# Effects of hyperon potentials and symmetry energy in quark deconfinement

Rajesh Kumar<EMAIL_ADDRESS>Krishna Aryal  Alexander Clevinger  Veronica
Dexheimer<EMAIL_ADDRESS>Department of Physics, Kent State University,
Kent, Ohio, 44242, USA

###### Abstract

In this letter we discuss how the results of recent nuclear experiments that
correspond to measurements at low densities can affect the equation of state
at large densities, changing the particle composition and ultimately
influencing deconfinement to quark matter. In particular, saturation values of
the hyperon potentials affect the hyperon content, while the symmetry energy
at saturation directly regulates how the stiffness of the equation of state
changes with isospin. We make use of a chiral model that describes nucleons,
hyperons, and quarks to show how astrophysical conditions, such as the ones in
neutron stars, present the ideal ground to study the effects of these two
quantities in dense matter. In this case, for small charge fraction/ large
isospin asymmetry, the couplings that reproduce different symmetry energy
slopes can significantly modify deconfinement, with quantitative changes in
the critical chemical potential depending on the deconfining potential. On the
other hand, different values of the parameter that controls the hyperon
potentials do not affect deconfinement significantly.

###### keywords:

Hyperons , Phase transition , Quark matter , QCD phase diagram , Deconfinement

##  1 Introduction

Recent nuclear experiments have reopened the discussion of the effects of
strangeness and isospin in dense matter. First, the updated Pb Radius
Experiment (PREX-II) reported on the neutron skin thickness $R_{\rm{skin}}$
[1]. This is directly related to the increase in energy for isospin-asymmetric
systems, such as neutron stars. The difference between the energy per nucleon
of fully asymmetric and symmetric matter at saturation density $n_{\rm{sat}}$,
is referred to as the symmetry energy $E_{\rm{sym}}$, or $S$. But, most
importantly, its derivative with respect to (baryon number) density, $L$,
determines how the symmetry energy increases as the system becomes more dense
with baryons or quarks, e.g., going towards the center of a neutron star.
There is a strong correlation between $R_{\rm{skin}}$ and $L$ [2] that allows
neutron skin thickness to be used to infer values of symmetry energy. One
issue is that, while PREX-II points to high values $L=(106\pm 37)$ MeV [3],
another recent PREX-II study of parity-violating asymmetry provides a much
smaller value of $L=(54\pm 8)$ MeV [4]. Furthermore, neutron skin thickness
calculated from the proton elastic scattering (RHIC) and ultra-relativistic
nuclear collisions (LHC) of 208Pb can also provide a constraint on $L$,
resulting in $L=60\pm 30$ MeV [5] and $L=79\pm 39$ MeV [6], respectively. In
fig. 5 of Ref. [7] the authors provide a compilation of terrestrial and
astrophysical analyses of $L$. Results for the neutron skin thickness also
relate to the tidal deformability $\tilde{\Lambda}$ of neutron stars, and
larger values of $R_{\rm{skin}}$ correspond to
$642\lessapprox\tilde{\Lambda}\lessapprox 955$ [3], in tension with values
estimated from LIGO (with minimal assumptions) $\tilde{\Lambda}\leq 630$ [8].

Different experiments taking place at CERN have recently brought back the
discussion of hyperon potentials. In relativistic models, the hyperon
potentials are defined as $U_{H}=$ vector interaction $+$ scalar interaction,
usually discussed at $n_{\rm{sat}}$ for isospin-symmetric matter. Since the
emulsion and bubble chamber experiments from the 1980’s, it has been known
that the $\Lambda$-nucleon potential is negative with $U_{\Lambda}\sim-28$ MeV
[9], with more recent estimates obtaining slightly lower values
$-32<U_{\Lambda}<-30$ MeV [10]. A couple of decades later, potentials for the
$\Sigma$ and $\Xi$ hyperons were also measured. It was found that the $\Sigma$
potential is repulsive, estimated using measurements at KEK (the High Energy
Accelerator Research Organization in Japan) to be $U_{\Sigma}=30\pm 20$ MeV
[11], and that the $\Xi$ potential is attractive, measured by KEK and J-PARC
(the Japan Proton Accelerator Research Complex) to be $U_{\Xi}=-21.9\pm 0.7$
MeV. The latter is much lower than what had been previously found, e.g.,
$U_{\Xi}=-14$ MeV [12, 11]. Finally, a much less attractive potential was
found from p-$\Xi$ correlation functions from the ALICE collaboration using
three-momenta measured at center of mass energy $s=\sqrt{13}$ TeV [13, 14]. No
error was provided, but their potential $U_{\Xi}=-4$ MeV agrees with the one
calculated by the HAL-QCD collaboration. The latter used hyperon interactions
extracted from (2+1)D lattice QCD in a Brueckner-Hartree-Fock (BHF)
calculation to obtain $U_{\Xi}=-4$ MeV, in addition to $U_{\Lambda}=-28$ MeV
and $U_{\Sigma}=+15$ MeV, with a statistical error of approximately $\pm 2$
MeV [15].

The effects of hyperon potentials [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34] and symmetry-energy slope [35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57] beyond
saturation have already been explored substantially. In this work we go beyond
prior analyses, by considering their effects on quark deconfinement at finite
temperature and out of weak ($\beta$) equilibrium. We make use of the Chiral
Mean Field (CMF) model, which includes a consistent treatment of deconfinement
to build multidimensional QCD (or high-energy) phase diagrams. We investigate
parametrizations and couplings that reproduce different hyperon potentials and
symmetry-energy slopes at saturation and determine what they imply for quark
deconfinement. We also explore different deconfinement potentials that have
different dependence on the density. We do this for different temperatures,
charge fractions, and strangeness constraints. The structure of our paper is
as follows. In Section 2, we briefly summarize the formalism adopted to
describe dense matter. In Section 3, we describe and discuss our results.
Finally, we present our conclusions in Section 4.

##  2 Formalism

To study dense matter, we employ the Chiral Mean Field (CMF) model. It is a
relativistic SU(3) effective chiral model, which has been modified to describe
quark degrees of freedom, in addition to baryons. As a consequence, it allows
for different strengths of the deconfinement phase transition to be
reproduced, including a smooth crossover at large temperatures [58]. The
Lagrangian density is given by

$\mathcal{L}=\mathcal{L}_{\rm{Kin}}+\mathcal{L}_{\rm{Int}}+\mathcal{L}_{\rm{Self}}+\mathcal{L}_{\rm{SB}}-U_{\Phi},$
(1)

where the first term is the hadron and quark kinetic-energy density. The
mesons do not contribute kinetically due to the use of the mean-field
approximation. The other terms include an interaction term between the baryons
or quarks and the (mean-field) mesons, a self-interaction term for the scalar
and vector mesons, an explicit symmetry-breaking term, and a potential for the
deconfining order parameter. The baryons included are nucleons and hyperons,
the quarks are the $3$ light ones, and the mesons are: the scalar-isoscalars
$\sigma$ and $\zeta$, scalar-isovector $\delta$, vector-isoscalars $\omega$
and $\phi$, and vector-isovector $\rho$. Both $\zeta$ and $\phi$ are strange
quark-antiquark states.

In particular, adding a separate mixed vector-isovector $\omega\rho$
interaction in the self-interaction term of vector mesons (first introduced in
a Walecka-type model [59]) can directly influence the symmetry-energy slope,
as demonstrated for zero temperature $\beta$-equilibrated hadronic matter for
the CMF and other relativistic models [60]. It has the form

$\mathcal{L}_{\omega\rho}=g_{\omega\rho}\omega_{\mu}\omega^{\mu}\rho_{\mu}\rho^{\mu},$
(2)

where $g_{\omega\rho}$ is the coupling term for $\omega\rho$ interaction. The
stronger the coupling, the softer the isospin-asymmetric equation of state
(EoS, pressure vs. energy density) at low/intermediate densities, and the
lower the symmetry-energy slope. This interaction also lowers the radii of
intermediate-mass ($\sim 1.4$ M⊙) neutron stars and their tidal
deformabilities, while not affecting their masses substantially [60]. One can
also add higher-order vector interactions of the form

$\mathcal{L}_{\omega^{4}}=g_{\omega^{4}}\omega^{4},$ (3)

where $g_{\omega^{4}}$ is a coupling constant. This correction can compensate
for the small decrease in neutron-star mass (resulting from the previous
interaction) by stiffening the EOS at large densities and, as a consequence,
increasing the mass of heavy neutron stars. This has been shown for the CMF
model for zero-temperature $\beta$-equilibrated matter [61].

In the CMF formalism, the degrees of freedom change from hadronic to quark as
the baryon chemical potential $\mu_{B}$ or temperature $T$ increase. This is
because the effective masses of baryons $B$ and quarks $q$, in addition to
being generated by the scalar mesons (or quark condensates), also depend on
the deconfinement order parameter $\Phi$

$\displaystyle M_{B}^{*}$
$\displaystyle=g_{B\sigma}\sigma+g_{B\delta}\tau_{3}\delta+g_{B\zeta}\zeta+M_{0_{B}}+g_{B\Phi}\Phi^{2},$
$\displaystyle M_{q}^{*}$
$\displaystyle=g_{q\sigma}\sigma+g_{q\delta}\tau_{3}\delta+g_{q\zeta}\zeta+M_{0_{q}}+g_{q\Phi}(1-\Phi),$
(4)

and on a mass correction $M_{0_{i}}$, which contributes to the masses of
nucleons, hyperons, and quarks as follows

$\displaystyle M_{0_{N}}$ $\displaystyle=m_{0}=50~{}\rm{MeV},\ \quad
M_{0_{H}}=m_{0}-m_{3}\left(\sqrt{2}\sigma_{0}+\zeta_{0}\right),$
$\displaystyle M_{0_{u}}$ $\displaystyle=M_{0_{d}}=5~{}\rm{MeV},\quad
M_{0_{s}}=150~{}\rm{MeV},$ (5)

with $m_{0}$ being a bare mass term for baryons, $M_{0_{q}}$ being the Higgs
contribution to the quarks masses, and $m_{3}$ being an explicit breaking
symmetry term (hereon referred to as ExpSB). Furthermore, similar to the
Polyakov loop dynamics [62], as $\Phi$ changes $0\to 1$, the effective masses
of baryons become very large and the quarks become low, signaling
deconfinement.

Figure 1: Hyperon potentials as a function of explicit symmetry-breaking
coupling for isospin-symmetric matter, $Y_{Q}=0.5$, at zero temperature at
saturation. Horizontal lines show constraint ranges discussed in the
introduction.

The original CMF potential for $\Phi$ is

$\displaystyle U_{\Phi}$
$\displaystyle=\left(a_{0}T^{4}+a_{1}\mu_{B}^{4}+a_{2}T^{2}\mu_{B}^{2}\right)\Phi^{2}$
$\displaystyle+a_{3}T_{0}^{4}\ln\left(1-6\Phi^{2}+8\Phi^{3}-3\Phi^{4}\right),$
(6)

where $a$’s and $T_{0}$ are parameters fitted to constraints from the QCD
phase diagram [58]. We also make use of a new potential

$\displaystyle U^{\prime}_{\Phi}$
$\displaystyle=\left(a^{\prime}_{0}T^{4}+a^{\prime}_{1}\mu_{B}^{2}+a^{\prime}_{2}T^{2}\mu_{B}^{2}\right)\Phi^{2}$
$\displaystyle+a_{3}T_{0}^{4}\ln\left(1-6\Phi^{2}+8\Phi^{3}-3\Phi^{4}\right),$
(7)

used with a different $g^{\prime}_{B\Phi}$, $g^{\prime}_{q\Phi}$.
$U^{\prime}_{\Phi}$ that was derived in Ref. [61] and later used in Ref. [63],
in both cases only at zero temperature for $\beta$-equilibrated matter. It was
shown that the weaker dependence on density or $\mu_{B}$ gives rise to a
weaker first-order phase transition at low temperatures and the vector
interactions for the quarks allows for pure quark matter to be present in
stable hybrid stars without resorting to mixtures of phases.

##  3 Results and Discussions

We start by illustrating for the first time the influence of the dimensionless
parameter ExpSB on the hyperon potentials for isospin-symmetric matter at
$n_{\rm{sat}}$. Usually, it is fixed to ExpSB $=0.86$ in order to produce
$U_{\Lambda}=-28$ MeV, but in this work we vary ExpSB from $0.76$ to $0.95$ in
order to vary $U_{\Lambda}$ by $\pm 5$ MeV. Figure 1 shows how all three
hyperon potentials change linearly with the change in ExpSB, with
$U_{\Lambda}$ and $U_{\Xi}$ always remaining negative, but with $U_{\Xi}$
changing sign. The values in Figure 1 lie within or are in the vicinity of
predictions from theory and experiments (indicated by horizontal lines in the
figure). Note that the only experimental value for $U_{\Sigma}$ has a very
large error bar, of $40$ MeV, because it is based on a compilation of
experimental and theoretical results. Its range has the same sign but it is
more repulsive than our results.

Figure 2: Temperature vs free energy phase diagram for matter without
constraints on strangeness shown for different charge fractions and explicit
symmetry-breaking couplings. The full lines overlap.

But, more importantly than studying specific values, we are interested in
determining if changes in ExpSB (that imply changes in hyperon potentials)
affect particle composition, specifically quark deconfinement. In addition, we
would like to, for the first time, study ExpSB effects at finite temperature.
To do so, we built QCD phase diagrams for different conditions (corresponding
to laboratory vs. astrophysics), easily identified by the degree of isospin
asymmetry or electric charge and strangeness. In particular, we work with
unitless fractions, such as the charge fraction (per baryon)

$Y_{Q}=\frac{Q}{B}=\frac{\sum_{i}Q_{i}n_{i}}{\sum_{i}Q_{B_{i}}n_{i}},$ (8)

where $Q_{i}$ is the electric charge, $n_{i}$ the density, and $Q_{B_{i}}$ the
baryon number of each particle (1 for baryons and 1/3 for quarks). In our
figures we plot the results as a function of free energy (per baryon) defined
for the cases we study as

$\tilde{\mu}=\mu_{B}+Y_{Q}\mu_{Q}+Y_{S}\mu_{S},$ (9)

where $\mu_{Q}$ is the charge (or equivalently the isospin) and $\mu_{S}$ the
strange chemical potential. $Y_{S}$ is the strangeness fraction (per baryon)

$Y_{S}=\frac{S}{B}=\frac{\sum_{i}S_{i}n_{i}}{\sum_{i}Q_{B_{i}}n_{i}},$ (10)

where $S_{i}$ is the strangeness of each particle. In the particular case of
highly isospin-asymmetric matter without any constraint on strangeness,
$Y_{Q}=0$ and $Y_{S}\neq 0$ (similar to astrophysical conditions, with
$\mu_{S}=0$), or isospin-symmetric matter with zero net strangeness,
$Y_{Q}=0.5$ and $Y_{S}=0$ (similar to laboratory conditions, with
$\mu_{Q}=0$), one has $\tilde{\mu}=\mu_{B}$.

Figure 3: Particle population as a function of free energy (or baryon chemical
potential) for zero-temperature, charge fraction $Y_{Q}=0$ matter with no
constraints on strangeness shown for different values of explicit symmetry-
breaking couplings. Quark densities were divided by $3$.

Starting with $Y_{S}\neq 0$, which is relevant for astrophysical time scales,
in Figure 2 we present the changes in the deconfinement phase coexistence line
that arise when ExpSB is varied within the range shown in Figure 1. The
difference is very small, only perceptible for the cases of low $Y_{Q}$, when
the hyperons are favored, and does not depend significantly on $T$. To
understand this result, we need to look at the particle content reproduced by
these parameters. Figure 3 depicts the particle population density for the
case of $Y_{Q}=0$ and $T=0$. In the hadronic phase only neutrons, protons,
$\Lambda$’s, and $\Sigma^{-}$’s appear, being the other hyperons suppressed by
the first-order phase transition to quark matter, with up- and down-quarks,
and later the strange one. Protons and $\Sigma^{-}$’s overlap due to imposing
charge neutrality. The EoS contributions from leptons are small, and therefore
not considered in this work. A larger value of ExpSB enhances the values of
the hyperon potentials, making the $\Lambda$’s and the $\Sigma$’s more
massive, more energetically costly to produce, and, as a consequence,
suppressed. Matter with fewer hyperons is stiffer and, consequently, the phase
transition takes place at larger energy densities, which correspond to
slightly lower baryon chemical potentials or free energies.

Figure 4: Temperature vs free energy (or baryon chemical potential) phase
diagram for charge fraction $Y_{Q}=0$ matter with ExpSB $=0.86$ shown for
matter with and without zero-net strangeness and different higher-order
couplings. Figure 5: Particle population as a function of free energy (or
baryon chemical potential) for zero-temperature, charge fraction $Y_{Q}=0$
matter with ExpSB $=0.86$ and no constraints on strangeness shown for
different higher-order couplings.

As the effect of ExpSB on the phase diagram is very small (it would be even
smaller for $Y_{S}=0$), we continue our analysis using the central value of
ExpSB $=0.86$. It reproduces $U_{\Lambda}=-28$ MeV, $U_{\Sigma}=5$ MeV,
$U_{\Xi}=-18$ MeV. Now we discuss the effects of different higher-order vector
couplings, $\omega\rho$ (isoscalar-isovector) and $\omega^{4}$ (isoscalar of
maximum order that does not violate causality, see the second footnote in Ref.
[61]). The $\omega^{4}$ coupling changes the hyperon potentials for isospin-
symmetric matter at saturation slightly to $U_{\Lambda}=-27$ MeV,
$U_{\Sigma}=6$ MeV, $U_{\Xi}=-17$ MeV. The $\omega\rho$ coupling does not
change isospin-symmetric matter. Figure 4 shows that the changes due to the
different couplings on deconfinement are larger at lower temperatures (see
different orange lines), where the coupling effects do not compete with pure
thermal effects. The $\omega\rho$ interaction pushes deconfinement to a larger
free energy, while $\omega^{4}$ has the opposite (and weaker) effect. Once
more, this is due to the matter softening (stiffening) caused by the
$\omega\rho$ ($\omega^{4}$) interactions.

In particular, the additional $\omega\rho$ interaction decreases the cost of
producing isospin asymmetry, having the direct effect of decreasing the
symmetry-energy slope $L$ from 88 to 75 MeV, thus increasing the isospin
asymmetry, and consequently modifying the hyperon population at low and
intermediate densities or $\mu_{B}$’s (see Figure 5). The $\omega^{4}$
interaction, on the other hand, only affects the EoS of matter at large
densities, but it does it directly, by contributing positively to the pressure
and negatively to the energy density. Small differences at low and
intermediate densities in this case are due to a slightly different $\omega$
coupling to the baryons used to fit saturation properties.

The effects of different higher-order vector couplings are also present (and
slightly larger) if we constrain matter to have zero net strangeness
($Y_{S}=0$); see black lines in Figure 4, in which case a non-zero strange
chemical potential $\mu_{S}$ is introduced (see also figure with discussion in
the Appendix). This zero temperature case does not allow for hyperons or
strange quarks and demonstrates that the effects of different higher-order
vector couplings go beyond strangeness effects. For a given coupling, allowing
for net strangeness ($Y_{S}\neq 0$) pushes the deconfinement phase transition
to large free energies and the difference is larger for larger (fixed)
temperatures, when the diagram bends leftwards (see extensive discussion in
Ref. [64]).

Figure 6: Same as Figure 4 but for a different charge fraction, $Y_{Q}=0.25$.

Figure 6 shows that for a larger charge fraction (or, equivalently, more
isospin symmetry) the pattern illustrated in Figure 4 persists, but with the
differences between the couplings and different choices of strangeness being
much smaller. As naturally expected, the isovector coupling is not as relevant
for more isospin-symmetric matter, as well as the amount of strangeness
present. Additionally, the small $\omega\rho$ difference present is now
approximately balanced by the opposite change from the $\omega^{4}$ coupling.
For even larger charge fractions (not shown here) the $\omega\rho$ interaction
has no effect, only the $\omega^{4}$.

We now consider the dependence of our results on the deconfining mechanism
itself; we study a different deconfining potential $U^{\prime}_{\Phi}$ and
discuss for the first time how this affects the QCD phase diagram. We start by
looking at the effects of ExpSB, which again are very small (not shown here).
Then, we consider different higher-order couplings for $Y_{Q}=0$, for which
the differences are very large. When comparing Figure 7 with Figure 4, we
immediately see that, when using $U^{\prime}_{\Phi}$, different couplings or
strangeness constraints generate much larger changes in the deconfining
$\tilde{\mu}$. The deconfinement temperatures are also much lower for
$U^{\prime}_{\Phi}$. Both facts are related to $U^{\prime}_{\Phi}$ producing a
much weaker first-order phase transition (than using the original potential
$U_{\Phi}$). The trend of having a larger deconfining $\tilde{\mu}$ for matter
with net strangeness (mostly) persists. On the other hand, the order of the
deconfining $\tilde{\mu}$’s for the different couplings depends on the $\Phi$
potential, with $\omega\rho$ and $\omega\rho+\omega^{4}$ exchanging order for
most of the temperature range analyzed.

Looking at the particle population for zero-temperature and $Y_{Q}=0$, Figure
8 shows that (when comparing with Figure 5) the differences are not in the
hadronic phase, but in the quark one. When using $U^{\prime}_{\Phi}$, there
are much fewer quarks and no strange quarks. The stiffening of the EoS, now
presenting quark vector interactions, pushes the phase transition to lower
energy densities (corresponding to lower free energies). The order of the
phase transition lines also changes, where for $U^{\prime}_{\Phi}$ the
$\omega\rho+\omega^{4}$ the phase transition takes place at higher energy
densities and free energies. This is due to how quark matter gets softer in
the presence of the additional $\omega^{4}$ interaction, now pressure being
lower as a function of both energy density and free energy (or baryon chemical
potential). This correspondence is unusual but not unphysical.

Figure 7: Same as Figure 4 but for a different deconfining potential
$U^{\prime}_{\Phi}$.

Overall, results for the $U^{\prime}_{\Phi}$ potential are much more dependent
on $Y_{Q}$, to the point that, for large $Y_{Q}$’s the first-order phase
transition disappears, as it becomes of higher order. To understand this, we
go back to $U_{\Phi}$, where a small change in temperature for the critical
point for different $Y_{Q}$’s had already been pointed out [65]. In this work,
we show that for $U^{\prime}_{\Phi}$ this feature is exacerbated, such that
the critical point goes down all the way to $T=0$. The values at which this
change of phase transition order takes place are summarized in Table 1.

Figure 8: Same as Figure 5 but for a different deconfining potential
$U^{\prime}_{\Phi}$.

##  4 Conclusion

Table 1: Values of charge fraction for which a first-order phase transition is found for $U^{\prime}_{\Phi}$ shown for different strangeness constraints and higher-order vector couplings. | std. |  std.+ $\omega\rho$ |  std. $+\omega\rho+\omega^{4}$
---|---|---|---
$Y_{S}\neq 0$ | $Y_{Q}=0-0.14$ | $Y_{Q}=0-0.04$ | $Y_{Q}=0-0.04$
$Y_{S}=0$ | $Y_{Q}=0-0.15$ | $Y_{Q}=0-0.07$ | $Y_{Q}=0-0.06$

The three pieces still missing from our understanding of dense matter are: the
quantification of the softness generated by the appearance of hyperons, the
quantification of the stiffness generated by isospin-asymmetric matter, and
how/where deconfinement to quark matter takes place. In this work, we
considered all these pieces, using one consistent description to study how
couplings that directly affect hyperon potentials (and their appearance in
dense matter) and how vector interactions that directly affect isospin can
change the free energy/temperature and strength of deconfinement to quark
matter. We do not assume weak ($\beta$) equilibrium but vary the charge
fraction and also study different possibilities for strangeness (either
constrained to be zero or not).

We find that hyperon potentials $U_{H}$ varied within or near the experimental
range (of $\sim 10$ MeV each) do not significantly affect deconfinement, but
higher-order interactions that modify the isospin sensitivity of the density,
$L$, by $7$ MeV do. The changes are more significant for low temperature, non-
constrained strangeness, and lower charge fraction $Y_{Q}$ (or very isospin-
asymmetric matter), pushing the deconfinement coexistence free energy by up to
$15$ MeV. These conditions are exactly the ones reproduced by neutron stars.
To verify the consistency of our results, we also investigate a different
deconfining potential $U^{\prime}(\Phi)$, which has a weaker dependence on
baryon chemical potential and reproduces, as a consequence, a weaker first-
order phase transition. As a result, using $U^{\prime}(\Phi)$, higher-order
vector interactions (that modify $L$) can increase the deconfinement
coexistence free energy much more, up to $185$ MeV.

Together, these results point to the importance of understanding recent
controversies surrounding the interpretation of PREXII results, especially for
astrophysics. While we were finalizing this work, Ref. [36] appeared online.
There, the authors study the effect of $L$ on $\beta$-equilibrated matter at
$T=0$ using a modified Walecka model combined with the vector-bag model. Their
results point to lower $L$ values being associated with a higher critical
baryon chemical potential for the phase transition (in agreement with our
results), thereby decreasing the quark content and increasing the maximum mass
of neutron stars.

##  5 Appendix

As discussed in Section 3, constraining matter (in our case to zero-net
strangeness, $Y_{S}$=0) gives rise to a non-zero $\mu_{S}$. In Figure A1 we
show $\mu_{S}$ as a function of free energy for two chosen temperatures for
the cases corresponding to the black lines in Figures 4, 6 and 7. In every
scenario, the absolute value of $\mu_{S}$ increases with increasing $\mu_{B}$,
and falls at (or around for large temperatures close to the critical point)
the phase transition. The cost to control strangeness is highest for
$U_{\Phi}$ with $Y_{Q}=0$, decreasing for $U_{\Phi}$ with $Y_{Q}=0.25$, and is
lowest for $U^{\prime}_{\Phi}$ with $Y_{Q}=0$. The difference in the hadronic
phase behavior between $U_{\Phi}$ and $U^{\prime}_{\Phi}$ is due to quarks
appearing in the hadronic phase at large temperatures [58].

Figure A1: Strange chemical potential as a function of free energy for all
cases discussed. Full lines show standard higher-order interactions, while
dashed also include $\omega\rho$ and dotted lines $\omega\rho$ and
$\omega^{4}$ terms.

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