Pion and kaon condensation at zero temperature in three-flavor $\chi$PT at nonzero isospin and strange chemical potentials at next-to-leading order

We consider three-flavor chiral perturbation theory ($\chi$PT) at zero temperature and nonzero isospin ($\mu_{I}$) and strange ($\mu_{S}$) chemical potentials. The effective potential is calculated to next-to-leading order (NLO) in the $\pi^{\pm}$-condensed phase, the $K^{\pm}$-condensed phase, and the $K^0/\bar{K}^0$-condensed phase. It is shown that the transition from the vacuum phase to these phases are second order and takes place when, $|\mu_I|=m_{\pi}$, $|{1\over2}\mu_I+\mu_S|=m_K$, and $|-{1\over2}\mu_I+\mu_S|=m_K$, respectively at tree level and remains unchanged at NLO. The transition between the two condensed phases is first order. The effective potential in the pion-condensed phase is independent of $\mu_K$ and in the kaon-condensed phases, it only depends on the combinations $\pm{1\over2}\mu_I+\mu_K$ and not separately on $\mu_I$ and $\mu_K$. We calculate the pressure, isospin density and the equation of state in the pion-condensed phase and compare our results with recent $(2+1)$-flavor lattice QCD data. We find that three-flavor $\chi$PT is in excellent agreement with lattice QCD near the second order phase transition at $\mu_{I}=m_{\pi}$ when using the central values of the low energy constants (LECs), which includes the vacuum pion (and kaon) masses, pion (and kaon) decay constants and the LECs ($L_{i}$) of the $\mathcal{O}(p^{4})$ $\chi$PT Lagrangian. For larger values of the isospin chemical potential (and zero strange quark chemical potential), while $\chi$PT and lattice QCD results are consistent for the observables including pressure, isospin density and energy density, the central values of LECs produce observables that are overestimates compared to the lattice results.


Introduction
Quantum chromodynamics (QCD), the theory of the strong force, is challenging to study due to its non-perturbative nature and the inability to use lattice QCD simulations in the phenomenologically most interesting regime, namely finite baryon density, due to the infamous fermion sign problem [1,2] present in (classical) Monte Carlo algorithms. As such, except for asymptotically large baryon chemical potentials, where QCD is expected to be in a color-flavor-locked phase [3,4] and can be studied due to asymptotic freedom which allows for the construction of an effective field theory, most of the phenomenologically relevant QCD phase diagram is inaccessible.
Recently, there has been renewed interest in a slightly different regime of QCD, one with a finite isospin chemical potential due to the possibility of a new form of compact stars known as pion stars, first discussed in Ref. [5]). This type of compact object could form in regions with large densities of neutrinos, which in turn leads to the production of pions and their subsequent condensation [6]. These pions under weak equilibrium lead to stable pion stars, which may be electromagnetically neutralized by either electrons/positrons or muons. They are expected to have radii and masses that are substantially larger than those of neutron stars and a mass-radius relationship that is extremely stiff [7]. Pion stars are also different from neutron stars in the sense that at T = 0 it is interactions that give rise to an (effective) equation of state, and not the statistics of its constituents.
QCD at finite isospin chemical potential was first studied by Son and Stephanov using chiral perturbation theory (χPT) [8][9][10][11][12] in their seminal paper [13] and since then have been studied extensively in other versions of QCD including two-color and adjoint QCD [14][15][16] and also in the NJL [17][18][19][20] and quark-meson models [21,22] but also through lattice QCD, where it does not suffer from the fermion sign problem (except at finite magnetic fields [23,24] due to the charge asymmetry of the up and down quarks). The first lattice QCD calculations of finite isospin QCD were done in Refs. [25,26] and a more recent, thorough analysis in Refs. [27][28][29]. They find as expected from chiral perturbation theory calculations that at zero temperature there is a second order phase transition at an isospin chemical potential, |µ I | = m π 1 , which remains largely unaltered at finite temperatures up to approximately 170 MeV beyond which quarks become deconfined [30]. Similarly, with increasing isospin chemical potentials the quarks in the pions become more loosely bound and occur in a BCS phase though owing to the fact that this phase has the same order parameter as the BEC phase, there is no real phase transition, only a crossover transition, with the size of the pion condensate decreasing substantially within a narrow isospin window. There have been a number of studies in recent years comparing (2 + 1) flavor lattice QCD results with both QCD models and effective theories. Recently, the NJL model (non-renormalizable) comparisons [31] were made that showed good agreement with the lattice while the quark-meson model [22] (which is renormalizable) largely agrees with the lattice. Furthermore, there have been other comparisons of lattice QCD with results from an effective field theory (and model-independent) description [32], which is valid for asymptotically large isospin chemical potentials [33], where the pions behave as a free Bose gas. A recent review can be found in Ref. [34].
The focus of this work is to compare the results of three-flavor χPT at finite density [35] with that of (2 + 1)-flavor lattice QCD of Refs. [27][28][29]. We previously studied two-flavor χPT at next-to-leading order (NLO) [36] and found that the NLO results are in better agreement with lattice QCD than the tree-level results though the pressure, isospin density and energy density were all found to be consistently smaller than lattice QCD values. This is not entirely unexpected since the lattice QCD observables included the effects of the sea strange quarks [37] while two-flavor χ PT does not. As such, we extend our previous work in NLO two-flavor χPT to include the effect of the strange quarks by using three-flavor χPT at finite isospin chemical potential and find that the observables near the second phase transition is in excellent agreement with lattice QCD. As a natural extension of our finite isospin study, we also construct the NLO, one-loop effective potential to study the effects of the simultaneous presence of both the isospin and strange quark chemical potential. 2 We find the second-order phase transition in the pion condensed phase remains at |µ I | = m π even with the inclusion of µ S and NLO corrections. 3 Similarly, the second order phase transition in the kaon condensed phase remains at |± 1 2 µ I +µ S | = m K where m K is the kaon mass. Furthermore the effective potential even in the presence of µ S in the pion condensed phase only depends on µ I and in the kaon condensed phase on the combination 1 2 µ I + µ S but not µ I and µ S separately.
The paper is organized as follows. In the next section, we discuss the Lagrangian of three-flavor chiral perturbation theory at finite isospin and strange chemical potentials at next-to-leading order in the low-energy expansion. In Sec. 3, we review the ground state of the theory and fluctuations in the different phases. In Sec. 4 the NLO effective potential in the three different phases of the theory is calculated. In Sec. 5, we present our results for the thermodynamic functions and discuss the phase diagram in more detail. We derive medium-dependent masses at tree level and the isospin density, the pressure, and the equation of state at NLO. In the pion-condensed phase, we compare our result with recent lattice simulations.

χPT Lagrangian at O(p 4 )
In this section, we briefly discuss the symmetries of three-flavor QCD as well the chiral Lagrangian to next-to-leading order in the low-energy expansion and its renormalization. The three-flavor Lagrangian of QCD is where m = diag(m u , m d , m s ) is the mass matrix, / D = γ µ ∂ µ − igA a µ t a is the covariant derivative, t a are the Gell-Mann matrices, g is the strong coupling, A a µ is the gauge field, and F a µν is the field-strength tensor. The global symmetries of massless three-flavor QCD is SU (3) L × SU (3) R × U (1) B , which is spontaneously broken down to SU (3) V × U (1) B in the vacuum. For two degenerate light quarks, i.e. in the isospin limit the symmetry is SU If we add a chemical potential for each of the quarks, the symmetry is U (1) In the present paper, we consider three-flavor QCD with two degenerate light quarks. The chiral Lagrangian then describes the octet of pseudo-Goldstone bosons consisting of the three pions π 0 , π ± , the four kaons K ± , K 0 , and the eta η. We begin with the chiral perturbation theory Lagrangian at O(p 2 ) [9] 4 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 iλ i φ i 2f and Σ = 1 is the vacuum. Moreover, λ i are the Gell-Mann matrices that satisfy Trλ i λ j = 2δ ij and φ i are the fields that parametrizes the Goldstone manifold (i = 1, 2, ..., 8). The covariant derivative at nonzero quark chemical µ q potentials (q = u, d, s) is defined as follows We can also express v µ in terms of the baryon, isospin and strangeness chemical potentials µ B , µ I , and µ S as We note that the µ B -dependent term in Eq. (2.11) commutes with Σ and Σ † in Eqs. (2.4)-(2.5) and so the baryon chemical potential drops completely out of the chiral Lagrangian. This reflects the fact that we have only included the mesonic octet, which has zero baryonic charge. We therefore set µ B = 0 in the remainder of the paper. It is well known that chiral perturbation theory encodes the interactions among the Goldstone bosons that arise due to the spontaneous breaking of chiral symmetry by the QCD vacuum, i.e.

Next-to-leading order Lagrangian
In order to perform calculations to NLO, we must go to next-to-leading order in the lowenergy expansion and consider the terms that contribute at O p 4 . There twelve operators in L 4 [10], but only eight of them are relevant for the present calculations. They are where L i and H i are unrenormalized couplings. The relations between the bare and renormalized couplings are Here Γ i and ∆ 2 are constants and Λ is the renormalization scale in the modified minimal substraction scheme MS. The renormalized couplings are running couplings and satisfy the renormalization group equations (2.19) These are obtained by differentiation of Eqs (2.16)-(2.17) noting that the bare parameters are independent of the scale Λ. The solutions are where Λ 0 is a reference scale. We note that the contact term H 2 Tr[χ † χ] gives a constant contribution to the effective potential which is the same in all phases. We keep it, however, since it is needed to show the scale independence of the final result for the effective potential.
In three-flavor QCD, the constants Γ i and ∆ 2 are (2.23) In writing the NLO Lagrangian above, we have ignored contributions at finite isospin through the Wess-Zumino-Witten (WZW) Lagrangian, which is of the form with the leading contribution at O(p 4 ). There is also a separate contribution at zero external field at the same order but neither of them contribute to the quantities we compute at NLO.

Ground state and fluctuations
In this section, we will discuss the classical ground state of the theory as a function of the chemical potentials µ I and µ S . We will also discuss how to parametrize the fluctuations above the ground state. The most general SU (3) matrix for the ground state can be written as where α is a rotation angle andφ i are variational parameters. However, depending on the chemical potentials, we expect that the ground state takes a certain form, i.e. that it is rotated in a specific way. For example, in the case µ S = 0, we expect pion condensation for |µ I | > m π [13] and that the two-flavor results carry over. We therefore briefly review the two-flavor case first. Here the ground state can be written as [13] Σ α = e iαφ i τ i = cos α + iφ i τ i sin α , (3.2) where τ i are the Pauli matrices and φ i are again variational parameters. In order to ensure the normalization of the ground state, Σ α Σ † α = 1, the coefficients must satisfy |φ i | 2 = 1. The static part of the O(p 2 ) Hamiltonian H 2 reads The first term favors α = 0, i.e. the vacuum state Σ 0 = 1, and it is clear that there is a competition between the two terms in Eq. (3.5). We notice that the static energy only depends on |φ 1 | 2 + |φ 2 | 2 , and it minimized by setting |φ 3 | = 0. Without loss of generality and for later convenience, we can chooseφ 1 = 1. The rotated vacuum Eq. (3.2) can then be written as Minimizing Eq. (3.5) with respect to α, we find two phases α = 0 for 2B 0 m < µ 2 I and cos α = 2B 0 I . The first phase is the vacuum phase and the second phase consists of condensate of charged pions 5 .
In analogy with the two-flavor case, we expect that the pion condensation in the threeflavor case can be captured by writing Eq. (3.1) as 6 The rotated ground state can also be conveniently written as which shows that the rotation does not affect the s-quark. We next consider kaon condensation in three-flavor χPT. Depending on the values of µ I and µ S , this can be either charged or neutral kaons that condense.
we expect K 0 to condense. In this case λ 4 and λ 5 replace λ 1 and λ 2 , respectively since we consider a condensate of charged kaons with u and anti-s quarks. We then write Σ K + α as The rotated ground state takes the form (3.11) Finally, using A K α = e i α 2 λ 7 , the rotated ground state in the case of K 0 condensation is (3.12) Consider next the more general state where the two matrices are The vacuum state is given by α = β = γ = 0, pion condensation to β = γ = 0, K + condensation to β = π 2 , γ = 0 and finally K 0 condensation to This is the same ansatz as in Ref. [34], which reduces for γ = 0 reduces to that of Ref. [35].
For β = γ = 0, the static Hamiltonian reduces to The minimum of the static Hamiltonian is The ground-state energy in the two phases is

21)
For β = π 2 , γ = 0, the static Hamiltonian reduces to The minimum of the static Hamiltonian is The ground-state energy in the two phases is

26)
Finally, the case β = γ = π 2 , which corresponds to condensation of neutral kaons. The results for this phase can be obtained from the results of the phase of condensed charged kaons by the substitution In order to find the global minimum, we must compare Eqs. (3.22) and (3.27) in the region |µ I | > m π and | 1 2 µ I +µ S | > m K . The boundary between the pion-condensed phase and the kaon-condensed phase is then given by equating these expressions. This yields where we have written m 2 π = 2B 0 m and m 2 K = B 0 (m + m s ). We will return to the phase diagram in the µ I − µ S plane in Sec. 5.1.

Parametrizing Fluctuations
Since we want to study the thermodynamics of the pion-condensed and kaon-condensed phases including leading-order quantum corrections, it is natural to expand the chiral perturbation theory Lagrangian around the relevant ground state. The Goldstone manifold as a consequence of chiral symmetry breaking is SU (3) L × SU (3) R /SU (3) V . We will focus on the pion-condensed phase for simplicity. The remarks below also apply to the kaoncondensed phases. Following Refs. [15,36], we write We emphasize that the fluctuations parameterized by L α and R α around the ground state depend on α since the broken generators (of QCD) need to be rotated appropriately as the condensed vacuum rotates with the angle α [15]. In the present case, U is an SU (3) matrix that parameterizes the fluctuations around the ground state: With the parameterizations stated above, we get This parameterization not only produces the correct linear terms that vanish when evaluated at the minimum of the static Hamiltonian O(p 2 ), the divergences of the one-loop vacuum diagrams also cancel using counterterms from the O(p 4 ) Lagrangian. Furthermore, the parametrization produces a Lagrangian that is canonical in the fluctuations and has the correct limit when α = 0, whereby as expected. If one expands the Lagrangian using the parametrization Σ = LΣ α R instead of Eq. (3.30), the kinetic terms of the Lagrangian are non-canonical. By a field redefinition that depends on the chemical potentials" these terms can be made canonical. However, calculating the leading corrections to the tree-level potential, it can be shown that the ultraviolet divergences can be eliminated by renormalization only at the minimum of the classical potential 7 Thus one cannot find the minimum of the next-to-leading order effective potential viewed as a function of α, showing that this parametrization is erroneous. Let us finally take a look at the rotated generators and consider the pion condensed phase for simplicity. An infinitesimal fluctuation can be written (to linear order in φ i ) as Using the (anti)commutator relations of the Gell-Mann matrices, Eq. (3.36) takes the form We notice that all the generators except λ 2 and λ 8 are rotated, some of them however, only by half the angle. Finally defining λ 1 = (cos αλ 1 + sin αλ 3 ) and so forth, we can write to all orders in α

Leading-order Lagrangian
Using the parameterization of Eq. (3.34) discussed above, we can write down the Lagrangian in terms of the fields φ a , which parametrizes the Goldstone manifold. The leading-order terms in the low-energy expansion are given by L 2 , which can be written as a power series in the fields where the ellipses indicates terms that are cubic or higher order in the fields. We restrict ourselves to the charged kaon-condensed phase as similar results can be obtained for the neutral kaon-condensed phase.

Normal Phase
In the normal phase, the different terms in Eq. (3.39) are The inverse propagator is block diagonal and can be written as where P = (p 0 , p) is the four-momentum and P 2 = p 2 0 − p 2 . The submatrices are (3.46) (3.47) The dispersion relations for the charges mesons are The tree-level masses of the pions, kaons, and the η are then given by m 2 π,0 = 2B 0 m, m 2 K,0 = B 0 (m + m s ), and m 2 η,0 = 2 3 B 0 (m + 2m s ).

Pion-condensed phase
In the pion-condensed phase, the different terms in Eq. (3.39) are We notice that the linear term vanishes at the maximum of L static 2 , i.e. at the minimum of the tree-level potential, as required. We get for the inverse propagator: The three different 2 × 2 matrices are given by (3.68) where the masses are The quasiparticle dispersion relations can be easily found and read

Charged kaon-condensed phase
In the kaon-condensed phase, the different terms in Eq. (3.39) are The inverse propagator is block diagonal and can be written as (3.90) -14 - The masses are  The quasiparticle dispersion relations can be easily found and read The linear terms in the condensed phases are given by Eqs. (3.63) and (3.85). By differentiation with respect to α, it is straightforward to see that the terms vanish at the extremum of the corresponding static Lagrangian. To show this at NLO, requires the calculation of the one-loop diagram that contribute to the one-point function, see Ref. [36] for details.

Next-to-leading order effective potential
In this section, we calculate the NLO effective potential in the three different phases we consider. At order p 2 , the contribution to the effective potential in each phase is given by minus the static Lagrangian L static 2 . At order p 4 , there are two contributions to the effective potential. The first is the Gaussian fluctuations about the ground state, i.e. the standard one-loop contribution. The second is given by minus the static Lagrangian L static 4 , which is found by setting Σ = 1 in Eq. (2.15) and evaluating the traces. The one-loop contribution is ultraviolet divergent and needs regularization. We regularize the ultraviolet divergences using dimensional regularization in d = 3 − 2 dimensions. The divergences are cancelled by renormalizing the coupling constants that multiply the operators in L 4 . The sum of the three contributions is the complete effective potential to order p 4 in χPT.
The one-loop contribution to the effective potential of a free massive boson is given by where m is the mass and the second integral is defined in d = 3 − 2 as dimensions and Λ is the renormalization scale associated with the modified minimal subtraction scheme (MS). Integrating over P 0 , one finds

Normal phase
The leading-order contribution to the effective potential is minus the static Lagrangian given in Eq. (3.40) The one-loop contribution to the effective potential is where the particle energies are given by Eqs.
The order-p 4 contribution from minus the static Lagrangian L static 4 is given by After renormalization, the effective potential is (4.8) Using the renormalization group equations (2.20) for the couplings, we find that the effective potential is independent of the renormalization scale Λ. We note that Eq. (4.8) is independent of the chemical potentials µ I and µ S , which shows that the renormalized effective potential has the Silver Blaze property [53].

Pion-condensed phase
The tree-level contribution to the effective potential is minus the static Lagrangian given in Eq. (3.62) The one-loop effective potential is where the energies are given by Eqs. (3.79)-(3.83). The one-loop contribution to the effective is divergent in the ultraviolet and needs renormalization. The contribution from E π 0 and E η 0 can be calculated analytically in dimensional regularization using Eq. (4.3). The remaining contributions require a little more work. Let us consider the contribution from the charged pions. In order to eliminate the divergences, their dispersion relations are expanded in powers of 1/p as To this order, the large-p behavior in Eq. (4.11) is the same as the sum E 1 + E 2 , where E 1 = p 2 + m 2 1 + 1 4 m 2 12 and E 2 = p 2 + m 2 2 + 1 4 m 2 12 . The integral over E π + +E π − −E 1 −E 2 is therefore convergent in the ultraviolet and the subtraction integrals of E 1 and E 2 can be done analytically in dimensional regularization. We can then write (4.14) The contributions from the kaons can be calculated analytically as follows. Consider first the contribution from the charged kaon which is given by which can be rewritten as Since m 4 = m 5 , the last term vanishes and the integrand can be factorized as Shifting integration variables in the two terms, p 0 → p 0 ∓ im 45 2 , the integral simplifies to The static part of the Lagrangian L 4 as a function of α and for β = γ = 0 is The renormalized one-loop effective potential V eff = V 0 + V 1 + V static 1 is given by the sum of Eqs. (4.9), (4.19), and (4.20) then reads (4.21)

Charged kaon-condensed phase
The tree-level contribution to the effective potential is The one-loop effective potential is The contributions from π ± , K ± , K 0 andK 0 can be treated as in the previous section and it is only the terms V 1,K ± that require a subtraction term. The relevant masses are defined asm The contribution from the mixed π 0 and η 0 is given by where the new masses are defined as This yields The static part of the Lagrangian L 4 as a function of α and for β = π 2 is After renormalization, the effective potential is

Results and discussion
In this section, we study the quasiparticle masses, isospin density, pressure and the equation of state. In order to evaluate these quantities, we need the numerical values of the lowenergy constants (L i ) as well as the meson masses and decay constants. The low-energy constants have been determined experimentally, with the following values and uncertainties at the scale µ = m ρ , where Λ 2 = 4πe −γ E µ 2 [12].
Since we are mainly interested in comparing our results to the predictions of the lattice simulations in Refs. [27], we will use their values for the pion and kaon masses as well as and the decay constants. With uncertainties, they are given by [56] m π = 131 ± 3MeV , m K = 481 ± 10MeV , (5.5) Since we have three parameters in the Lagrangian, B 0 m, B 0 m s , and f , we need to pick three observables from the set above, and we choose m π , m K , and f π . The relevant meson masses and the the pion decay constants f π are given by Eqs.

Phase diagram
In order to show that the transitions from the vacuum phase to the Bose-condensed phases at a critical chemical potential, we expand the effective potential in a power series in α around α = 0 up to order α 4 to obtain an effective Landau-Ginzburg energy functional [36,51], As pointed out before, in the charged pion-condensed phase, V eff and therefore the coefficients are independent of µ S . Similarly, in the charged kaon-condensed phase, they only depend on the combination 1 2 µ I + µ S , and in the neutral kaon-condensed phase, only on the combination − 1 2 µ I + µ S , Using the expressions for the pion mass m π (A.1) and the piondecay constant f π , (A.3), it can be shown that in the pion-condensed phase (see Ref. [36] for details) The critical isospin chemical potential µ c I is defined by the vanishing of a 2 (µ I ), and Eq. (5.12) shows that |µ c I | = m π . Moreover, using the techniques in Ref. [51] it can be shown that a 4 (µ c I ) > 0, implying that the the transition from the vacuum phase to a pion-condensed phase is second order located at µ c I = ±m π . 8 Similarly, in the charged kaon-condensed phase, we find where m K is the physical kaon mass, whose one-loop expression is given by Eq. (A.2). The critical chemical potential is again given by the vanishing of a 2 , i.e. | 1 2 µ I + µ S | = 8 If a4(µ c I ) < 0, the transition is first order.
m K The coefficient of the order α 4 term can be shown to be positive when evaluated at 1 2 µ I + µ S = m K . This shows there is a second-order transition to a kaon-condensed phase at 1 2 µ I + µ S = ±m K . For the transition to a neutral kaon-condensed phase, we have − 1 2 µ I + µ S = ±m K . While the transitions from the vacuum to either a pion-condensed phase or a kaoncondensed phase are second order, the transition between the two Bose-condensed phases is first order. At leading, this is straightforward to see. For example the pion and kaon condensates are given by I , µ i > m π (5.14) For any µ I > m π and 1 2 µ I + µ S > m K , these condensates jump discontinuously to zero as we cross the phase line. The transition line itself is given by the equality of the pressures in the two phases and at tree level given by Eq. (3.29). In Fig. 2 we show the tree-level phase diagram in the µ I -µ S plane. The vacuum phase is in the region bounded by the straight lines µ I = ±m π , µ S = ±( 1 2 µ I + m K ), and µ S = ±(− 1 2 µ I + m K ). The corners from where the first-order lines emerge are located at (µ I , µ S ) = (±131, ±415.5) MeV. The solid lines represent second-order transitions while the dashed line indicates a first-order transition. In the vacuum phase, the thermodynamic functions are independent of the isospin and strange chemical potentials. This is an example of the so-called Silver Blaze property [53].

Medium-dependent masses
In this subsection, we will briefly discuss the medium-dependent masses. We restrict ourselves to a leading-order calculation, i.e. we consider the tree-level dispersion relations evaluated at p 2 = 0. In the pion-condensed phase, they are given by Eqs. (3.79)-(3.83).
In the kaon-condensed phase, they are given by Eqs. (3.103)-(3.107). In the left panel of Fig. 3, we show the medium-dependent masses as a function of the isospin chemical potential µ I for fixed strange chemical potential µ S = 200 MeV. For µ I = 0, we are in the normal phase, the pion masses take on their vacuum values, while the kaons are degenerate in pairs. The mass of π + decreases as we increase µ I and vanishes when µ I = m π and enter the pion-condensed phase. At µ I = m π , the masses vary continuously reflecting the second-order nature of the transition. We also note that the mass of η 0 is independent of µ I . which follows directly from Eq. (3.83). Finally, for asymtotically large values of µ I , the kaons and pions are pairwise degenerate. In the right panel of Fig. 3, we show the medium-dependent masses as a function of isospin chemical potential µ I for fixed strange chemical potential µ S = 460 MeV. At µ I = 0, we are in the vacuum phase. The kaons are again degenerate in pairs, the pions are also degenerate taking on their vacuum values. We enter the kaon-condensed phase at µ I = 42 MeV, which is a second-order transition. In this phase. K + is the Goldstone mode associated with the spontanous breakdown of the U (1) S -symmetry. As we increase the isospin chemical potential past µ I = 220? MeV, we enter the pion-condensed phase. In this phase, π + is the Goldstone mode associated with the spontanous breakdown of the U (1) I 3 -symmetry. This first-order nature of the transition can be seen by the jumps in the quasiparticle masses.

Pressure and equation of state
In this section, we discuss the pressure, isospin density and the equation of state in the pion condensed phase and compare our results to lattice QCD results of Refs. [27][28][29]. We begin with the pressure, which is defined as with the effective potential evaluated at its minimum. We also normalize the pressure such that it is zero in the normal phase. In Fig. 4, we plot the pressures from three-flavor χPT, lattice QCD and also from two-flavor χPT. We note that the pressure agrees strongly with lattice QCD results up to an isospin chemical: µ I ∼ 1.3m π , beyond which the pressure from three-flavor χPT is consistently larger than lattice QCD. It is also worth noting that the pressure from two-flavor χPT is consistently smaller than pressures from both threeflavor χPT and (2 + 1)-flavor lattice QCD. This is not surprising since the contribution to the pressure from the strange quarks is absent in the two-flavor case. We note that the light-green band includes uncertainties from all the LECs and is dominated by the uncertainties in L i . When the uncertainties in L i s are excluded we are left with much smaller uncertainties shown by the dark-green band. Next, we discuss the isospin density, We note that the result is consistent with the fact that the normal vacuum has zero isospin density, i.e. n I = 0 when α = 0. In Fig. 5, we plot the isospin density as a function of the isospin chemical potential, in particular µ I /m π . As with the pressure, the three-flavor χPT results are in extremely good agreement with the lattice results but only up to isospin chemical potentials of µ I ∼ 1.3m π , beyond which the isospin density from three-flavor χPT is larger than that of both lattice QCD and two-flavor χPT. The light-green bands in Fig. 5 represent the uncertainties in the isospin density due to the uncertainties in all the LECs, while the dark green bands only encode uncertainties from the masses and the decay constant. Interestingly, the isospin density from two-flavor χPT is in much better agreement with lattice results even though two flavor χPT ignores the vacuum contribution from the strange quarks. Finally, we present the equation of state (EoS), i.e. energy density as a function of pressure, in Fig.6. The energy density is defined as the Legendre transform better agreement with the lattice. Though it also worth noting that the uncertainties in the energy density for the three-flavor case is quite large due to the uncertainties in the LECs including the meson masses and the decay constants. We note that the uncertainty is mostly dominated by the uncertainties in the L i s and are quite large as shown by the light green bands. With the inclusion of these uncertainties the three-flavor results are consistent with lattice QCD for higher pressures though the central value estimates are consistently larger.