Fluctuations in cool quark matter and the phase diagram of Quantum Chromodynamics

We consider the phase diagram of hadronic matter as a function of temperature, T , and baryon chemical potential, mu. Currently the dominant paradigm is a line of first order transitions which ends at a critical endpoint. In this work we suggest that spatially inhomogenous phases are a generic feature of the hadronic phase diagram at nonzero mu and low T . Familiar examples are pion and kaon condensates. At higher densities, we argue that these condensates connect onto chiral spirals in a quarkyonic regime. Both of these phases exhibit the spontaneous breaking of a global U(1) symmetry and quasi-long range order, analogous to smectic liquid crystals. We argue that there is a continuous line of first order transitions which separate spatially inhomogenous from homogenous phases, where the latter can be either a hadronic phase or a quark-gluon plasma. While mean field theory predicts that there is a Lifshitz point along this line of first order transitions, in three spatial dimensions strong infrared fluctuations wash out any Lifshitz point. Using known results from inhomogenous polymers, we suggest that instead there is a Lifshitz regime. Non-perturbative effects are large in this regime, where the momentum dependent terms for the propagators of pions and associated modes are dominated not by terms quadratic in momenta, but quartic. Fluctuations in a Lifshitz regime may be directly relevant to the collisions of heavy ions at (relatively) low energies, sqrt(s)/A : 1 to 20 GeV.


I. INTRODUCTION
The phases of Quantum Chromodynamics (QCD), as a function of temperature, T , and the baryon (or equivalently, quark) chemical potential, µ, are of fundamental interest [1].
At zero chemical potential, numerical simulations on the lattice indicate that there is no true phase transition, just a crossover, albeit one where the degrees of freedom increase dramatically [2][3][4]. At a nonzero chemical potential, however, a crossover line may meet a line of first order transitions at a critical endpoint [5][6][7][8][9][10]. These lines separate two phases: a hadronic phase in which chiral symmetry is spontaneously broken, and a (nearly) chirally symmetric phase of quarks and gluons.
At densities between a dense hadronic phase and deconfined quarks, cool quark matter is quarkyonic: while the pressure is (approximately) perturbative, the excitations near the Fermi surface are confined [102]. The Fermi surface of quarks, which starts out as isotropic, breaks up into a set of patches, with the longitudinal fluctuations governed by a Wess-Zumino-Novikov-Witten Lagrangian [103][104][105]. The transverse fluctuations, however, are of higher order in momenta: they are not quadratic, but quartic. Among condensed matter systems the closest analogy are smectic liquid crystals, which consist of elongated molecules periodically ordered in just one direction [13]. The color and flavor quantum numbers which quarks carry makes the analogous state more involved. As for pion/kaon condensates [34][35][36][37][38][39][97][98][99][100], at nonzero temperature there are long range correlations in the inhomogeneous phase, and there is no true order parameter. At nonzero temperature in 3+1 dimensions, the fluctuations exhibit complicated patterns. The propagators of two-quark operators decay exponentially with a temperature dependent correlation length, and propagators of spin and flavor singlet operators, composed of 2N f quarks, fall off as a power law.
We then consider how a quarkyonic phase matches onto the usual hadronic phase at low temperature and density. We first discuss how as the density decreases, the chiral spirals in a quarkyonic phase transform naturally into the pion/kaon condensates of hadronic nuclear matter. We show that the phase diagram valid in mean field theory -two lines of second order phase transitions, meeting a line of first order transitions at a Lifshitz point -is dramatically altered by fluctuations. As demonstrated first by Brazovskii [89,[107][108][109], the line of transitions between the symmetric phase and that with chiral spirals becomes a line of first order transitions. Most importantly, the infrared fluctuations about a Lifshitz point are so strong that there is, in fact, no true Lifshitz point [15][16][17][23][24][25]. This is known to occur in inhomogenous polymers, both from experiment and numerical simulations [18][19][20][21][22].
We suggest that in QCD, infrared fluctuations also wipe out the Lifshitz point, leaving just a line of first order transitions separating the region with inhomogeneous phases from those without. What remains is a Lifshitz regime, which is illustrated in Figs. 2 and 3 below.
The Lifshitz regime is manifestly non-perturbative, as the momentum dependence for the propagators for pions (or the associated modes of a chiral spiral) are dominated by terms quartic, instead of quadratic, in the momenta. This change in the momentum dependence generates large fluctuations, which may be related to known [106,[110][111][112][113][114][115] and possible [116,117] anomalies in the collisions of heavy ions at center of mass energies per nucleon √ s/A : 1 → 20 GeV.

II. THE MODEL OF QUARKYONIC PHASE
A quarkyonic phase only exists when quark excitations near the Fermi surface are (effectively) confined. While for three colors numerical simulations of lattice gauge theories at nonzero quark density are afflicted by the sign problem for three colors, they are possible for two colors [118][119][120][121][122][123]. Although the original argument for a quarkyonic phase was based upon the limit of a large number of colors [102], these simulations show that even for two colors, the expectation value of the Polyakov loop is small, indicating confinment, up to large values of the quark chemical potential [118][119][120][121][122][123]. Directly relevant to our analysis are the results of Bornyakov et al., who find that the string tension decreases gradually from its value in vacuum, to essentially zero at µ q ∼ 750 MeV: see Fig. 3 of Ref. [123]. This suggests that a quarkyonic phase may dominate for a wide range of chemical potential, and indeed, for all values relevant to hadronic stars [124,125].
There is an elementary argument for why there could be such a large region in which cool quark matter is quarkyonic. Consider computing a scattering process in vacuum. By asymptotic freedom, this is certainly valid at large momentum. As the momentum decreases, non-perturbative effects enter, and a perturbative computation is invalid. This certainly occurs by momenta Λ scat ∼ 1 GeV, if not before.
Now consider computing the pressure perturbatively. The scattering processes which contribute involve the scattering of quarks and holes with momenta whose magnitude is on the order of the Fermi momentum. This suggests that the pressure can be computed perturbatively down to a scale which is typical of that where perturbative computations are valid: that is, on the order of Λ scat ∼ 1 GeV. This argument is clearly qualitative: the estimate of Λ scat as applying to the perturbative computation of the pressure could well vary by a factor of two. Furthermore, our argument applies only to the pressure: excitations about the Fermi surface involve much smaller energies than the chemical potential, so that one expects a transition from the perturbative, to the quarkyonic, regime [102]. What is important is that the momentum scale Λ scat does not depend strongly upon the number of colors. This elementary argument may explain why there is a large quarkyonic regime even for two colors [118][119][120][121][122][123].
This estimate differs from that at zero quark density and a nonzero temperature, T . In computing the pressure at T = 0 and µ q = 0, the dominant momentum scale is naively the first Matsubara frequency, = 2πT [126,127]. Detailed computations to two loop order [128,129] show that the precise value is a bit larger, but this estimate is qualitatively correct.
At temperatures ∼ 150 MeV, this is about ∼ 1 GeV, which is the same as we estimate for T = 0 and µ q = 0.
We note that a quantitative measure of what momentum scale perturbative computations of the pressure are valid will be provided by computations to high order, such as ∼ g 6 , as are currently underway [130,131].
Returning to the quarkyonic phase, previously we assumed that the chiral symmetry was restored [103][104][105]. However, for the reasons of convenience we will work with the nonrelativistic limit of the quark Hamiltonian which is formally justified if the the constituent quark mass is nonzero. At nonzero quark mass, we work with the nonrelativistic limit of the quark Hamiltonian: where We assume that there is single gluon exchange, where D 00 is a confining propagator, and D ⊥ is perturbative, Near the Fermi surface antiquarks can be ignored, with ψ + α,f,n and ψ α,f,n being creation and annihilation operators of nonrelativistic fermions carrying spin (not Dirac spinors) about the Fermi surface. Here α = ±1/2 is a spin projection, a, b . . . In the first approximation we neglect the asymmetry introduced by differences in q f and m f and by the magnetic interaction V 2 . Then we can treat j = (α, f ) as a united set of indices, so the theory (1) is invariant under a larger symmetry of SU (2N f ). Then realistic values are 2N f = 4 (low density, no strange quarks) or 2N f = 6 for high density.
If the chiral symmetry is broken, m f are renormalized quark masses. In fact, exact definitions of m f do not affect the qualitative side of our arguments since for excitations near the Fermi surface the only difference between massless and massive quarks is the difference in the Fermi velocity, v F , which follows from the relationship between energy and momentum.
We also neglect the frequency dependence of the gluon propagator D 00 , and set c = 1.
The extended symmetry of SU (2N f ), instead of SU (N f ), is due to a doubling from the spin degrees of freedom, and is no longer respected once magnetic interactions (3) are included. The same symmetry has been discussed, both in the hadronic spectrum and at nonzero quark density, in Refs. [132][133][134][135].
As was demonstrated previously [104] the quarkyonic phase supports a collection of spinflavor density waves. Since in the nonrelativistic limit spin and flavor are treated on equal footing, a wave with wave vector Q is characterized by the slow order parameter field in the form of a 2N f × 2N f matrix field U Q such that at long distances the spin-flavor density can be decomposed as where ρ 0 is the average density. The set of wave vectors Q i and amplitudes of the matrix fields related to det U are determined by the matter density, which follows from the value of the chemical potential. In condensed matter systems a density wave with wave vector Q usually forms when the parts of the Fermi surface connected by Q can be superimposed on each other, which is the nesting condition. Once this condition is fulfilled, the susceptibility acquires a singularity at Q signifying a possible instability. However, for cold quarks a perfect nesting is unnecessary due to the singular character of the confining potential in Eq. (2) [104]. As a result quarks can scatter with one another only at small angles and still remain near the edge of the Fermi surface. So it is sufficient to fulfill the nesting condition just on limited patches of the Fermi surface whose area, Λ 2 , is determined by the interplay between the string tension σ and the curvature of the Fermi surface. Within these patches the problem is essentially one dimensional, and can be treated by non-Abelian bosonization and conformal embedding [50,136].
In the first approximation when we neglect the magnetic interaction ( to SU (N f ).
The minimal possible number of patches is six, as a cube embedded into a spherical Fermi surface. When the density increases it becomes energetically advantageous to form triangular patches at the corners of the cube, so another eight patches, with fourteen in all.
We estimate the number of patches as follows. The effective current-current interaction in (1) scales to strong coupling giving rise to a characteristic energy scale, ∆. This can be estimated from the self consistency condition in the rainbow diagram with one gluon and one quark propagator: where v F is the Fermi velocity and |Q| = 2k F . The size of the patch is estimated setting the transverse part of the quarks' kinetic energy equal to this scale: From (9) we find the following estimate for the number of patches: where n is the density of quarks. Since σ 0 is essentially zero above some value of the chemical potential, for massive quarks the number of patches first grows and than sharply decreases with density. The estimate of Eq. (10) also shows the dependence of the number of patches on the quark mass m. For massless quarks, p ⊥ ∼ ∆, and Λ 2 ∼ σ 0 . Thus chiral symmetry breaking decreases the number of patches.
The quantum action for an individual patch describes the baryonic strange metal described in Ref. [50]: the quarks are confined, but the baryons remain gapless and incoherent.
The only coherent excitations are the bosonic collective modes, so that the corresponding phase can be characterized as a Bose metal.

III. GINZBURG-LANDAU DESCRIPTION OF QUARKIONIC CRYSTAL PHASE
In this Section we return to the problem of description of the Quarkyonic Crystal phase.
We show that at nonzero temperature there is no long range order and that this state resembles smectic liquid crystals, which are ordered periodically only in one direction, although with some significant caveats.
As we have mentioned above, the action for the fluctuations normal to the Fermi surface is given by the sum of WZNW actions for each patch. For the static components of the order parameter fields one can omit the Wess-Zumino terms, leaving just terms from the gradient expansion. The corresponding Ginzburg-Landau (GL) free energy for the fields U was written in [104], but it requires correction. It is known [34][35][36][37][38][39][97][98][99][100] that the fluctuations tangential to the Fermi surface must have a zero stiffness, since the orientation of the entire set of Q i 's is arbitrary and for spherical Fermi surface any such rotation costs zero energy. Therefore at nonzero temperature where all fluctuations are classical, the free energy density similar to that in smectic liquid crystal [97]: where U −Q = U + Q and V is the local potential which fixes the amplitudes of these matrix fields. We normalize the Q's as unit vectors. Eq. (11) can be formally derived from Eq. (1).
The first term was derived in our previous paper [104], while the second originates from the fusion of the two perturbing operatorŝ Then the estimates of the parameters: When the Fermi vectors of up, down and strange quarks are different, the GL theory should be augmented by the term where the first matrix in the tensor product acts in flavor space, and the second in spin space. This contribution is the non-Abelian bosonization of the above fermionic term. This extra contribution can be removed by the redefinition of the matrix field: So the shift of the Fermi level of strange quarks does not break the SU(6) symmetry, at least in the leading approximation in δm. A magnetic field does, which is taken into account later.
The matrix U can be parametrized as where the amplitude A Q is fixed by the potential V (11), and G Q is a SU (2N f ) matrix.
Omitting the massive fluctuations of the amplitude A Q we get from (11) the free energy density for the soft modes. It is divided into two parts, Abelian and non-Abelian sigma models:

IV. FLUCTUATIONS AND ORDER
We next show that because the transverse stiffness vanishes, no symmetry is broken at nonzero temperature. In that respect the Quarkyonic Phase resembles the lamellar phases of liquid crystals, which have the same bare fluctuation spectrum.
where the correlation length ξ is where C ∼ 1. Thus the non-Abelian sector is disordered, although at low temperature, the correlation length is exponentially large. The Abelian action in Eq. (18) is a free theory. The corresponding observables are complex exponents of φ Q and as such are periodic functionals of φ Q . In two spatial dimensions vortices of the φ Q field would also enter, but in three spatial dimensions these vortices are extended objects with an energy proportional to their length, and so can be ignored. Thus at nonzero temperature, there is a phase transition of second order into a phase characterized by long range correlations of the fields φ Q .
An order parameter can be constructed for this critical phase. It cannot directly involve correlations of ρ jk , because the correlations of such fields decay exponentially. However, since the charge phase φ Q is a free field over large distances, we can construct an operator which exhibits quasi long range order. Since det G = 1 the order parameter includes 4N f fermions: At distances ξ when fluctuations of G can be treated as massive one can replace B by some fixed amplitude. The average O Q = 0, but its correlations fall off as powers of the relative distance: So at finite temperatures the quarkyonic crystal melts into an Abelian critical phase with wave vectors 2N f times greater than the ones established by the energetics at zero temperature. It is essentially a density wave of a quasi-condensate of 4N f bound states of quarks.
These bound states are spin and flavor singlets.

V. MAGNETIC FIELD
A sufficiently strong magnetic field [137] has a profound effect on the structure of the quarkyonic phase. We will start our analysis at zero temperature when the SU (2N f ) symmetry is spontaneously broken. In this case the matrix G Q can be approximated as where G 0 is some constant matrix andT F are generators of the SU (4) algebra. Then to leading order, instead of Eq. (19) to quadratic order we obtain V 0 ∼ ∆ 2 /µ is proportional to the square of amplitude of A Q in Eq. (16).
For N f = 2 it is convenient to represent the generators T a in terms of the Pauli matrices acting in the spin and the flavor spaces: The magnetic field splits the dispersion of the Goldstone modes. In particular, the mode (I ⊗ τ z ) is not affected by the magnetic field and remains gapless. The three (σ a ⊗ I) modes and three (τ z ⊗ σ a ) are affected only by the Zeeman term. Their spectrum is The only modes affected by the orbital magnetic field are the eight modes with off-diagonal τ ± Pauli matrices. To simplify the discussion of their spectrum we will consider two limiting cases.
1. B Q ẑ, A x = By, A y = A z = 0. Then the spectrum is determined by the equation Here the real vector t includes the modes corresponding to generators (I ⊗ τ ± ), (σ a ⊗ τ ± ).
We need just to shift y by ±p x /B after which p x drops out of the eigenvalue equation. The general solution of Eq. (28) is given by The spectrum is We can determine function g 1 (n) at large quantization numbers n 1 using semiclassical approximation: Then the spectrum follows from the Bohr-Sommerfeld quantization condition: so that at n 1 2. B ⊥ Q and ŷ, A x = Bz, A y = A x = 0.
The spectrum is given by where using the semiclassical approximation at n 1 we get To summarize: in the presence of a magnetic field, the spectrum is divided into three groups. There are eight gapped modes which include components along the τ ± generators; their gaps depend strongly on the direction of the magnetic field. The second group consists of the modes which include only spin operators and τ z ; they have much smaller gaps ∼ B(∆/µ). The third group includes the mode (I ⊗ τ z ) which remains gapless.
Hence there are three regimes of temperature. When temperature is so high that the correlation length is smaller than the magnetic length the influence of the magnetic field is small: There is an intermediate interval of temperature when the magnetic field suppresses some modes, leaving as nominally gapless the modes (I ⊗ τ z ), (σ a ⊗ I) and (σ a ⊗ τ z ). In this region one can approximate the order parameter field as a product of SU(2) matrix g and U(1) (2) At yet smaller temperatures the SU(2) part g is gapped by the magnetic field, through quantum effects. In both cases the Abelian modes α, φ (the total charge and the diagonal flavor one) survive as gapless. Being Abelian they remain long ranged even if fluctuations are taken into account. As a consequence by gapping out all non-Abelian modes the magnetic field instigates quasi long range order in the σ ρ (f,σ),(f,σ) which is quadratic in quark creation and annihilation operators.

A. General effective Lagrangian
In the previous section we considered chiral spirals in a quarkyonic phase in QCD. This is relevant at chemical potentials above that for hadronic nuclear matter, but below those where perturbative QCD applies.
Moving up in chemical potential, the transition from a quarkyonic phase to the perturbative regime appears straightforward. The width of a patch with a chiral spiral is proportional to the square root of the string tension, Eq. (9). As discussed in Sec. (II), numerical simulations for two colors show that the string tension decreases with increasing chemical potential, and is essentially zero by µ q ∼ 750 MeV, Fig. 3 of Ref. [123]. In a perturbative regime the interactions near the Fermi surface lead to color superconductivity symmetrically over the Fermi surface, instead of leading to formation of chiral spirals in patches. On the quarkyonic side the only order parameter is the U (1) phase of the determinantal operator, detρ Q of Eq. (22). On the color superconductivity side the broken symmetry is completely different and therefore it is natural to expect that the phase transition is of first order, as between charge density wave and superconductivity in condensed matter systems. There are also other order parameters for color superconductivity which may enter [138]. For three colors the precise value at which this transition occurs for three colors cannot be fixed by our qualitative arguments, but following the discussion at the beginning of Sec. (II), it is presumably somewhere around µ q ∼ 1 GeV.
This suggests that there is a direct relation between the pion/kaon condensates of hadronic nuclear matter and those of the quarkyonic regime. We argued previously that the only true order parameter for the quarkyonic regime was an overall phase factor of U (1). Such a phase clearly arises for the field of Eq. (39), which we denote by the phase φ. The physical origin of this phase is obvious: at a given point along the Q direction, the condensate points entirely in a given direction, say along π 3 . Where this point is just an overall shift in the phase, though.
Thus pion/kaon condensates have a U (1) phase, which is sometimes termed the "phonon" mode [96,98]. This is then a strict order parameter which distinguished hadronic nuclear matter, without a pion/kaon condensate, from that with. If this transition is not of first order, then it must be of second order, in the universality class of U (1).
As the chemical potential increases further, it is very plausible that it is not possible to rigorously distinguish between the pion/kaon condensate of Eq. (39) and chiral spirals of the quarkyonic regime. The only difference is that there are N 2 f − 1 very light modes for a pion/kaon condensates, and 4N 2 f − 1 light modes for a quarkyonic chiral spiral. It is natural to assume that the additional 3N 2 f modes of the latter become lighter as µ q increases. As discussed previously, there may be phase transitions as the number of patches increases, although this is not really necessary. In that vein, we note that in the original discussion of pion condensates by Overhauser [26], it was explicitly stated that the simplest solution in three dimensions is that with six patches.
The relation between pion/kaon condensates and chiral spirals also suggests a less trivial speculation. For static quantities, at high density the effective theory for the light modes However, we argued that for non-static quantities, there is also a WZNW term, of level 3 for SU (2N f ), and level 6 for SU (N f ). This suggests that there might be a WZNW term for pion/kaon condensates, with level 6. This is not obvious. The original theory has a WZNW term in four dimensions, but this involves derivatives in all four dimensions [139]. Perhaps a WZNW term arises in two dimensions from the variation of the patches in the transverse directions.
The similarity between chiral spirals and pion/kaon condensates has been recognized previously, at least implicitly [93][94][95]97]. The appearance of a U (1) order parameter, and the possible appearance of a WZNW term, is novel.
To support the above considerations we establish a formal correspondence between the relativistic and the nonrelativistic versions of the QCD Hamiltonian. We start with the Dirac Hamiltonian: where τ a act in the the chiral basis (R, L) and σ a act in the spin space. We neglect the quark mass. Then under the transformation   q Rσ (p) where the Hamiltonian Eq. (40) becomes In what follows we will drop the antiparticles η.
Now let us consider the chiral order parameter ρ selecting in it only the part associated with particles: where the basis vectors e are defined in (41) and the ellipses include the contributions of antiparticles. This shows that the chiral order parameter ρ, which is uniform at µ = 0, naturally acquires oscillatory terms at µ = 0. These can be either pion/kaon condensates or quarkyonic chiral spirals. As we neglected antiparticles, this only happens for sufficiently large µ q . These arguments do not show how large µ q must be, but they do establish that both chiral symmetry breaking and the formation of inhomogenous phases can be described within the same effective model.
To discuss the phase diagram we then consider a SU (N f ) × SU (N f ) field ρ, taking a customary linear sigma model, In four spacetime dimensions ρ has dimensions of mass. The first two terms are standard kinetic terms. The coefficient of the second term, with two spatial derivatives, can have an arbitrary coefficient Z. Implicitly we consider systems at nonzero temperature and quark density, and so there is a preferred rest frame. Then Lorentz symmetry is lost, and Z = 1 is allowed. In particular, we allow Z to be negative. Stability then requires the addition of a positive term with four spatial derivatives; the coefficient of that term is ∼ 1/M 2 , where M is some mass scale derived from the underlying theory.
While we only consider static quantities, and so can ignore the first term with two time derivatives, we add it to emphasize that the only higher derivatives considered are those in the spatial coordinates. It is well known that higher order derivatives in time lead to acausal behavior, which should not occur in an effective theory. This is not dissimilar to causal theories of higher derivative gravity, such as Horava-Lifshitz gravity [140][141][142][143].
When the coefficient of the term with two spatial derivatives is positive, Z > 0, the phase diagram is standard. For positive mass squared, m 2 > 0, the theory is in a symmetric phase, with ρ = 0. For positive quartic coupling, λ > 0, the global flavor symmetry is broken when the mass squared is negative, m 2 < 0, but not the translational symmetry. At zero mass, m 2 = 0, there is a second order transition in the appropriate universality class.
The hexatic couplings, such as κ, are assumed to be positive, so the quartic coupling λ can be negative. It is easy to show that there is then a first order transition at positive mass squared, m 2 > 0.
In the plane of the mass squared, m 2 , and the quartic couplings λ, the phase diagram Since there is more than one quartic coupling, even if the quartic couplings are originally positive, in the infrared limit they can flow to negative values, and so generate a fluctuation induced first order transition [144][145][146][147]. For this Lagrangian in 4 − dimensions, to leading order in this happens when N f > √ 2 [148]. This is only valid to leading order in . For two flavors, the symmetry is SU We next turn to the case of m 2 < 0, when ρ has an expectation value ρ 0 = 0. Spatially inhomogenous condensates arise when Z is negative. We take where Q = Q 0ẑ .
The ansatz of Eq. (46) is only a caricature of the full solution. The detailed form of the condensate differs depending upon whether the broken symmetry is discrete or continuous. For our qualitative analysis, though, the precise form of the condensate is secondary. We then minimizing the terms with spatial derivatives with respect to Q 0 , For Q 0 to be real, Z has to be negative. Plugging this back into the Lagrangian, we obtain where λ = N 2 f λ 1 + N f λ 2 , and Henceforth we ignore the hexatic coupling ∼ κ, as we uniformly assume that the quartic coupling λ is positive.
Consider a given value of m 2 < 0, crossing from positive to negative values of Z. For Z > 0, the theory is in a broken phase; for negative Z, in the phase with the chiral spiral of Eq. (46). Ignoring κ, the potential energy at the minimum is Assuming that the variation in Z is linear in the appropriate thermodynamic variable, such as the chemical potential or temperature T , then it is trivial to show that while the potential, or free energy, is continuous at T = T 0 , the first derivative, related to the energy density, is discontinuous. This is natural: except for Goldstone bosons, the correlation lengths are nonzero in both phases. Further, there is an order parameter which distinguishes the phases: φ is constant when Z is positive, while with the chiral spiral of Eq. (46), the spatial average of φ vanishes when Z < 0.
Consider next the case when Z is negative and the original mass squared, m 2 , is positive.
Then the effective mass m eff vanishes when m 2 = Z 2 M 2 /4, and one expects a second order phase transition.
For the assumed parameters, the φ propagator is When Z < 0, there is a minimum for nonzero spatial momentum. Expand Inserting Eq. (53) into Eq. (52) and expanding, the terms proportional to k z vanish if Q 0 satisfies Eq. (47). Then where m 2 eff is that of Eq. (49). Notice that the terms quadratic in the transverse momenta, k tr , vanish, although there are terms of higher order in k tr . This is similar to the behavior of Goldstone bosons in a chiral spiral.
Consequently, an integral over virtual fields is dominated by fluctuations in the direction of k z . When the effective mass is small, the correction to the mass term is Similarly, the correction to the quartic coupling is Both of these results follow because the fluctuations for small m eff are those for a theory in one dimension, along k z . Because the infrared divergences of Eqs. (55) and (56)  transition between the two phases, but it is necessarily of first order, where m eff is always nonzero in each phase.
This has been termed a "fluctuation induced first order" transition [89,[107][108][109], but the terminology is somewhat misleading. In theories with several coupling constants, couplings can flow to negative values [144][145][146][147], and so generate a first order transition. This depends upon how the coupling constants flow under the renormalization group in the infrared limit, and so depends both upon the symmetry group, and the dimensionality of space-time. Carignano [95]. Instead of m 2 and Z, the physical phase diagram is a function of temperature, T , and the baryon (or quark) chemical potential, µ. In the NJL model, for the specific interaction assumed, the Lifshitz point coincides with the critical endpoint, but this is an artifact of the simplest model to one loop order [151].
Consider the theory at the Lifshitz point. The static propagator is At leading order the leading correction to the mass is This develops a logarithmic divergence in the infrared in four dimensions, which is then the lower critical dimension [15-17, 23, 24]. Corrections to the quartic coupling begin at one loop order as This is logarithmically divergent in eight dimensions, which is the upper critical dimension [15][16][17]. This is contrast to an ordinary critical point: for a propagator ∆(k) = 1/k 2 , where the lower and upper critical dimensions are two and four, respectively.
At the Lifshitz point in four spatial dimensions, in the infrared the logarithmic divergences always disorder the theory. This is stronger at nonzero temperature, when d = 3 and the infrared divergences are power like ∼ 1/m. Consequently, once fluctuations are included, there cannot be a true Lifshitz point.
Inhomogenous polymers provide an example of the absence of a Lifshitz point in three spatial dimensions [18][19][20][21][22]. The simplest case is a mixture of oil and water. These separate into droplets of oil or water, but by adding a surficant to alter the interface tension, other phases emerge. A related example is a mixture of two different polymers, formed from monomers of type A and type B. To this are added A-B diblock copolymers, which are long sequences of type A, followed by type B. These A-B copolymers localize at the interfacial boundaries separating phases with only A or B homopolymers, and act to decrease the interface tension; at sufficiently high concentrations, the interface tension changes sign, and is negative.
By varying the temperature and the concentration of diblock copolymers one can form three different phases. At high temperature A, B, and A-B polymers mingle to form a homogeneous phase, analogous to the symmetric phase of a spin system. At low temperature and low concentrations of A-B copolymers, the system separates into droplets of A, B and A-B polymers, which is like the broken phase of a spin system. At low temperature and high concentration of A-B copolymers, the interface tension becomes negative, and there is an inhomogenous phase, as the system forms a lamellar state with alternating layers of A and B polymers. This is similar to a smectic liquid crystal, albeit without orientational order.
Mean field theory predicts that there is a Lifshitz point where these three phases meet.
In contrast, both experiment and numerical simulations with self consistent field theory indicate that there is no Lifshitz point [18][19][20][21][22]: see, e.g., Fig. 3 of Ref. [21]. Instead, the symmetric phase enlarges, and includes a bicontinuous microemulsion, which exhibits nearly isotropic fluctuations in composition with large amplitude. In this regime the surface tension is essentially zero, and there is a spongelike structure with large entropy.
The absence of the Lifshitz point can be understood by analogy. Consider a spin system, with a continuous symmetry, in two or fewer dimensions. The symmetry cannot be spontaneously broken as that would generate massless Goldstone bosons, which are not possible in such a low dimensionality. Instead, fluctuations generate a mass non-perturbatively.
What happens in the Lifshitz regime, when the number of spatial dimensions is four or less, is similar. We can tune either the coefficient of the term quadratic in momenta to vanish, Z = 0, or the mass, m 2 , to vanish, but not both. If m 2 = 0, then Z = 0 is generated non-perturbatively; alternately, if Z = 0, then m 2 = 0 is generated non-perturbatively. For the latter, the propagator is not Eq. (57), but where m 2 = 0 is non-perturbative. We cannot conclude anything about the size of the Lifshitz regime, only that it exists. For inhomogeneous polymers, the Lifshitz regime includes a bicontinuous microemulsion, where Z ≈ 0 and m 2 = 0; see, e.g., Fig. 2 of Ref. [21].
A possible phase diagram which incorporates fluctuations is that of Fig. 2. There is a strict order parameter which distinguishes the broken and symmetric homogeneous phases, The Lifshitz critical endpoint C is not of this form. The simplest possibility is that at C, a term quadratic in the momenta, Z > 0, is generated non-perturbatively, with m 2 = 0. This implies that the universality class of the Lifshitz critical endpoint C is the same as along the line of second order transitions.
Consider moving away from the Lifshitz critical endpoint C, down in Z into the inhomogeneous phase. Since mean field theory indicates that an inhomogeneous phase only arises when Z is negative, the appearance of an inhomogeneous phase infintesimally below C must be due to strong, non-perturbative fluctuations.
Alternately, consider moving away from the Lifshitz critical endpoint to the right, for increasing m 2 . Doing so, one will enter a region where Z is very small, but the mass squared m 2 is nonzero and positive. This region is directly analgous to a bicontinuous microemulsion [18][19][20][21][22]. For inhomogeneous polymers, this region is seen to be an enlargement of the sym- Before continuing to the implications for the phase diagram of QCD, we remark that our analysis is valid for nonzero temperature in three spatial dimensions. At zero temperature, by causality there must always be terms quadratic in the energy. The integral analogous to Eq. (58) then becomes As m → 0 this is infrared convergent in more than two spatial dimensions, d > 2. Thus we expect that the infrared fluctuations are well behaved at low temperature. Further, the dynamic behavior near the Lifshitz critical endpoint, C, differs from that for a typical critical endpoint, C.

VII. RELATION TO QCD
As we have discussed, the phase diagram is a function of at least three parameters: the mass squared, quartic coupling(s), and the spatial wave function renormalization Z. At the outset, we assume that the quartic couplings λ A caricature of the possible phase diagram in QCD is illustrated in Fig. 3. The Lifshitz point is wiped out by strong infrared fluctuations, leaving a Lifshitz regime. We denote this by the shaded regime in Fig. 3, but it is not a precisely defined region. The infrared fluctuations in the Lifshitz regime are dominated by massive modes whose momentum dependence is dominated by quartic terms.
Of particular interest is the highest temperature at which there is a spatially inhomoge- Since the pressure is continuous at a first order phase transition, by taking derivatives of the pressure with respect to µ, we find This implies that even though there is a first order transition at T 0 , the densities are equal.
This is known in thermodynamics as a point of equal concentration. Since the transition is of first order, the entropies between the two phases differ at T 0 .
We assume that the crossover line terminates at T 0 , so in the chiral limit, T 0 coincides with the Lifshitz critical endpoint, C. We cannot prove that T 0 is the shadow of C, but it is a most natural conjecture.
What are the possible signals of the phase diagram in Fig. 3? In heavy ion collisions, assuming that the system thermalizes, it starts at high temperature and then cools down.
The trajectory in the plane of temperature T , and quark chemical potential, µ, is model dependent, but the point at which the system freezes out of equilibrium is found by fits to the spectra for different particle species, and gives values for the final T and µ. (The baryon chemical potential is three times that for the quarks.) The fits to thermal models [106,[110][111][112][113][114][115] demonstrate that one enters a region where the quark chemical potential at freezeout is significant.
The standard picture [5][6][7][8][9][10] assumes the crossover line for small µ meets a line of first order transitions at a critical endpoint as µ increases and T decreases. At a critical endpoint, in infinite volume and in thermal equilibrium, there are divergent fluctuations for the critical mode, which is associated with the σ meson. There should also be large fluctuations for modes which couple to the σ meson, including pions, kaons, and nucleons. It is not possible to measure the fluctuations for σ's directly, but as we discuss below, it is possible to measure that for protons. Measuring the fluctuations for pions and kaons is experimentally very challenging, but we argue is essential in order to distinguish between different models. As the lighter particle, near a critical endpoint C the fluctuations for pions should be greater than for kaons.
If the phase diagram does not have a critical endpoint, but instead has an unbroken line of first order transitions as in Fig. 3, then the signals depend upon the trajectory in the T − µ plane. One obvious difference is that with Fig. 3, it is possible to cross two first order lines before hadronization occurs.
We first discuss the case in which the system enters the Lifshitz regime but is still in the symmetric phase, before it crosses the line of first order transitions. In principle it is necessary to include a nonzero chemical potential for up and down quarks and to impose the condition that the net strangeness vanishes. We leave these details to future study to make the following elementary point.
Consider a particle with the usual dispersion relation, E = √ k 2 + m 2 . In the limits of high and low temperature the average momentum is In the ultra-relativistic limit the average momentum is necessarily independent of mass and is proportional to the only mass scale, which is the temperature. In the non-relativistic limit the average momentum is proportional to the square root of the mass, times the temperature.
Now consider a particle in the Lifshitz regime, assuming that the coefficient of the term quadratic in the spatial momentum is essentially zero. The dispersion relation is then The mass scale M ensures that the term quartic in the spatial momentum, Eq. (52), has the correct mass dimension. As discussed above, this dispersion relation is analogous to the bicontinuous microemulsion phase of inhomogenous polymers [18][19][20][21][22]. For such a dispersion relation, the average momentum in the limits of high and low temperature is Again, in the ultra-relativistic limit the average momentum is independent of the mass m, but now it is only proportional to the square root of temperature, with M making up the remaining mass scale. In the limit of low temperature, the average momentum is proportional not to the square root of the mass, but to the fourth root thereof.
In heavy ion collisions, the freezeout temperature is near the pion mass, so for simplicity we assume that the pions are ultra-relativistic. For kaons, we assume that they are nonrelativistic. Of course this is a gross simplification, but it not difficult to carry out a more careful analysis in a thermal model.
Because the dispersion relation in the Lifshitz regime differs fundamentally from the usual relation, the relative abundance of kaons to pions must change when these particles are in the Lifshitz regime. In particular, the mass dependence for heavy particles, such as kaons, is less sensitive to mass, k ∼ m 1/4 , Eq. (66), versus k ∼ m 1/2 in Eq. (66). Thus in the Lifshtiz regime, the ratio of kaons to pions is greater than a fit with a standard thermal model.
The difference between pions and kaons persists once spatially inhomogeneous condensates develop. In the simplified discussion of Sec. (VI A), we did not distinguish between pions and kaons. This valid in the strict chiral limit, but not in QCD. Moving down in temperature at fixed µ, presumably a pion condensate develops before that for kaons. Indeed, pion condensates are naturally chiral spirals, rotating between σ and a given direction for the pions. In contrast, kaons presumably develop a kink crystal first, oscillating about a given expectation value for ss. As the chemical potential increases at a fixed, small value of the temperature, these condensates then evolve into a chiral spiral of the quarkyonic phase, and approach the SU (3) symmetric limit in flavor. In any case, the effective masses for fluctuations are given by Eq. (49), and differ markedly from those of free particles.
For each particle species, in a phase with spatially inhomogenous condensates the fluctuations are concentrated not about zero momentum but about the momentum for the condensate, Q 0 in Eq. (47). This should be measurable by measuring the fluctuations in different bins in momenta. This is challenging experimentally, as any condensate is with respect to the local rest frame, which is boosted by hydrodynamic expansion to a significant fraction of the speed of light.
Depending upon the trajectory in the plane of T and µ, it may be possible to cross not just one, but two lines of first order transitions before the system hadronizes. Lastly, the point T 0 is of especial interest, although it is not clear whether trajectories naturally flow into it.
Before discussing heavy ion experiments, we note that Andronic et al. [106] argued that there is a triple point in the T − µ plane. The Lifshitz regime can be considered as an explicit way of generating this phenomenon.
There are two notable anomalies in the collisions of heavy ion at relatively low energies.
The first is a strong departure from thermal models. In heavy ion collisions, there is a peak in the ratio of K + /π + and Λ/π at energies ∼ 10 GeV [106,[110][111][112][113][114][115]. Deviations from thermal behavior for the ratio of kaons to pions is suggestive of the Lifshitz regime, Eqs.
(64) and (66) above. However, the ratio K − /π − shows no such deviation. Clearly a more careful analysis, including the condition of zero net strangeness, is essential.
The second anomaly concerns fluctuations in net protons. Experimentally it is possible to measure cumulants, which are related to the derivatives of the pressure with respect to the chemical potential, c n (T, µ) ∼ ∂ ∂µ n p(T, µ) .
The results from numerical simulations on the lattice appears to agree remarkably well with the predictions of lattice gauge theory except at the lowest energies [152][153][154][155][156][157][158]. There, unpublished data from the Beam Energy Scan with the STAR experiment at RHIC suggests a possible anomaly at √ s/A ∼ 8 GeV [116,117]. In the Lifshitz regime, pions and kaons behave strongly non-pertubatively, and this feeds into the fluctuations of protons. This could explain this possible anomaly.
To distinguish between a Lifshitz regime and a critical endpoint, it is essential to measure the fluctuations of pions and kaons. Near a critical endpoint the fluctuations for pions are larger than for kaons. This is not true in the Lifshitz regime, either in the symmetric or chiral spiral phase. It is very possible that for a certain range of energies, that the fluctuations for kaons exceed those for pions.
Of course evidence for crossing not just one, but two phase transitions, would also be exceptional. Nevertheless, without model calculations we cannot estimate how strong the first order transitions are.
In conclusion, it is surprising that there are such close analogies between the phase transitions in condensed matter systems, such as smectics and inhomogenous polymers, and those of QCD. While our analysis is a first step, it may directly impact our understanding of the collisions of heavy ions at low energies.