QCD at finite isospin density: chiral perturbation theory confronts lattice data

We consider the thermodynamics of three-flavor QCD in the pion condensed phase at nonzero isospin $\mu_I$ and vanishing temperature using chiral perturbation theory in the isospin limit. The transition from the vacuum phase to a superfluid phase with a Bose-Einstein condensate of charged pions is shown to be second order and takes place at $\mu_I=m_{\pi}$. We calculate the pressure, isospin density, and energy density to next-to-leading order in the low-energy expansion. Our results are compared with recent high-precision lattice simulations as well as previously obtained results in two-flavor chiral perturbation theory. The agreement between the lattice results and the predictions from three-flavor chiral perturbation theory is excellent for $\mu_I<200$ MeV. For larger values of $\mu_I$, the agreement between lattice data and the two-flavor predictions is surprisingly good.


Introduction
QCD in extreme conditions, i.e. high temperature and density has received a lot of attention in the past decades due to its relevance to the early universe, heavy-ion collisions, and compact stars [1][2][3]. For example, QCD at finite baryon density µ B is of significant interest since the equation of state (EoS) is used as input for calculating the macroscopic properties of neutron stars. However, lattice QCD cannot be applied to QCD at nonzero baryon density due to the sign problem: integrating out the fermions in the path integral for the partition function gives rise to a functional determinant that can be considered part of the probability measure. At µ B = 0, this determinant is complex and the standard Monte Carlo techniques cannot be applied. A way to circumwent this problem, for high temperatures and small chemical potentials, is by Taylor expanding the thermodynamic quantities about zero µ B [4]. For small T and large µ B , this is obviously hopeless. Due to asymptotic freedom, we expect that we can use weak-coupling techniques at very high densities [5,6]. In the weak-coupling expansion the series is now known to order α 2 s for massive quarks [7] and α 3 s log 2 α s for massless quarks [8]. For lower densities, where weak-coupling techniques do not apply, we have to use lowenergy models of QCD, see Ref. [9] for a recent review.
There are theories that do not suffer from the sign problem. These include two-color QCD [10], three-color QCD with fermions in the adjoint representation [11], QCD in an external magnetic field [12], and three-color QCD at finite isospin [13][14][15][16][17]. The absence of the sign problems implies that one can simulate these systems on the lattice and compare the results with low-energy models and theories. In the case of QCD at finite isospin, one finds at T = 0, a transition from the vacuum to a pion-condensed phase at a critical isospin chemical potential µ c I = m π . The mechanism of pion condensation and the transition to a pion superfluid phase out of the vacuum is simply that it is energetically favorable to form such as condensate for µ I ≥ µ c I . Moreover, with increasing isospin chemical potential, it is expected that there is a crossover to a BCS phase. Since the order parameter in the BCS phase has the same quantum numbers as a charged pion condensate, this is not a true phase transition, but associated with the formation of a Fermi surface and subsequent condensation of Cooper pairs. A very recent review on meson condensation can be found in Ref. [18]. Chiral perturbation theory (χPT) is a low-energy effective theory of QCD based only on the symmetries and the degrees of freedom, and the predictions of χPT are therefore model independent [19][20][21][22]. It has been remarkably successful in describing the phenomenology of the pseudo-Goldstone bosons that result from the spontaneous breakdown of chiral symmetry in the QCD vacuum. χPT at finite isospin was first considered by Son and Stephanov in their seminal paper two decades ago [23], in which all the leading order results were derived.
In this letter, we calculate the effective potential in chiral perturbation theory at next-to-leading (NLO) order in the lowenergy expansion for three flavors at finite isospin chemical potential. While the phase diagram as functions of isospin and strange chemical potentials (µ S ) has been mapped out and leading order (LO) thermodynamic functions have been known for two decades [23,24], the leading quantum corrections at finite µ I are presented here for the first time, however, see Ref. [25] for some partial NLO results in two-color QCD. We derive the pressure, isospin density, and equation of state (EoS), and compare these quantities with recent lattice results as well earlier results from two-flavor χPT [26]. Results on the thermody-namics of the kaon condensed phases at finite µ S and µ I as well as calculational details can be found in an accompanying long paper [27].

Chiral perturbation theory
As mentioned above, χPT is an effective low-energy theory of QCD based solely on its symmetries and low-energy degrees of freedom. In massless three-flavor QCD, the sym- If we add a quark chemical potential for each flavor, the symmetry is In three-flavor QCD, we keep the octet of mesons, which implies that chiral perturbation theory is not valid for arbitrarily large chemical potential. Considering the hadron spectrum, one naively expects that the expansion is valid for |µ u | = |µ d | < 300 MeV [24]. Chiral perturbation theory has a well defined power counting scheme, where each derivative as well as each factor of a quark mass counts as one power of momentum p. At leading order in momentum, O(p 2 ), there are only two terms in the chiral Lagrangian where f is the bare pion decay constant, χ = 2B 0 M , and is the quark mass matrix and Σ = U Σ 0 U , where U = exp iλiφi 2f and Σ 0 = 1 is the vacuum. Here λ i are the Gell-mann matrices that satisfy Trλ i λ j = 2δ ij and φ i are the fields parametrizing the Goldstone manifold (i = 1, 2..., 8). In the remainder we work in the isospin limit, m = m u = m d . By expanding the Lagrangian (1) to second order in the fields, we obtain the terms needed for our NLO calculation. 1 The covariant derivative and its Hermitian conjugate at nonzero quark chemical µ q (q = u, d, s) potentials are defined as follows with It turns out that the Lagrangian is independent of µ B which reflects the fact that all degrees of freedom, namely the meson octet, have zero baryon number. Since we are focusing on pion condensation and want to compare with lattice data, we set µ S = 0 such that v 0 = 1 2 µ I λ 3 . Based on the twoflavor case [23], the ground state in the pion-condensed phase is parametrized as [27] Σ α = e iα(φ1λ1+φ2λ2) = cos α + (iφ 1 λ 1 +φ 2 λ 2 ) sin α , (6) where α is a rotation angle and i |φ i | 2 = 1 to ensure that the ground state is normalized, Σ † α Σ α = 1. From Eq. (1), we find the static Hamiltonian where the first term can be written as 1 There is a competition between the two terms in Eq. (7): The first term favors Σ α in the λ 1 and λ 2 directions, while the second terms favors Σ α prefers the vacuum direction 1 [23]. It turns out the that the former only depends on |φ 1 | 2 + |φ 2 | 2 and so we chooseφ 1 = 0 without loss of generality. The matrix λ 2 generates the rotations and the rotated vacuum is given by , and Σ 0 = 1. The rotated vacuum can then be written in the form Here the rotation in the subspace of the u and the d-quark is evident and at tree level, we have ψ ψ 2 + π + 2 = ψ ψ 2 vac , i.e. the quark condensate is rotated into a pion condensate. This interpretation does not hold beyond leading order [26].
The fluctuations around the condensed or rotated vacuum must also be parametrized and this requires some care. Naively, one would write the field as Σ = U Σ α U , where U = exp iλiφi 2f . However, this parametrization is incorrect since it can be shown that one cannot renormalize the effective potential at next-toleading order using the standard renormalization of the lowenergy constants appearing in the NLO Lagrangian (see below). One way of understanding the failure of this parametrization is to realize that the generators of the fluctuations about the ground state must also be rotated since the vacuum itself has been rotated. The field must therefore be written as where This parametrization reduces to the standard parametrization for α = 0 and has none of the flaws of the naive parametrization.
The tree-level effective potential V 0 = H static 2 = −L static is now evaluated to be At next-to-leading order in the low-energy expansion, there are twelve operators. Not all of them are relevant for the present calculations, in fact only eight contribute to the effective potential. They are In writing the NLO Lagrangian above, we have ignored the Wess-Zumino-Witten terms since they do not contribute to the quantities in the present paper. The last term in Eq. (11) is a contact term, which is needed to renormalized the vacuum energy and to show the scale independence of the final result for the effective potential in each phase. Their contribution to H static In a next-to-leading order calculation, we need to renormalize the couplings L i and H i to eliminate the ultraviolet divergences that arise from the functional determinants. The relations between the bare and renormalized couplings are with λ = Λ −2 2(4π) 2 1 + 1 . Here Γ i and ∆ i are constants [21] and Λ is the renormalization scale associated with the modified minimal substraction scheme MS. Taking the derivative of Eqs. (13)- (14) and using the fact that the bare couplings are scale independent, one finds the renormalization group equations for the renormalized couplings, The contact term H 2 Tr(M M † ) makes a constant contribution to the effective potential which is independent of the chemical potential. and therefore same in both phases. We keep it, however, in the final expression for the NLO effective potential since H r 2 (Λ) is running and needed to show the scale independence of V eff . The renormalized NLO effective potential where L r i (Λ) are the renormalized coupling constants and the masses arẽ Finally, V fin eff,π ± , V fin eff,K ± , and V fin eff,K 0 are finite subtraction terms, to which we return shortly. The couplings are running in such a way that their Λ-dependence cancel against the explicit Λdependence of the chiral logarithms in Eq. (17), implying that Λ dV eff dΛ = 0, cf. Eqs. (15)- (16). In order to obtain Eq. (17), we must isolate the ultraviolet divergences from the functional determinants. This is done by adding and subtracting a divergent term that we calculate analytically in dimensional regularization. The subtracted term is then combined with the original one-loop expression for the effective potential giving finite terms V fin eff,meson that can be easily computed numerically. The divergences are finally remove them by renormalization of the L i s according to Eq. (13). The details of the subtraction and renormalization procedure can be found in Ref. [27] and the NLO effective potential in the two-flavor case can be found in Ref. [26].
Thermodynamic quantities can be calculated from the effective potential Eq. (17), for example the pressure P = −V eff , the isospin density n I = − ∂V eff ∂µ I , and the energy density = −P + n I µ I . All these quantities are evaluated at the value of α that minimizes the effective potential, i.e. satifies ∂V eff ∂α = 0.

Results
The expressions for the effective potential, isospin density, pressure, and energy density are all expressed in terms of the isospin chemical potential, the parameters B 0 m, B 0 m s , and f of the chiral Lagrangian as well as the renormalized couplings L r i . In order to make predictions, we need to determine the parameters of the chiral Lagrangian in terms of the physical meson masses and the pion decay-constant f π . In χPT, one can calculate the pole masses of the mesons and f π systematically in the low-energy expansion. The results are expressed in terms of B 0 m, B 0 m s , f , and L i . These equations can be easily solved to find the parameters of the chiral Lagrangian and thereby numerically evaluated the effective potential. The treelevel values of m π and m K can be expressed in terms of B 0 m and B 0 m s as m 2 π,tree = 2B 0 m and m 2 K,tree = B 0 (m + m s ). Since we want to compare our predictions with the results of the lattice simulations [32], we use their values [28], The low-energy constants have been determined experimentally, with the following values and uncertainties at the scale Since we have three parameters, we must choose three of the four physical quantities from Eqs. (22)- (23). For the results that we present below we use m π , m K , and f π . Using the oneloop χPT expression for the f K , we obtain f K = 112.1 MeV for the central value, which is off which is off by approximately 5% compared to the lattice value of f K = 150 √ 2 = 106.1 MeV. In order to gauge the uncertainty associated with the choice of physical quantities that we reproduce, we have checked that the results are essentially the same using the set m π , m K , and f K . In this case, the central value of the pion decay constant is f π = 81.4 MeV compared to the central value on the lattice f π = 130 √ 2 = 90.5 MeV.
The uncertainties in L r i , m π , m K , and f π translate into uncertainties in the parameters B 0 m, B 0 m s , and f , It turns out that the uncertainties in these parameters in the three-flavor case is completely dominated by the uncertainties in the LECs. In the two-flavor case, they are dominated by the uncertainties in the pion mass and the pion decay-constant. We therefore simplify the analysis and add the uncertainties. This yields The equation ∂V eff ∂α = 0 has two solutions. For µ I < m π , the solution is α = 0. In this case it is straightforward to show that the effective potential and therefore the thermodynamic functions are independent of µ I . We refer to this phase as the vacuum phase, which exhibits the Silver Blaze property [35], namely that the thermodynamic functions are independent of µ I up to a critical value µ c I = m π . For µ I > m π , we have a nonzero condensate of π + , which breaks the U (1) I3 symmetry of the chiral Lagrangian and a nonzero value for α. In Fig. 1, we show the solution α gs to ∂V eff ∂α = 0 as a function of µ I mπ . For asymptotically large values of the isospin chemical, α gs approaches π 2 . By expanding the effective potential around α to fourthorder, we obtain a Ginzburg-Landau energy function of the form The vanishing of a 2 defines the critical chemical potential µ c I . Since a 2 = f 2 π (µ 2 I −m 2 π ), we have µ c I = m π . The onset of Bose condensation at µ c I = m π is an exact result. Moreover, since the coefficient a 4 (µ c I ) > 0, the transition to a pion-condensed phase is of second order, with mean field critical exponents. These results are in agreement with lattice simulations [15][16][17] as well as model calculations [33].
In Fig. 2, we show the isospin n I divided by m 3 π as a function of µ I /m π . The red solid line is the LO result. Note that the LO result is the same in the two -and three-flavor cases for all thermodynamic quantities. We have used the central values for the low-energy constantsl i in the two-flavor case to obtain the blue dashed line as explained in Ref. [26]. The blue band is obtained by including their uncertainties. The light green band is the result of the three-flavor calculation with the minimum, central, and maximum values of the parameters given above, while the dark green band is is from using the central values of L i with uncertainties coming from the lattice parameters only. Finally, the data points are from the lattice calculations of Refs. [15][16][17]. The two-flavor band is very small compared to the three-flavor In conclusion, we have calculated the isospin density, pressure, and equation of state for two and three-flavor QCD using chiral perturbation at finite µ I and vanishing temperature at next-to-leading order. For low values of the isospin chemical potential, up to approximately 200 MeV, the three-flavor NLO results is in very good agreement with the results obtained on the lattice. For larger values, the two-flavor results at NLO is in much better agreement with lattice data. The wide bands in the three-flavor case are due to the uncertainties in the L r i s and a more accurate determination of them would be desirable.