Quark and pion condensates at finite isospin density in chiral perturbation theory

In this paper, we consider two-flavor QCD at zero temperature and finite isospin chemical potential ($\mu_I$) using a model-independent analysis within chiral perturbation theory at next-to-leading order. We calculate the effective potential, the chiral condensate and the pion condensate in the pion-condensed phase at both zero and nonzero pionic source. We compare our finite pionic source results for the chiral condensate and the pion condensate with recent (2+1)-flavor lattice QCD results and find that they are in excellent agreement.


Introduction
Quantum Chromodynamics (QCD) has a rich phase structure as a function of temperature and quark chemical potentials [1,2,3]. The phases are characterized by their symmetry and symmetry-breaking properties. The QCD vacuum breaks chiral symmetry, a symmetry which is unbroken at the level of the Lagrangian itself (for massless quarks). The order parameter for chiral symmetry breaking of the QCD vacuum is the chiral condensate, a zero-momentum (spatially homogeneous) state analogous [5] to the energetically favored Cooper pairing due to the attractive phonon interactions in the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity [4]. The analogy between chiral symmetry breaking and a e-mail: prabal.adhikari@wellesley.edu b e-mail: andersen@tf.phys.ntnu.no Cooper-pair formation was first pointed out by Nambu and Jona-Lasinio [5], a physical picture that is affirmed by the presence of Goldstone modes, which are the low energy excitations around the chiral symmetry-broken QCD vacuum. The Goldstone modes are the three pions (π ± , π 0 ) in QCD, whose symmetries are consistent with Goldstone's theorem [6] assuming the following symmetry breaking pattern for N f = 2. The symmetry group of the Lagrangian has 2(N 2 f −1) generators and that of the vacuum has N 2 f −1 generators, which leads to exactly N 2 f − 1 Goldstone modes.
Surprisingly, while the evidence for chiral symmetry breaking is convincing, the chiral condensate itself is not a physical observable as is evident through the leading-order Gell-Mann-Oakes-Renner (GOR) relation [7], valid at zero temperature and density where m π is the pion mass, f π is the pion decay constant, m u and m d are the up and down quarks masses respectively and q = u, d. The GOR relation shows that only the product of the quark mass and the chiral condensate can be measured indirectly through a measurement of the pion mass m π and the pion decay constant f π . Furthermore, in the chiral limit, the pion mass is zero confirming Nambu's physical picture of chiral symmetry breaking. The strength of chiral symmetry breaking, as measured by the magnitude of the chiral condensate, changes depending on the physical environment. In the presence of a magnetic field particles are largely restricted to moving in the direction of the magnetic field, an effect known as dimensional reduction [8]. This leads to the strengthening of the quark-antiquark pairing in the chiral condensate channel, an effect analogous to the guaranteed presence of bound states for any potential well in one-dimensional quantum mechanics, (i.e. Cooper's theorem). The chiral condensate for the upquark-up-antiquark pairing is more enhanced than that of the down-quark-down-antiquark pairing.
On the other hand, thermal fluctuations, due to the presence of heat bath, have an opposite effect on the strength of the chiral condensate. Lattice calculations show that chiral symmetry is "restored" at a temperature of approximately T χ c = 155 MeV though strictly speaking the transition is only a crossover. This temperature is slightly less than the crossover temperature for the deconfinement transition T decon c = 170 MeV. A model-independent analysis within next-to-leading order chiral perturbation theory (χPT) shows that with the chiral condensate decreasing quadratically with temperature (T ) assuming T 4πf π , the regime of validity of χPT, and the coefficient depending on the number of flavors (N f ) [9,10,11].
Furthermore, the presence of matter can also have an effect on the chiral condensate. For instance, within nucleons, the valence quarks can expel the chiral condensate as has been shown (in a model-independent calculation) using the Feynman-Hellman theorem [12]. The physics is quite intuitive -the gluons that couple quarks or quarks and antiquarks, favor the formation of protons and neutrons when the quark chemical potential is approximately a third of the proton mass (nucleon density at saturation). As more gluons become confined in protons and neutrons, fewer are confined within the chiral condensate leading to its reduction. The deviation from the vacuum value of the chiral condensate ψ ψ 0 at low nuclear densities ρ N is where σ N is the pion-nucleon sigma term determined empirically to be σ N ≈ 45 MeV and ρ N is the nucleon density. Then the nucleon density at complete expulsion is In this paper, we focus on the nature of condensates within next-to-leading order, finite isospin χPT, which is the effective field theory of QCD valid at energies much lower than the typical hadronic scales, i.e.
where p χ is a parameter with mass dimension 1. The quantities relevant for this paper include momentum p, the isospin chemical potential µ I and a pseudoscalar, pionic source j [13].
We will focus not only on the behavior of the chiral condensate but also on the pion condensate which while zero in the QCD vacuum and for isospin chemical potentials smaller than the critical value, i.e. |µ I | ≤ µ I,c ≡ m π becomes finite. It is further known that pion condensates due to their electromagnetic charge form currents in a superconducting phase when a weak external magnetic field is present [14]. For larger magnetic fields, the pion condensate attains a spatially inhomogeneous structure in the form of a single vortex or a triangular vortex lattice similar in nature to the vortex lattice in type-II superconductors [15] explained by BCS theory [4]. Chiral perturbation theory at tree-level shows that the decrease in the size of the chiral condensate that occurs due to the formation of pion condensates is exactly compensated for by an increase in the pion condensate. In particular, At low isospin chemical potentials, the behavior of the chiral condensate in the pion-condensed phase relative to the normal vacuum from model-independent and tree-level calculations within χPT [16] is where n I is the tree-level isospin density, which at low densities scales linearly with the isospin chemical potential. It is worth noting that the ratio of the medium to vacuum chiral condensates (due to the expulsion of the chiral condensate by the formation of the pion condensed phase) is analogous in structure to the ratio found in nucleons due to expulsion of the chiral condensate through the pairing of the valence quarks qqq ρ .
Recently there have been lattice computations of finite isospin QCD [17], which does not suffer from the fermion sign problem. This is due to the complex phase cancellation between the up and down quark which have equal and opposite isospin numbers. Lattice QCD shows that the chiral structure of Eq. (9) is not preserved away from the critical isospin chemical potential. This violation is also observed in model-dependent calculations within the Nambu-Jona-Lasinio (NJL) model [18]. In this paper, we perform model-independent calculations of the chiral and pion condensates in the pioncondensed phase at next-to-leading order within χPT. Currently, lattice condensates are only available in the presence of a finite pionic source.
The paper is organized as follows. In the next section, we briefly discuss the chiral Lagrangian and the ground state in the presence of a nonzero isospin chemical potential. In Sec. 3, we derive the effective potential at next-to-leading order in χPT including a pionic source. In Sec. 4, we calculate the zero temperature quark and pion condensates at finite µ I . In Sec. 5, we plot the quark and pion condensates using lattice QCD parameters. At finite pionic source, we compare our results with the available lattice QCD data.

χPT Lagrangian
The Lagrangian of massless two-flavor QCD has a local SU (2) gauge symmetry in addition to the global For nonzero quark masses in the isospin limit, i.e for m u = m d , the symmetries are SU (2) V × U (1) B . Adding a quark chemical potential µ q for each quark, the symmetry is In the pion-condensed phase, the U (1) I3 symmetry is broken and either the π + or the π − is the associated Goldstone boson depending on the sign of µ I . Chiral perturbation theory is a low-energy effective theory for QCD based on the symmetries and degrees of freedom [19,20,21,22]. In two-flavor QCD, the degrees of freedom are the pions, while for three-flavor QCD we have additionally the charged and neutral kaons as well the eta. In the low-energy expansion of the Lagrangian in χPT, each covariant derivative counts as order p, while a quark mass term counts as order p 2 . We begin with the chiral Lagrangian in the isospin limit at O(p 2 ) where f is the bare pion decay constant, with M = diag(m, m) being the quark mass matrix and we have introduced a pionic source j, which is necessary for calculating the pion condensate. τ i represent the Pauli matrices and the covariant derivative is defined as with where µ I = µ u − µ d is the isospin chemical potential. We have also set µ B = 3 2 (µ u + µ d ) = 0 for the purpose of this paper. 1 In the two-flavor case [16], the ground state in χPT is parametrized as (17) where α at tree level can be interpreted as a rotation angle andφ 2 1 +φ 2 2 = 1 to ensure the normalization of the ground state, i.e. Σ † α Σ α = 1. In the remainder of the paper we chooseφ 1 = 1 andφ 2 = 0, without loss of generality. The matrix τ 2 generates the rotations and we can write the rotated vacuum as and Σ 0 = 1 is the trivial vacuum. We also need to parametrize the fluctuations around the condensed vacuum, which requires some care. Since the vacuum is rotated, we must also rotate the generators of the fluctuations in the same manner. This was discussed in Ref. [24] and an explicit example was given in [25]. The field Σ is written as with Here U is the SU (2) matrix that parametrizes the fluctuations around the ground state Σ 0 = 1 Combining Eqs. (18)-(20), the expression for Σ is which reduces to Σ = U 2 for α = 0 as required.
In order to calculate the effective potential and the condensates to NLO, we need to evaluate the path integral in the Gaussian approximation. In order to do so, we must expand the Lagrangian L 2 in the fields φ a as 1 In the pion-condensed phase, physical quantities are independent of µ B [26].
where the terms we need are where the source-dependent masses are We then get for the inverse propagator: where P = (p 0 , p), P 2 = p 2 0 −p 2 , and the 2×2 submatrix is given by (33) Here the off-diagonal mass is defined as At next-to-leading order in the low-energy expansion, there are ten different operators in the Lagrangian [21]. The terms relevant for the present calculations are Here l i and h i are bare couplings. The relations between the bare and renormalized couplings l r i (Λ) and h r i (λ) are [21] where Λ is the renormalization scale in the modified minimal subtraction (MS) scheme. The constants γ i and δ i are [21] Taking the derivative of Eqs. (36)-(37) with respect to Λ and using that the bare couplings are independent of the scale, one finds that the running couplings satisfy the equations, These equations can be easily solved for the running couplings l r i and h r i , The relations between the running couplings and the so-called low-energy constantsl i and h i in two-flavor χPT are Up to a prefactor, the low-energy constants are the running couplings evaluated at the scale Λ = 2B 0 m. We return to this in the Sec. 5.

Effective potential
At tree level, the effective potential V 0 is given by −L static The value of α that minimizes the tree-level potential V 0 is given by ∂V0 ∂α = 0 or 2B 0mj − µ 2 I sin α cos α = 0. The linear term L linear 2 in Eq. (25) then vanishes at the minimum of the tree-level potential, as required.
At next-to-leading order, there are two contributions to the effective potential, namely the static term V static 1 = −L static 4 and the one-loop contribution V 1 from L 2 . The static part of the NLO effective potential is V static acts as counterterms in the NLO calculation.
The one-loop contribution to the effective potential in Euclidean space of a free massive boson is given by where now P 2 = p 2 0 + p 2 and the integral is defined as (46) We use dimensional regularization to regulate ultraviolet divergences. With dimensional regularization, the momentum integral is generalized to d = 3 − 2 dimensions. The integral in Eq. (45) is The contribution from π 0 can be calculated analytically in dimensional regularization using Eq. (47), The contribution from the charged pions requires a little more work. Using Eq. (47), we obtain where the energies E π ± are found by calculating the zeros of the inverse propagator D −1 12 and read 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. (50) is the same as the sum E 1 +E 2 , where the energies and masses are E 1,2 = p 2 + m 2 1,2 + 1 4 m 2 12 = p 2 +m 2 1,2 ,m 2 1 = m 2 3 andm 2 2 = 2B 0 m j . We can then write The divergent integrals in Eq. (52) can be done analytically in dimensional regularization and the subtraction integral (53) is finite and can be computed numerically. Using Eq. (47), the divergent part of the one-loop contribution can be written as Renormalization is now carried out by adding Eqs. (43), (44), and (54), using Eqs. (36)-(37). Using Eqs. (41)-(42) the renormalized effective potential is For zero pionic source, j = 0, Eq. (55) reduces to the result of Ref. [26] after subtracting the constant term proportional toh i . We note that since h r 2 does not run due to Eq. (42), we have definedh 1 = (4π) 2 h r 1 = (4π) 2 h 1 . The term proportional toh 1 in the effective potential is independent of α and does not affect the ground state.

Quark and pion condensates
In Refs. [25,26], we studied the thermodynamic properties of the pion-condensed phase of QCD at T = 0 at next-to-leading by calculating the first quantum correction to the tree-level potential. It was shown that the transition from the vacuum phase to a pion-condensed phase is second order and takes place at a critical isospin chemical potential µ c I = m π , where m π is the physical pion mass. We continue the study of the pion-condensed phase by calculating the quark and pion condensates.
In the isospin limit, the quark condensates ūu and d d are equal and in the following we denote each of them by ψ ψ . The quark and pion condensates at finite isospin are then defined as 2 At tree level, the condensates are given by the partial derivatives of V 0 , which yields 2 Note that in the finite isospin lattice QCD simulation of Ref. [17], ψ ψ = ūu + d d but in our notation ψ ψ = ūu = d d . Consequently, there is an explicit factor of 1 2 in our definition of ψ ψ . Additionally, compared to Ref. [17], we define the pion condensate with an extra factor of 1 2 . The pionic source λ in Ref. [17] corresponds exactly to j in this paper.
where ψ ψ tree 0 = −f 2 B 0 denotes the quark condensate in the vacuum phase. Eqs. (57)-(58) show that we can interpret α as a rotation angle such that the quark condensate is rotated into a pion condensate. As we shall see below, this interpretation is not valid at next-to-leading order and is not seen on the lattice. At next-to-leading order in the low-energy expansion, the quark condensate is In the limit of vanishing source j and α = 0, Eq. (59) is independent of the isospin chemical potential and are consistent with expressions given in Refs. [20,21]. At next-to-leading order in the low-energy expansion, the pion condensate is which vanishes in the normal vacuum with α = 0.

Results and discussion
In this section, we present our numerical results for the chiral condensate and the pion condensate both at zero and non-zero pionic source. We compare the non-zero pionic source results with lattice simulations for which lattice data are available. Finite isospin QCD on the lattice is studied by adding an explicit pionic source since spontaneous symmetry breaking in finite volume is forbidden. Obtaining the chiral and pion condensate then requires not just taking the continuum limit but also extrapolating to a zero external source, which is technically challenging on the lattice The quark condensate is given by Eqs. (59), while the pion condensate is given by Eq. (60). The value of α in the equations is found by extremizing the effective potential, i.e. solving of Eq (55), ∂V eff ∂α = 0.

Definitions and choice of parameters
The chiral condensate depends on the low-energy constanth 1 of two-flavor χPT, which is unphysical and undeterminable within χPT [27,28]. Furthermore,h 1 is scale-independent and does not affect the ground state value of α. Consequently, we define the quark and pion condensate deviations relative to the values of the respective condensates at zero isospin and zero pionic source. The definitions of the condensate deviations 3 are [29] where m is the degenerate mass of the up and down quarks, m π is the pion mass, and f π is the pion decay constant. O µ I is the value of the condensate O at an isospin chemical potential µ I and a pionic source j. ψ ψ j=0 0 is the value of the chiral condensate when µ I = 0 and j = 0. The definition of the chiral condensate deviation, Σψ ψ , ensures that it is equal to 1 when µ I = 0 and j = 0 and the definition of the pion condensate deviation does not contain a trivial subtraction of the pion condensate at zero pionic source and zero isospin, π + j=0 0 , since it equals zero. Furthermore, the definitions ensure that the following constraint is satisfied at tree level including for any pionic source j Σ treē which is consistent with Eqs. (57) and (58). However the constraint is not satisfied at next-to-leading order as will be evident. For our calculation of the condensate deviations, we choose the following values of the quark masses [30] The LECs of two-flavor χPT and their respective uncertainties are defined at the scale Λ = 2B 0 m through Eq. (42) [32]  The physical pion mass m π and the physical pion decay constant f π can be calculated within χPT at NLO [20], Given the values m π , f π ,l 3 , andl 4 , we can calculate the parameters f and 2B 0 m appearing in the chiral Lagrangian: where m 2 π,0 ≡ 2B 0 m. Using this relation, we can calculate B 0 , which also depends on the tree-level pion mass and the continuum value of the quark mass.

Condensates at j = 0
In Fig. 1, we plot the T = 0 quark condensate deviation (which is normalized to 1) and the pion condensate deviation defined in Eqs. (61) and (62) respectively. In the left panel of Fig. 1, we plot the tree level chiral condensate deviation in red and the next-to-leading order deviation in dashed (blue). Similarly, in the right panel of Fig. 1, we plot the tree-level pion condensate deviation in red. The light blue shaded regions in the two panels of Fig. 1 represent the uncertainty in the condensate deviations due to the uncertainty in the values of the pion mass and the pion decay constant in the lattice and the uncertainty in the LECs, which arises due to experimental uncertainties. We note that the uncertainty in the condensate deviations is dominated by the uncertainties in the pion mass and pion decay constant with the uncertainties in the LECs not contributing significantly.
We find that relative to the tree-level condensate deviations, the next-to-leading condensate deviations are significantly larger for the chiral condensate and significantly smaller for the pion condensate. The magnitude of the chiral condensate at next-to-leading order decreases more slowly and the magnitude of the pion condensate increases more slowly compared to their respective tree-level values. Furthermore, the tree-level pion condensate deviation asymptotes to 1 very efficiently, a behavior which is absent at next-to-leading order.

Finite-j condensates and comparison with lattice simulations
In this section, we plot the chiral and pion condensate deviations at T = 0 with a non-zero pionic source (j = 0) and compare our results with lattice QCD [17,31]. We note that while there is no lattice QCD data available for comparison at j = 0, the comparison of finite-j condensate deviations from χPT with the lattice allows us to gauge the quality of our j = 0 results calculated at next-to-leading order in χPT. A non-zero j is required to stabilize lattice simulations and consequently j = 0 results are "cumbersome" to generate [17].
In Fig. 2, we show the chiral condensate deviation on the left panel and pion condensate deviation on the right panel. The deviations are calculated at j = 0.00517054m π , which is the smallest value of the pionic source for which lattice QCD data is available at T = 0. In order to perform this comparison fairly, it is important to know the exact quark masses in the continuum since this determines the χPT parameter, B 0 , on which the condensates depend. Continuum quark masses have not been calculated in the lattice QCD study. Conse- quently, the comparison performed here is only meant to be suggestive, a more thorough comparison requires using the exact value of the continuum quark masses. For the purposes of our comparison here, we use the lattice continuum quark masses from a separate lattice QCD simulation [30].
Firstly, we note that due to the presence of an external pionic source, the ground state explicitly breaks isospin symmetry. Consequently, there is no second order phase transition as there is in the absence of the pionic source. Instead, the transition is a crossover involving a range of isospin chemical potentials within which the chiral and pion condensates change significantly.
The condensate deviations in Fig. 2 shows excellent agreement with the lattice for isospin chemicals potential up to µ I ≈ 1.5m π . For larger isospin chemical potentials, the lattice chiral condensate deviation is slightly smaller than the corresponding deviation from χPT at next-to-leading order and the lattice pion condensate deviation is slightly larger than the corresponding deviation from χPT at next-to-leading order. For all values of the isospin chemical potential, the next-toleading order χPT results are a significant improvement on the tree-level results. The difference between the tree level pion condensate deviation and the corresponding lattice QCD deviation is more significant.
Finally, in Fig. 3, we show the chiral condensate deviation on the left panel and the pion condensate deviation on the right panel for j = 0.0129263m π , including χPT results at tree-level, next-to-leading order and lattice QCD including uncertainties. As with the previous figure, the results at next-to-leading order χPT are a significant improvement over tree level deviations. The improvement is most significant in the pion condensate deviation, which shows a qualitatively different asymptotic behavior -the next-to-leading order pion condensate deviation does not asymptote to 1 as the tree-level result does. The agreement of the deviations with lattice QCD is excellent especially for lower values of isospin chemical potential consistent with the fact that χPT is an effective theory with systematic corrections that increase with the isospin chemical potential. We also note that the discrepancy between the conden-sate deviations at larger isospin chemical potentials is larger for j = 0.0129263m π than j = 0.00517054m π , which is again consistent with expectations for an effective theory.
In conclusion, we have performed a calculation of the quark and pion condensates at next-to-leading order χPT in the absence of an external pionic (pseudoscalar) for the first time -the results presented here can be used to gauge the quality of future lattice calculation of the chiral and pion condensate at zero source, a calculation that is currently quite challenging to perform. We have also calculated the condensates at finite pionic source and performed a qualitative comparison with the lattice which shows a significantly improved agreement after we include next-to-leading order corrections. While this suggests the importance of performing a next-to-leading χPT calculation of condensates, at this stage it is not possible to perform a fully quantitative comparison with lattice QCD due to the absence of continuum quark mass values.