Remarks on nuclear matter: how an $\omega_0$ condensate can spike the speed of sound, and a model of $Z(3)$ baryons

I make two comments about nuclear matter. First, I consider the effects of a coupling between the $O(4)$ chiral field, $\vec{\phi}$, and the $\omega_\mu$ meson, $\sim + \, \omega_\mu^2 \, \vec{\phi}^{\, 2}$; for any net baryon density, a condensate for $\omega_0$ is unavoidably generated. I assume that with increasing density, a decrease of the chiral condensate and the effective $\omega_0$ mass gives a stiff equation of state (EoS). In order to match that onto a soft EoS for quarkyonic matter, I consider an $O(N)$ field at large $N$, where at nonzero temperature quantum fluctuations disorder any putative pion"condensate"into a quantum pion liquid (Q$\pi$L) (arXiv:2005.10259). In this paper I show that the Q$\pi$L persists at zero temperature. If valid qualitatively at $N=4$, the $\omega_0$ mass goes up sharply and suppresses the $\omega_0$ condensate. This could generate a spike in the speed of sound at high density, which is of relevance to neutron stars. Second, I propose a toy model of a $Z(3)$ gauge theory with three flavors of fermions, where $Z(3)$ vortices confine fermions into baryons. In $1+1$ dimensions this model can be studied numerically with present techniques, using either classical or quantum computers.

In this paper I make two comments about nuclear matter. The first is a suggestion as to how ω 0 [2][3][4]6] and pion [1,[103][104][105][106][107][108][109] condensates can affect the EoS, and generate non-monotonic behavior for the speed of sound as the density increases well above n sat . The second is a toy model in which fermion fields, analogous to quarks, are confined into baryons by a Z(3) gauge field.

SPIKING THE SPEED OF SOUND
In the past few years astronomical observations of neutron stars  have provided significant insight into the nuclear EoS at densities above n sat . This includes quantities such as their mass, radius, and tidal deformability. The EoS is given by the pressure, p, as a function of the energy density, e. Analyses with piecewise polytropic EoS are useful [19][20][21][22].
However, a more sensitive probe of the EoS is given by the speed of sound squared: c 2 s = ∂p/∂e. Free, massless fermions have c 2 s = 1/3, which is termed soft. In contrast, several studies of neutron stars find that it is essential for the nuclear EoS to have a region in which the EoS is stiff, where c 2 s significantly larger than 1/3 [23][24][25][26][27][28][29][30][31][32][33]. For example, consider the analysis of Drischler et al. [27], who extrapolate up from n sat using chiral effective field theory. To obtain neutron stars with masses above two solar masses, they find that there is a region of density in which the EoS is stiff: if there is a neutron star of 2.6 solar masses, at some n B , c 2 s ∼ 0.55. To agree with small tidal deformability from GW170817, though, the EoS of nuclear matter must be soft until n B ∼ 1.5 − 1.8n sat .
That is, there is a "spike" in the speed of sound, with a relatively narrow peak at a density significantly above n sat : see, e.g., Fig. (1) of Greif et al. [24].
As the density n B → ∞, by asymptotic freedom the EoS approaches that of an ideal gas of dense quarks and gluons, and so is soft. In Quantum ChromoDynamics (QCD), corrections to the quark EoS have been computed in part up to four loop order [110][111][112][113][114]. However, perturbation theory is only useful down to densities much larger than n sat . At very high densities, excitations near the Fermi surface are dominated by color superconductivity [34,35,37].
Going down in density, nuclear matter becomes quarkyonic [49][50][51][52][53][54][55][56][57][58][59][60][61]. The free energy is close to that of QCD perturbation theory, but the excitations near the Fermi surface are confined, and so baryonic. A quarkyonic regime is inescapable for a SU (N color ) gauge theory as N color → ∞, as then quark loops are suppressed by ∼ 1/N color . This does not seem to be special to large N color , though. In lattice gauge theory, when N color ≥ 3 the sign problem prevents classical computers from computing at zero temperature and nonzero quark density [115]. Two colors, however, is free of the sign problem, and while it has unique features -notably, since baryons are bosons there is no Fermi sea -lattice simulations find a broad quarkyonic region [116][117][118][119][120][121]. This suggests the same applies to QCD, where N color = 3.
I assume that the quarkyonic EoS is soft. Bedaque and Steiner [122] have argued, from a variety of examples, the any quasiparticle model is soft. While some authors propose that quarks can give a stiff EoS [29][30][31], for simplicity I do not.
To match a nuclear onto a quarkyonic EoS, McLerran and Reddy take a quark EoS up to some Fermi momentum k F Q , which is then surrounded by a baryonic shell of width ∆, Fig. (1) of Ref. [55]. As the baryon density increases, k F Q grows, and ∆ shrinks. Taking an ideal EoS for both quarks and baryons, an appropriate choice of the width ∆ generates a spike in the speed of sound, Fig. (2) of Ref. [55]. At densities near k F Q , though, neither equation of state is close to ideal. The Quantum HadroDynamics (QHD) of Serot and Walecka [3,4,6] can be used for baryons, although to be capable of modeling a confined but chirally symmetric phase, all chiral partners of the nucleons and mesons must be included in a paritydoubled QHD (PdQHD) [64][65][66][67][68][69][70][71][76][77][78][79][80]. The quark EoS can be modeled by coupling quarks and gluons to a linear sigma model for mesons. Such a PdQHD was considered by Cao and Liao [79].
My purpose here is to discuss, in an entirely qualitative manner, of how an ω 0 condensate, and strong fluctuations in a pion "condensate" [1,[103][104][105][106][107][108][109] could affect the EoS in PdQHD. My discussion is admittedly speculative, because given the wealth of experimental data, it is not easy to describe the EoS of nuclear matter both near n sat and at n B n sat . In QHD, saturation results from a balance between repulsion from the ω µ meson and attraction from the σ meson [3,4,6]. That the ω µ meson could generate a stiff EoS was first noted by Zel'dovich [2]. Given the coupling of the ω µ to a nucleon ψ as ∼ g ω ψ ω µ γ µ ψ, then at any nonzero baryon density, ψγ 0 ψ = n B = 0, a condensate for ω 0 is automatically generated [123]: If only these terms matter, then the EoS is as stiff as possible, with the speed of sound equal to that of light, c 2 s = 1. Son and Stephanov showed that QCD at nonzero isospin density provides a precise example of this [124].
Of course in QHD, Eq. (1) is not the only term which matters. Integrating over nucleon loops at nonzero density, there is an infinite series of terms in ω 0 which are generated at n B = 0, including those ∼ ω 2 0 , ∼ ω 3 0 , and so on. Similarly, the nucleon couples to the σ, whose properties also change with n B . These effects have been computed to one loop order [3,4,6], but even for strong g ω , do not dramatically alter the EoS.
The ω µ Lagrangian is F µν = ∂ µ ω ν − ∂ ν ω µ is the field strength for ω µ , m ω a mass term, and there is a quartic coupling ∼ κ 2 between ω µ and φ, where φ is the O(4) chiral field for two light flavors, φ = (σ, π). The coupling κ 2 must be positive to ensure stability for large values of the ω µ and φ fields.
The anomalous interactions, though, all involve at least three derivatives, which for the ω 0 meson, are all spatial derivatives. This is why the coupling ∼ κ 2 in Eq. (2) is so important, as the only renormalizable, non-derivative coupling which the ω µ has with the chiral field φ.
My principal assumption is that for some n B > n 1 > n sat , that one enters a region dominated by the ω 0 condensate. Notably, if Z decreases with increasing n B , the effective mass squared of the σ increases as ∼ 1/Z, while if κ = 0, the ω 0 becomes light as the chiral symmetry is restored. By Eq. (1), L B ω = −g 2 ω n 2 B /(2m 2 ω ), and a heavy σ, with a light ω 0 , could generate a stiff EoS for n B > n 1 .
Assuming that a light ω 0 gives a stiff EoS, then how can the ω 0 condensate evaporate to match onto a soft quarkyonic EoS? Presumably the couplings of the ω µ with nucleons behave smoothly with density. That leaves the couplings of the ω µ to the chiral field φ, but as demonstrated above, these are limited. This question does assume that a hadronic phase matches onto quarkyonic matter. It is possible to simply paste a stiff hadronic EoS onto a soft quark EoS through what is presumably a strongly first order transition. This is not consistent, however, with the analyses for either N color → ∞ [49][50][51][52][53][54][55][56][57][58][59][60][61] or lattice results for N color = 2 [116][117][118][119][120][121], which indicate a quarkyonic regime. Nor why the nuclear EoS appears to be soft near n sat , and only stiff when n B ∼ 1.5 − 1.8n sat [27].
I stress that reducing the contribution of the ω 0 condensate at large chemical potential, µ, and low temperature, µ T , has no analogy to the more familiar case, at nonzero temperature and low density. When T µ, it is easy matching the EoS of hadronic matter, with a relatively few degrees of freedom, onto a quark-gluon plasma, with many. This is precise in the limit of a large number of colors, N color → ∞, where the pressure in the hadronic phsae is ∼ N 0 color , versus ∼ N 2 color in the deconfined phase. Similarly, the contribution of the chiral condensate is only ∼ N 1 color , and decreases as T increases. In contrast, at µ T , the pressure is always ∼ N 1 c , in both the hadronic and quark-gluon phases.
At nonzero density, the appearance of a condensate for ω 0 is special to the ω µ meson: there is no other hadron which couples directly to the net baryon density. This assumes that the only net charge is for baryon number. When there is a net isospin charge, a condensate for the ρ µ meson is generated, ∼ ρ 3 0 . In this case, terms such as φ 2 ρ 2 µ , amongst others [70,71,75,76], need to be included; further, couplings between the ω µ and ρ µ mesons, ∼ ω 2 µ ρ 2 µ , must be added [73]. It is then very difficult to fit the EoS of an ω 0 condensate onto that of cold quarks: either the coupling of the ω 0 becomes small, or the mass of the ω 0 becomes large. Since the coupling of the ω 0 is strong in vacuum, the former is most implausible. Thus the mass of the ω 0 must increase, although this does not occur in mean field theory [151]. I now argue that the mass of the ω 0 increases sharply due to large quantum fluctuations.
Returning to the Lagrangian in Eq. (4), it is standard except for the term quartic in the spatial derivatives, ∼ 1/M 2 [152]. Causality implies that only terms with two time derivatives enter. With the term ∼ 1/M 2 to ensure stability, it is possible to allow the coefficient of the term with two spatial derivatives, Z, to be negative.
While in vacuum Z = 1 by Lorentz covariance, this is not true in a medium. If Z is negative, classically a condensate is generated: which is a pion condensate in the z direction [1,[103][104][105][106][107][108][109]153]. In 1 + 1 dimensions, such chiral spiral condensates are ubiquitous at low temperature and nonzero density [106][107][108], although in general the solutions are more involved. Given these examples, it is natural to assume that in QCD, at low temperature Z < 0 for some range in density above n sat . Most discussions of a pion condensate use a nonlinear Lagrangian, in which the σ meson does not explicitly appear. The advantage of using a linear Lagrangian is that it is much easier studying how the symmetric phase is approached. Following Ref. [1] I generalize from O(4) to O(N ), where the solution is direct as N → ∞ [142,154].
The solution at large N is standard, and proceeds by introducing the a field ξ = φ 2 , and a constraint field, , L cons = i (ξ − φ 2 )/2. I only seek the solution for the symmetric phase, although the solution in the broken phase can also be determined [1]. Using this constraint, the φ and ξ fields are integrated out to give the effective action I expand about a stationary point in and ω 0 , = i + q and ω 0 = ω 0 + ω q 0 , where q and ω q 0 are quantum fluctuations. The effective mass m 2 eff = m 2 0 + + κ 2 ω 2 0 . To have a well defined limit for large N , as N → ∞ all terms in the action should scale as ∼ N , so I take λ, κ 2 ∼ 1/N , g ω ρ B , ω 0 ∼ √ N , and m 2 ω , M 2 , Z, m 2 0 , , m 2 eff ∼ N 0 . Remember that N is just a fictitious parameter, and is not related to the number of colors or flavors.
Requiring that the effective action is a stationary point in q and ω q 0 fixes and ω 0 , The solution for general values of the parameters is involved, so to make a qualitative point I only consider the limit of Z → −∞, where classically the condensate of Eq. (5) dominates. Instead, in perturbation theory one finds that would be Goldstone bosons have a double pole at non-zero momentum, about k c [1]. Such a double pole generates a logarithmic infrared divergence at zero temperature, and a power law divergence at nonzero temperature.
The solution at large N shows how these infrared divergences are avoided. As Z → −∞, at N = ∞ take m eff ≈ −ZM/2 + δm eff . About k ≈ k c , The loop integral is dominated by k ≈ k c + δk, and to leading logarithmic order becomes Solving Eq. (8) for , as Z → −∞, where # is a positive, nonzero number. It is worth contrasting this solution with that at nonzero temperature [1]. Then the integral over ω is a discrete sum, and the zero energy mode is the most important. It generates a power law divergence, with the solution δm eff ≈ 1/Z 4 , Eq. (58) of Ref. [1]. The statement in Ref. [1] that δm eff vanishes at zero temperature is incorrect: it is just that δm eff is suppressed exponentially in 1/ √ −Z, instead of by a power. I refer to this disorder as a quantum pion liquid, QπL [155].
I have neglected the equation for ω 0 in Eq. (8). While δm 2 eff is very different at zero and nonzero temperature, though, what matters there is the value of the loop integral. Since ≈ m 2 eff ∼ Z 2 M 2 /4, by Eq. (8) tr∆ As φ 2 = N tr∆, by Eq. (3) the ω 0 mass increases sharply, and by Eq. (8) suppresses the ω 0 condensate, ω 0 ∼ 1/Z 2 . Note that the presence of the coupling ∼ κ 2 is essential for this to occur. When Z is negative, classically one expects a pion condensate to form, but the solution at large N shows that instead a quantum pion liquid (QπL) forms . While this is rigorous at large N , as it arises from the double pole at k c = 0 for the would be Goldstone modes, it is very likely that there is a QπL for all N > 2 [1]. I also assume that the quantum fluctuations are sufficiently strong so that a QπL forms for massive pions.
My suggestion is thus the following. For n B > n 1 > n sat , the theory enters a phase dominated by the ω 0 condensate, which stiffens the EoS. When n B > n 2 , it is approximately described by a QπL: both the σ and ω 0 are heavy, which suppresses ω 0 . In total, the enhancement and then suppression of the ω 0 condensate generates a spike in the speed of sound.
Clearly a detailed analysis is required to determine the dependence of the various parameters with density, or more properly for thermodynamics, with the baryon chemical potential, µ B . This includes the µ B dependence of the wave function renormalization constant Z, the mass parameter M (which is of some hadronic scale), m 0 , λ, and so forth.
The most direct approach is to use PdQHD, with a selfconsistent one loop approximation for the nucleons, the chiral fields φ, and the ω 0 . While involved, I comment that it is far simpler to look for a QπL -which is just a non-monotonic dispersion relation, Eq. (9) -than for a pion condensate, which is not spatially homogeneous [106].
As quantum computers are (very) far from computing the properties of cold, dense QCD [115], to proceed from first principles requires the functional renormalization group (FRG) [91-93, 133-140, 156-159]. Ref. [91] use a chiral effective model up to ∼ 2n sat , matching onto QCD perturbation theory with a Fierz complete FRG [137][138][139] at intermediate n B . They find evidence for a spike in the speed of sound at ∼ 10n sat [91], which is much higher than Ref. [27]. The ultimate goal is to use the parameters determined by the FRG in vacuum [156][157][158] to compute the EoS for nuclear matter. Fu, Pawlowski, and Rennecke [159] find that Z < 0 at rather high T and µ B = 0, Fig. (21) of [159]. A complete FRG analysis should certainly see a quantum pion liquid, if it exists.
The pion is not an exact Goldstone boson, but I assume it is so light that the QπL phase wins over a pion condensate. The same may not be true for strange quarks [160]. When at some n B the Fermi sea spills over to form one of strange quarks, if the pion Z is negative, by SU (3) flavor symmetry that for kaons will be as well. As the strange quark is much heavier than up and down quarks, instead of a quantum kaon liquid, a kaon condensate might form [105]. This would be a crystal of real kinks, where ss oscillates about a constant, nonzero value Bringoltz [153] showed that this happens for the 't Hooft model in 1 + 1 dimensions [161].
Admittedly my analysis is merely a sketch of how a spike in the speed of sound might arise in nuclear matter. It appears inescapable, though, that the interaction of the ω 0 and the chiral fields plays an essential role.

A MODEL OF Z(3) BARYONS
Some properties of nuclear matter, such as those discussed above, are surely special to QCD. It would be useful, however, to have the simplest possible model which exhibits the confinement of some type of "quarks" into baryons. A SU (N color ) gauge theory in 1 + 1 dimensions [161] has baryons [153,[162][163][164], but as N color → ∞, there are ∼ N 2 color degrees of freedom. There are also models in 1 + 1 dimensions which are soluble about the conformal limit [107,108], but these do not generalize to higher dimensions.
An understanding of confinement from Z(N color ) vortices in a SU (N color ) gauge theory was proposed by 't Hooft [165,166]; for recent work, see [167][168][169] and references therein. I suggest discarding the non-Abelian degrees of freedom in SU (N color ) to retain just those of Z(N color ). A Z(3) gauge theory is constructed following Krauss, Wilczek, and Preskill [170][171][172]: where F µν = ∂ µ A ν − ∂ ν A µ is the field strength for an Abelian gauge field; χ is a complex valued scalar, and there are three degenerate types, or "flavors" of fermions, q i , with equal mass m q . The q i have unit charge, D µ = ∂ µ − ieA µ , but I choose the scalar to have charge three, D χ µ = ∂ µ − 3ieA µ . I consider the case of 1+1 dimensions first, and assume that the fermions are heavy. (Light fermions, m q e, may undergo spontaneous symmetry breaking, which because of the lack of Goldstone bosons in 1+1 dimensions, complicates the analysis [107,108,[162][163][164], and is really secondary to my desire to construct a theory for nuclear matter.) If m 2 χ < 0, spontaneous symmetry breaking occurs, and the photon becomes massive. For large distances, > 1/(3e|m χ |), naively one expects that there is no interaction from the photons, and the fermions propagate freely. Besides perturbative fluctuations, there are also vortices, which in two (Euclidean) dimensions are like pseudoparticles, localized at a given point. The vacuum is a superposition of vortices, where each vortex has an action S v ∼ (m χ ) 4 /λ χ . If χ had unit charge, the propagation of fermions is affected only when they are near a vortex, and the vortices are relatively inconsequential.
When χ has charge three, however, a vortex can carry a Z(3) charge, which greatly affects the propagation of the fermion. If a fermion of unit charge encircles a single vortex, it picks up an Aharonov-Bohm phase of exp(±2πi/3). With a vacuum composed of an infinite number of vortices, these phases confine [173], the fermions entirely through these random phases, exactly analogous to how Z(3) vortices in a SU (3) gauge theory confine [168].
While a state such as q 3 1 is neutral under Z(3), this vanishes, as q 1 is a fermion field which anti-commutes with itself. This is different from QCD, where the antisymmetric tensor in color space can be used to form a baryon with one flavor, ∼ abc q a 1 q b 1 q c 1 . Consequently, in a Z(3) model to obtain (simple) baryons it is necessary to take three flavors, so the baryon ∼ q 1 q 2 q 3 , is neutral under Z(3). The mesons form an octet in flavor, which is (presumably) lighter than the singlet meson (plus higher excitations, of course).
In weak coupling the action of a single vortex is small, vortices are dilute, and confinement occurs over large distances, ∼ exp(−S v ). The fermions interact over distance ∼ 1/m χ , but at long distances, only interact through the Z(3) phases generated by the vortex ensemble in vacuum. These Z(3) baryons are weakly bound over large distances, so that in any scattering experiment, it would be obvious that they have composite substructure. This is in contrast to QCD, where baryons have weak attraction at large distances, but a strong repulsive core at short distances.
That is, in QCD it is hard prying the quarks out of a baryon. This would occur if the density of vortices is large. In the effective model above, this requires strong coupling, which cannot be studied analytically. However, this limit can be studied on the lattice, and just produces a Z(3) gauge theory [174] coupled to three flavors of degenerate fermions.
In 1 + 1 dimensions, as for the U (1) gauge theory [175,176], the Z(N ) gauge theory confines. On a lattice, classical computers have been used to study the properties in vacuum of Z(2) [177][178][179][180][181] and Z(3) [178,181] gauge theories with a single flavor. The behavior of a U (1) theory with two flavors was computed at non-zero density in Ref. [182]. Thus classical computers can be used to compute the properties of a Z(3) gauge theory with three degenerate flavors at nonzero density. This can then provide a benchmark to compare against computing the free energy at nonzero density using quantum computers. The great advantage of a Z(3) gauge group is that only two qubits are needed to describe a group element, as opposed to many more for any continuous gauge group.
In 2 + 1 dimensions the vortices sweep our lines in space-time, and cylinders in 3 + 1 dimensions. Assuming that Z(3) vortices confine in QCD, these models should exhibit confinement as well. It would be interesting analyzing the behavior of Z(3) nuclear matter at strong coupling as a counterpoint to that in QCD. Centers under the award for the "Co-design Center for Quantum Advantage". After this work was completed, the nature of the QπL regime was developed further with A. Tsvelik [183]; the signatures of a QπL regime in heavy ion collisions were analyzed with F. Rennecke [184].
[129] R. Amorim and J.  [125] or BF [126][127][128][129][130] formalisms. With the BF formalism, auxiliary two-index gauge potentials B i αβ could be introduced, and defined to transform under 1-form gauge transformations λ i α as B i αβ → B i αβ − ∂αλ i β + ∂ β λ i α . Adding to the action a 3-index field strength tensor for B i αβ , integration over B i αβ generates the coupling ∼ κ 2 if the BF term is chosen as ∼ κ αβµν φ i B i αβ Fµν . However, such a BF term is gauge invariant only for constant φ i , and not for a dynamical field, where ∂αφ i = 0. This is unremarkable, given that mass terms for non-Abelian fields, such as the ρ and a1, also cannot be introduced in a form which respects gauge invariance and unitarity [125,130].