# Remarks on nuclear matter: how an $\omega_{0}$ condensate can spike the
speed of sound, and a model of $Z(3)$ baryons

Robert D. Pisarski  Department of Physics, Brookhaven National Laboratory,
Upton, NY 11973

###### Abstract

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) [1]. 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.

To determine the equation of state (EoS) for nuclear matter, effective models
can be used for baryon densities, $n_{B}$, up to and above that for nuclear
saturation, $n_{\rm sat}$ [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Neutron stars
probe densities $n_{B}>n_{\rm sat}$. In the past few years, the observation of
neutron stars with masses above two solar masses [12, 13] and astronomical
observations, especially of their mergers in binary systems [14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], have provided
a wealth of data. Many models apply above $n_{\rm sat}$ [34, 35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,
58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76,
77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95,
96, 97, 98, 99, 100, 101, 102], but to date, consensus is lacking.

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

##  I Spiking the speed of sound

In the past few years astronomical observations of neutron stars [12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]
have provided significant insight into the nuclear EoS at densities above
$n_{\rm sat}$. 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: $c_{s}^{2}=\partial p/\partial e$. Free, massless fermions have
$c_{s}^{2}=1/3$, 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 $c_{s}^{2}$ 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 $n_{\rm sat}$ 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 $n_{B}$, $c_{s}^{2}\sim 0.55$. To agree with small tidal
deformability from GW170817, though, the EoS of nuclear matter must be soft
until $n_{B}\sim 1.5-1.8n_{\rm sat}$.

That is, there is a “spike” in the speed of sound, with a relatively narrow
peak at a density significantly above $n_{\rm sat}$: see, e.g., Fig. (1) of
Greif et al. [24].

As the density $n_{B}\rightarrow\infty$, 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 $n_{\rm sat}$. 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 $SU(N_{\rm color})$
gauge theory as $N_{\rm color}\rightarrow\infty$, as then quark loops are
suppressed by $\sim 1/N_{\rm color}$. This does not seem to be special to
large $N_{\rm color}$, though. In lattice gauge theory, when $N_{\rm
color}\geq 3$ 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 $N_{\rm color}=3$.

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, 31, 30], 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 $k_{FQ}$, which is then surrounded by a baryonic
shell of width $\Delta$, Fig. (1) of Ref. [55]. As the baryon density
increases, $k_{FQ}$ grows, and $\Delta$ shrinks. Taking an ideal EoS for both
quarks and baryons, an appropriate choice of the width $\Delta$ generates a
spike in the speed of sound, Fig. (2) of Ref. [55].

At densities near $k_{FQ}$, 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 parity-doubled 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
$\omega_{0}$ condensate, and strong fluctuations in a pion “condensate” [103,
104, 105, 106, 107, 108, 109, 1] 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 $n_{\rm sat}$ and
at $n_{B}\gg n_{\rm sat}$.

In QHD, saturation results from a balance between repulsion from the
$\omega_{\mu}$ meson and attraction from the $\sigma$ meson [3, 4, 6]. That
the $\omega_{\mu}$ meson could generate a stiff EoS was first noted by
Zel’dovich [2]. Given the coupling of the $\omega_{\mu}$ to a nucleon $\psi$
as $\sim g_{\omega}\,\overline{\psi}\,\omega_{\mu}\gamma^{\mu}\,\psi$, then at
any nonzero baryon density,
$\langle\overline{\psi}\gamma^{0}\psi\rangle=n_{B}\neq 0$, a condensate for
$\omega_{0}$ is automatically generated 111As a vector field, an expectation
value for $\omega_{0}$ is generated whenever there is net baryon charge in the
system, and is also why the energy density increases. This is exactly like QED
with net charge, except that here the $\omega_{\mu}$ meson is massive. In
contrast, for the chiral condensate the energy density is lowered when it has
a nonzero expectation value. I thank L. McLerran for discussions on this
point.:

${\cal
L}_{\omega}^{B}=-g_{\omega}n_{B}\omega_{0}+\frac{m_{\omega}^{2}\omega_{\mu}^{2}}{2}\Rightarrow\langle\omega_{0}\rangle=\frac{g_{\omega}}{m_{\omega}^{2}}\,n_{B}\;.$
(1)

If only these terms matter, then the EoS is as stiff as possible, with the
speed of sound equal to that of light, $c_{s}^{2}=1$. 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
$\omega_{0}$ which are generated at $n_{B}\neq 0$, including those
$\sim\omega_{0}^{2}$, $\sim\omega_{0}^{3}$, and so on. Similarly, the nucleon
couples to the $\sigma$, whose properties also change with $n_{B}$. These
effects have been computed to one loop order [3, 4, 6], but even for strong
$g_{\omega}$, do not dramatically alter the EoS.

The $\omega_{\mu}$ Lagrangian is

${\cal L}_{\omega}=\frac{{\cal
F}_{\mu\nu}^{2}}{4}+\frac{1}{2}\left(\widetilde{m}_{\omega}^{2}+\kappa^{2}\vec{\phi}^{\,2}\right)\omega_{\mu}^{2}\;;$
(2)

${\cal F}_{\mu\nu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}$ is
the field strength for $\omega_{\mu}$, $\widetilde{m}_{\omega}$ a mass term,
and there is a quartic coupling $\sim\kappa^{2}$ between $\omega_{\mu}$ and
$\vec{\phi}$, where $\vec{\phi}$ is the $O(4)$ chiral field for two light
flavors, $\vec{\phi}=(\sigma,\vec{\pi})$. The coupling $\kappa^{2}$ must be
positive to ensure stability for large values of the $\omega_{\mu}$ and
$\vec{\phi}$ fields. (Incidentally, while the term
$\sim\widetilde{m}_{\omega}^{2}$ can be written in gauge invariant, unitarity
form [125, 126, 127, 128, 129, 130], that $\sim\kappa^{2}$ cannot [131].)

Although the coupling $\kappa^{2}$ violates vector meson dominance [132, 133,
134, 135, 136, 137, 138, 139, 91, 140, 92, 93], this is relatively innocuous.
For the $\rho_{\mu}$ meson, in vacuum a similar term
$\sim\kappa^{2}\vec{\rho}_{\mu}^{\,2}$ just shifts the $\rho_{\mu}$ mass, and
doesn’t alter its electromagnetic couplings.

In general the complete mass for the $\omega_{\mu}$ meson is

$m_{\omega}^{2}=\widetilde{m}_{\omega}^{2}+\kappa^{2}\,\langle\vec{\phi}^{\,2}\rangle\;.$
(3)

Spontaneous symmetry breaking occurs in the QCD vacuum,
$\langle\phi^{i}\rangle=(\sigma_{0},0)$, where for two flavors,
$\sigma_{0}=f_{\pi}$, the pion decay constant. Thus in vacuum, in mean field
theory the mass squared of the $\omega_{\mu}$ meson is
$m_{\omega}^{2}=\widetilde{m}_{\omega}^{2}+\kappa^{2}\sigma_{0}^{2}$. Large
$\widetilde{m}_{\omega}$ favors small $\kappa$, and vice versa. For massless
pions, at tree level $\widetilde{m}_{\omega}=0$ when
$\kappa^{2}=m_{\omega}/f_{\pi}\sim 8.4$.

Couplings similar to $\kappa^{2}$ have appeared before. In Refs. [8] and [36],
a chirally asymmetric term $\sim\omega_{\mu}^{2}\sigma^{2}$ was added to the
Lagrangian, but the generalization to a chirally symmetric term is obvious.
Ref. [73] introduced mixing the $\omega_{\mu}$ and $\vec{\rho}_{\mu}$ mesons,
$\sim\omega_{\mu}^{2}\vec{\rho}_{\mu}^{\;2}$, and note the possibility of
terms $\sim(\omega_{\mu}^{2})^{2}$, etc.. The implications of these terms for
neutron stars were computed in Refs. [36, 72, 74, 73, 97]. Refs. [70, 75, 71,
76] denote $\kappa^{2}$ as $h_{1}$, but neglect it, as $\kappa^{2}\sim
1/N_{\rm color}$ is small for a large number of colors. Ref. [79] denote
$\kappa^{2}$ as $g_{SV}$; for massive pions they find $\kappa\sim 9.0$, which
seems large compared to the upper bound of $\kappa^{2}<8.4$ for massless
pions.

As the density increases, chiral symmetry breaking becomes weaker, and
$\sigma_{0}$ decreases. Thus Eq. (3) implies that $m_{\omega}$ decreases with
$\sigma_{0}$, at least as long as
$\langle\vec{\phi}^{\,2}\rangle=\sigma_{0}^{2}$. This is reminiscent of the
scaling of Brown and Rho [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48].

For the $\vec{\phi}$ Lagrangian I take [1]

${\cal
L}_{\phi}=\frac{(\partial_{0}\vec{\phi})^{2}}{2}+\frac{(\partial_{i}^{2}\vec{\phi})^{2}}{2M^{2}}+\frac{Z(\partial_{i}\vec{\phi})^{2}}{2}+\frac{m_{0}^{2}\vec{\phi}^{\,2}}{2}+\frac{\lambda(\vec{\phi}^{\,2})^{2}}{4}\;,$
(4)

and work in the chiral limit, so there is no term linear in $\vec{\phi}$.
Notice that the $\omega_{\mu}$ meson does not appear [62, 63, 70, 71, 76, 77,
78, 79]. This is because the $\omega_{\mu}$ corresponds to the $U(1)_{B}$ of
baryon number, and with $q$ the quark fields, $\phi\sim\overline{q}q$ is
invariant under $U(1)_{B}$. Conversely, when $\vec{\rho}_{\mu}$ and
$\vec{a}_{1}^{\mu}$ mesons are added, they do appear in Eq. (4), since
$\vec{\phi}$ transforms non-trivially under $SU(2)_{L}\times SU(2)_{R}$. The
$\omega_{\mu}$ meson does interact with pions through anomalous interactions
generated by the Wess-Zumino-Witten Lagrangian, such as
$\omega_{\mu}\rightarrow 3\pi$ [63, 70, 71, 76, 75, 77, 78, 79, 80, 81, 82,
83]. These interactions survive in the chirally symmetric phase, and typically
become $\omega\rightarrow\sigma\pi\pi\pi$, Eq. (15) of [63].

The anomalous interactions, though, all involve at least three derivatives,
which for the $\omega_{0}$ meson, are all spatial derivatives. This is why the
coupling $\sim\kappa^{2}$ in Eq. (2) is so important, as the only
renormalizable, non-derivative coupling which the $\omega_{\mu}$ has with the
chiral field $\vec{\phi}$.

This assumes that processes which violate the axial $U(1)_{A}$ symmetry
survive at densities far above $n_{\rm sat}$, as indicated by an analysis
using a dilute gas of instantons [141]. If at zero temperature and nonzero
baryon density the axial $U(1)_{A}$ symmetry remains strongly broken by
topologically nontrivial configurations even when the $O(4)=SU(2)_{L}\times
SU(2)_{R}$ chiral symmetry is restored, then as the $\omega_{\mu}$ meson is a
chiral singlet, it need not become degenerate with its parity partner, which
is presumably the $f_{1}(1285)$ [75]. This is unlike mesons which carry
flavor, such as the $\rho_{\mu}$ and $a_{1}^{\mu}$, which are degenerate in a
chirally symmetric phase. Similarly, this is why I can restrict the chiral
symmetry to be $O(4)=SU(2)_{L}\times SU(2)_{R}$ and not $U_{A}(1)\times
SU(2)_{L}\times SU(2)_{R}$ [142].

Of course the $\omega_{\mu}$ meson interacts directly with nucleons [3, 4, 6,
81, 82, 83]. The CBELSA/TAPS experiment found that the mass of the
$\omega_{\mu}$ does not shift significantly at nuclear densities [143, 144],
although its width is over thirty times larger than in vacuum [145, 146, 147,
148, 149, 150]. This does not concur with QHD [3, 4, 6] nor Refs. [38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48], where the $\omega_{\mu}$ mass decreases by
$n_{\rm sat}$.

This does not exclude changes as the baryon density exceeds $n_{\rm sat}$. For
the usual analyses of QHD [3, 4, 6], Refs. [38, 39, 40, 41, 42, 43, 44, 45,
46, 47, 48], and PdQHD [64, 65, 66, 67, 68, 69, 70, 71, 76, 75, 77, 78, 79,
80], the $\sigma$ and $\omega_{0}$ masses both decrease, as the balance
between $\sigma$ attraction and $\omega_{0}$ repulsion gives a soft EoS.

My principal assumption is that for some $n_{B}>n_{1}>n_{\rm sat}$, that one
enters a region dominated by the $\omega_{0}$ condensate. Notably, if $Z$
decreases with increasing $n_{B}$, the effective mass squared of the $\sigma$
increases as $\sim 1/Z$, while if $\kappa\neq 0$, the $\omega_{0}$ becomes
light as the chiral symmetry is restored. By Eq. (1), ${\cal
L}_{\omega}^{B}=-g_{\omega}^{2}n_{B}^{2}/(2m_{\omega}^{2})$, and a heavy
$\sigma$, with a light $\omega_{0}$, could generate a stiff EoS for
$n_{B}>n_{1}$.

Assuming that a light $\omega_{0}$ gives a stiff EoS, then how can the
$\omega_{0}$ condensate evaporate to match onto a soft quarkyonic EoS?
Presumably the couplings of the $\omega_{\mu}$ with nucleons behave smoothly
with density. That leaves the couplings of the $\omega_{\mu}$ to the chiral
field $\vec{\phi}$, 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 $N_{\rm color}\rightarrow\infty$ [49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61] or lattice results for $N_{\rm
color}=2$ [116, 117, 118, 119, 120, 121], which indicate a quarkyonic regime.
Nor why the nuclear EoS appears to be soft near $n_{\rm sat}$, and only stiff
when $n_{B}\sim 1.5-1.8n_{\rm sat}$ [27].

I stress that reducing the contribution of the $\omega_{0}$ condensate at
large chemical potential, $\mu$, and low temperature, $\mu\gg T$, has no
analogy to the more familiar case, at nonzero temperature and low density.
When $T\gg\mu$, 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, $N_{\rm
color}\rightarrow\infty$, where the pressure in the hadronic phsae is $\sim
N_{\rm color}^{0}$, versus $\sim N_{\rm color}^{2}$ in the deconfined phase.
Similarly, the contribution of the chiral condensate is only $\sim N_{\rm
color}^{1}$, and decreases as $T$ increases. In contrast, at $\mu\gg T$, the
pressure is always $\sim N_{c}^{1}$, in both the hadronic and quark-gluon
phases.

At nonzero density, the appearance of a condensate for $\omega_{0}$ is special
to the $\omega_{\mu}$ 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
$\vec{\rho}_{\mu}$ meson is generated, $\sim\rho_{0}^{3}$. In this case, terms
such as $\vec{\phi}^{2}\vec{\rho}_{\mu}^{\;2}$, amongst others [70, 75, 71,
76], need to be included; further, couplings between the $\omega_{\mu}$ and
$\vec{\rho}_{\mu}$ mesons, $\sim\omega_{\mu}^{2}\vec{\rho}_{\mu}^{\;2}$, must
be added [73].

It is then very difficult to fit the EoS of an $\omega_{0}$ condensate onto
that of cold quarks: either the coupling of the $\omega_{0}$ becomes small, or
the mass of the $\omega_{0}$ becomes large. Since the coupling of the
$\omega_{0}$ is strong in vacuum, the former is most implausible. Thus the
mass of the $\omega_{0}$ must increase, although this does not occur in mean
field theory [151]. I now argue that the mass of the $\omega_{0}$ 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, $\sim 1/M^{2}$ [152]. Causality implies
that only terms with two time derivatives enter. With the term $\sim 1/M^{2}$
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:

$\vec{\phi}=\sigma_{0}(\cos(k_{c}z),\sin(k_{c}z),0,0)\;;\;k_{c}^{2}=\frac{-ZM^{2}}{2}\;,$
(5)

which is a pion condensate in the $z$ direction [103, 104, 105, 153, 106, 107,
108, 109, 1]. 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 $n_{\rm sat}$.

Most discussions of a pion condensate use a nonlinear Lagrangian, in which the
$\sigma$ 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 $N\rightarrow\infty$ [142, 154].

The solution at large $N$ is standard, and proceeds by introducing the a field
$\xi=\vec{\phi}^{\,2}$, and a constraint field, $\epsilon$, ${\cal L}_{\rm
cons}=i\epsilon(\xi-\vec{\phi}^{\,2})/2$. 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 $\vec{\phi}$ and $\xi$ fields are
integrated out to give the effective action

${\cal S}_{\rm eff}=\frac{N}{2}{\rm tr}\log\Delta^{-1}+\int
d^{4}x(\frac{\epsilon^{2}}{4\lambda}+\frac{\widetilde{m}_{\omega}^{2}\omega_{\mu}^{2}}{2}-g_{\omega}\omega_{0}\rho_{B});$
(6)

$\Delta^{-1}$ is the inverse propagator for the $\vec{\phi}$ field, which in
momentum space is $\Delta^{-1}(\omega,\vec{k})=\omega^{2}+E(k)^{2}$, where

$E(k)^{2}=\frac{(\vec{k}^{\,2})^{2}}{M^{2}}+Z\vec{k}^{\,2}+m_{\rm eff}^{2}\;,$
(7)

I expand about a stationary point in $\epsilon$ and $\omega_{0}$,
$\epsilon=i\widehat{\epsilon}+\epsilon_{q}$ and
$\omega_{0}=\widehat{\omega}_{0}+\omega_{0}^{q}$, where $\epsilon_{q}$ and
$\omega_{0}^{q}$ are quantum fluctuations. The effective mass $m_{\rm
eff}^{2}=m_{0}^{2}+\widehat{\epsilon}+\kappa^{2}\widehat{\omega}_{0}^{2}$. To
have a well defined limit for large $N$, as $N\rightarrow\infty$ all terms in
the action should scale as $\sim N$, so I take $\lambda,\kappa^{2}\sim 1/N$,
$g_{\omega}\rho_{B},\widehat{\omega}_{0}\sim\sqrt{N}$, and
$\widetilde{m}_{\omega}^{2},M^{2},Z,m_{0}^{2},\widehat{\epsilon},m_{\rm
eff}^{2}\sim N^{0}$. 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 $\epsilon_{q}$
and $\omega_{0}^{q}$ fixes $\widehat{\epsilon}$ and $\widehat{\omega}_{0}$,

$\widehat{\epsilon}=\lambda N{\rm
tr}\Delta\;\;;\;\;\widehat{\omega}_{0}=\frac{g_{\omega}\rho_{B}}{\widetilde{m}_{\omega}^{2}+\kappa^{2}N{\rm
tr}\Delta}\;.$ (8)

The solution for general values of the parameters is involved, so to make a
qualitative point I only consider the limit of $Z\rightarrow-\infty$, 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
$Z\rightarrow-\infty$, at $N=\infty$ take $m_{\rm eff}\approx-ZM/2+\delta
m_{\rm eff}$. About $k\approx k_{c}$,

$E(k\approx
k_{c})^{2}\approx\frac{1}{M^{2}}((k^{2}-k_{c}^{2})^{2}-ZM^{3}\delta m_{\rm
eff}+\ldots)\;.$ (9)

The loop integral is dominated by $k\approx k_{c}+\delta k$, and to leading
logarithmic order becomes

$\displaystyle{\rm tr}\Delta$ $\displaystyle\approx$
$\displaystyle\frac{\sqrt{-Z}M^{2}}{2^{5/2}\pi^{2}}\int\frac{d\delta
k}{\sqrt{(\delta k)^{2}+M\delta m_{\rm eff}/2}}$ (10) $\displaystyle\approx$
$\displaystyle\sqrt{-Z}\frac{M^{2}}{2^{7/2}\pi}\log\left(\frac{\\#\sqrt{-Z}M}{\delta
m_{\rm eff}}\right)\;.$

Solving Eq. (8) for $\widehat{\epsilon}$, as $Z\rightarrow-\infty$,

$\delta m_{\rm
eff}\approx\\#\sqrt{-Z}M\exp\left(-\;\frac{2^{3/2}\pi^{2}}{\lambda
N}(-Z)^{3/2}\right)\;,$ (12)

where $\\#$ is a positive, nonzero number.

It is worth contrasting this solution with that at nonzero temperature [1].
Then the integral over $\omega$ is a discrete sum, and the zero energy mode is
the most important. It generates a power law divergence, with the solution
$\delta m_{\rm eff}\approx 1/Z^{4}$, Eq. (58) of Ref. [1]. The statement in
Ref. [1] that $\delta m_{\rm eff}$ vanishes at zero temperature is incorrect:
it is just that $\delta m_{\rm eff}$ is suppressed exponentially in
$1/\sqrt{-Z}$, instead of by a power. I refer to this disorder as a quantum
pion liquid, Q$\pi$L [155].

I have neglected the equation for $\widehat{\omega}_{0}$ in Eq. (8). While
$\delta m_{\rm eff}^{2}$ is very different at zero and nonzero temperature,
though, what matters there is the value of the loop integral. Since
$\widehat{\epsilon}\approx m_{\rm eff}^{2}\sim Z^{2}M^{2}/4$, by Eq. (8)

${\rm tr}\Delta\approx Z^{2}\;\frac{M^{2}}{4\lambda
N}\;,\;Z\rightarrow-\infty\;,$ (13)

As $\langle\vec{\phi}^{\,2}\rangle=N{\rm tr}\Delta$, by Eq. (3) the
$\omega_{0}$ mass increases sharply,

$m_{\omega}^{2}=\widetilde{m}_{\omega}^{2}+Z^{2}\;\frac{\kappa^{2}}{4\lambda}\;M^{2}\;,$
(14)

and by Eq. (8) suppresses the $\omega_{0}$ condensate,
$\widehat{\omega}_{0}\sim 1/Z^{2}$. Note that the presence of the coupling
$\sim\kappa^{2}$ 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 (Q$\pi$L)
forms . While this is rigorous at large $N$, as it arises from the double pole
at $k_{c}\neq 0$ for the would be Goldstone modes, it is very likely that
there is a Q$\pi$L for all $N>2$ [1]. I also assume that the quantum
fluctuations are sufficiently strong so that a Q$\pi$L forms for massive
pions.

My suggestion is thus the following. For $n_{B}>n_{1}>n_{\rm sat}$, the theory
enters a phase dominated by the $\omega_{0}$ condensate, which stiffens the
EoS. When $n_{B}>n_{2}$, it is approximately described by a Q$\pi$L: both the
$\sigma$ and $\omega_{0}$ are heavy, which suppresses $\widehat{\omega}_{0}$.
In total, the enhancement and then suppression of the $\omega_{0}$ 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, $\mu_{B}$. This includes the $\mu_{B}$ dependence
of the wave function renormalization constant $Z$, the mass parameter $M$
(which is of some hadronic scale), $m_{0}$, $\lambda$, and so forth.

The most direct approach is to use PdQHD, with a self-consistent one loop
approximation for the nucleons, the chiral fields $\vec{\phi}$, and the
$\omega_{0}$. While involved, I comment that it is far simpler to look for a
Q$\pi$L - 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) [133, 134, 135, 136, 137, 138, 139, 91, 140, 92,
93, 137, 138, 139, 156, 157, 158, 159]. Ref. [91] use a chiral effective model
up to $\sim 2n_{\rm sat}$, 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 $\sim 10n_{\rm sat}$ [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
$\mu_{B}\neq 0$, 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
Q$\pi$L 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
$\langle\overline{s}s\rangle$ 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 $\omega_{0}$ and the chiral fields plays an essential role.

##  II 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
$SU(N_{\rm color})$ gauge theory in $1+1$ dimensions [161] has baryons [162,
163, 164, 153], but as $N_{\rm color}\rightarrow\infty$, there are $\sim
N_{\rm color}^{2}$ 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 $Z(N_{\rm color})$ vortices in a
$SU(N_{\rm color})$ 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 $SU(N_{\rm color})$ to retain just those
of $Z(N_{\rm color})$. A $Z(3)$ gauge theory is constructed following Krauss,
Wilczek, and Preskill [170, 171, 172]:

$\displaystyle{\cal L}_{Z(3)}$ $\displaystyle=$
$\displaystyle\frac{F^{2}_{\mu\nu}}{4}+|D_{\mu}^{\chi}\chi|^{2}+m_{\chi}^{2}|\chi|^{2}+\lambda_{\chi}(|\chi|^{2})^{2}$
(15) $\displaystyle+$
$\displaystyle\sum_{i=1}^{3}\overline{q}_{i}(D_{\mu}+m_{q})q_{i}\;,$

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field
strength for an Abelian gauge field; $\chi$ 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,
$D_{\mu}=\partial_{\mu}-ieA_{\mu}$, but I choose the scalar to have charge
three, $D^{\chi}_{\mu}=\partial_{\mu}-3ieA_{\mu}$.

I consider the case of $1+1$ dimensions first, and assume that the fermions
are heavy. (Light fermions, $m_{q}\ll e$, may undergo spontaneous symmetry
breaking, which because of the lack of Goldstone bosons in $1+1$ dimensions,
complicates the analysis [162, 163, 164, 107, 108], and is really secondary to
my desire to construct a theory for nuclear matter.) If $m_{\chi}^{2}<0$,
spontaneous symmetry breaking occurs, and the photon becomes massive. For
large distances, $>1/(3e|m_{\chi}|)$, 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
$S_{v}\sim(m_{\chi})^{4}/\lambda_{\chi}$. If $\chi$ had unit charge, the
propagation of fermions is affected only when they are near a vortex, and the
vortices are relatively inconsequential.

When $\chi$ 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
$\exp(\pm 2\pi i/3)$. With a vacuum composed of an infinite number of
vortices, these phases confine 222My language is common but imprecise: since
the fermions carry $Z(3)$ charge, any flux string can always break by the
production of fermion anti-fermion pairs. By confinement I mean that there is
a mass gap for all particles in the spectrum, which by necessity are gauge
singlets., 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 $q_{1}^{3}$ 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 anti-symmetric tensor in color space can be used to form a
baryon with one flavor, $\sim\epsilon^{abc}q_{1}^{a}q_{1}^{b}q_{1}^{c}$.
Consequently, in a $Z(3)$ model to obtain (simple) baryons it is necessary to
take three flavors, so the baryon $\sim q_{1}q_{2}q_{3}$, 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, $\sim\exp(-S_{v})$. The fermions
interact over distance $\sim 1/m_{\chi}$, 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.

###### Acknowledgements.

I thank S. Carignano, L. Classen, V. Dexheimer, T. Izubuchi, Y. Kikuchi, Y.-L.
Ma, L. McLerran, F. Rennecke, M. Rho, D. Rischke, F. Tartaglia, A. Tsvelik,
and A. Tomiya for discusions; also Brian Serot, albeit some time ago, about
the ineluctable virtues of cockatiels and QHD; and S. Reddy, for discussions
on the role of an $\omega_{0}$ condensate. This research was supported by the
U.S. Department of Energy under contract DE-SC0012704; the work in Sec. I, by
B.N.L. under the Lab Directed Research and Development program 18-036; the
work in Sec. II, by the U.S. Department of Energy, Office of Science National
Quantum Information Science Research Centers under the award for the “Co-
design Center for Quantum Advantage”. After this work was completed, the
nature of the Q$\pi$L regime was developed further with A. Tsvelik [183]; the
signatures of a Q$\pi$L regime in heavy ion collisions were analyzed with F.
Rennecke [184].

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