# Large $N_{c}$ QCD phase diagram at $\mu_{B}=0$.

T.D. Cohena, L. Ya. Glozmanb a Department of Physics and Maryland Center for
Fundamental Physics, University of Maryland, College Park, MD 20742 USA
b Institute of Physics, University of Graz, A-8010 Graz, Austria

###### Abstract

Lattice studies suggest that at zero baryon chemical potential and increasing
temperature there are three characteristic regimes in QCD that are connected
by smooth analytical crossovers: a hadron gas regime at $T<T_{ch}\sim 155$
MeV, an intermediate regime, called stringy fluid, at $T_{ch}<T<\sim 3T_{ch}$,
and a quark-gluon plasma regime at higher temperatures. These regimes have
been interpreted to reflect different approximate symmetries and effective
degrees of freedom. In the hadron gas the effective degrees of freedom are
hadrons and the approximate chiral symmetry of QCD is spontaneously broken.
The intermediate regime has been interpreted as lacking spontaneous chiral
symmetry breaking along with the emergence of new approximate symmetry, chiral
spin symmetry, that is not a symmetry of the Dirac Lagrangian, but is a
symmetry of the confining part of the QCD Lagrangian. While the high
temperature regime is the usual quark-gluon plasma which is often considered
to reflect “deconfinement” in some way. This paper explores the behavior of
these regimes of QCD as the number of colors in the theory, $N_{c}$, gets
large. In the large $N_{c}$ limit the theory is center-symmetric and notions
of confinement and deconfinement are unambiguous. The energy density is ${\cal
O}(N_{c}^{0})$ in the meson gas, ${\cal O}(N_{c}^{1})$ in the intermediate
regime and ${\cal O}(N_{c}^{2})$ in the quark-gluon plasma regime. In the
large $N_{c}$ limit these regimes may become distinct phases separated by
first order phase transitions. The intermediate phase has the peculiar feature
that glueballs should exist and have properties that are unchanged from what
is seen in the vacuum (up to $1/N_{c}$ corrections), while the ordinary dilute
gas of mesons with broken chiral symmetry disappears and approximate chiral
spin symmetry should emerge.

###### keywords:

QCD phase diagram; chiral spin symmetry; stringy fluid; quark gluon plasma

††journal: Nuclear Physics B

##  1 Introduction

In the early days of QCD there was a widespread belief that QCD had a
deconfinement phase transition at the Hagedorn temperature in which the system
went from a dilute hadron resonance gas to a weakly interacting quark-gluon
plasma[1, 2, 3]. However, it was recognized long ago that this picture is too
simple. Moreover, RHIC experiments have established that the matter created
within a fireball while very different from a dilute hadron gas, is also very
different from a weakly coupled quark-gluon plasma (QGP)[4, 5, 6, 7, 8]. These
properties have been reconfirmed at higher temperatures at the LHC. A
prominent characteristic of this matter is a small value $\eta/s\sim 0.2$
(where $\eta$ is the shear viscosity and $s$ the entropy density) both at RHIC
and LHC temperatures [9, 10]; this implies that this matter behaves as a
strongly coupled highly collective liquid with a very small mean free path of
the effective constituents. Another indication of the strongly-coupled nature
of the matter within the fireball is a modification of a jet that propagates
through the fireball[11]. It was often assumed that this matter is a system of
deconfined strongly interacting quark and gluon quasiparticles which was
reflected in the name sQGP.

In parallel with the experimental activity, hot QCD matter in the equilibrium
at a given temperature was investigated on the lattice. At zero chemical
potential a smooth analytic crossover around the chiral restoration
pseudocritical temperature $T_{ch}\sim 155$ MeV [12, 13] was established by a
few groups[14, 15] and it was widely believed that a deconfinement crossover
occurs in the same temperature region. However, it has become increasing clear
that this belief was erroneous [16].

In retrospect, it is hardly surprising that the old belief that deconfinement
and chiral restoration happened together was erroneous: general theoretical
principles suggest that there is no rigorous way that the two can be directly
connected. Note that both spontaneous chiral symmetry breaking and
deconfinement are associated with symmetries that are exact only in particular
limits of the the theory and that the limits associated with the two are
incompatible. Chiral symmetry becomes exact in the limit of massless quarks.
In contrast, the only strict definition of deconfinement at our disposal is
QCD with infinitely heavy quarks (where QCD degenerates into Yang-Mills
theory); in that case the action has an $Z_{N_{c}}$ center symmetry. This
$Z_{N_{c}}$ center symmetry becomes broken in the high temperature phase; its
breaking is associated with the disappearance of a linearly-rising confining
effective potential between fundamental color charges and can be taken as a
clean definition for deconfinement. The difficulty is that in the limit where
chiral symmetry is cleanly defined, there is no clean way to define
deconfinement and conversely. This strongly suggests that at theoretical level
there is no unambiguous way to relate the two phenomena. Of course in real QCD
the quarks are neither massless nor infinitely heavy and both the notion of
spontaneous chiral symmetry breaking and deconfinement are somewhat fuzzy
notions.

In this respect, the large $N_{c}$ limit of QCD with massless quarks may help
clarify issues theoretically since the theory is manifestly chirally symmetric
and at the same time, explicit violations of $Z_{N_{c}}$ center symmetry
become increasingly suppressed as the $N_{c}\rightarrow\infty$ is approached;
in the limit $Z_{N_{c}}$ center symmetry becomes exact and morphs into a
continuous $U(1)$ symmetry rather than a discrete symmetry. This paper will
explore this large-$N_{c}$ theory for insights into what is going on.

Before exploring the large $N_{c}$ world, it is important to look in a bit
more detail at the behavior of real-world QCD with $N_{c}=3$ and small, but
not zero quark mass. It has become increasingly clear[16] that the crossovers
associated with chiral restoration (as indicated, for example, by a rapidly
dropping chiral order parameter, leading to a maximum in the chiral
susceptibility at $T_{ch}$) and the cross-over into a QGP regime occur at
essentially different temperatures; the latter above $\sim 3T_{ch}$. Moreover
there appears to be very interesting behavior in the intermediate regime.

Tools to understand this behavior were developed starting almost ten years
ago. The initial hint involved an artificial truncation of the near-zero modes
of the Dirac operator in lattice calculations [17, 18, 19, 20]; these studies
contained evidence of an approximate emergent symmetry [21, 22] in this
artificial regime. It is emergent in the sense that while there is evidence
for the symmetry in this regime, the symmetry is larger than the symmetry of
the QCD Lagranigan. This emergent symmetry is the $SU(2)_{CS}$ “chiral spin”
symmetry; it has as a subgroup $U(1)_{A}$. Its flavor extension to $N_{f}$
light flavors is $SU(2N_{F})$, which includes $SU(N_{F})_{L}\times
SU(N_{F})_{R}\times U(1)_{A}$. These are symmetries of the color charge and of
the electric part of the QCD Lagrangian [16].

Since the confining dynamics of non-abeliean gauge theories has often been
associated with the formation of color-electric flux tubes, it is not
unnatural to identify the color-electric interaction of QCD as the “confining
interaction”. In the remainder of this paper, we will refer to this part of
QCD Lagrangian or Hamiltonian in this way. We recognize that it is possible to
object to this identification as treating the complex ideas associated with
confinement in a very simple way. Nevertheless it is helpful to make this
identification, minimally for ease of communication. Moreover, it is useful
conceptually to take seriously the identification and see where it leads. This
identification of the confining interaction is interesting since the confining
interaction, so identified, is invariant under chiral spin—while the color-
magnetic interaction as well as the quark kinetic terms are not.

A clear implication of these truncation studies is that the confining
interaction in QCD is distributed among all modes of the Dirac operator, not
only in the near zero modes that are responsible for $SU(2)_{L}\times
SU(2)_{R}\times U(1)_{A}$ breaking[16, 21, 22]. To the extent that the
identification of the confining interaction is valid, this result strongly
suggests that confinement and spontaneous chiral symmetry breaking are
distinct phenomena that are not directly related, which, as noted above, was
typically assumed in the early days of QCD.

To the extent that truncation studies capture a general feature of QCD, there
are obvious implications for hot QCD. Recall that as the system is heated, the
density of near-zero Dirac modes is suppressed. The Banks-Casher relation[23]
relates the chiral condensate to this density as a direct proportionality so
that above the chiral symmetry restoration crossover where this density
becomes suppressed, one might naturally expect the system to behave much like
the system in which the near-zero Dirac modes were removed by hand. Moreover,
one might expect that along with chiral restoration, an approximate chiral
spin symmetry and its extensions could naturally emerge since it was seen in
the artificial regime [24]. This scenario was verified to occur in lattice
simulation in $N_{F}=2$ QCD with the domain wall Dirac operator[25, 26, 27],
where distinct multiplets of the above groups were observed in spatial and
temporal correlators above $T_{ch}$ but below roughly $3T_{ch}$. Similar
conclusions have been obtained very recently in $N_{F}=2+1+1$ QCD [28]. Since
the confining part of the QCD interaction respects these symmetries there is
no reason to believe that confinement dynamics need also be suppressed in this
regime. Thus one would expect that although (approximate) chiral symmetry has
been restored to good approximation, the system would still be in a confining
regime.

From these lattice studies, three regimes of QCD were identified with clearly
distinguishable approximate symmetries: below $T_{ch}$ spontaneously broken
chiral symmetry , chiral restoration with an approximate emergent $SU(2)_{CS}$
(as well as $SU(2N_{f})$ symmetries) at $T_{ch}<T<\sim 3T_{ch}$ and with just
the usual chiral symmetries (without spontaneous breaking) at higher
temperatures. A simple way to interpret this pattern, is to suggest that after
the initial chiral crossover, the dominant interaction is the confining one,
which respects these enlarged symmetries. These enlarged symmetries are only
approximately realized in this regime since the confining interaction is
dominant but other interactions and quark kinetic terms are present; but to
the extent the confining interaction dominates the patterns associated with
the enlarged symmetries are clearly discernible. As the temperature increases
towards a deconfined regime the effect of confining interaction weakens
relative to the remainder and the enlarged symmetries cease to be observable.

One possible way to interpret these observations in terms of a physical
picture is to suggest that at $T_{ch}<T<\sim 3T_{ch}$, the effective degrees
of freedom act like chirally symmetric quarks that are bound by a electric
field into color-singlet hadron-like objects. Consequently this regime was
dubbed a stringy fluid. Note that these hadron-like clusters are densely
packed so that the quark interchanges between them required by Pauli principle
are significant. Still it is a system with confinement. At higher temperatures
the confining electric field gets screened and one observes a smooth
transition to the QGP. Note that the physical picture of a “stringy fluid” is
simply a heuristic way to view things. The overall pattern of approximate
multiplets for correlation functions that are consistent with the symmetries
outlined is a physical phenomenon that exists regardless of whether this
heuristic picture ultimately turns our to be accurate.

There is additional evidence that the regime $T_{ch}<T<\sim 3T_{ch}$, is not
described by a simple description in which (quasi)partons are the effective
degrees of freedom: screening masses and equation of state [29]. Direct
evidence for effective hadron degrees of freedom above $T_{ch}$ can be seen
from the pion spectral function extracted from the spatial correlators [30,
31]. These indicate that while $\pi$ and $\pi^{\prime}$ excitations become
broader with temperature, clear and discernible hadron-like structures persist
in the regime; these structures cease to be discernible above around
$3T_{ch}$. Other direct evidence of hadronic degrees of freedom above $T_{ch}$
is the bottomonuim spectrum with the $1S,2S,3S,1P,2P$ states with similar
masses like in vacuum remain in this regime; they simply become broader with
temperature [32]. For review of symmetries and lattice evidences of hadronic
origin of the stringy fluid regime see Ref. [16].

There is also experimental evidence for hadron degrees of freedom above
$T_{ch}$ obtained at CERN SPS: dimuon spectra coming from the fireball
demonstrate a clearly discernible $\rho$ structure; it is broader than in
vacuum [33]. An extraction of the fireball temperature at SPS via the black
body radiation yields $T\sim 205$ MeV, which is clearly above $T_{ch}$ [34].

All this evidence suggests three different regimes in QCD that are connected
by smooth crossovers.

This paper will use analysis based on the large $N_{c}$ limit of QCD and
$N_{c}$ counting rules to attempt to gain insight into the phenomena by asking
how one would expect the phenomena to evolve with $N_{c}$ as $N_{c}$ gets
large. There is a realistic prospect that the physics becomes significantly
more transparent at large $N_{c}$.

There are two related reasons that large $N_{c}$ could provide insight. The
first was noted above: as $N_{c}\rightarrow\infty$, the effects of explicit
$Z_{N_{c}}$ center symmetry breaking in QCD diminishes and becomes an exact
$U(1)$ symmetry. Thus for QCD with massless quarks and in the large $N_{c}$,
it becomes possible in principle to simultaneously and unambiguously identify
whether the system is in a regime that is chiral restored and whether it is
deconfined; this makes it possible to ask well-posed questions about the
nature of these regimes without recourse to heuristic and imprecise
arguments.111In the large Nc limit there are two independent clear cut order
parameters, one associated with confinement (the Polyakov loop) and one for
chiral symmetry (the quark condensate). We also note that the ’t Hooft anomaly
matching conditions [36] and the Coleman-Witten theorem for chiral symmetry
breaking [37] do not connect the notion of confinement to center symmetry.
Importantly the Coleman-Witten theorem exploits Lorentz invariance and so by
construction is restricted only to zero temperature. Similarly, the ’t Hooft
anomaly matching is valid at $T=0$ and its generalization to $T>0$ is unclear.
The second is that thermodynamic observables such as the energy density,
$\epsilon$, pressure, $P$ or entropy density, $s$ naturally scale differently
with $N_{c}$ in the various regimes: In a hadronic gas,

$\epsilon_{\rm had}\sim N_{c}^{0}\;\;,\;\;P_{\rm had}\sim N_{c}^{0}\;,\;s_{\rm
had}\sim N_{c}^{0}\;,$ (1a) in a quark-gluon-plasma regime, $\epsilon_{\rm
QGP}\sim N_{c}^{2}\;\;,\;\;P_{\rm QGP}\sim N_{c}^{2}\;,\;s_{\rm QGP}\sim
N_{c}^{2}$ (1b) and assuming that an intermediate regime that is confined (in
the sense of having a non-zero string tension and unbroken center symmetry)
but without spontaneous chiral symmetry breaking, survives in the large
$N_{c}$ limit $\epsilon_{\rm int}\sim N_{c}^{1}\;\;,\;\;P_{\rm int}\sim
N_{c}^{1}\;,\;s_{\rm int}\sim N_{c}^{1}\;.$ (1c)

As the $N_{c}\rightarrow\infty$ limit is approached these regimes become
infinitely separated and thus in the limit must correspond to distinct phases
separated by phase transitions. This in turn makes the regime in question
unambiguous.

Before proceeding, it is worth noting that a priori there are four main
scenarios for the QCD phase diagram at $\mu_{B}=\mu_{I}=0$ (where $\mu_{B}$
and $\mu_{I}$ are the baryon and isospin chemical potentials respectively) for
the large $N_{c}$ limit with massless quarks that appear to be consistent with
what is known about the large $N_{c}$ world based on theoretical scaling
arguments[39, 40, 41] and lattice studies for systems with large but finite
$N_{c}$.

  1. 1.

There are two distinct phases, the hadron resonance gas and the QGP, separated
by a single first-order phase transition at large $N_{c}$; the latent heat is
of order $N_{c}^{2}$ [38]. The existence of three qualitatively different
regimes observed at $N_{c}=3$ does not persist as $N_{c}$ gets large, with the
intermediate regime vanishing in the large $N_{c}$ limit.

  2. 2.

There are two distinct phases separated by a single first-order phase
transition at large $N_{c}$; the latent heat is of order $N_{c}^{2}$. The low
temperature phase is a conventional hadron phase, while the high temperature
phase restores chiral symmetry and spontaneously breaks center symmetry at all
temperatures in the regime. However the high temperature phase does not act
like weakly interacting plasma until well above the phase transition and
hadron-like structures (as would be seen in correlation functions) would be
present at temperatures near the transition. There could be a smooth cross-
over to a regime in which these hadron-like” structures cease to be
discernible as temperature increase.

  3. 3.

There are three distinct phases at large $N_{c}$ with the low-temperature
hadronic phase separated from the intermediate phase by a second-order phase
transition; the intermediate phase separated from the high-temperature QGP
phase by a first order transition with a latent heat of order $N_{c}^{2}$.

  4. 4.

There are three distinct phases at large $N_{c}$ with the low-temperature
hadronic phase separated from the intermediate phase by a first-order phase
transition, with a latent heat of order $N_{c}$; the intermediate phase is
separated from the high-temperature QGP phase by a first order transition with
a latent heat of order $N_{c}^{2}$.

As will be discussed below the plausibility of the various depends in part on
the assumptions one makes about the how QCD dynamics manifests itself in the
various regimes.

##  2 Large $N_{c}$ counting rules

Before discussing the various scenarios it is useful to briefly remind the
reader of standard results of large $N_{c}$ analysis with an emphasis on the
implications for thermodynamics.

Most studies of large $N_{c}$ QCD are based on studies of correlation
functions of local color singlet operators that (too leading order in
$1/N_{c}$) cannot be decomposed into more than one color-singlet[40, 41, 39].
The $N_{c}$ of these correlators can be deduced from scaling of Feynman
diagrams that contribute to them. Two key rules can be deduced that control
this scaling. The first is that connected diagrams composed solely of gluon
lines (and sources that connect to gluon lines) scale as $N_{c}^{2}$ if the
diagram is planar and slower than $N_{c}^{2}$ if they are non-planar. The
second is that planar diagrams that contain quarks loops that bound the
diagram are suppressed by a factor of $1/N_{c}$ per quark loop relative to
diagrams composed entirely of gluons with addition suppression if the diagram
is non-planar or the quark loop fails to bound the diagram.

From the $N_{c}$ counting of the correlators along with the additional
assumption of confinement ( in the sense that all physical states of the
theory are color singlets) it is straightforward to deduce the following rules
for the $N_{c}$ scaling of properties[40, 41, 39] of mesons and glueballs:

  * 1.

At large $N_{c}$, properties of mesons and glueballs can be described via an
effective tree-level theory; at leading order vertices scale with $N_{c}$
according to $N_{c}^{2-n_{\rm g}-\frac{1}{2}n_{\rm m}}$, for a vertex with
$n_{\rm m}$ meson lines and $n_{\rm g}$ gluon lines at the vertex which means
that at leading order

  * 2.

Mesons and glueballs masses scale as $N_{c}^{0}$.

  * 3.

Meson decay amplitudes scale asymptotically as $N_{c}^{-1/2}$ and their widths
as $N_{c}^{-1}$. Glueball decay amplitudes scale as $N_{c}^{-1}$ and their
widths as $N_{c}^{-2}$.

    * (a)

Mesons and glueballs are stable at large $N_{c}$.

  * 4.

These hadrons interact weakly.

    * (a)

Scattering amplitudes for meson-meson interactions, glueball-glueball
interactions and meson-glueball scale as $1/N_{c}$, $1/N_{c}^{2}$, and
$1/N_{c}^{2}$ respectively.

  * 5.

For every allowed quantum number an infinite number of distinct mesons and
glueballs exist. [40].

  * 6.

In terms of $N_{c}$ counting, hybrid mesons with exotic quantum numbers scale
at large $N_{c}$ the same way as ordinary mesons [35].

  * 7.

At large $N_{c}$ The OZI rule [43, 44, 45] is exact .

    * (a)

The effects of the axial $U(1)$ anomaly are suppressed.

    * (b)

Glueball-meson mixing is suppressed.

As noted above meson and glueball masses are independent of $N_{c}$ at large
$N_{c}$. In contrast, as discussed by Witten [40, 39] baryon masses scale as
$N_{c}^{1}$ for the simple reason that a baryon contains at least $N_{c}$
quarks. This means that at temperatures of $N_{c}^{0}$–the regime of interest
in this paper, baryons are suppressed by factors that are exponential in
$N_{c}$ and hence negligible.

As the role of chiral symmetry plays an essential role in the analysis it is
useful to note the $N_{c}$ scaling of several quantities that are central to
chiral physics, the pion decay constant—$f_{\pi}$, the chiral
condensate—$\langle\overline{q}q\rangle$, the $\sigma$ term for the $j^{\rm
th}$ species of meson—$\sigma_{m_{j}}$ and the $\sigma$ term for the $k^{\rm
th}$ species of glueball—$\sigma_{m_{g}}$

$f_{\pi}\sim N_{c}^{1/2}\;\;,\;\;\langle\overline{q}q\rangle\sim
N_{c}^{1}\;,\;\;\sigma_{m_{j}}\sim N_{c}^{0}\;\;\sigma_{m_{g}}\sim
N_{c}^{0}\;;$ (2)

the $\sigma$ term of hadron $h$ is given by $m_{q}\partial m_{h}/\partial
m_{q}$ where $m_{q}$ is the average of the light quark masses and $m_{h}$ is
the mass of the hadron and corresponds to the shift in the matrix element of
$m_{q}\langle\overline{q}q\rangle$ in the hadron state relative to the vacuum.
Note that the $\sigma$ term cannot be defined unambiguously for a resonance
rather than a stable particle, but at large $N_{c}$ mesons and glueballs are
stable.

Related to chiral symmetry at large $N_{c}$ the $\eta^{\prime}$ is an ordinary
pseudo-Goldstone boson since the effects of the axial $U(1)$ anomaly are
suppressed.

One aspect of large $N_{c}$ analysis that plays a key role in what follows is
the fact that properties of glueballs and mesons scale in qualitatively
different ways. A critical difference is that while meson-meson scattering
amplitudes scale as $N_{c}^{-1}$, meson-glueball (and glueball-glueball)
scattering amplitudes scale as $N_{c}^{-2}$. Thus although both mesons and
glueballs are hadrons with masses of order $N_{c}^{0}$ that couple weakly to
other hadrons, glueballs couplings are qualitatively weaker. This can have a
profound effect on propagation in a medium with a moderate density of hadrons:
the medium will affect meson propagation substantially more than glueballs and
this leads to the possibility that an intermediate phase could exist with
glueball degrees of freedom unaffected by the phase (neglecting $1/N_{c}$
corrections) while meson degrees of freedom could be radically altered.
Ultimately the cause of the different scalings can be traced to the fact that
local gauge invariant operators with meson quantum numbers must couple to
quarks and the number of distinct colors for quarks is $N_{c}$ while operators
that couple to glueballs couple to gluons of which there are $N_{c}^{2}-1\sim
N_{c}^{2}$

This may seem somewhat counter intuitive if one regards both glueballs and
mesons as generic hadrons; in that case one might think that they ought to
behave similarly. Of course, in practice the community has had little if any
experience with the behavior of glueballs in the physical world of $N_{c}=3$,
since there are few or no states in nature that are clearly glueballs. (The
most plausible reason for this is simply that possible glueballs tend to be
heavy.) Thus any intuition one might have that glueballs are ordinary hadrons
similar to mesons is not based on significant experience in the $N_{c}=3$
world.

##  3 The phases at large $N_{c}$

###  3.1 The hadron gas phase

The large $N_{c}$ limit greatly simplifies the nature of the hadronic
resonance gas phase. Since the phase exists only at temperatures of order
$N_{c}^{0}$, baryons can be neglected. Moreover at large $N_{c}$ hadrons
(mesons and glueballs) do not interact so the phase should be accurately
modeled at any given temperature consistent with the phase by a gas of
multiple species of non-interacting hadrons with their various masses in
thermal equilibrium[42]. Each species of density and momentum distribution are
given by the appropriate Bose-Einstein distribution in the same manner used
phenomenologically in hadron resonance gas models[46, 47]. This allows a
straightforward calculation of thermodynamic properties such as energy
density, entropy density and pressure as a function of temperature as well as
the number density of each species of hadron.

The number density of species $k$, $n_{k}(T)$, is given by
$n_{k}(T)=(2S_{k}+1)(2I_{k}+1)\int\frac{{\rm
d}^{3}p}{(2\pi)^{3}}\,\frac{1}{e^{\sqrt{p^{2}+m_{k}^{2}}/T}-1}\;,$ (3a) where
$s$ and $I$ are the spin and of the species. The energy density and pressure
of the system are given by $\epsilon_{\rm
had}(T)=\sum_{k}(2S_{k}+1)(2I_{k}+1)\int\frac{{\rm
d}^{3}p}{(2\pi)^{3}}\,\frac{\sqrt{p^{2}+m_{k}^{2}}}{e^{\sqrt{p^{2}+m_{k}^{2}}/T}-1}$
(3b) $P_{\rm
had(T)}=\sum_{k}(2S_{k}+1)(2I_{k}+1)\left(\frac{m_{k}^{2}T^{2}}{2\pi^{2}}\right)\,\sum_{j=1}^{\infty}\,K_{2}\left(\frac{jm_{k}}{T}\right)$
(3c) where the sum over $k$ represents species of hadron (glueballs and
mesons) and $K_{2}$ is a modified Bessel function. The entropy density is
given by $s_{\rm had(T)}=\frac{\epsilon+P}{T}$ (3d)

It is easy to see that the $\epsilon_{\rm had}(T)$, $P_{\rm had}(T)$ and
$S_{\rm had}(T)$ scale as in Eq. 1a and that he number density of each species
of hadrons also scales as $N_{c}^{0}$.

In the hadronic regime,the fractional change in the chiral condensate at
temperature $T$ relative to the vacuum is given by [48]

$\frac{\langle\overline{q}q\rangle_{T}}{\langle\overline{q}q\rangle_{\rm
vac}}=\frac{m_{\pi}^{2}f_{\pi}^{2}-\sum_{j}n_{j}(T)\sigma_{j}}{m_{\pi}^{2}f_{\pi}^{2}}=1-{\cal
O}(N_{c}^{-1})$ (4)

where the sum is over hadron (meson and glueball) species, $n_{j}$ is the
density of the species and $\sigma_{j}$ is the sigma term for that species.
Note there is an implicit $m_{q}\rightarrow 0$ limit on the right hand side of
the expression; it is well defined in the chiral limit since $m_{\pi}^{2}$ and
the sigma term are each proportional to the quark mass. The key point is that
since $n_{j}$, $\sigma_{j}$ and $m_{\pi}^{2}$ are each of order $N_{c}^{0}$
while $f_{\pi}^{2}$ is of order $N_{c}$, it is apparent that at large $N_{c}$
, $\frac{\langle\overline{q}q\rangle_{T}}{\langle\overline{q}q\rangle_{\rm
vac}}=1$ and to leading order, the chiral condensate is unchanged from its
vacuum value.

###  3.2 The intermediate phase

From the perspective of large $N_{c}$ analysis and given current knowledge
there is no guarantee that an intermediate phase that is confined ( in the
sense of having a non-zero string tension and unbroken center symmetry) but
without spontaneous chiral symmetry breaking exists. Indeed of the four
scenarios considered in this paper, only two have such a phase. However, it is
important to focus on such a phase as it is extremely interesting. In
particular, if such a phase does exist one can deduce key properties that the
phase must have at large $N_{c}$. The most startling of these is that in this
phase glueballs exist and they must interact weakly with themselves and with
the quark degrees of freedom.

Recall that all diagrams containing quark loops are suppressed by factors of
$1/N_{c}$. Thus the difference between every correlation function of local,
gauge-invariant sources containing only gluon fields in QCD and the same
correlation function in Yang-Mills is of order $1/N_{c}$ and vanishes in the
limit: at large $N_{c}$ these correlation functions are the same in QCD and
Yang-Mills. However, Yang-Mills in the regime where center-symmetry is
unbroken is in a confining phase where all correlation functions are unchanged
from their vacuum values (up to corrections of relative order $1/N_{c}$).
These correlators describe a gas of glueballs that neither interact among
themselves nor with any other degrees of freedom in the system. Thus, without
further analysis one knows that if such an intermediate phase exists in the
large $N_{c}$ limit it contains a gas of glueballs that neither interact with
each other or with degrees of freedom containing quarks.

On the other hand, by hypothesis the intermediate phase is chirally restored
which implies that n-point functions of local gauge-invariant color-singlet
sources that include quark-bilenears are radically different from those in the
low-temperature hadronic gas phase. Consider, for example the chiral
condensate, the one-point functions of $\overline{q}q$. Its matrix element is
generically of order $N_{c}^{1}$ and its vacuum value is indeed of that order.
In the ordinary hadron gas phase, Eq. (4) implies that the change in the
chiral condensate relative to the vacuum at $T=0$ is known to be suppressed by
a factor of $1/N_{c}$. We know that this cannot be the case in the
intermediate phase where the spontaneously symmetry breaking does not occur:
the chiral condensate has shifted from its vacuum value by an amount of order
$N_{c}^{1}$ (i.e. from order $N_{c}^{1}$ to zero). This implies that in the
putative intermediate phase, the condition restricting shifts in connected
n-point functions of local gauge-invariant single-color trace sources to being
of relative order $1/N_{c}$ compared to the vacuum does not apply: in the
intermediate phase shifts of relative order unity in a $1/N_{c}$ expansion are
permitted.

However, the quark kinetic energy term, a 1-point function of the quark
bilinear $\overline{q}\mbox{\boldmath$D$\unboldmath}\\!\\!\\!\\!/q$, is a
contribution to the energy density. If it is shifted of relative order unity
relative to the vacuum (absolute order $N_{c}^{1})$, then there will be shifts
in the energy of order $N_{c}^{1}$ as seen in Eq. (1c).

There is a potential caveat to that conclusion. It remains logically possible
(although quite implausible) that the chiral condensate shifts from its vacuum
by amounts of absolute order $N_{c}^{1}$ but other n-point functions of
connected n-point functions of local gauge-invariant single-color trace
sources do not. However, there is a strong evidence that this not the case and
that shifts of order $N_{c}^{1}$ are generic. Consider in particular,
correlators of the quark bilinear sources seen in chiral spin symmetry. The
connected two-point functions of these operators are all of order $N_{c}^{1}$
in the vacuum and are generically different from one another in the vacuum and
the hadron gas phase (excepting correlators connected by isopin). However in a
phase with approximate chiral spin symmetry which, by hypothesis, the
intermediate phase has many of these correlators become (nearly)
identical—indicating a shift of absolute order $N_{c}^{1}$ relative to the
vacuum and hadron gas phases. This strongly suggests that shifts of connected
n-point functions of local gauge-invariant single-color trace sources are
generically of order $N_{c}^{1}$.

Moreover, the pressure is related to energy density via the speed of sound:
$\left.\frac{P}{\epsilon}\right|_{s}=1/v_{s}^{2}$. Since the speed of sound is
bounded from above by unity, one sees that the pressure in such a phase must
scale as $N_{c}$ as well.

There remains an open question regarding chiral spin symmetry in the putative
intermediate phase that large $N_{c}$ scaling arguments do not resolve. It has
been assumed that in the zero quark masses this phase is chirally restored and
the chiral condensate vanishes in the combined large $N_{c}$ and chiral limits
of the theory. This is an unambiguous notion since chiral symmetry is a
symmetry of the theory in that combined limit. However, chiral spin symmetry
is not an explicit symmetry of the theory in that limit, it is a symmetry of
only its confining part. Thus, the issue arises as to whether it is an exact
emergent symmetry that is present in this phase or whether it is an
approximate symmetry as it is in the physical world of $N_{c}=3$ and small but
nonzero quark masses. At this point there is no known dynamical mechanism by
which chiral spin symmetry would need to emerge as an exact symmetry in the
intermediate phase of the theory in the combined limit.

At the energy density ${\cal O}(N_{c}^{1})$ the glueballs do not interact, as
was discussed above. At the same time usual mesons with valence quarks
interact strongly and their interaction appears at the same order as meson
mass in vacuum, ${\cal O}(N_{c}^{0})$. This would naturally imply a finite
widths of those mesons, the property, that is observed at $N_{c}=3$. The
intermediate phase could thus be considered as a dense system of strongly
interacting mesons—with the caveat that these mesons are not the usual ones
seen in the vacuum.

###  3.3 Quark-gluon plasma phase

For the purposes of this work, the quark-gluon plasma phase is identified as
the phase in which center symmetry is broken and the string tension goes to
zero—conditions that become unambiguous at large $N_{c}$. The phase is
generically characterized by an energy density and pressure that are of order
$N_{c}^{2}$. This is precisely the scaling that one would have with a plasma
of non-interacting quarks and gluons and would naturally apply to a weakly-
interacting plasma as well. However, the large $N_{c}$ analysis also implies
that the system cannot be a weakly interacting interacting plasma of
deconfined quarks and gluons throughout the entire QGP phase.

It is easy to see that a non-interacting plasma of massless quarks and gluons
has the following scaling relations

$\epsilon\sim N_{c}^{2}T^{4}\;\;\;P=\frac{\epsilon}{3}\;;$ (5)

the energy relation holds since the plasma is dominated by gluons and the
number of gluon colors is $N_{c}^{2}-1$ each of which contributes an energy
density proportional to $T^{4}$, while the relationship of energy density to
pressure is characteristic of non-interacting massless particles. In a weakly
interacting plasma of quarks and gluons one would expect Eq. (5) to hold
approximately. Moreover, we know that at high temperatures asymptotic freedom
implies that coupling constants will be small so that the the plasma will be
weakly interacting and will approach the scaling in Eqs. 5 to hold. Note that
the running of the ‘t Hooft coupling $gN_{c}^{1/2}$ is described by a $\beta$
function that is independent of $N_{c}$ for large $N_{c}$ so that perturbative
corrections to the non-interacting plasma become small (in a logarithmic
sense) at temperatures of order $\Lambda_{\rm QCD}$ times factors of order
$N_{c}^{0}$, establishing the scaling in (5).

However, regardless of which of the four scenarios enumerated in the
introduction is correct, Eq. (5) cannot be true in the neighborhood of the
phase boundary. Whether the transition is first-order or second-order, the
pressure is continuous across the transition boundary. Thus, in scenarios 1
and 2 the pressure in the QGP phase just above the transition is of order
$N_{c}^{0}$ while in the scenarios 3 and 4 the pressure is order $N_{c}^{1}$,
in either case it is incompatible with Eq. (5), even qualitatively. This is
consistent with the generic expectation that the pressure is generically of
order $N_{c}^{2}$ in the phase; the behavior at the phase boundary is not
generic and the coefficient of order $N_{c}^{2}$ term simply goes to zero at
the boundary.

The important conclusion to draw from this is that the thermodynamics of this
phase cannot be interpreted as arising from quarks and gluons moving quasi-
freely through this phase. To the extent that the term “quark-gluon plasma” is
appropriate throughout this phase, one must conclude that it is a strongly-
interacting QGP—at least at the lower end of the phase. This in turn implies
that the mere fact that quarks are “deconfined” in the sense that the string
tension goes to zero is not sufficient to deduce that quarks and gluons are
quasi-free.

##  4 The scenarios

The introduction enumerated four scenarios as to how the qualitative features
observed in the world of $N_{c}=3$ could evolve as $N_{c}\rightarrow\infty$.
In this section the plausibility of each of these will be considered.

Scenario 4, is interesting in that the intermediate regime seen in lattice
studies with $N_{c}=3$ becomes a distinct phase at large $N_{c}$.

A novel feature of that phase is that while the properties of degrees of
freedom associated with quarks—as seen through n-point functions of local
gauge invariant quark bilinears are radically different from those in the
hadron gas phase, all n-point functions of purely gluonic operators are
identical to what they were in the hadron gas phase at $T=0$ or in Yang-Mills
theory are up to corrections of relative order ${\cal O}(1/N_{c})$. This means
that at large $N_{c}$ ordinary glueballs exist in this phase as they do in the
hadron gas phase, and they interact neither with each other or with quark
degrees of freedom.

While this behavior is quite natural in the context of large $N_{c}$ analysis,
it is also quite remarkable given the binary way that regimes in QCD are
usually viewed. Traditionally one thinks in terms of a regime where the system
acts as a gas of hadrons or a QGP regime where the quarks and gluons are the
effective degrees of freedom. In this putative intermediate phase glueballs
remain as ordinary hadrons but do so in a phase where other hadronic degrees
of freedom—the mesons with valence quarks—have their properties radically
altered. This can occur because the $N_{c}$ scaling of meson with valence
quarks and glueballs is characteristically different.

One might worry that a phase that is so different from the common
understanding of how QCD behaves is intrinsically implausible given a half
century of experience with the theory. However, the speculation about the
emergence of such a phase is driven by strong numerical evidence that an
intermediate regime exists at $N_{c}=3$ and it is by no means far fetched to
speculate that such a regime would become a distinct phase at large $N_{c}$.
It is also worth noting that something analogous happens at large $N_{c}$ in
the quarkyonic phase of confined but chirally restored matter which is
postulated to exist at large baryon chemical potential and has been justified
with large $N_{c}$ arguments[38].

Scenario 3 is similar to scenario 4 in having an intermediate phase with the
same qualitative properties noted there. It differs solely in the order of the
transition between hadronic gas matter and the intermediate phase; it is
second order in scenario 3.

Assuming that an intermediate phase exists, it is useful to attempt to
distinguish between scenarios 3 and 4, i.e. to determine the order of the
transition between the hadron gas and intermediate phases. It is typically
very difficult to determine the order of a transition by theoretical analysis
and this problem is no different. However, there is a plausibility argument
suggesting that scenario 4 is more likely.

The argument is based on the fact that a second order phase transition has a
diverging specific heat at the phase transition temperature. Starting from the
hadron gas phase, at large Nc the scaling of the energy density as a function
of temperature is given by Eq. (3b). It is easy to show from Eq. (3b), the
specific heat cannot diverge unless $N(m)$, the number of hadrons (including
spin and flavor degeneracy factors with mass less than $m$) grows
exponentially with $m$, i.e. $N(m)=p(m)\exp\left(m/T_{H}\right)$ where $T_{H}$
is a parameter known as the Hagedorn temperature[1] and $p(m)$ is a
subexponential prefactor. (Note, that while the existence of a Hagedorn
spectrum at large $N_{c}$ is necessary for a second-order transition out of
the hadron gas phase, it is not sufficient. For example, a first-order
transition can occur at temperatures below $T_{H}$.)

Plausibility arguments independent of string-like description of QCD exist
that large $N_{c}$ QCD ought to have a Hagedorn spectrum[49, 50]. However a
natural dynamical explanation for a Hagedorn spectrum is that highly excited
states correspond to states with highly excited, and therefore long, flux
tubes—which at large $N_{c}$ cannot break. To the extent that the flux tubes
are much longer than they are wide (as one would expect with highly excited
states), they ought to act as strings and string theories are known to have
Hagedorn spectra[51]. While this string-theory based explanation is a natural
picture for why a Hagedorn spectrum could exist and with it the possibility of
a second-order transition, string theories should have Hagedorn spectra for
both open strings (mesons) and closed strings (glueballs) with the same
$T_{H}$ for both. This is incompatible with scenario 3 which would require a
Hagedorn spectrum for mesons but also requires that glueballs either do not
have a Hagedorn spectrum or have one at a higher temperature. This suggests
that scenario 4 seems more likely than scenario 3

In scenario 2 the three distinct regimes at $N_{c}=3$ persist at large
$N_{c}$, but these regimes do not all correspond to distinct phases. Rather,
in this scenario there are two phases: a confined hadronic phase with energy
densities and pressures of order $N_{c}^{0}$ and a deconfined phase with a
vanishing string tension and with energy densities and pressures generically
of order $N_{c}^{2}$. In this scenario, hadron-like structures could be
discerned in correlation function at temperatures near the transition even
though it is the deconfined nature of the phase; such correlators would
reflect an approximate chiral spin symmetry. As in the $N_{c}=3$ world, there
could then be a smooth cross-over to a regime in which these hadron-like
structures cease to be discernible as temperature increase. However, it would
be difficult to explain then why hadron-like structures appear in the
deconfined phase at the same masses as in vacuum, i.e. in the confining phase.
This property is clearly seen in lattice simulations.

Scenario 1 is simply that the existence of an intermediate regime that is seen
in lattice studies of the $N_{c}=3$ world has no analog in the large $N_{c}$
world. Such a scenario is uninteresting in the sense that large $N_{c}$
analysis provides no insight into the phenomena, but this scenario cannot be
ruled out on any general grounds.

In this context it is worth recalling that the degree to which phenomena in
the $N_{c}=3$ world and the large $N_{c}$ world correspond, at least
qualitatively, varies with the observable being considered. A concrete example
of this alluded to earlier: glueballs. In the large $N_{c}$ world glueballs
must exist and are clearly discernible from mesons while in the physical world
of $N_{c}=3$ there are few if any unambiguous glueball states. Given the
possibility that the large $N_{c}$ world and the physical world can have
signinificant qualitative differences, one cannot rule out the possibility
that the emergence of an approximate chiral spin symmetry is special to
$N_{c}=3$ and could disappear at large $N_{c}$.

The question arises about the width of the intermediate phase. Both
deconfinement, $T_{d}$ and chiral restoration, $T_{ch}$ temperatures are known
to scale as $N_{c}^{0}$. Consequently by construction the width of the
intermediate phase vanishes as $N_{c}^{-1}$ in scenarios 1 and 2, while this
width is of the order $N_{c}^{0}$ in scenarios 3 and 4.

To conclude this discussion it would be important to mention a result of the
large $N_{c}$ lattice studies of the quark condensate at $T=0$, see [52, 53,
54, 55, 56, 57] and references therein. These studies demonstrate that the
quark condensate in the large $N_{c}$ world is quite close to its value at
$N_{c}=3$. This suggests that for the spontaneous breaking of chiral symmetry
the $1/N_{c}$ corrections are small and the physics of this phenomenon in the
large $N_{c}$ world and in the real world $N_{c}=3$ is likely to be similar.
Thus, it is plausible that the chiral symmetry restoration phase transition in
the large $N_{c}$ world—the temperature of the phase transition (in the chiral
limit)— should be close to the temperature of the chiral restoration phase
transition at $N_{c}=3$. The latter temperature is known from the lattice
calculations to be around $T\sim 130$ MeV [59].

In the large $N_{c}$ world the quark loops are suppressed and the physics of
confinement should be quailitatively similar to that of pure Yang-Mills theory
or in quenched QCD. In pure Yang-Mills and in quenched QCD the deconfinement
phase transition of the first order is known to happen at $T\sim 270-300$ MeV
for the case of $N_{c}=3$. Thus, it is plausible that in the large $N_{c}$
world there exist two distinct temperatures, one for the chiral restoration
phase transition (around $T^{largeN}_{ch}\sim 130$ MeV) and for the
deconfinement transition at a temperature around 300 MeV. This qualitative
prediction could and should be checked in large $N_{c}$ lattice simulations.
If correct, it would be direct verification that either scenario 3 or scenario
4 is realised.

##  5 Conclusions

In this paper we have considered the QCD phase diagram at zero baryon chemical
potential and established $N_{c}$ scalings of different regimes. This analysis
is potentially important since in the large $N_{c}$ limit the center symmetry
and confinement notions are well defined and consequently allow one to
unambiguously discus phases that are characterized by confinement or
deconfinement as well as with explicit chiral symmetry or its spontaneous
breaking. We have identified four scenarios as to how the qualitative features
seen in the physical world can be generalized to large $N_{c}$. Lattice
calculations at large $N_{c}$ would help to distinguish which of these is
correct. While lattice studies for QCD with $N_{c}>3$ have been done[58] the
values of Nc have been modest ($N_{c}\leq 5$) and calculations at larger
$N_{c}$ would be needed to determine which scenario is correct. Given what is
known about critical temperatures in the real world and on the magnitude of
the quark condensate at $T=0$ at large $N_{c}$, we have argued that scenarios
in which there are distinct temperatures of chiral and deconfinement phase
transitions in the large $N_{c}$ world are plausible.

We note that the analysis here could be extended to the inclusion of chemical
potentials. Probably the simplest extension is to the inclusion of an isospin
chemical (since unlike baryon chemical potentials which couple to degrees of
freedom with a minimum energy of order $N_{c}$ the isospin chemical potential
couples to excitations witn a minimum excitation of $N_{c}^{0}$). The isospin
chemical potential also has the virtue that it can be simulated in lattice
studies without a sign problem[60, 61]. The study of large $N_{c}$ QCD with a
nonzero chemical potential can lead to the exploration of a very rich phase
diagram.

## Acknowledgements

The authors thank Owe Philipsen for interesting and insightful comments. They
also thank Aleksey Cherman for noting the issue of the Coleman-Witten theorem.
The work of TDC was supported in part by the the U.S. Department of Energy,
Office of Nuclear Physics under Award Number DE-FG02-93ER40762.

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