Pion condensation and phase diagram in the Polyakov-loop quark-meson model

We use the Polyakov-loop extended two-flavor quark-meson model as a low-energy effective model for QCD to study the phase diagram in the $\mu_I$--$T$ plane where $\mu_I$ is the isospin chemical potential. In particular, we focus on the Bose condensation of charged pions. At $T=0$, the onset of pion condensation is at $\mu_I={1\over2}m_{\pi}$ in accordance with exact results. The phase transition to a Bose-condensed phase is of second order for all values of $\mu_I$ and in the $O(2)$ universality class. The chiral critical line joins the critical line for pion condensation at a point whose position depends on the Polyakov-loop potential and the sigma mass. For larger values of $\mu_I$ these curves are on top of each other. The deconfinement line enters smoothly the phase with the broken $O(2)$ symmetry. We compare our results with recent lattice simulations and find overall good agreement.


I. INTRODUCTION
The phases of QCD as functions of the baryon chemical potential µ B or the quark chemical potential µ = 1 3 µ B , and temperature T have been studied in detail since the first phase diagram was proposed more than fourty years ago [1][2][3]. At vanishing baryon chemical potential, it is possible to perform lattice simulations to calculate the thermodynamic functions and the transition temperature associated with chiral symmetry restoration and deconfinement. For physical quark masses and two flavors, the transition is a crossover at a temperature of approximately 155 MeV [4][5][6][7].
At nonzero baryon chemical potential, however, Monte Carlo simulations are hampered by the so-called sign problem, namely that the fermion determinant becomes complex. Being complex, the usual interpretation of it as part of a probability distribution can no longer be upheld. The sign problem in QCD at finite baryon density has spurred the interest in QCD-like theories free of this problem. This includes QCD with quarks in the adjoint representation [8], two-color QCD [9], QCD at finite isospin µ I [10], and QCD in a magnetic field B [11]. These theories are all interesting in their own right; QCD at finite isospin and QCD in a magnetic field are also relevant for compact stars. In addition, the application of Monte-Carlo methods allows a direct test of various model approaches in the cases mentioned above. Such a confrontation of model calculations with lattice simulations of QCD in a magnetic field has been very * adhika1@stolaf.edu † andersen@tf.phys.ntnu.no ‡ patrick.kneschke@uis.no fruitful in understanding their strengths and limitations [12,13]. Lattice simulations of QCD at finite isospin have been performed in e.g. Refs. [14][15][16][17][18] with particular emphasis on Bose condensation of charged pions for isospin chemical potentials above the zero-temperature critical value µ c I = 1 2 m π . Chiral perturbation theory (ChPT) [10,[19][20][21][22], which is a model-independent low-energy theory for QCD valid at low densities has been used to study pion condensation. ChPT predicts a second-order transition, which is in agreement with lattice simulations. There have also been a number of other approaches and model calculations studying various aspects of the QCD phase diagram at finite isospin density, including the resonance gas model [23], random matrix models [24], the Nambu-Jona-Lasinio (NJL) model [25][26][27][28][29][30][31][32][33][34][35][36][37], the quarkmeson (QM) model [38][39][40][41] 1 , and effective theory at asymptotically high isospin [42].
Finally, we mention that one expects another phase transition at large isospin chemical potential. In perturbation theory, one-gluon exchange gives rise to an effective attractive interaction between u andd quarks leading to the formation of Cooper pairs [10]. The transition from a Bose-Einstein condensate (BEC) to a Bardeen-Cooper-Schrieffer (BCS) state is expected to be an analytical crossover as the symmetry-breaking pattern is the same.
As pointed out in Ref. [38], there is a mapping of the quark-meson model at finite isospin and the corresponding two-color quark-meson-diquark model at finite baryon chemical potential. The neutral pion π 0 is replaced by an isovector triplet π. The charged pions π ± are replaced by a diquark-antidiquark pair ∆ and ∆ * , which instead of being coupled to µ I is now coupled to a baryon chemical potential µ B . 2 Since the gauge groups SU (2) and SU (3) are fundamentally different, this mapping is valid for the matter sector; once we couple the QM model to the Polyakov loop, this identification is lost.
In the present paper, we study the QCD phase diagram at finite temperature and isospin density using the PQM model. The main conclusions of our work are 1. The second order transition to a BEC state. The transition is in the O(2) universality class. At T = 0, the transition is exactly at µ I = 1 2 m π . 2. The BEC and chiral transition lines meet at a point (µ meet I , T meet ) and coincide for larger isospin chemical potentials µ I . 3. The deconfinement and chiral transition lines coincide in the non-condensed phase for a logarithmic Polyakov-loop potential and a sufficiently low sigma mass.
4. The deconfinement line penetrates smoothly into the symmetry-broken phase.
These results are in agreement with the recent lattice simulations of Refs. [16][17][18]. The paper is organized as follows. In Sec. II, we briefly discuss the quark-meson model and in Sec. III we calculate the effective potential in the mean-field approximation. In Sec. IV, we discuss the coupling to the Polyakov loop, while in Sec. V, we present the phase diagram in the µ I -T plane and compare it to recent lattice results. In Appendix A, we list a few integrals needed in the calculations, while Appendix B provides the reader with some details of how the parameters of the quark-meson model are determined.

II. QUARK-MESON MODEL
The Lagrangian of the two-flavor quark-meson model in Minkowski space is 2 The diquarks are the baryons of two-color QCD.
where ψ is a color N c -plet, a four-component Dirac spinor as well as a flavor doublet and µ f = diag(µ u , µ d ), where µ u and µ d , are the quark chemical potentials, µ I is the isospin chemical potential, τ i (i = 1, 2, 3) are the Pauli matrices in flavor space, π = (π 1 , π 2 , π 3 ), and π ± = 1 √ 2 (π 1 ± iπ 2 ). Apart from the global SU (N c ) symmetry, the Lagrangian (1) has a U (1) B × SU (2) L × SU (2) R symmetry for h = 0 and a U (1) B × SU (2) V symmetry for h = 0. When µ u = µ d , this symmetry is reduced to The number density associated with a chemical potential µ A is where V is the effective potential. The baryon and isospin densities can be expressed in terms of the quark densities n u and n d as Eqs. (4)-(5) together with the chain rule can be used to derive relations among the baryon and isospin chemical potentials and the quark chemical potentials. We have This yields Similarly, we find ∂µu ∂µ B = ∂µ d ∂µ B = 1 3 . From this, we find the following relations among the chemical potentials Introducing the quark chemical potential µ = 1 3 µ B and inverting the relations (8)-(9), we find

III. EFFECTIVE POTENTIAL
The expectation values of the fields are written as where φ 0 and π 0 are constant in space. The former is the usual chiral condensate, while the latter represents a homogeneous pion condensate. A pion condensate breaks the U I3L (1)×U I3R (1) symmetry to U I3V (1) or the U I3 (1) symmetry. Introducing ∆ = gφ 0 and ρ = gπ 0 , the treelevel potential in Euclidean space can be written as Expressing the parameters in Eq. (1) terms of the sigma mass m σ , the pion mass m π , the pion decay constant f π , and the constituent quark mass m q , we find Inserting these relations, we can write the tree-level potential as The quark energies can be read off from the zeros of the determinant of the Dirac operator. One finds where we have defined Note that the quark energies explicitly depend on µ I . In the following we choose µ I > 0, but similar results are obtained for µ I < 0.
The one-loop contribution to the effective potential at where the integral is in d = 3 − 2 dimensions (See Appendix A). The integral in Eq. (20) is ultraviolet divergent and in order to isolate the divergences, we need to expand the energies in powers of µ I to the appropriate order. This yields The remainder V fin is finite and reads Note that V fin can be evaluated directly in d = 3 dimensions. In the present case, V fin must be evaluated numerically. Using the expressions for the integrals listed in Appendix A, we can write the unrenormalized one-loop effective which contains poles in . These poles are removed by mass and coupling constant renormalization. In the MS scheme this is achieved by making the substitutions The renormalized one-loop effective potential then reads where the subscript MS indicates that the parameters are running with the renormalization scale Λ. Using Z g 2 in Eq. (24) and the wavefunction renormalization factor Z φ = 1 − 4Ncg 2 (4π) 2 , it is seen that the fields ∆ and ρ do not run. In Appendix B, we discuss how one can express the parameters in the MS scheme in terms of physical masses and couplings. Using Eqs. (B14)-(B17), the final expression for the one-loop effective potential in the large-N c limit becomes The finite-temperature part of the one-loop effective potential at µ B = 0 is The complete one-loop effective potential in the QM model in the large-N c limit is then the sum of Eqs. (26) and (27). Note that Eq. (27) vanishes at T = 0 and that the only µ I -dependence of V 1−loop is line three of Eq. (26).

IV. COUPLING TO THE POLYAKOV LOOP
In a pure gauge theory, the Polyakov loop is an order parameter for deconfinement, as first discussed in Refs. [43,44]. In QCD with dynamical quarks, it is an approximate order parameter. This is analogous to the quark condensate which is an exact order parameter for chiral symmetry for massless quark but only an approximate order parameter for massive quarks. The Polyakov loop is defined as the trace of the thermal Wilson line, where the thermal Wilson line L is given by where A 4 = iA 0 is the temporal component of the gauge field in Euclidean space, A 0 = t a A a 0 , t a = 1 2 λ a are the generators of SU (3) c gauge group, λ a are the Gell-Mann matrices, and P denotes path ordering. The background field A 4 in the Polyakov gauge is where A 3 4 and A 8 4 are time independent fields. Substituting Eq. (29) into Eq. (28), the Wilson line becomes where we have defined φ 1 = 1 2 βA 3 4 and φ 2 = 1 2 √ 3 βA 8 4 .

Introducing the Polyakov loop variables
the finite-temperature fermion contribution can then be written as Eq. (32) reduces to Eq. (27) upon setting Φ =Φ = 1, i.e. we obtain the finite-temperature part of the effective potential in the quark-meson model. The Polyakov loop has now been coupled to the quark sector of the model; we next need to include the contribution to the free energy density from the gauge sector. This is a phenomenological potential, which is a function of Φ andΦ, and is required to reproduce the pressure for pure-glue QCD calculated on the lattice for temperatures around the transition temperature. There are several potentials on the market [45][46][47][48] with similar properties. We will first be using the polynomial potential of Ref. [45] U where the constants are with a 0 = 6.75, a 1 = −1.95, a 2 = 2.625, and, a 3 = −7.44. 3 We express the various contributions to the effective potential in terms of Φ andΦ, although they are equal in the present case.
We will also use the logarithmic Polyakov-loop potential of Ref. [46] U with a = 3.51 − 2.47 The temperature T 0 is defined by where we have modeled the µ I -dependence in the same way as the µ B -dependence in [47] b The parameter T τ = 1.77 GeV and α 0 = 0.304 are determined such that the transition temperature for pure glue at µ I = 0 is T 0 = 270 MeV [49]. The curvature of the deconfinement transition in µ I direction is governed by b µ I , which is chosen as The full thermodynamic potential is now given by the sum of Eqs. (26), (32), and (33) or (37) respectively. From Eqs. (17)- (18), it is easy to see that Eq. (32) is real, thus there is no sign problem at µ B = 0. We also note that Eqs. (32), (33) and (37)  In Fig. 1, we show the normalized chiral condensate ∆ mq (blue lines) and the expectation value of the Polyakov loop Φ as functions of the temperature T at µ B = µ I = 0 using the polynomial potential (33). The blue dashed line is the chiral condensate obtained in QM model while the blue solid line is obtained in the PQM model, i.e. with the coupling between the order parameters. Similarly, the red dashed line is obtained using the pure-glue potential for Φ (with the N f dependent T 0 = 208 MeV), while the red solid line is obtained in the PQM model. We notice that the critical temperature for the chiral transition moves to the right, i.e. to higher temperatures while the transition temperature for deconfinement moves to the left. They are now within a few MeV of each other, with the deconfinement transition occurring at slightly lower temperature than the chiral transition.

V. PHASE DIAGRAM
In this section, we discuss the phase diagram in the µ I -T plane. In the numerical work below, we set N c = 3, m π = 140 MeV, and f π = 93 MeV. We vary m σ between 500 and 600 MeV.
In Fig. 2, we show the chiral (blue line) and pion condensates (red line) as functions of µ I at zero temperature. We notice the onset of pion condensation which takes place at exactly µ I = 1 2 m π as we will discuss in some detail below. Moreover, the quark condensate de-creases with µ I once the pion condensate is nonzero. Finally, all physical quantities, are independent of µ I from µ I = 0 all the way up to µ I = 1 2 m π . For example, the effective potential is independent of µ I , implying via Eq.
(3) that the isospin density vanishes. This is an example of the Silver Blaze property [50] and was discussed in detail in the context of pion condensation in Refs. [38,41]. We refer to this region as the vacuum phase. The onset of pion condensation at T = 0 is for µ I = 1 2 m π , which is guaranteed by the way we have determined the parameters in the Lagrangian. This was explicitly demonstrated in Ref. [41]. We can understand this result by considering the energy of a zeromomentum pion in the vacuum phase is m π − 2µ I . If condensations of pions is a second order transition, it must take place exactly at a point where the (mediumdependent) mass of the pion drops to zero, because in the condensed phase there is a massless Nambu-Goldstone mode associated with the breaking of a U (1) symmetry. If one uses matching at tree level, there will be finite corrections to this relation. Likewise, if one uses the effective potential itself to define the pion mass, one uses the pion self-energy at zero external momentum and so the pole of the propagator is not at the physical mass. Again there will be finite corrections and in some cases, the deviation from the exact result can be significant [38]. The condensation of pions is always a second-order transition with mean-field critical exponents. The order of the transition to a BEC is in agreement with the functional renormalization group application to the QM model in Ref. [38]. The critical isospin chemical potential is fairly constant for temperatures up to approximately T = 100 MeV, after which it rapidly increases. For large values of µ I the critical temperature for pion condensation stays at T ρ ≈ 187 MeV. We also notice that the chiral transition temperature line T χ meets the critical temperature line for pion condensation T ρ at (µ meet I , T meet ) ≈ (105, 159) MeV, and coincide for larger values of µ I .
As we have seen, we enter the BEC phase when µ I exceeds 1 2 m π . As µ I increases the quark mass ∆ decreases as shown in Fig. 2. Once µ I > ∆, the u-quark and d-quark energies, Eqs. (17) and (18), are no longer minimized for p = 0, but for p = µ 2 I − ∆ 2 . This change is a signal of the BEC-BCS crossover. Although the BEC-BCS crossover is not particularly sharp, it is typically defined by the condition µ I > ∆ [33,51]. The crossover starts at µ I = 113 MeV for T = 0 and is almost independent of the temperature, as can be seen from the Figure. In Fig. 4, we show the phase diagram in the µ I -T plane at zero baryon chemical potential with the Polyakov loop and U T 4 given by (33). The green line is the critical line for Bose-Einstein condensation of charged pions, the red line is the critical line for deconfinement, and the blue line is the critical line for the chiral transition. Finally, the black dotted line indicates the BEC-BCS transition line. The blue and red lines are defined by the inflection point of the order parameters ∆ and Φ as functions of T for fixed µ I . As The gap between the chiral and deconfinement line can be reduced by using a logarithmic Polyakov potential (37) instead of Eq. (33) and decreasing the sigma mass. For m σ = 500 MeV the two lines basically coincide at T = 0 as seen in Fig. 5. The chiral and deconfinement transition line also meet the pion-condensed line at a point for a smaller value of µ I as compared to For completeness, we show in Fig. 6 the phase diagram in the standard mean-field approximation (sMFA), which is a common approximation used in the literature, where one ignores the loop corrections to the vacuum potential, i.e. uses Eq. (16) instead of Eq. (26). We find the critical temperature for pion condensation to be smaller than for the one-loop potential in Fig. 4. We also find a first-order transition of the pion condensate above a critical isospin chemical potential µ I ≈ 86 MeV, indicated by the black dot in the figure. This critical point is absent once we go beyond the sMFA, at least in the region of µ I considered here. Our phase diagram is in qualitative good agreement with that obtained by Brandt, Endrődi, Schmalzbauer using lattice simulations [16][17][18], in particular if we use a logarithmic Polyakov loop potential and choose a lower sigma mass of 500 MeV. We believe that the quantitative differences (essentially the temperature dependence of the various transition lines) can mainly be attributed to the fact that we have two light flavors, while they consider 2+1 flavors; for example the deconfinement transition temperature decreases with the number of quarks and our transition line is consistently higher. 4 They find that chiral and BEC transition lines meet at a triple point, beyond which they coincide. The latter transition is again found to be second order for all values of µ I and the scaling analysis is consistent with the O(2) universality class. They computed contour lines of the expectation values of the renormalized Polyakov loop Φ for values 0.2, 0.4, 0.6, 0.8, and 1.0. Given their renormalization prescription for the Polyakov loop, developed in [52], a possible choice for T dec is Φ = 1, which implies that it coincides with T χ within errors [18]. Finally, we mention that the deconfinement line penetrates smoothly into the BEC phase and that they identify this line with the BEC-BCS transition.
are chosen such that they exactly cancel the loop corrections to the self-energies and couplings evaluated on the mass shell, and such that the residues evaluated on shell are unity. Consequently, the renormalized parameters are independent of the renormalization scale and satisfy the tree-level relations [54][55][56]. In the MS scheme, the counterterms are chosen so that they cancel only the poles in of the loop corrections. The bare parameters are the same in the two schemes, which means that we can relate the corresponding renormalized parameters. The running parameters in the MS scheme can therefore be expressed in terms of the physical masses m σ , m π , and m q as well as the pion decay constant f π . In Ref. [53] we found where A(m 2 q ), B(p 2 ), and B (p 2 ) are integrals in d = 4 − 2 dimensions in Minkowski space. Going to Euclidean space, they can be straightforwardly computed and read where we have defined The solutions to Eqs. (B10)-(B13) are where m 2 0 , g 2 0 , λ 0 and h 0 , are the values of the running parameters at the scale Λ 0 . We choose Λ 0 to satisfy log Λ 2 0 m 2 q + F (m 2 π ) + m 2 π F (m 2 π ) = 0 .