor coefficient we can calculaten B to different orders. Using only the c 2 coefficient is a calculation up to O(μ B ) as the pressure has to be integrated over μ B compared ton B .
Since we tuned our simulations to fulfill n S = 0 and n Q = 0.4 n B , it is interesting to know how μ Q and μ S behave at real chemical potential for different values of μ B . The results are shown in figure 4. These results can also be used in phenomenological calculations. For example some considerations that can be done with dμQ dT are presented in [6].

Acknowledgements
C.R. would like to thank Volker Koch, Jacquelyn Noronha-Hostler, Jorge Noronha and Bjorn Schenke for fruitful discussions. This project was funded by the DFG grant SFB/TR55. This material is based