QCD at finite isospin density: chiral perturbation theory confronts lattice data
QQCD at finite isospin density: chiral perturbation theory confronts lattice data
Prabal Adhikari a,b , Jens O. Andersen b,c, ∗ a Wellesley College, Department of Physics, 106 Central Street, Wellesley, MA 02481, United States b Department of Physics, Norwegian University of Science and Technology, Høgskoleringen 5, N-7491 Trondheim, Norway c Niels Bohr International Academy, Blegdamsvej 17, DK-2100 Copenhan, Denmark
Abstract
We consider the thermodynamics of three-flavor QCD in the pion-condensed phase at nonzero isospin chemical potential ( µ I ) andvanishing temperature using chiral perturbation theory in the isospin limit. The transition from the vacuum phase to a superfluidphase with a Bose-Einstein condensate of charged pions is shown to be second order and takes place at µ I = m π . We calculatethe pressure, isospin density, and energy density to next-to-leading order in the low-energy expansion. Our results are comparedwith recent high-precision lattice simulations as well as previously obtained results in two-flavor chiral perturbation theory. Theagreement between the lattice results and the predictions from three-flavor chiral perturbation theory is very good for µ I < MeV. For larger values of µ I , the agreement between lattice data and the two-flavor predictions is surprisingly good and better thanwith the three-flavor predictions. Finally, in the limit m s (cid:29) m u = m d , we show that the three-flavor observables reduce to thetwo-flavor observables with renormalized parameters. The disagreement between the results for two-flavor and three-flavor χ PTcan largely be explained by the differences in the measured low-energy constants.
Keywords:
Chiral perturbation theory, Finite isospin, QCD
1. Introduction
QCD in extreme conditions, i.e. high temperature and den-sity has received a lot of attention in the past decades due to itsrelevance to the early universe, heavy-ion collisions, and com-pact 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 ofneutron stars. However, lattice QCD cannot be applied to QCDat nonzero baryon density due to the sign problem: integratingout the fermions in the path integral for the partition functiongives rise to a functional determinant that can be consideredpart of the probability measure. At µ B (cid:54) = 0 , this determinantis complex and standard Monte Carlo techniques cannot be ap-plied. A way to circumvent this problem, for high tempera-tures and small chemical potentials, is by Taylor expanding thethermodynamic quantities about zero µ B [4]. For small T andlarge µ B , this is obviously hopeless. Due to asymptotic free-dom, we expect to be able to use weak-coupling techniques atvery high densities [5, 6]. In the weak-coupling expansion theseries is now known to order α s for massive quarks [7] and α s log α s for massless quarks [8]. For lower densities, whereweak-coupling techniques do not apply, we have to use low-energy models of QCD, see Ref. [9] for a recent review.There are variants of QCD that do not suffer from the signproblem. These include two-color QCD [10], three-color QCDwith fermions in the adjoint representation [11], zero densityQCD in an external magnetic field [12], and three-color QCD at ∗ Corresponding author
Email address: [email protected] (Jens O. Andersen) finite isospin [13, 14, 15, 16, 17]. The absence of the sign prob-lem implies that one can simulate these systems on the latticeand compare the results with low-energy models and theories.In the case of QCD at finite isospin chemical potential, one findsat T = 0 , a transition from the vacuum to a pion-condensedphase at a critical isospin chemical potential µ cI = m π . Themechanism of pion condensation and the transition to a pionsuperfluid phase out of the vacuum is simply that it is energeti-cally favorable to form such a condensate for µ I ≥ µ cI . More-over, with increasing isospin chemical potential, it is expectedthat there is a crossover to a BCS phase. Since the order pa-rameter in the BCS phase has the same quantum numbers asa charged pion condensate, this is not a true phase transition,but associated with the formation of a Fermi surface and sub-sequent condensation of Cooper pairs. A very recent review onmeson condensation can be found in Ref. [18].Chiral perturbation theory ( χ PT) is a low-energy effectivetheory of QCD based only on its global symmetries and thedegrees of freedom, and the predictions of χ PT are, therefore,model independent [19, 20, 21, 22]. It has been remarkably suc-cessful in describing the phenomenology of the pseudo-Goldstonebosons that result from the spontaneous breakdown of chiralsymmetry in the QCD vacuum. χ PT at finite isospin was firstconsidered by Son and Stephanov in their seminal paper twodecades ago [23], in which all the leading order results werederived.In this letter, we calculate the effective potential in chiralperturbation theory at next-to-leading (NLO) order in the low-energy expansion for three flavors at finite isospin chemical po-tential. While the phase diagram as functions of isospin and
Preprint submitted to Physics Letter B June 1, 2020 a r X i v : . [ h e p - ph ] M a y trange chemical potentials ( µ S ) has been mapped out and lead-ing order (LO) thermodynamic functions have been known fortwo 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 and Refs. [26,27, 28, 29, 30, 31, 32] for various aspects of χ PT for three-color QCD including some NLO effects. Finite isospin systemshave also been studied in the context of low-energy effectivemodels including the non-renormalizable Nambu-Jona-Lasiniomodel [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47],and the renormalizable quark-meson model [48, 49, 50, 51].We derive the pressure, isospin density, and equation ofstate, and compare these quantities with recent lattice resultsas well earlier results from two-flavor χ PT [52]. In the large- m s limit, the three-flavor result is matched onto the two-flavorresult of Ref. [52] with renormalized parameters. The disagree-ment between the two-flavor and three-flavor results are dis-cussed and shown to be related to the differences in the ex-perimental values of the low-energy constants. Results on thethermodynamics of the kaon-condensed phases at finite µ S and µ I as well as calculational details can be found in an accompa-nying long paper [53].
2. Chiral perturbation theory
As mentioned above, χ PT is an effective low-energy theoryof QCD based solely on its global symmetries and low-energydegrees of freedom. In massless three-flavor QCD, the sym-metry is SU (3) L × SU (3) R × U (1) B , which in the vacuum isbroken down to SU (3) V × U (1) B . For two degenerate lightquarks, the symmetry is SU (2) I × U (1) Y × U (1) B . If weadd a quark chemical potential for each flavor, the symmetry is U (1) I × U (1) Y × U (1) B . In three-flavor QCD, we keep theoctet of mesons, which implies that chiral perturbation theoryis not valid for arbitrarily large chemical potential. Consideringthe hadron spectrum, one naively expects that the expansion isvalid for | µ u | = | µ d | < MeV [24]. χ PT has a well definedpower counting scheme, where each derivative as well as eachfactor of a quark mass counts as one power of momentum p . Atleading order in momentum, O ( p ) , there are only two terms inthe chiral Lagrangian L = f (cid:2) ∇ µ Σ † ∇ µ Σ (cid:3) + f (cid:2) χ † Σ + χ Σ † (cid:3) , (1)where f is the bare pion decay constant, χ = 2 B M , M = diag( m u , m d , m s ) (2)is the quark mass matrix and Σ = U Σ U , where U = exp iλ i φ i f and Σ = is the vacuum. Here λ i are the Gell-mann matricesthat satisfy Tr λ i λ j = 2 δ ij and φ i are the fields parametriz-ing the Goldstone manifold ( i = 1 , ..., ). In the remainderwe work in the isospin limit, m = m u = m d . The covariantderivative and its Hermitian conjugate at nonzero quark chemi-cal potentials, µ q ( q = u, d, s ), are defined as follows ∇ µ Σ ≡ ∂ µ Σ − i [ v µ , Σ] , (3) ∇ µ Σ † = ∂ µ Σ † − i [ v µ , Σ † ] , (4) with v µ = δ µ diag( µ u , µ d , µ s )= δ µ diag( µ B + µ I , µ B − µ I , µ B − µ S ) , (5)where µ B = ( µ u + µ d ) , µ I = µ u − µ d , and µ S = ( µ u + µ d − µ s ) . It turns out that the Lagrangian is independent of µ B which reflects the fact that all degrees of freedom, namelythe meson octet, have zero baryon number. Since we are fo-cusing on pion condensation and want to compare with latticedata, we set µ S = 0 such that v = µ I λ . By expanding theLagrangian (1) to second order in the fields, we obtain the termsneeded for our NLO calculation. Based on the two-flavor case [23], the ground state in thepion-condensed phase is parametrized as [53] Σ α = e iα ( ˆ φ λ + ˆ φ λ ) = cos α + ( i ˆ φ λ + ˆ φ λ ) sin α , (6)where α is a rotation angle and ˆ φ + ˆ φ = 1 to ensure that theground state is normalized, Σ † α Σ α = . From Eq. (1), we findthe static Hamiltonian H static2 = f v , Σ α ][ v , Σ † α ] − f B Tr[ M Σ α + M Σ † α ] , (7)where the first term can be written as f Tr[ v , Σ α ][ v , Σ † α ] = f µ I Tr[ λ Σ α λ Σ † α − λ ] . There is a competition betweenthe two terms in Eq. (7): The first term favors Σ α in the λ and λ directions, while Σ α in the second terms prefers the normalvacuum, [23]. It turns out the that the former only dependson ˆ φ + ˆ φ and so we choose ˆ φ = 1 without loss of generality.The matrix λ generates the rotations and the rotated vacuumis given by Σ α = A α Σ A α where A α = e i α λ , and Σ = .The rotated vacuum can then be written in the form Σ α = 1 + 2 cos α iλ sin α + cos α − √ λ = cos α sin α − sin α cos α
00 0 1 . (8)Here the rotation in the subspace of the u and the d -quark isevident and at tree level, we have (cid:104) ¯ ψψ (cid:105) + (cid:104) π + (cid:105) = (cid:104) ¯ ψψ (cid:105) ,i.e. the quark condensate is rotated into a pion condensate.The fluctuations around the condensed or rotated vacuummust also be parametrized and this requires some care. Naively,one would write the field as Σ = U Σ α U , where U = exp iλ i φ i f .However, this parametrization is incorrect since it can be shownthat one cannot renormalize the effective potential at next-to-leading order using the standard renormalization of the low-energy couplings appearing in the NLO Lagrangian. One wayof understanding the failure of this parametrization is to realizethat the generators of the fluctuations about the ground state A covariant derivative and a mass term both count as order p in the low-energy expansion. Σ = L α Σ α R † α , (9)where L α = A α U A † α and R α = A † α U † A α . This parametriza-tion reduces to the standard parametrization for α = 0 and hasnone of the flaws of the naive parametrization.The tree-level effective potential V = H static2 = −L static is now evaluated to be V = − f B (2 m cos α + m s ) − f µ I sin α . (10)At next-to-leading order in the low-energy expansion, there aretwelve operators. Not all of them are relevant for the presentcalculations, in fact only eight contribute to the effective poten-tial. They are L = L (cid:0) Tr (cid:2) ∇ µ Σ † ∇ µ Σ (cid:3)(cid:1) + L Tr (cid:2) ∇ µ Σ † ∇ ν Σ (cid:3) Tr (cid:2) ∇ µ Σ † ∇ ν Σ (cid:3) + L Tr (cid:2) ( ∇ µ Σ † ∇ µ Σ)( ∇ ν Σ † ∇ ν Σ) (cid:3) + L Tr (cid:2) ∇ µ Σ † ∇ µ Σ (cid:3) Tr (cid:2) χ † Σ + χ Σ † (cid:3) + L Tr (cid:2) ( ∇ µ Σ † ∇ µ Σ) (cid:0) χ † Σ + Σ † χ (cid:1)(cid:3) + L (cid:2) Tr (cid:0) χ † Σ + χ Σ † (cid:1)(cid:3) + L Tr (cid:2) χ † Σ χ † Σ + χ Σ † χ Σ † (cid:3) + H Tr[ χ † χ ] . (11)In writing the NLO Lagrangian above, we have ignored theWess-Zumino-Witten terms since they do not contribute to thequantities in the present paper. The last term in Eq. (11) is acontact term, which is needed to renormalize the vacuum en-ergy and to show the scale independence of the final result forthe effective potential in each phase. The contribution from theterms in Eq. (11) to H static4 = −L static4 = V static1 is V static1 = − (4 L + 4 L + 2 L ) µ I sin α − L B (2 m cos α + m s ) µ I sin α − L B mµ I cos α sin α − L B (2 m cos α + m s ) − L B (2 m cos 2 α + m s ) − H B (2 m + m s ) . (12)In a next-to-leading order calculation, we need to renormalizethe couplings L i and H i to eliminate the ultraviolet divergencesthat arise from the functional determinants. The relations be-tween the bare and renormalized couplings are L i = L ri (Λ) − Γ i λ , (13) H i = H ri (Λ) − ∆ i λ , (14)with λ = Λ − (cid:15) π ) (cid:2) (cid:15) + 1 (cid:3) . Here Γ i and ∆ i are constants [21] Γ = 332 , Γ = 316 , Γ = 0 , Γ = 18 , (15) Γ = 38 , Γ = 11144 , Γ = 548 , ∆ = 524 , (16) and Λ is the renormalization scale associated with the modi-fied minimal substraction scheme MS . Taking the derivativeof Eqs. (13)–(14) and using the fact that the bare couplings arescale independent, one finds the renormalization group equa-tions for the renormalized couplings, Λ dL ri (Λ) d Λ = − Γ i (4 π ) , (17) Λ dH ri (Λ) d Λ = − ∆ i (4 π ) . (18)The contact term H Tr[ χ † χ ] makes a constant contribution tothe effective potential which is independent of the chemicalpotential and therefore the same in both phases. We keep it,however, in the final expression for the NLO effective potentialsince H r (Λ) is running. It is needed to show the scale inde-pendence of V eff . The renormalized NLO effective potential V eff = V + V + V static1 is given by V eff = − f B (2 m cos α + m s ) − f µ I sin α − (cid:20) L r + 4 L r + 2 L r + 116(4 π ) (cid:18)
92 + 8 log Λ m + log Λ ˜ m (cid:19)(cid:21) µ I sin α − (cid:20) L r + 12(4 π ) (cid:18)
12 + log Λ ˜ m (cid:19)(cid:21) × B (2 m cos α + m s ) µ I sin α − (cid:20) L r + 12(4 π ) (cid:18)
32 + 4 log Λ m − log Λ ˜ m (cid:19)(cid:21) × B mµ I cos α sin α + B m sin α [16 L r − H r ] − (cid:20) L r + 8 L r + 4 H r + 1(4 π ) (cid:18) ˜ m + 49 log Λ m (cid:19)(cid:21) B m s − (cid:20) L r + 1(4 π ) (cid:18)
119 + 2 log Λ ˜ m + 49 log Λ m (cid:19)(cid:21) × B mm s cos α − (cid:20) L r + 16 L r + 8 H r + 1(4 π ) (cid:18) ˜ m ++2 log Λ m + log Λ ˜ m + 19 log Λ m (cid:19)(cid:21) B m cos α + V fin1 ,π + + V fin1 ,π − , (19)where L ri (Λ) are the renormalized coupling constants and themasses are ˜ m = 2 B m cos α , (20) m = 2 B m cos α + µ I sin α , (21) ˜ m = B ( m cos α + m s ) + 14 µ I sin α , (22) m = 2 B ( m cos α + 2 m s )3 . (23)3inally, V fin1 ,π ± are finite subtraction terms which depend on B and m but are independent of m s . For details, see Ref. [53].The couplings are running in such a way that their Λ -dependencecancel against the explicit Λ -dependence of the chiral loga-rithms in Eq. (19), implying that Λ dV eff d Λ = 0 , cf. Eqs. (17)–(18). In order to obtain Eq. (19), we must isolate the ultravioletdivergences from the functional determinants. This is done byadding and subtracting a divergent term that we calculate an-alytically in dimensional regularization. The subtracted termis then combined with the original one-loop expression for theeffective potential giving finite terms V fin1 ,π ± that can be easilycomputed numerically. The divergences are finally removed byrenormalization of the L i s according to Eqs. (13)–(14). Thedetails of the subtraction and renormalization procedure can befound in Ref. [53] and the NLO effective potential in the two-flavor case can be found in Ref. [52].Thermodynamic quantities can be calculated from the ef-fective potential Eq. (19), for example the pressure P = − V eff ,the isospin density n I = − ∂V eff ∂µ I , and the energy density (cid:15) = − 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 .For sufficiently large values of m s , we expect using effective-field theory arguments, that all degrees of freedom that containan s -quark freeze and decouple. Thus we expect that the kaonsand eta decouple from the low-energy dynamics involving thepions. Formally, this is the limit B m (cid:28) B m s (cid:28) (4 πf π ) .The system is then described in terms of two-flavor chiral per-turbation theory where the effects of the s -quark shows up inthe renormalization of the coupling constants l i of the form log Λ ˜ m K, and log Λ ˜ m η, , where the masses are ˜ m K, , = B m s and ˜ m η, = B m s . Expanding the effective potential Eq. (19)in inverse powers of m s , we obtain V eff = − f ˜ B m cos α − f B m s −
12 ˜ f µ I sin α − (cid:20) l r + 4 l r + 1(4 π ) (cid:18)
32 + log Λ ˜ m +2 log Λ m (cid:19)(cid:21) B m cos α − (cid:20) l r + 1(4 π ) (cid:18)
12 + log Λ m (cid:19)(cid:21) × B mµ I cos α sin α − (cid:20) l r + l r + 12(4 π ) (cid:18)
12 + log Λ m (cid:19)(cid:21) µ I sin α +4( − h r + l r ) B m − [16 L r + 8 L r + 4 H r + 1(4 π ) (cid:32) ˜ m K, + 49 log Λ ˜ m η, (cid:33)(cid:35) B m s + V fin1 ,π + + V fin1 ,π − , (24)where we have defined the combinations of the renormalized couplings l ri and h r as well as renormalized ˜ f and ˜ B as l r + l r = 4 L r + 4 L r + 2 L r + 116(4 π ) (cid:34) log Λ ˜ m K, − (cid:35) , (25) l r + l r = 16 L r + 8 L r + 14(4 π ) (cid:34) log Λ ˜ m K, − (cid:35) + 136(4 π ) (cid:34) log Λ ˜ m η, − (cid:35) , (26) l r = 8 L r + 4 L r + 14(4 π ) (cid:34) log Λ ˜ m K, − (cid:35) , − h r + l r = 4 L r − H r , (27) ˜ f = f (cid:20) B m s f (16 L r + 1(4 π ) log Λ ˜ m K, (cid:33)(cid:35) , (28) ˜ B = B (cid:20) − B m s f (16 L r − L r − π ) log Λ ˜ m η, (cid:33)(cid:35) . (29)Several comments are in order: The terms in Eq. (24) that areproportional to powers of m s are independent of α and µ I .They can be interpreted as a constant renormalized contributionto the vacuum energy from the s -quark and can be omitted. Theconstant term proportional to B m can be omitted for simi-lar reasons. The relations between the renormalized couplings l ri , h ri and the low-energy constants ¯ l i , ¯ h i in two-flavor χ PT are l ri (Λ) = γ i π ) (cid:20) ¯ l i + log 2 B m Λ (cid:21) , (30) h ri (Λ) = δ i π ) (cid:20) ¯ h i + log 2 B m Λ (cid:21) , (31)where γ = , γ = , γ = − , γ = 2 , and δ = 2 [20]. Therenormalization group equations are then Λ dl ri (Λ) d Λ = − γ i (4 π ) .Given the renormalization group equations for l ri , h ri , L ri , H ri ,one verifies that the Λ -dependence of the left - and right-handside in Eqs. (25)–(27) is identical. Moreover, the parameters ˜ f and ˜ B are independent of the scale. Eqs. (25)–(29) are inagreement with the original calculations of Ref. [21], where re-lations among the renormalized couplings in two - and three-flavor χ PT were derived. This agreement is a nontrivial checkof our calculations. Inserting these relations using (31) into4q. (24), we finally obtain V eff = − f ˜ B m cos α −
12 ˜ f µ I sin α − π ) (cid:20) − ¯ l + 4¯ l + log (cid:18) B m ˜ m (cid:19) +2 log (cid:18) B mm (cid:19)(cid:21) B m cos α − π ) (cid:20)
12 + ¯ l + log (cid:18) B mm (cid:19)(cid:21) × B mµ I cos α sin α − π ) (cid:20)
12 + 13 ¯ l + 23 ¯ l + log (cid:18) B mm (cid:19)(cid:21) × µ I sin α + V fin1 ,π + + V fin1 ,π − . (32)In the limit B m s (cid:28) (4 πf π ) , B in the NLO terms can beidentified with ˜ B using Eq. (29) and the result reduces to thatof two-flavor χ PT in Ref. [52].
3. Results and discussion
The expressions for the effective potential, isospin density,pressure, and energy density are all expressed in terms of theisospin chemical potential, the parameters B m , B m s , and f of the chiral Lagrangian as well as the renormalized couplings L ri . In order to make predictions, we need to determine theparameters of the chiral Lagrangian using the physical mesonmasses and the decay constants. In χ PT, one can calculate thepole masses of the mesons and the decay constants ( f π , f K )systematically in the low-energy expansion. At one loop, theresults are expressed in terms of B m , B m s , f , and L ri .These equations can be solved to find the parameters of the chi-ral Lagrangian and thereby numerically evaluate the effectivepotential. The tree-level values of m π, and m K, can be ex-pressed in terms of B m and B m s as m π, = 2 B m and m K, = B ( m + m s ) . Since we want to compare our predic-tions with the results of the lattice simulations [54], we use theirvalues for the meson masses and decay constants [55], m π = 131 ± , m K = 481 ±
10 MeV , (33) f π = 128 ± √ , f K = 150 ± √ . (34)The low-energy constants have been determined experimen-tally, with the following values and uncertainties at the scale µ = m ρ , where m ρ is the mass of the ρ meson and Λ =4 πe − γ E µ [56] L r = (1 . ± . × − L r = (1 . ± . × − (35) L r = ( − . ± . × − L r = (0 . ± . × − (36) L r = (1 . ± . × − L r = (0 . ± . × − (37) L r = (0 . ± . × − . (38) All the relevant relationships between bare and physical quantities (massesand decay constants) are stated in Ref. [53].
Since we need to determine three parameters in the effectivepotential, we must choose three of the four physical quanti-ties from Eqs. (33)–(34). For the results that we present be-low, we use m π , m K , and f π . Using the one-loop χ PT ex-pression for f K , we obtain f K = 113 . MeV for the centralvalue, which is off by approximately 7% compared to the lat-tice value of f K = √ = 106 . MeV. The uncertainties in L ri , m π , m K , and f π translate into uncertainties in the param-eters B m , B m s , and f . It turns out that the uncertainties inthese parameters in the three-flavor case are completely domi-nated by the uncertainties in the LECs. In the two-flavor case,they are dominated by the uncertainties in the pion mass andthe pion decay constant. Furthermore, for the lowest values ofLECs obtained using the largest uncertainties in Eq. (38), the η mass becomes imaginary and therefore unphysical. Conse-quently, we are forced to restrict the smallest value of the LECsused to ones obtained using of the total uncertainty. Wetherefore simplify the analysis and add the uncertainties. Thisyields m cen π, = 131 .
28 MeV m cen K, = 520 .
65 MeV (39) m low π, = 148 .
45 MeV m low K, = 617 .
35 MeV (40) m high π, = 115 .
93 MeV m high K, = 437 .
84 MeV (41) f cen = 75 .
16 MeV (42) f low = 79 .
88 MeV (43) f high = 70 .
44 MeV . (44)Given that the effective potential derived in three-flavor χ PT ofEq. (19) reduces to the result in two-flavor χ PT, in the limit oflight up and down quarks, it is worthwhile comparing the pre-dictions from two-flavor χ PT from Ref. [52] using the N f = 2 LECs from the literature and those obtained by using Eqs. (25)–(27). The N f = 2 LECs have the following values [56] ¯ l ( N f = 2) = − . l ( N f = 2) = 4 . (45) ¯ l ( N f = 2) = 2 . l ( N f = 2) = 4 . . (46)The three-flavor LECs L ri are the running couplings evaluatedat the scale m ρ and we use their renormalization group equa-tions to run them to the scale m π, , where the two-flavor LECs( ¯ l i ), defined in Eq. (31), are evaluated according to Eqs. (25)–(27). We then get the following central values ¯ l ( N f = 3) = 14 . l ( N f = 3) = 6 . (47) ¯ l ( N f = 3) = 4 . l ( N f = 3) = 4 . . (48)The disagreement is most significant in ¯ l which have signsthat are opposite in the two-flavor versus the three-flavor case.The differences in the other LECs are less significant but stillnon-trivial except for ¯ l . In order to evaluate the effect of thesediscrepancies on physical observables in the pion-condensed We note that it is standard practice to quote the LECs in two-flavor χ PTusing ¯ l i defined through Eq. (31). On the other hand, for three-flavor χ PT,quoting L ri at the scale µ equal to the ρ mass ( m ρ ) is standard. ONLO ( N f = ) NLO ( N f = N f = ) NLO ( N f = ) μ I / m π α g s Figure 1: α gs as a function of µ I /m π at LO (red), at NLO with two flavors(blue), NLO with three flavors (green), and NLO with two flavors and three-flavor LECs (brown). See main text for details. phase, we have generated the isospin density, pressure, and theequation of state using the two-flavor LEC values generated us-ing three-flavor LECs, which we discuss at the end of this sec-tion.The equation ∂V eff ∂α = 0 has two types of solutions. For µ I < m π , the solution is α = 0 , where it is straightforwardto show that the effective potential and therefore the thermody-namic functions are independent of µ I . We refer to this phaseas the vacuum phase, which exhibits the Silver Blaze prop-erty [58], namely that the thermodynamic functions are inde-pendent of µ I up to a critical value µ cI = m π . For µ I > m π ,we have a nonzero condensate of π + , which breaks the U (1) I symmetry of the chiral Lagrangian, and a nonzero value for α .In Fig. 1, we show the solution α gs to the equation ∂V eff ∂α = 0 as a function of µ I m π at LO. For asymptotically large values ofthe isospin chemical, α gs approaches π .We next expand the effective potential around α = 0 toobtain a Ginzburg-Landau energy functional that can be usedto determine the order of the phase transition. This expansionis valid close to the phase transition where α (cid:28) . To fourth-order, we obtain V LGeff = a ( µ I ) + a ( µ I ) α + a ( µ I ) α . (49)The vanishing of a defines the critical chemical potential µ cI .Since a = f π ( µ I − m π ) , we have µ cI = m π . The onset of Bosecondensation at µ cI = m π is an exact result. Moreover, sincethe coefficient a ( µ cI ) > , the transition to a pion-condensedphase 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 [51].In Fig. 2, we show the isospin n I divided by m π as a func-tion of µ I /m π . The red solid line is the LO result. Note thatthe LO result is the same in the two and three-flavor cases forall thermodynamic quantities. We have used the central valuesfor the low-energy constants ¯ l i in the two-flavor case to obtainthe blue dashed line as explained in Ref. [52]. The blue band At LO, the two and three-flavor results for α coincide. LONLO ( N f = ) NLO ( N f =
2, N f = ) NLO ( N f = ) Lattice0.0 0.5 1.0 1.5 2.0 2.50.00.51.01.5 μ I / m π n I / m π Figure 2: Normalized isospin density as a function of µ I /m π at LO (red), atNLO with two flavors (blue), NLO with three flavors (green), and NLO withtwo flavors and three-flavor LECs (brown). See main text for details. is obtained by including their uncertainties. The light greenband is the result of the three-flavor calculation with the mini-mum, central, and maximum values of the parameters discussedabove, while the dark green band is from using the central val-ues of L ri with uncertainties coming from the lattice parametersonly.The data points shown in Fig. 2 are from the lattice calcu-lations of Refs. [15, 16, 17]. The two-flavor band is very smallcompared to the three-flavor band reflecting the large uncer-tainty in the three-flavor L ri s. The central line in the three-flavorcase is in very good agreement with lattice data up to approx-imately µ I ∼ MeV. After this, the curve overshoots andfor larger values the two-flavor central curve is in much betteragreement with lattice data.
LONLO ( N f = ) NLO ( N f = N f = ) NLO ( N f = ) Lattice1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.40.00.51.01.5 μ I / m π P / m π Figure 3: Pressure normalized by m π as a function of µ I /m π at LO (red), atNLO with two flavors (blue), NLO with three flavors (green), and NLO withtwo flavors and three-flavor LECs (brown). See main text for details. In Fig. 3, we show the pressure P divided by m π as a func-tion of µ I /m π . Note that we have subtracted the pressure inthe vacuum phase which is given by evaluating the negative ofEq. (19) for α = 0 . The red line is the LO result. The bluedashed line is again the result from two-flavor χ PT using thecentral values of ¯ l i , while the band is obtained by including their6ncertainties. Similarly, the dashed-dotted line corresponds tothe central values of the L ri s in the three-flavor case, while thelight green band is obtained by including their uncertainties.Finally, by including only the uncertainties from the lattice pa-rameters we obtain the much narrower dark green band. Here,the LO and the two-flavor results very close in the entire rangeand systematically slightly below the lattice data. The three-flavor curve is in very good agreement with the results of theMonte Carlo simulations up to µ I = 200 MeV, after which itoverestimates the pressure.
LONLO ( N f = ) NLO ( N f = N f = ) NLO ( N f = ) Lattice0.0 0.1 0.2 0.3 0.4 0.5 0.60.00.51.01.5 P / m π ϵ / m π Figure 4: Energy density as a function of pressure, both normalized by m π , atLO (red), at NLO with two flavors (blue), NLO with three flavors (green), andNLO with two flavors and three-flavor LECs (brown). See main text for details. In Fig. 4, we show the energy density (cid:15) divided by m π asa function of pressure P divided by m π . For all values of Pm π three-flavor χ PT overestimates the energy density compared tolattice data though for values of Pm π up to approximately . ,the discrepancy is quite small. On the other hand, two-flavor χ PT underestimates the energy density as a function of pres-sure for values of Pm π up to . . For values larger than ap-proximately . , two-flavor χ PT agrees very well with latticeresults.Given the results shown in Figs. 2, 3 and 4 above, in par-ticular the large differences between the results in two-flavorand three-flavor χ PT and the results in lattice QCD comparedto three-flavor χ PT, it is important to explain this large discrep-ancy. The naive expectation is that the loop effects from thestrange quarks in three-flavor χ PT are small since the effectis sub-leading in the chiral expansion. Furthermore, their ef-fects should be suppressed since strange quark masses are con-siderably larger than the masses of the up and down quarks.While this picture is correct, it ignores the significant differ-ences between the low energy constants of two-flavor χ PT andthe ones that are extracted from three-flavor χ PT after integrat-ing out the effect of the strange quarks. We list the values inEqs. (45) and (47) noting significant discrepancies between thetwo sets. In each of the figures (2, 3 and 4), we incorporatean additional result in two-flavor χ PT using three-flavor LECsshown using brown and dashed lines. We note that even two-flavor χ PT using three-flavor LECs overestimates the isospindensity, pressure and the energy density compared to lattice QCD results. For isospin chemical potential near the secondorder phase transition up to approximately µ I m π ∼ . , the dif-ferences in the LECs fully explains the discrepancy. For largervalues of isospin chemical potential, the role of strange quarkloops becomes more significant – our results suggests that theyhave a negative effect on the pressure and isospin density com-pared to the effects of the up and down quarks.
4. Acknowledgments
The authors would like to thank B. Brandt, G. Endr˝odi, andS. Schmalzbauer for useful discussions as well as for providingthe data points of Ref. [54].
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