Polyakov loop potential at finite density

The Polyakov loop potential serves to distinguish between the confined hadronic and the deconfined quark-gluon plasma phases of QCD. For Nf=2+1 quark flavors with physical masses we determine the Polyakov loop potential at finite temperature and density and extract the location of the deconfinement transition. We find a cross-over at small values of the chemical potential running into a critical end-point at mu/T>1.


Introduction
In recent years much progress has been made in our understanding of the phase structure of QCD at finite temperature and density. This understanding has been achieved with a variety of methods ranging from first principle lattice and continuum computations to elaborate model studies.
At vanishing density all these methods by now converge quantitatively leaving only a few open fundamental questions, e.g. the order of the phase transitions in different regions of the Columbia plot. In turn, at finite density, progress has been hampered by several intricate problems. On the lattice one has to face the sign problem which so far has made it impossible to access chemical potentials with µ/T > 1 [1,2]. First principle continuum computations with functional methods, such as Dyson-Schwinger equations (DSE) and functional renormalisation group (FRG) equations, are based on an expansion of the theory in terms of quark-gluon correlation functions. Hence at finite density they have to cope with the increasingly complicated ground state structure of QCD in terms of these correlation functions. Finally, low energy effective models are usually anchored and benchmarked at the vacuum and thermal physics at vanishing density. In turn, the more important the density fluctuations get, the less quantitative are the results.
Facing these problems, it is apparent that progress in our understanding of QCD at finite temperature and density is probably best achieved by a combination of the different methods at hand. In the present work we push forward the functional continuum approach towards the phase diagram of QCD supplemented with results from lattice QCD. We determine, for the first time, the Polyakov loop potential at finite temperature and real chemical potential.

The phase diagram with functional methods
In the past decade continuum quark and gluon correlations functions have been computed with the help of functional equations for the effective action of QCD. These works have been mostly performed in (background) Landau gauge, whereĀ is chosen to be the expectation value of the gauge field,Ā = A . The present work also utilizes the gauge (1). Correlation functions in ordinary Landau gauge are directly related to those in background Landau gauge by simply substituting plain momentum p 2 with background covariant momentum, p 2 → −D 2 [3,4]. In this approach the Polyakov loop variable in the fundamental representation, evaluated at the minimum of the Polyakov loop effective potential V [A 0 ], is an order parameter for confinement [3,5]. The effective potential is defined from the effective action Γ, evaluated at constant background fields A const 0 and vanishing gauge fields, The minimum of V [A 0 ] singles out the expectation value of the gauge field in the background Landau gauge, A 0 . The related order parameter satisfies within an appropriate (re)normalization of L[A 0 ] , see [3][4][5][6]. This inequality holds true for both, Yang-Mills theory and fully dynamical QCD. In the presence of a phase transition both sides vanish at T c and the inequality (4) is saturated below T c . In turn, in the presence of a cross-over we expect the cross-over temperature computed from L[ A 0 ] to be lower than the one computed from L[A 0 ] . The effective potential, or its A 0 -derivative, can be computed from the functional DSE and FRG equations, see Fig. 1 and Fig. 2, respectively. For the FRG this has been put forward in Yang-Mills theory, [3][4][5][6], and in QCD at finite temperature and imaginary chemical potential in [7]. There, the effective potential V [A 0 ] is computed solely from the scale-dependent propagators. More recently, a similar computation of the Polyakov loop potential has also been performed in Coulomb gauge, [8,9]. Related lattice computations can be found in [10][11][12][13]. Large circles indicate dressed propagators and vertices, and S stands for the classical action, see [4]. In turn, the DSE-formulation has been put forward in [4]. It is apparent from Fig. 1, that the effective potential V [A 0 ] can be computed from the DSE once the ghost, gluon and quark propagators as well as the three-gluon vertex and ghost-gluon vertex are known. In the present work we utilize the observation in Ref. [4] that within an optimized renormalisation scheme the two-loop terms in Fig. 1 are sub-leading at temperatures about T c . This has been thoroughly tested for Yang-Mills theory within a comparison of the DSE results from Fig. 1 with the FRG results from Fig. 2. For temperatures about T c the results agree quantitatively. The inclusion of the quarkloop in full QCD does not change this picture. Moreover, we neglect the A 0 -dependence of the back-reaction of the Polyakov loop potential to the chromo-electric propagator in terms of ∂ 2 A0 V [A 0 ]. While these back-reaction effects may be crucial for the critical scaling of the chromoelectric component of the gluon propagator close to the phase transition of pure Yang-Mills theory [4,14], we expect its influence on the QCD transition to be small. This needs to be verified in future work.
Within this approximation the A 0 -dependence solely originates from the shifted Matsubara frequencies p 0 + gA 0 . The diagrams in Fig. 1 and Fig. 2 can be diagonalized in color space leaving us with where ϕ m are the eigenvalues of βgA 0 /(2π), depending on the representation. For example, for two-color QCD, the constant temporal gauge field can be rotated into the Cartan subalgebra, A 0 = 2πT ϕ/g τ 3 , with Cartan generator τ 3 . We have the eigenvalues in the adjoint and fundamental representation respectively. The factors 1/2 in ϕ fund carry information on the explicit center-symmetry breaking of the quarks.
In the physical case of SU (3) we restrict ourselves to 2πT ϕ/g τ 3 in the Cartan subalgebra generated by τ 3 , τ 8 . 1 The corresponding eigenvalues are given by for more details see [4,6]. Then, the shifted Matsubara frequencies p 0 + gA 0 read after diagonalization, 2πT (n + ϕ ad ) , and 2πT n + for ghost, gluon in the adjoint representation and the quark in the fundamental representation respectively. The additive nature of the loop representation in Fig. 1 and Fig. 2 leads to the simple form Here, V glue , contains all contributions from the gluon and ghost diagrams in the DSE and FRG, see Figs.1,2.
In the present approximation, i.e. without the backreaction of V [A 0 ] to the chromo-electric gluon, all diagrams contributing to V glue involve only traces and contractions in the adjoint representation, and hence the eigenvalues ϕ ad in (6), (7). In turn, the matter contribution, V quark , involves only traces and contractions in the fundamental representation, and hence the eigenvalues ϕ fund in (6), (7). With (8) this leads to the periodicities We observe that the periodicity of V quark is independent of N c in contrast to that of the glue part. The latter dependence reflects the fact that V glue is center-symmetric and hence invariant under Z Nc -transformations. For the simple case of N c = 2 the Cartan is one-dimensional and a center transformation entails ϕ → 1 − ϕ with centersymmetric point ϕ = 1/2. Evidently this is not the symmetry of the quark potential V quark due to its periodicity, see (11). The Polyakov loop in the fundamental representation in SU (2) reads and vanishes at the center-symmetric point ϕ = 1/2. For N c = 3 (and higher N c ) a center transformation is a rotation in the Cartan. Accordingly, the explicit centerbreaking in the quark potential is only visible for general gauge fields in the Cartan sub-algebra, i.e. A 0 = A 3 0 τ 3 + A 8 0 τ 8 , which are not considered here. Interestingly, for SU (3) the quark potential has the same periodicity w.r.t. ϕ as the glue potential in contradistinction to SU (2). This may be a helpful property for model applications at finite density, [15][16][17][18][19][20][21][22], and shall be studied elsewhere. The Polyakov loop in three-color QCD reads and vanishes at the confining values ϕ = 2/3, 4/3 in the fundamental period ϕ ∈ {0, 2}. This gives us direct access to an order parameter potential for the confinement-deconfinement phase transition in a DSEapproach to the phase structure of QCD as put forward in [23,24]. In the following we will exploit this approach at finite temperature and density thus providing first insights into the Polyakov loop potential at finite density.

DSE for the quark and gluon propagators
In order to determine the N f = 2 + 1 quark and gluon propagators at finite temperature and chemical potential we have solved their corresponding DSEs given diagrammatically in Figs. 3 and 4. In the gluon DSE we work with an approximation neglecting unquenching effects in the Yang-Mills part of the equation. Consequently this part can be replaced by the inverse quenched propagator denoted by the diagram with the box labelled 'YM' in Fig. 4. This approximation is valid on the few percent level [24]. For the quenched gluon propagator one may use corresponding lattice results [14,[26][27][28] or input from a FRG calculation within Yang-Mills theory [4,29]. We have checked that our results for the potential and the respective critical temperatures are hardly affected by this choice. This is a direct consequence of the inheritance of the above-mentioned renormalisation scheme in the quenched case [4], allowed by the absence of two-loop diagrams in the matter sector of the DSE. To make contact with the results of Ref. [24] in the following we use the  (14), compared to gauge-fixed unquenched lattice data from [25].
lattice results of Ref. [26] as input. The only other unknown quantity in our system of DSEs is the fully dressed quark-gluon vertex. Since no reliable calculations of this quantity at finite temperature are available, we resort to the model ansatz of Refs. [23,24]. There, the vertex is constructed utilizing information from the Slavnov-Taylor identity of the vertex as well as constraints due to the perturbative RG running of the vertex. It has been shown in [24] that such an ansatz is sufficient to deliver results for the chiral condensate at finite temperature in good agreement with lattice gauge theory [30]. A further justification of our quark-gluon interaction is given in Fig. 5. In the thermal medium, the color-diagonal gluon propagator is given by where the dressing functions Z L and Z T represent the parts with longitudinal and transversal orientation with respect to the heat bath and the P T,L µν are the corresponding projectors. For three different temperatures these dressing functions are plotted in Fig. 5. The dashed lines are fits to the quenched lattice data of [26]. The unquenched results (solid lines), predicted in the DSE framework [24], are compared with very recent unquenched lattice results from Ref. [25]. We observe large unquenching effects in the longitudinal part of the propagator and somewhat smaller effects in the magnetic part. For both dressing functions the prediction from the functional framework is nicely matched by the lattice data. We believe these results provide solid justification for the vertex construction and the truncation of the gluon DSE used in our work.

Results
The DSE for the potential depicted in Fig. 1 is used to compute ∂ ϕ V (ϕ). Upon ϕ-integration this yields the Polyakov loop potential V (ϕ) as a function of temperature and chemical potential. In Fig. 6 and Fig. 7 we show the dimensionless potential V (ϕ)/p SB with V (0) = 0 and p SB = 19π 2 36 T 4 + 3 2 T 2 µ 2 + 3 4π 2 µ 4 . The pressure is hidden in the integration constant [4] and will be discussed elsewhere.
We have computed the Polyakov loop potential V (ϕ) in 2+1 flavor QCD at the physical pion mass. The confining minimum with vanishing Polyakov loop, L(ϕ) = 0, is at ϕ = 2/3, see (13). In turn, for ϕ = 0 we have L(ϕ = 0) = 1. One clearly sees the transition from the confining regime at low temperature/small chemical potential to the deconfined phase at high temperature/large chemical potential. The sharper cross-over transition as a function of chemical potential with fixed T = 115 MeV reflects the proximity of the critical endpoint. tion is a smooth cross-over. There is no unique definition of the cross-over temperature T conf . In the present work we use the inflection point of the Polyakov loop, i.e., the maximum of the thermal derivative. Other definitions include the inflection point of the expectation value A 0 , and that of the dual chiral condensate as computed in [24] for 2+1 flavors. In [24] the cross-over temperature is computed from the susceptibility and differs slightly from the dual T conf computed here. Also the quark masses have been slightly larger than the physical ones; this has been corrected in the present work. The cross-over sharpens with increasing chemical potential and finally turns into a first order transition at (T * , µ * ) = (101 MeV, 174 MeV). Note that the critical point (T * , µ * ) as well as the first order line does not depend on the definition of the cross-over temperature. In Fig. 9 we show T conf together with the chiral transition temperature T χ which is obtained from the inflection point of the light-quark condensate. The shaded area shows the width of the deconfinement cross-over defined by 80% of the inflection peak. Interestingly, all transition temperatures, T conf and T χ agree within this width for the whole phase diagram. Since definitions of T conf with either Polyakov loop potential or dressed Polyakov loop are based on different properties of the quark and gluon correlation functions, this provides a highly non-trivial check of the self-consistency of the present approximation. Nevertheless, at very large chemical potential the present scheme may not be sufficient, see Ref. [24] for a more detailed discussion. In this work we presented the first results for the Polyakov loop potential at finite chemical potential in QCD with N f = 2 + 1, evaluated from a combination of functional and lattice methods. Besides providing input for model applications, our results serve as a benchmark prediction for future evaluations of the potential with different methods.