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Chapter 1 QCD Phase Structure at Finite Baryon Density H.-T. Ding��W. J. Fu�� F. Gao�� M. Huang�� X. G. Huang�� F. Karsch��J. F. Liao�� X. F. Luo�� B. Mohanty�� T. Nonaka�� P. Petreczky��K. Redlich�� C. D. Roberts�� and N. Xu 1.1��Strong Interaction at Finite Temperature and Baryon Density 1.1.1��Path Integral Formulation of QCD Thermodynamics The equilibrium thermodynamics of elementary particles interacting only through the strong force is controlled by Quantum ChromoDynamics. Its partition function can be expressed in terms of a Euclidean path integral. The grand canonical partition function����is given by an integral over the fundamental quark and gluon (Au) fields. In addition to its dependence on volume (V)�� temperature (T) and a set of Nf chemical potentials�� the partition function implicitly depends on the masses�� of the Nf different quark flavors. In Euclidean space-time�� which is obtained from the Minkowski formulation by substituting t ir with r g R�� the QCD Lagrangian is given by (1.1) where Greek letters are spinor indices��is the color index�� Nc is the number of colors�� and is the mass of quarks with flavor.The covariant derivative 0Eand the field strength tensor are given by (1-2) (1.3) Here are the gauge fields�� are the quark (antiquark) fields�� Aa are the generators of�� fabc are the corresponding structure constants��are the Euclidean Dirac matrices obeying�� and g is the bare coupling constant. In the Euclidean path integral formalism�� the partition function of QCD is then given by (1.4) with the Euclidean action (1.5) Here we have suppressed the dependence of the Euclidean Lagrangian and action on the fields but have stressed explicitly their dependence on the various quark chemical potentials that couple to the conserved quark number currents (1-6) The thermal expectation value of physical observables O can be obtained through (1.7) Basic thermodynamic quantities like the pressure (P)�� energy density (e)�� or net-quark number density n f can be obtained from the logarithm of the partition function using standard thermodynamic relations�� (1.8) (1.9) (1.10) where the chemical potentials are introduced in units of temperature�� In fact�� the QCD partition function depends on chemical potentials only through these dimensionless combinations�� which are the logarithms of the fugacities. The numerical analysis of thermodynamic observables at non-vanishing chemical potential�� is difficult in lattice regularized QCD. A viable approach that circumvents the so-called sign problem in lattice QCD at�� suitable for moderate values of the chemical potentials�� is to consider Taylor expansions for thermodynamic observables. The starting point for such an analysis is the Taylor expansion of the pressure�� it reads (1.11) where the expansion coefficients are dimensionless�� generalized susceptibilities that can be evaluated at = 0�� (1.12) The set of hadron chemical potentials�� related to conserved quantum numbers�� and the set of quark flavor chemical potentials are easily related to each other�� (1-13) With this it is straightforward to rewrite the Taylor series given in Eq. (1.11) in terms of quark chemical potentials�� also in terms of baryon number (B)�� electric charge (Q)�� and strangeness (5) chemical potentials�� (1.14) where the expansion coefficients can again be evaluated at x = 0�� (1.15) Fluctuations of and correlations among conserved charges�� i.e.�� baryon numbers (B)�� electric charge (Q)�� and strangeness (S)�� have been long considered as sensitive observables to probe the QCD phase structure. Experimental proxies�� like proton�� kaons are used for measurements of mean�� variance�� skewness�� and kurtosis of conversed charges. Expressions for the skewness�� kurtosis�� hyper-skewness�� and hyper-kurtosis ratios of conserved charges x = B��Q and S are listed as follows: (1.16) (1.17) (1.18) (1.19) where is the generalized susceptibility defined as the n-th order derivative of pressure with respect to chemical potential .On the lattice can be computed by Taylor expanding the pressure obtained either at zero baryon chemical potential or at non-zero imaginary chemical potential up to a certain order in This renders the comparison of QCD results with experimental measurement possible. Although many effects�� e.g.�� non-equilibrium effects�� detector effects�� etc.can make the comparison between theory and experiment difficult in practice�� a baseline for thermally equilibrated QCD�� provided by Lattice QCD computations as well as from��functional renormalization group/Dyson-Schwinger equation method�� clearly is important. In order to provide input on bulk thermodynamics and cumulants of conserved charge distributions�� that is suitable for the comparison of lattice QCD calculations to heavy-ion collision experiments�� it is important to set up fin