condensation in a superfluid

Tom´aˇsBrauner1, ∗ and Xu-Guang Huang2, † 1Faculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway 2Physics Department and Center for Physics and Field Theory, Fudan University, Shanghai 200433, China We revisit the suggestion that charged ρ-mesons undergo Bose-Einstein condensation in - rich nuclear . Using a simple version of the Nambu-Jona-Lasinio (NJL) model, we conclude that ρ-meson condensation is either avoided or postponed to isospin chemical potentials much higher than the ρ-meson as a consequence of the repulsive interaction with the preformed pion con- densate. In order to support our numerical results, we work out a linear sigma model for and ρ-mesons, showing that the two models lead to similar patterns of medium dependence of meson . As a byproduct, we analyze in detail the mapping between the NJL model and the linear sigma model, focusing on conditions that must be satisfied for a quantitative agreement between the models.

PACS numbers: 12.38.Aw, 12.39.Fe, 14.40.Be Keywords: Chiral Lagrangians, Bose-Einstein condensation, ρ-meson

I. INTRODUCTION isospin symmetry is spontaneously broken in this phase and hence the ρ-meson mass no longer depends linearly The phase structure of nuclear and matter at low on the chemical potential. Unfortunately, the issue can- temperatures and high densities remains one of the ma- not be settled within the model-independent approach of jor unresolved problems in contemporary physics. While chiral perturbation theory (χPT) [5] since it addresses direct Monte Carlo simulations of Quantum ChromoDy- physics outside of χPT’s range of validity. A holographic namics (QCD) at high density are inhibited by model for QCD at nonzero isospin density was proposed the notorious sign problem, this does not affect QCD at in Ref. [6], concluding that, indeed, ρ-mesons condense nonzero isospin density [1]. Although the latter is not for sufficiently high values of µI (µI & 1.7mρ in their entirely physical, since a medium with nonzero isospin setup). This work is, nevertheless, based on QCD in the but zero net baryon density does not seem to exist in na- large-Nc limit and in the chiral limit, where the theory ture, the study of QCD under such conditions may shed possesses a continuous axial U(1) symmetry. some light on ordinary nuclear matter. A preliminary lattice study of QCD with two flavors The fact that the lightest carrying isospin, the of Wilson with focus on the meson spectrum pion, is a pseudo-Nambu-Goldstone (NG) of spon- was carried out in Ref. [7]. It was found that at suf- taneously broken chiral symmetry, allows to draw rigor- ficiently high values of µI, the charged ρ-meson mass ous conclusions about the QCD phase diagram at nonzero drops, within the numerical accuracy, to zero, which sug- isospin density. At zero temperature, charged pions un- gests BEC. However, their results are at odds with other dergo Bose-Einstein condensation (BEC) once the isospin model-independent predictions of χPT. The latter im- chemical potential µI exceeds the pion mass, mπ [2]. In plies that within the pion superfluid phase, the mass of the opposite extreme of very high isospin chemical po- the neutral pion equals µI. In fact, this is now understood tential, one expects a crossover from meson-dominated to be an exact consequence of spontaneous breaking of to quark-dominated matter. Such high-isospin-density isospin by the charged pion condensate: the neutral pion matter is characterized by a first-order confinement- is the massive NG boson of the isospin symmetry [8]. deconfinement transition at temperatures much lower This is in contrast to the findings of Ref. [7], according

arXiv:1610.00426v2 [hep-ph] 5 Nov 2016 than the QCD scale [1,3]. What happens at interme- to which the neutral pion mass starts to decrease with diate isospin densities remains unclear though. chemical potential upon the onset of BEC at µI = mπ. Since the next-to-lightest particle in terms of the mass- Motivated by these works, the objective of the present to-isospin ratio is the ρ-meson, it was suggested early on paper is to provide new insight into the question whether that at sufficiently high µI, charged ρ-mesons will un- ρ-mesons condense at sufficiently high isospin chemical dergo BEC as well [4]. This would have far-reaching con- potential. Our primary tool is the Nambu-Jona-Lasinio sequences for the structure of isospin-rich nuclear matter (NJL) model. Being merely a model, this, of course, does as it would, among others, imply spontaneous breaking not allow us to make rigorous conclusions about the QCD of rotational invariance. However, the fate of ρ-mesons phase diagram. Still, the NJL model is known to work in the pion superfluid phase is not that obvious, since reasonably for vacuum physics [9] and its advantage is a relatively low number of free parameters, all of which can be fixed by a fit to vacuum observables. The analysis is carried out in Sec.II, starting with a detailed discus- ∗ [email protected] sion of the condensates in the pion superfluid phase, and † [email protected] proceeding to the meson spectrum therein. Treating the 2 vacuum ρ-meson mass as a free parameter, we show that where the modified Dirac operator of the theory reads if present at all, ρ-meson BEC is postponed to chemical ~ potentials far beyond the scope of the NJL model. iD iD/ m σ iγ5~π ~τ + ~ρ/ ~τ + a/ ~τγ5. (3) ≡ − − − · · · In order to gain deeper insight into the of this result, we complement the NJL analysis by one using Integrating out the quark degrees of freedom then leads the linear sigma model in Sec.III. It turns out that the to the fully bosonized effective action repulsive interaction between pions and ρ-mesons results Z 2 2 2 2 ! 4 σ + ~π ~ρµ + ~aµ in postponing the onset of ρ-meson BEC. While the latter Seff = d x + i Tr log D. (4) always occurs in the linear sigma model, for arbitrary − 4G 4GV − model parameters and sufficiently high µ , it can indeed I This effective action is the starting point for the calcula- be pushed to values µI mρ. Apart from our main results, some aspects of the pre- tion of both the thermodynamic potential (and hence the sented work might be useful in a broader context for those phase structure of the model) and the meson spectrum. working on effective model description of quark matter. First, to the best of our knowledge, this is the first time A. Vacuum physics that the condensates of the temporal vector and axial vector field in the pion superfluid phase have been cal- culated self-consistently using the NJL model. We show In the QCD vacuum, isospin is not spontaneously bro- how the results obtained in the NJL model match the pre- ken [11]. Hence only the σ field can acquire a vacuum dictions of χPT. For the reader’s convenience, the χPT expectation value, which is found self-consistently from analysis is summarized in AppendixA. Second, we un- the gap equation δSeff/δσ = 0. After a simple manipula- dertake a systematic study of the mapping between the tion, the gap equation can be brought to the form NJL model and the linear sigma model. To what extent σ = 16GN MI , (5) the predictions of the NJL model and the matched linear h i c 1 sigma model agree is discussed in AppendixC. where I1 denotes the tadpole momentum integral, Z d4k 1 I i , (6) II. NAMBU-JONA-LASINIO MODEL ANALYSIS 1 ≡ (2π)4 k2 M 2 − The basic degrees of freedom of the NJL model are the and we explicitly highlighted the dependence on the num- ber of colors, N = 3. In both expressions, M m + σ . Their detailed dynamics depends on the choice c ≡ h i of interaction which is to some extent arbitrary. We want denotes the dynamical (constituent) quark mass. While to take advantage of the simplicity of the mean-field ap- the vacuum expectation value of σ is an observable spe- proximation, hence we must include interaction channels cific to the present NJL model, it can be directly related with the quantum numbers of the pions and ρ-mesons. to the expectation value of the scalar operator composed A minimal NJL-type model that fulfills this requirement of the quark fields, and at the same time respects the full chiral symmetry of δSeff σ QCD is the two-flavor model defined by the Lagrangian ψψ¯ = = h i, (7) h i − δm − 2G  2 2 L = ψ¯(iD/ m)ψ + G (ψψ¯ ) + (ψ¯iγ ~τψ) − 5 where the gap equation for σ has already been used.  2 2 (1) h i GV (ψγ¯ µ~τψ) + (ψγ¯ µγ5~τψ) . This, together with Eq. (5), constitutes one of the match- − ing relations used to fix our model parameters. (A similar model was used in Ref. [10] to study ρ-meson The other observables used to fix the model parameters condensation in the QCD vacuum in a magnetic field.) are related to the meson spectrum in the vacuum. This Here ψ is the isospin-doublet quark field and ~τ the Pauli can be extracted from the meson polarization functions, matrices in isospin space. The covariant derivative in- defined by the second functional derivative of the effective τ3 cludes the isospin chemical potential via D0 ∂0 iµI 2 . action (4), symbolically The model is defined completely by four parameters:≡ − the 2 current quark mass m (assumed for simplicity to be the δ Seff χ(AB)(x y) , (8) same for both quark flavors), the scalar and vector cou- − ≡ δφA(x)δφB(y) pling G and GV , and the ultraviolet cutoff Λ. We follow the usual bosonization procedure wherein where φA runs over all the meson fields of the model and the four-point interaction is decoupled by introducing a the multi-index includes the field type as well as the isospin and LorentzA indices. As a consequence of conser- set of collective bosonic fields, σ, ~π, ~ρµ and ~aµ, and sub- sequently carrying out the Hubbard-Stratonovich trans- vation of isospin and [12] in the QCD vacuum, the formation. This brings the Lagrangian (1) to the form polarization function matrix will take a block-diagonal form. The simplest block is that of the sole isospin sin- 2 2 2 2 σ + ~π ~ρµ + ~aµ glet of the model, σ. We will not discuss it in detail here L = + + ψ¯iDψ, (2) − 4G 4GV as it is of no interest for our purposes though. 3

The other meson modes are all isospin triplets. The po- field ~ρµ does not describe a propagating degree of free- larization function must clearly be diagonal in the isospin dom. The three physical degrees of freedom of the mas- space. Denoting isospin indices as a, b, it can be writ- sive vector meson are all contained in the transverse part, (AB) (AB) ten as χab (p) χ (p)δab upon Fourier transform ≡ T 2 1 16 8 2 2 2 to momentum space, where the multi-index A now only χ(ρ)(p ) = NcI1 Nc(p +2M )I2(p ). (13) labels the field type and the Lorentz index. The polar- 2GV − 3 − 3 (AB) ization functions χ can in turn be obtained by eval- This equation can be used to determine the ρ-meson mass uating the right-hand side of Eq. (8), which is equivalent in the vacuum once all the parameters of the model are to a one-loop integral with two external meson legs, known. Z 4 The case of the pion and the axial vector meson is (AB) (AB) d k somewhat more complicated [13, 14]. Namely, the pion χ (p) = 2G + 2iNc 4 (2π) and the longitudinal component of the ~aµ field carry the  (A) (B)  (9) trD Γ (k/ + p/ + M)Γ (k/ + M) same quantum numbers and thus can mix. The polar- . × (k + p)2 M 2(k2 M 2) ization function in this sector forms a nontrivial 2 2 − − matrix, represented by the components × (AB) Here G is a constant diagonal matrix defined by (ππ) 1 2 2 (ππ) (ρρ) (aa) χ (p) = + 8NcI1 + 4Ncp I2(p ), G = 1/(4G) and Gµν = Gµν = gµν /(4GV ). Also, −2G − the matrices Γ(A) carry information about quantum num- (πa) 2 χµ (p) = 8ipµNcMI2(p ), (14) (π) (ρ) bers of the modes, and read Γ = iγ5,Γµ = γµ and 1 (a) − L 2 2 2 χ(a)(p ) = + 16NcM I2(p ). Γµ = γµγ5. The trace in Eq. (9) is to be carried out 2GV over the− space of Dirac matrices only. Upon some alge- braic manipulation, all the polarization functions can be The transverse part of ~aµ is characterized by the polar- expressed in terms of two basic momentum integrals: the ization function, defined similarly to Eq. (12), tadpole I1 (6) and the one-loop integral 1 16 8 χT (p2) = N I N (p2 4M 2)I (p2), (15) Z 4 (a) 2G − 3 c 1 − 3 c − 2 2 d k 1 V I2(p ) i . (10) ≡ − (2π)4 (k + p)2 M 2(k2 M 2) − − which in principle allows to determine the mass of the axial vector meson (the a1-meson). Note that the trans- Since we are going to regulate the divergent integrals verse and longitudinal polarization functions of the vec- using a sharp three-momentum cutoff, we give here also tor and axial vector meson satisfy the sum rule [13] suitable expressions for both integrals after the frequency integration has been carried out, χT (p2) χT (p2) = χL (p2) χL (p2). (16) (a) − (ρ) (a) − (ρ) Z d3k 1 We are now ready to put together the set of equations I1 = 3 , (2π) 2k that can be used to fix the parameters of our model. (11) As already mentioned above, there are altogether four Z d3k 1 2 parameters: m, G, G and the cutoff Λ. We therefore I2(p ) = 3 2 2 , V (2π) k(4 p ) k − need four observables on input, and it is most common to choose three of these as the chiral condensate ψψ¯ , pion where  √k2 + M 2 is the dis- h i k mass mπ and decay constant fπ. The fourth one is nat- persion relation≡ in the vacuum. urally provided by the ρ-meson mass mρ. However, for We next focus on the ρ-meson polarization function. the sake of convenience, we shall trade the chiral conden- Being a symmetric rank-two tensor, this can be decom- sate for the constituent quark mass M. When combined posed into a longitudinal and transverse part, with mπ and mρ, this has the advantage of giving us a   direct control over the threshold for the (unphysical) de- (ρρ) L 2 pµpν T 2 pµpν cay of the ρ-meson into a quark-antiquark pair, ρ qq¯, χµν (p) χ (p ) +χ (p ) gµν . (12) ≡ (ρ) p2 (ρ) − p2 as well as over the threshold for the (physical but absent→ at mean-field level) two-pion decay, ρ ππ. All four L 2 → The longitudinal piece is constant, χ(ρ)(p ) = 1/(2GV ), equations needed for parameter fixing can thus be put reflecting the fact that the longitudinal component of the together in the following compact form, 4

σ = 16GN MI (chiral condensate), h i c 1 2 2 m 4NcmπI2(mπ) = 2 2 (pion mass), 2GM 1 + 32GV NcM I2(mπ) (17) 2 m σ fπ = h i2 (pion decay constant), 2Gmπ

1 σ 8 2 2 2 h i = Nc(mρ + 2M )I2(mρ)(ρ-meson mass). 2GV − 3GM 3

The condition for the pion mass was obtained from the The unbroken symmetries of the pion superfluid phase mixing matrix (14) by integrating out the longitudinal consist of spacetime translations, spatial rotations and 1,2 component of ~aµ, and setting the resulting effective pion the modified parity. Apart from π , condensates of polarization function to zero. Also, both here and in the temporal vector fields are alsoh consistenti with these the condition for the ρ-meson mass, the gap equation (5) symmetries and therefore have to be included in our anal- 3 was used to simplify the expressions. This makes it clear ysis, namely the neutral vector meson condensate ρ0 1,2h i that the pion becomes massless in the chiral limit where and the charged axial vector meson condensate a0 . m 0. Finally, we note that the condition for the pion From the effective action (4), we infer the mean-fieldh i → decay constant is not a result of an actual NJL model thermodynamic potential density Ω. Remarkably, even calculation; for the sake of simplicity, we used instead the in presence of the pion and vector condensates, the de- Gell-Mann-Oakes-Renner relation [9] rewritten in terms terminant of the Dirac operator (3) can still be factorized of σ using Eq. (7). This is expected to give an accurate analytically in terms of quasiparticle dispersion relations h i ± approximation to fπ as long as the pion mass is well below Ek , given by the characteristic scale of spontaneous chiral symmetry ± 2 2 2 2 2 2 breaking in the QCD vacuum. (Ek ) = k + M +µ ˜ + ~π + ~a As we show below, the tendency towards ρ-meson BEC p (20) 2 (Mµ˜ ~π ~a)2 + k2(˜µ2 + ~a2). depends very sensitively on the vacuum mass mρ. Taking ± − × its physical value, m 775 MeV, would require us to ρ Here ~π ~a is understood as π1a2 π2a1 andµ ˜ µ /2+ρ, go to a range of µ way≈ beyond the scope of the NJL 0 0 I I where ρ×denotes the vector mean− field ρ3. Likewise,≡ the model. We therefore choose a different approach. Unless 0 Lorentz index on the axial vector mean field ~a is dropped explicitly stated otherwise, we will use the following set, 0 and it is denoted simply by ~a, in addition to the pion M = 300 MeV, mπ = 140 MeV, fπ = 92.4 MeV. mean field ~π. The mean-field thermodynamic potential (18) density then acquires the usual form, consisting of a con- For illustration purposes, we tune the ρ-meson mass to densate contribution and a fermionic quasiparticle con- an artificially low value, mρ = 500 MeV. This results, by tribution, means of Eq. (17), in the parameter set 2 2 2 2 Z 3 −2 Ω σ + ~π ~ρ + ~a X d k G = 2.92 GeV , Λ = 817 MeV, = 2N (19) V 4G − 4G − c (2π)3 −2 V s=± GV = 3.12 GeV , m = 3.30 MeV. (21) h s −βEs i E + 2T log 1 + e k . × k B. Pion superfluid phase: condensates How do we in practice determine the expectation val- ues of the mean fields? A straightforward solution would The presence of the isospin chemical potential µI ex- be to simultaneously solve a set of gap equations, ob- plicitly breaks the SU(2) isospin symmetry down to its tained by differentiating the thermodynamic potential Abelian subgroup U(1) , generated by τ . When µ > I3 3 I with respect to the mean fields. However, this may not mπ, charged pions undergo BEC, which can be repre- be the most convenient way in case there are several solu- 1,2 sented by nonzero expectation value of (one of) the π tions; one needs to compare their free energies in order to fields. Such a condensate further breaks the remaining determine the actual equilibrium state. A common alter- continuous isospin symmetry. At the same time, the pion native is to look for the absolute minimum of the thermo- condensate spontaneously breaks parity. However, there dynamic potential. Here we have to exercise some care is a discrete Z2 symmetry which remains intact, gener- though. The vector mean fields effectively play the role ated by simultaneous parity transformation and a U(1) I3 of Lagrange multipliers for the corresponding charge den- τ iπ 3 rotation by 180 degrees, e 2 = iτ3. Although we cannot sities, and the thermodynamic potential is thus a nega- prove rigorously that such modified parity is not sponta- tive definite function of their deviations from equilibrium. neously broken in the pion superfluid phase, we take this (Its second derivatives measure, up to a sign, the fluctu- as a plausible starting point [15]. ations of the charge densities.) The way out is to first 5

agreement with χPT regarding the results for the vector 300 condensates, see Eq. (A8) in AppendixA. The predic- tions of χPT and their comparison to the NJL model do not involve any free parameters, and the agreement there- fore confirms the consistency of our approach, including 200 the mixing of pions with axial vector mesons. [MeV] i π h , i C. Pion superfluid phase: mass spectrum σ h 100 There are two types of excitations that our mean- field analysis allows us to address: elementary fermionic (quark) and collective bosonic (meson). The dispersion 0 0 relation of the fermionic is fully determined by the values of the condensates and Eq. (20). On the microscopic level, the formation of the pion condensate 10 corresponds to pairing of u-quarks and d-antiquarks, at − least at high isospin density where a Fermi sea of quarks is formed. It is therefore instructive to find the correspond- 20 ing gap in the quasiparticle dispersion relation, ∆ [17]. [MeV] − −

i This is determined by minimization of Ek with respect a

h to momentum, and a short calculation yields , 30 i

ρ − h µ˜ ~π M ~a ∆ = p| | − | | . (22) 40 µ˜2 + ~a2 − Obviously, due to the presence of the axial vector con- 50 densate ~a, the quasiparticle gap is not given by the mag- − 0.0 0.5 1.0 1.5 2.0 2.5 3.0 nitude of the pion condensate ~π as usual. To assess the µI/mπ difference of ∆ and ~π more quantitatively, we note that for our parameter set,| | the vector condensates are numer- FIG. 1. The values of various condensates at T = 0 as a func- ically much smaller than the chemical potential, hence to tion of the isospin chemical potential. Top panel: σ (dashed) first order in ~a and ~ρ we can write and π (solid). Bottom panel: ρ (solid) and a (dashed). For  M ~a  comparison, we show the results both of the NJL model (thick ∆ ~π 1 | | . (23) ≈ | | − µ ~π lines) and of χPT (thin lines). The dimension-one conden- | | sates ρ and a are related to the χPT prediction (A8) by a Using the χPT prediction for the condensates, Eqs. (A6) factor of 4GV . The numerical results were obtained with the and (A8), we then get the approximate expression parameter set (19).  8G f 2  ∆ ~π 1 V π , (24) ≈ | | − x4 solve the set of gap equations for ~ρ and ~a, treating them where x µI/mπ. For the parameter set (19), we have as constraints on the charge densities. Once the solution 2 ≡ 8GV fπ 0.21. The crossover from the BEC phase to the is plugged back to Ω, it can be subsequently minimized Fermi sea≈ of quarks occurs roughly at x 1.6 [18], and with respect to σ and ~π as usual. at this point the fermionic quasiparticle≈ gap is reduced The analysis is further simplified by noting that the just by about 3% compared to the pion condensate ~π. axial vector condensate is necessarily orthogonal to the Moreover, the two rapidly converge to each other as the pion condensate in the isospin space. This is easy to un- chemical potential further increases. derstand within χPT (see AppendixA), but can also be Next we focus on the meson fluctuations. Their spec- proven directly within the NJL model (see AppendixB trum is again determined by Eq. (8). However, the func- for details). In addition, we can always use the symme- tional derivatives are now to be taken in the equilibrium try to choose the orientation of the pion condensate at state that features all the condensates discussed in the 1 3 2 will. Finding thermodynamic equilibrium therefore boils previous subsection, σ , π , ρ0 and a0 . The ex- down to determining the values of four mean fields, cho- plicit expression for theh i isospin-singleth i h i polarizationh i func- 1 3 2 sen without loss of generality as σ, π , ρ0 and a0. tion (9) then has to be appropriately generalized, The numerical results for the condensates at zero tem- Z d4k perature as a function of µ are shown in Fig.1. While (AB) (AB) I χ (p) = 2G + iNc 4 the results for the (pseudo)scalar condensates are hardly (2π) (25) surprising [16], we would like to emphasize the perfect tr Γ(A)S(k + p)Γ(B)S(k), × D,f 6

600 20

500 15 400 π

300 /m [MeV] 10 ρ I,crit m 200 µ 5 100

0 0 0123456 100 200 300 400 µI/mπ mρ [MeV]

FIG. 2. Dependence of the in-medium charged ρ-meson mass FIG. 3. Dependence of the critical isospin chemical potential, on the isospin chemical potential. The solid, dashed and dot- at which the mass of the lightest vector degree of freedom ted line corresponds respectively to the vacuum ρ-meson mass drops to zero, on the vacuum ρ-meson mass. The other phys- of 250 MeV, 400 MeV, and 550 MeV. In each case, the pa- ical input variables were fixed according to Eq. (18). rameters of the model were adjusted in order to keep the other physical variables, listed in Eq. (18), fixed. modes that are of particular interest to us, that is, pions and ρ-mesons. First of all, as already said above, one of where the trace is now to be taken over the Dirac and the charged pions becomes exactly massless in the pion flavor (isospin) space. The fermion propagator entering superfluid phase as a consequence of the Goldstone theo- the polarization function includes the chemical potential rem. In addition, the neutral pion acquires a mass equal and all the condensates, to µI, that is, it is the massive NG boson of the isospin SU(2) symmetry [8]. Both of these are exact properties −1 1 1 S(k) k/ + µIτ3γ0 m σ i π τ1γ5 ≡ 2 − − h i − h i (26) of the pion superfluid phase, guaranteed by the isospin 3 2 symmetry, which are moreover easy to verify numerically. + ρ τ3γ0 + a τ2γ0γ5. h 0i h 0i Finally, we focus on the lightest vector degree of free- As in the vacuum phase, the polarization function will dom, which corresponds to a mixture of the two charged take a block-diagonal form, the structure of the blocks ρ-meson fields. The dependence of its in-medium mass on being determined by the unbroken symmetry in the equi- the chemical potential is shown in Fig.2; the three curves librium. The 22 physical degrees of freedom contained in correspond to different choices of the vacuum mass mρ the fields σ, ~π, ~ρµ and ~aµ then split into the following while keeping the other physical variables (18) fixed. The sectors, differing by their transformation properties un- figure reveals two characteristic properties of the vector der spatial rotations and the modified parity: mass spectrum. First of all, just after the onset of pion BEC, the mass The σ, π1,2, ρ3, a1,2 sector: 3 propagating scalar • { 0 0 } increases as a function of µI. Note that the same behavior modes, one of which is the NG boson of the spon- was predicted using effective field theory for charged vec- taneously broken isospin U(1) invariance. I3 tor mesons in two-color QCD [20], although the evidence from direct lattice simulations of the same theory seems The π3, ρ1,2, a3 sector: 1 propagating pseudo- • { 0 0} somewhat inconclusive so far [21]. Second, at sufficiently scalar mode of mass µI, corresponding to the mas- high chemical potential, a maximum occurs and the mass sive NG boson of the isospin symmetry [8, 19]. starts dropping again. Provided that the ρ-meson is suf- 3 1,2 ficiently light in the vacuum, the in-medium mass drops The ρi , ai sector: 9 propagating vector modes. • { } to zero at some µI within the range of validity of the NJL 1,2 3 model. This is a signature of ρ-meson BEC. The ρi , ai sector: 9 propagating axial vector • modes.{ } To appreciate how high µI has to be for us to observe ρ- meson BEC, we plot in Fig.3 the value of such “critical” The detailed mass spectrum of the meson excitations has µI as a function of the vacuum mass mρ. In QCD, the ρ- to be determined numerically by finding the zeros of the meson mass cannot be tuned freely. The parameter space determinant of the polarization matrix in each sector. of the NJL model is larger than that of QCD though. While it is in principle no problem to determine the The way to think about Fig.3 is as follows. We start full meson spectrum numerically, here we focus on the in a region of the NJL model parameter space where the 7 onset of ρ-meson BEC is under theoretical control. Then thought of as having been obtained from the NJL model we tune mρ, trying to extrapolate towards the physical by integrating out quarks, dropping the scalar and axial surface within the parameter space, which matches QCD. vector degrees of freedom, and truncating the Lagrangian It is obvious that such an extrapolation cannot be done to operators of dimension four or less. To what extent reliably since already for values of mρ shown in Fig.3, the parameters of the linear sigma model can actually be the critical chemical potential lies way beyond the NJL determined by this procedure is discussed in AppendixC. model cutoff. However, we can conclude with confidence Here we will treat g and λ as free parameters. that should ρ-meson BEC occur in QCD at all, it has to set in at a value of µI much higher than the vacuum ρ-meson mass. A.

From our discussion of the NJL model, we expect that III. LINEAR SIGMA MODEL ANALYSIS once the isospin chemical potential µI exceeds the pion mass mπ, a pion condensate appears, accompanied by a In order to get some insight in the results obtained temporal vector condensate. The orientation of the pion in the previous section, we now turn to a much simpler condensate can be chosen at will; in this section we will description of the meson spectrum, based on the linear use the following notation to distinguish the condensates sigma model. While this leads to a further extension of from the dynamical fields, the parameter space, it has the great advantage of incor- 1 3 porating the physical degrees of freedom as elementary π0 π , ρ0 ρ . (29) ≡ h i ≡ h 0i excitations rather than bound states of quarks. Since the mean-field approximation in the NJL model resums all The condensates are determined by extremization of the quark loops but treats the bosonic fields as classical, the static part of the Lagrangian (27), which leads to a set tree-level approximation is the appropriate equivalent in of two coupled gap equations, the linear sigma model. λπ2 (µ + gρ )2 + m2 = 0, In order to keep our discussion as simple as possible, 0 − I 0 π 2 2 (30) we only include in the model those modes which are of g(µI + gρ0)π0 + mρρ0 = 0. most interest to us, that is, pions and ρ-mesons. This is reasonable since we expect that at sufficiently high An analytic solution of this set of equations would require chemical potentials, (some of) these will be the lightest solving a cubic equation. We choose to gain insight by degrees of freedom. Therefore, the model discussed below examining various special cases rather than by writing can be thought of as a low-energy effective theory for the down a fully general solution. lightest modes in the pion superfluid phase. As such, First, the special case g = 0 is well known. Namely, it cannot possess full chiral symmetry, not in a linear the pion and ρ-meson sectors decouple in this limit. Con- realization at least. We will instead impose the isospin sequently, there is no vector condensate, while the pion SU(2) symmetry alone. condensate acquires the usual value, To that end, we note that the determinant of the Dirac r operator (3) can be made invariant under gauged isospin µ2 m2 π = I − π . (31) SU(2) transformations, provided that the meson fields 0 λ are assigned suitable transformation rules. In this case, Now we switch on the coupling g but focus on the limit of the ~ρµ field plays the role of a gauge field. Noting that such gauge invariance is broken explicitly only by the very high chemical potential µI. Then we can neglect the quadratic, condensate part of the action (4), we consider mass term in the first of the gap equations (30), which the following linear sigma model, results in a set of equations that is easy to solve by a series expansion, 1 2 1 2 2 λ 2 2 1 µν 1 2 µ = (D ~π) m ~π (~π ) F~ F~ + m ~ρ ~ρ . 1/3 L µ π µν ρ µ 2 ! 2 −2 − 4 −4 · 2 · µImρ −1/3 (27) π0 = + (µI ), ~ ˜ ˜ ˜ ˜ g2√λ O Here Fµν ∂µ~ρν ∂ν ~ρµ + g~ρµ ~ρν is the isospin SU(2) (32) ≡ − × ˜ !1/3 field-strength tensor and Dµ~π ∂µ~π+g~ρµ ~π the covari- 2 ≡ × µI 1 λµImρ −1/3 ant derivative of the pion field. Again motivated by the ρ0 = + + (µ ). − g g g2 O I form of the Dirac operator (3), the vector meson field is shifted inside the gauge-invariant part of the Lagrangian according to The asymptotic behavior of the pion condensate as a function of µI is very different in the cases where g is ˜ 1 2 3 zero and nonzero. However, it is interesting to note that g~ρµ (gρµ, gρµ, gρµ + δµ0µI). (28) ≡ the difference disappears once the chemical potential is The shift does not affect the ρ-meson mass term in traded for the isospin density nI, defined by the deriva- Eq. (27). Altogether, the linear sigma model (27) can be tive of the static part of the Lagrangian (27) with respect 8 to µI. Namely, both Eq. (31) and Eq. (32) reduce to Inserting this back into Eq. (36), we find that just after the onset of pion BEC, the vector meson mass equals  2 1/6 nI π0 (33) 2 ≈ λ g mπ λ 2mρ − mρ+ mρ mπ + δµI 2 2 . (39) 2g mπ to the leading order in powers of nI. ≈ − λ + 2 mρ

We observe that the vector meson mass increases with µ B. Meson spectrum I for µI & mπ, just as in Fig.2, provided that the gauge coupling g is strong enough. The meson dispersion relations can be determined by Next, we shift our attention to the region of high chem- shifting the fields by the above-found condensates and ical potentials. It is not a priori obvious from Eq. (36) subsequently expanding the Lagrangian to the second or- whether or not the lightest mass mρ+ drops to zero at der in the field fluctuations. This is a completely routine sufficiently high µI, which would signal BEC. However, procedure and we therefore omit details, merely provid- it is easy to see that this indeed does happen when µI is ing a summary of the results. For the sake of simplicity so high that the following condition is satisfied [22], we set the spatial momentum to zero and just overview 2 2 the mass spectrum. (µI + gρ0) = mρ. (40) The neutral π sector: the mass of the neutral pion Solving this together with the gap equations (30), we find • is found to be µ , in accord with the prediction I that the mass of the lightest vector meson mode drops to based on the exact isospin symmetry. zero for arbitrary (finite and nonzero) values of the model The charged π sector: we find one gapless mode as parameters at a critical chemical potential determined by • expected, and one gapped mode with the mass 2  2  g mπ p 2 2 µ = m + m . (41) m − = 6(µ + gρ ) 2m , (34) I,crit ρ ρ π I 0 − π λ − mρ where ρ is determined implicitly by Eq. (30). 0 This is not so surprising, for a nonzero expectation value The neutral ρ sector: the mass of neutral ρ equals of a non-Abelian charge density (in this case isospin) is • q known to act as a chemical potential for BEC of the as- 2 2 2 mρ0 = mρ + g π0, (35) sociated gauge [23]. Eq. (41) makes it clear that the critical chemical poten- where π0 is likewise determined by Eq. (30). tial is always larger than the vacuum mass mρ (as long as 2 The charged ρ sector: we find two gapped modes mρ > mπ). Moreover, by tuning the ratio g /λ suitably, • with masses given by it can in principle be made arbitrarily large. Regarding the dependence of the linear sigma model predictions on  2 2 1 2 2 the unknown parameters g and λ, it is useful to note that m ± = m + (µ + gρ ) + g π (36) ρ ρ I 0 2 0 all the in-medium meson masses depend only on the vac- r 1/2 uum masses m and m , the chemical potential µ , and 1  π ρ I g4π4 + 2(µ + gρ )2(2m2 + g2π2) . the ratio g2/λ. Together with Eq. (41), this implies that ∓ 4 0 I 0 ρ 0 the dependence of the in-medium masses on µI within the linear sigma model is completely fixed once the values of We would now like to see to what extent this result can m , m and µ are given. reproduce the behavior we found using the NJL model, π ρ I,crit see Fig.2. We first focus on the behavior of the lightest vector mode just after the onset of pion BEC. To that IV. CONCLUSIONS AND DISCUSSION end, we define the nonrelativistic chemical potential for the charged pion as We have investigated the possibility that vector mesons δµI µI mπ. (37) undergo BEC in QCD at sufficiently high isospin density ≡ − and zero temperature. Using a combination of arguments Similarly to the asymptotic solution (32) at very high µ , I based on the NJL and the linear sigma model, we con- it is now possible to solve the gap equations (30) by a cluded that condensation of ρ-mesons is disfavored by series expansion in δµ . The result reads I the preformed condensate of charged pions. As a conse- v u quence, if present at all, ρ-meson BEC is postponed to u 2mπδµI 3/2 π0 = 2 2 + (δµI ), much higher chemical potentials than what would follow t 2g mπ O λ + m2 from a naive estimate based on the vacuum mass m . ρ (38) ρ We were able to determine the critical chemical po- δµI 2 ρ0 = 2 + (δµI ). tential for ρ-meson BEC for model parameter sets corre- − λmρ O 1 + 2 2 2g mπ sponding to artificially light ρ-mesons. Unfortunately, an 9 extrapolation to the physical ρ-meson mass is not possi- 1. Ground state at nonzero isospin density ble within our model setup since it leads far beyond the range of validity of the effective models. Ultimately, the Introducing the isospin chemical potential µI is equiv- question has to be settled by direct lattice simulation, alent to setting the background fields Lµ and Rµ to which is possible at least in principle since QCD with nonzero isospin chemical potential does not suffer from 1 L = R = δ µ τ . (A3) the sign problem [1]. ν ν 2 ν0 I 3 Finally, let us make a comment regarding the stability Likewise, the 2 2 unitary matrix Σ can always be de- of the ρ-mesons in the pion superfluid medium. While in × the vacuum, ρ-mesons decay predominantly into two pi- composed into a linear combination of the unit and Pauli matrices; the presence of the scalar and pseudoscalar con- ons (as long as mρ > 2mπ), the possible decay channels in densates σ and π1 then corresponds to the parame- the pion superfluid phase are constrained by the modified h i h i kinematics and by the modified parity invariance. As a terization consequence, a charged ρ-meson can decay into one neu- Σ = cos θ + iτ sin θ (A4) tral pion and one charged pion, but not into two charged 1 pions. (The latter is not forbidden by conservation of with coordinate-independent angle θ. This angle is found or isospin, which is spontaneously broken by minimization of the dimensionless potential by the pion condensate.) Since the mass of the neutral pion equals µI, it follows that at sufficiently high µI, the L 1 V (θ) = = x2 sin2 θ cos θ, (A5) lightest charged ρ-meson will be protected against de- 2 2 −fπmπ −2 − cay into two pions by kinematics. Barring emergence of new decay channels in the superfluid medium, we can where we introduced the dimensionless parameter x ≡ make the model-independent conclusion that the lightest µI/mπ. A pion condensate appears for x > 1 and is charged ρ-meson will become stable, hence it makes sense given implicitly by cos θ = 1/x2. This corresponds to the to speak of its BEC in the first place. vacuum expectation value r 1 1 Σ = 11 + iτ 1 , (A6) ACKNOWLEDGMENTS h i x2 1 − x4

T. B. would like to thank Denis Parganlija, Owe which determines the dependence of the scalar and pseu- Philipsen, and Dirk H. Rischke for helpful discussions doscalar condensates on µI up to an overall factor. at various stages of this project. X.-G. H. thanks Gao- qing Cao, Lianyi He, and Swagato Mukherjee for use- 2. Vector and axial vector condensates ful discussions. We also appreciate correspondence with Yoshimasa Hidaka pertinent to rigorous constraints on ρ-meson condensation. X.-G. H. is supported by the The pion condensate, defining the superfluid ground 1000 Young Talents Program of China and by NSFC with state, induces secondary condensates, admitted by the Grant No. 11535012. unbroken symmetry. These are in particular the vector and axial vector condensate. Within χPT, they can be calculated by introducing additional infinitesimal back- Appendix A: Chiral perturbation theory treatment ground fields V~µ and A~µ such that 1 1 In this appendix we briefly review how the condensates Lµ = (V~µ A~µ) ~τ, Rµ = (V~µ + A~µ) ~τ. (A7) in the pion superfluid phase, discussed in Sec.IIB, can be 2 − · 2 · addressed using χPT. The pseudo-NG boson degrees of The condensates can be obtained by taking a derivative freedom of χPT are encoded in a unitary matrix variable of the Lagrangian with respect to the backgrounds fields, Σ, in terms of which the leading-order χPT Lagrangian evaluated in the ground state. A straightforward manip- reads [1] ulation leads to 2   fπ  † µ 2 †  ∂L 1 L = tr D Σ D Σ + m (Σ + Σ ) . (A1) ¯ τa 0 2 µ π ψ 2 γ ψ a = δa3mπfπx 1 4 , 4 h i ≡ ∂V0 − x r (A8) Under a chiral transformation, defined by the left- and ∂L 1 1 ψ¯ τa γ0γ ψ = δ m f 2 1 . right-handed unitary matrices UL and UR, the field Σ 2 5 a a2 π π 4 h i ≡ ∂A0 x − x transforms as Σ U ΣU † . Therefore, it couples to → L R background matrix gauge fields Lµ and Rµ of the chiral These expressions completely fix the dependence of the group via the covariant derivative vector and axial vector condensates in the pion superfluid phase on the chemical potential. In the vacuum phase DµΣ ∂µΣ iLµΣ + iΣRµ. (A2) (x 1) both condensates are zero. ≡ − ≤ 10

Appendix B: Orthogonality of the pion and axial isospin space. This fact was used to simplify the numer- vector condensates ical calculation. In this appendix we give a proof of this claim within the NJL model. First, we write down the In Sec.IIB we claimed that the axial vector conden- gap equations for σ and ~π derived from the thermody- sate is necessarily orthogonal to the pion condensate in namic potential (21),

Z 3 " # X d k 1 Mµ˜ ~π ~a s σ 4GNc M + sµ˜ − × [1 2f(E )] = 0, 3 s p 2 2 2 2 k − (2π) Ek (Mµ˜ ~π ~a) + k (˜µ + ~a ) − s=± − × Z 3 " # 1 X d k 1 1 2 Mµ˜ ~π ~a s π 4GNc π sa − × [1 2f(E )] = 0, (B1) 3 s 0 p 2 2 2 2 k − (2π) Ek − (Mµ˜ ~π ~a) + k (˜µ + ~a ) − s=± − × Z 3 " # 2 X d k 1 2 1 Mµ˜ ~π ~a s π 4GNc π + sa − × [1 2f(E )] = 0, 3 s 0 p 2 2 2 2 k − (2π) Ek (Mµ˜ ~π ~a) + k (˜µ + ~a ) − s=± − × where f(x) 1/[1 + exp (x/T )] is the Fermi-Dirac distribution. If we introduce the abbreviations, ≡ X Z d3k 1 x 1 4GN [1 2f(Es )] , ≡ − c (2π)3 Es − k s=± k (B2) Z 3 X d k s Mµ˜ ~π ~a s y 4GNc − × [1 2f(E )] , 3 s p 2 2 2 2 k ≡ (2π) Ek (Mµ˜ ~π ~a) + k (˜µ + ~a ) − s=± − ×

the above gap equations can be cast as The derivative expansion of the logarithm of the Dirac operator D is most conveniently carried out using the xM m yµ˜ = 0, method of covariant symbols [24]. First we drop the − − π1 a2  x (B3) scalar and axial vector fields in order to keep just the 0 = 0, π2 a1 y field content of the model analyzed in Sec.III. Using − 0 the fact that the determinant of the Dirac operator D is gauge-invariant, we can actually keep the pion field only; From the first line in Eq. (B3) we can conclude that x the dependence on ~ρ can subsequently be restored by and y cannot vanish simultaneously as long as the current µ imposing the gauge invariance. This leads to the linear quark mass m is nonzero. Then the second line requires sigma model Lagrangian the determinant of the coefficient matrix to be zero which 2 2 2 2 implies ~π ~a = 0. This orthogonality of the pion and axial L = d1(Dµ~π) + d2~π + d3(~π ) vector condensates· in the isospin space is consistent with (C1) + d F~ F~ µν + d ~ρ ~ρµ, the expectation from the χPT. 4 µν · 5 µ · where the effective couplings are given by 1 Appendix C: Matching the NJL and linear sigma d = 2N I , d = 2N I , d = , 1 c 2 3 − c 2 5 4G models V (C2) 1 Nc 1 2 d2 = + 4NcI1, d4 = (I2 + 2 M I3). The linear sigma model analyzed in Sec.III was moti- −4G − 6 vated by the NJL model, yet its couplings were treated The coefficients In stand for a set of Euclidean loop in- as free parameters. In this appendix we wish to take a tegrals, generalizing Eqs. (6) and (10), closer look at the extent to which the linear sigma model Z 4 parameters can be determined from the NJL model. d kE 1 In 4 2 2 n . (C3) As far as the correlators of the meson interpolating ≡ (2π) (kE + M ) fields are concerned, the bosonized action (4) is equiva- Those with n 2 are ultraviolet divergent and thus re- lent to the original NJL model (1). The corresponding quire and explicit≤ cutoff. Integration by parts shows that approximate linear sigma model can then be obtained by the integrals satisfy the recursive relation expanding the action in powers of the meson fields and   their derivatives, and keeping only operators with canon- 2 In I = 1 . (C4) ical dimension four or less. n+1 − n M 2 11

800 the original NJL model already in the vacuum, especially 700 regarding the ρ-meson mass. It is therefore hopeless to use such a model to gain a quantitative insight into the 600 spectrum of mesons in the pion superfluid phase. The source of the problem is twofold. First, the cou- 500 plings di of the Lagrangian (C1) suffer from an ambi- guity which arises from the ultraviolet divergence of the

[MeV] 400

π momentum integrals involved. This can be seen already

m 300 from the fact that the coupling d4 differs from what one 200 would expect based on the transverse part of the ρ-meson polarization tensor (13). As a matter of fact, naive appli- 100 cation of the recursive relation (C4) would suggest that I3 = 0, which would remove the discrepancy. Of course, 0 the integral I3 itself is ultraviolet-finite and by its defini- 1000 tion (C3) nonzero and positive. This is not a problem of the method of covariant symbols or even of the effective 900 Lagrangian (C1). Namely the same ambiguity affects the meson polarization functions in the NJL model; it is well 800 known that the ultraviolet cutoff is a part of the defini- tion of the model. 700 The second problem is the truncation of the expansion [MeV]

ρ in powers of derivatives, wherein only the leading, two- m 600 derivative kinetic term is kept. To study the effect of this truncation, we consider the pole conditions for the pion and ρ-meson mass, given in Eq. (17). Modulo the am- 500 biguity just discussed, expanding the meson polarization functions to first order in squared momentum and solv- 400 ing for the pole should give the same mass spectrum as 0 2 4 6 8 10 1/2 the linear sigma model (C1). We can then subsequently √m0 [MeV ] improve upon the linear sigma model by adding higher order terms to the series expansion of the meson polar- FIG. 4. The pion and ρ-meson vacuum mass as a function of ization functions. (This is equivalent to adding bilinear the current quark mass. The solid lines denote the pole masses terms with four or more derivatives to the linear sigma obtained from Eq. (17). The dotted, dashed, dash-dotted and model Lagrangian.) dotted lines correspond to the approximate masses, obtained by expanding the pion and ρ-meson polarization functions to The result of this procedure is shown in Fig.4. Ob- the first, second, third and fourth order in squared momen- viously, the linear sigma model gives an accurate predic- tum, respectively. The numerical results were obtained with tion for the pion mass unless the quarks are very heavy. all model parameters but m fixed according to Eq. (19). Moreover, the upper plot in Fig.4 confirms numerically the Gell-Mann-Oakes-Renner relation: the squared pion mass is proportional to the current quark mass. However, Once we know the coefficients in Eq. (C1) in terms of the prediction for the ρ-meson mass is much worse. This the NJL model couplings, it is a matter of a simple field is natural since the ρ-meson is heavier than the pion and redefinition to match this Lagrangian to that of Eq. (27), thus the series expansion of its polarization tensor con- leading to the matching relations verges more slowly at the pole. While the values shown in the lower plot of Fig.4 exhibit a still relatively accept- d d m2 = 2 , m2 = 5 , able error of roughly 20%, for the physical ρ-meson mass π −d ρ −8d the matched linear sigma model clearly cannot be taken 1 4 (C5) d 1 quantitatively seriously. This is the reason why in the λ = 3 , g = . 2 discussion in Sec.III, we prefer to treat the couplings λ −d1 2√ d4 − and g as free parameters; using Eq. (C5) instead might Unfortunately, it turns out that the predictions of such give a false impression of having the linear sigma model a matched linear sigma model can be far from those of predictions under numerical control.

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