Amine Chabane

# Towards the phase diagram of cold and dense heavy QCD

Owe Philipsen

###### Abstract

The thermodynamics of QCD with sufficiently heavy dynamical quarks can be
described by a three-dimensional Polyakov loop effective theory, obtained
after a truncated character and hopping expansion. We investigate the
resulting phase diagram for low temperatures by mean field methods. Taking
into account chemical potentials for both baryon number and isospin, we obtain
clear signals for a liquid-gas type transition to baryon matter at $\mu_{I}=0$
and a Bose-Einstein condensation transition at $\mu_{B}=0$, as well as for
their connection when both chemical potentials are non-zero.

##  1 Introduction

The QCD phase structure in the $(T,\mu_{B})$ plane is important to understand
many physical phenomena for various disciplines in modern physics. The
interest in neutron stars and their mergers moreover motivates studies of non-
vanishing isospin chemical potential $\mu_{I}\neq 0$, and in particular the
mixed situation with both $\mu_{B},\mu_{I}\neq 0$. While ordinary Monte Carlo
simulations are possible for $(\mu_{B}=0,\mu_{I}\neq 0)$, cf. [1] and
references therein, a strong sign problem causes their breakdown whenever the
baryon chemical potential $\mu_{B}\neq 0$. Most methods to circumvent the
obstacle introduce further approximations, and are therefore restricted to the
region with $\mu_{B}/T\leq 3$ (see for example Refs. [2, 3, 4]).

Our approach is to study $3D$ effective Polyakov loop lattice theories, which
are obtained from Wilson’s lattice QCD when integrating over all spatial links
after a truncated strong coupling and hopping parameter expansion, and
describe QCD with very heavy quarks [5]. Here we investigate the phase diagram
of the simplest of these effective theories with a mean field approach, and
compare with previous solutions obtained by series expansions or Monte Carlo
simulations. After successful tests with the finite temperature deconfinement
transition we consider $T\rightarrow 0$, where we identify both the onset
transition to cold baryon matter at $\mu_{B}\approx m_{B},\mu_{I}=0$, as well
as the transition to a pion condensate at $\mu_{B}=0,\mu_{I}=m_{\pi}/2$.
Finally, we switch on both chemical potentials and follow the critical lines
into the $(\mu_{B},\mu_{I})$-plane. We find them to connect in a branching
point, where the vacuum, baryon and pion phases meet.

##  2 The effective theory

In this section we summarise the main features of the effective theory to be
analysed below, for details see [6, 5]. Starting point is the path integral

$\displaystyle Z$ $\displaystyle=\int\quantity[\differential
U_{\mu}]~{}\Big{(}\prod^{N_{f}}_{f=1}\det
Q_{f}\quantity[U_{\mu}]\Big{)}e^{-S_{g}\quantity[U_{\mu}]}=\int\quantity[\differential
U_{0}]e^{-S_{\text{eff}}[U_{0}]},$ (1)

with

$\displaystyle-S_{\text{eff}}[U_{0}]\equiv\ln\int\quantity[\differential
U_{i}]\quantity[\Big{(}\prod^{N_{f}}_{f=1}\det
Q_{f}\quantity[U_{\mu}]\Big{)}~{}e^{-S_{g}[U_{\mu}]}].$ (2)

An approximation of the effective action Eq. (2) can be obtained after a
truncated expansion of the gauge action in terms of the fundamental character
coefficients $u(\beta)$ and a hopping expansion of the fermion determinants in
terms of the hopping parameters $\kappa_{f}$ [7],

$u(\beta)=\beta/18+\beta^{2}/216+\ldots,\quad\kappa_{f}=(2am_{f}+8)^{-1},$ (3)

followed by analytic integrating over the spatial gauge links. Because of the
hopping expansion, the resulting effective theory is only valid for large bare
quark masses $m_{f}$. Here we specialise to mass-degenerate quarks, for which
$m_{f}=m,\kappa_{f}=\kappa$ for all flavours $f$. The remaining dependence on
the temporal gauge links is in terms of untraced Wilson lines,
$W(\mathbf{x})=\prod^{N_{\tau}}_{\tau=1}U_{0}\quantity(\mathbf{x},\tau)$, and
Polyakov loops, $L\quantity(\mathbf{x})=\Tr W(\mathbf{x})$. The result after
these steps can be written as,

$\displaystyle Z$ $\displaystyle\approx\int
DW\underbrace{\prod_{\expectationvalue{\mathbf{x},\mathbf{y}}}\quantity[1+\lambda_{1}\quantity(L^{\dagger}_{\mathbf{x}}L_{\mathbf{y}}+L_{\mathbf{x}}L^{\dagger}_{\mathbf{y}})]}_{\text{pure
gauge}}$ (4)
$\displaystyle\times\underbrace{\prod_{\mathbf{x}}\prod^{N_{f}}_{f=1}\quantity(1+h_{1f}L_{\mathbf{x}}+h_{1f}^{2}L^{\dagger}_{\mathbf{x}}+h_{1f}^{3})^{2}~{}\quantity(1+\overline{h}_{1f}L^{\dagger}_{\mathbf{x}}+\overline{h}_{1f}^{2}L_{\mathbf{x}}+\overline{h}_{1f}^{3})^{2}}_{\text{static}}$
$\displaystyle\times\prod_{\expectationvalue{\mathbf{x},\mathbf{y}}}\underbrace{\quantity(1-2h_{2}\sum^{N_{f}}_{f=1}\quantity[W^{f}_{11}\quantity(\mathbf{x})-\overline{W}^{f}_{11}\quantity(\mathbf{x})]\quantity[W^{f}_{11}\quantity(\mathbf{y})-\overline{W}^{f}_{11}\quantity(\mathbf{y})])}_{\text{kinetic}}$

with the effective couplings [6]

$\displaystyle\lambda_{1}$ $\displaystyle=$ $\displaystyle
u^{N_{\tau}}\exp\quantity[N_{\tau}\quantity(4u^{4}+12u^{5}-14u^{6}-16u^{7}+\frac{295}{2}u^{8}+\frac{1851}{10}u^{9}+\frac{1055797}{5120}u^{10})]$
$\displaystyle h_{1f}$ $\displaystyle=$ $\displaystyle
2\kappa\exp\quantity[N_{\tau}a\mu_{f}]\exp\quantity[6N_{\tau}\kappa^{2}u\quantity(\frac{1-u^{N_{\tau}-1}}{1-u}+4u^{4}-12\kappa^{2}+9\kappa^{2}u+4\kappa^{2}u^{2}-4\kappa^{4})]$
$\displaystyle\overline{h}_{1f}$ $\displaystyle=$ $\displaystyle
2\kappa\exp\quantity[-N_{\tau}a\mu_{f}]\exp\quantity[6N_{\tau}\kappa^{2}u\quantity(\frac{1-u^{N_{\tau}-1}}{1-u}+4u^{4}-12\kappa^{2}+9\kappa^{2}u+4\kappa^{2}u^{2}-4\kappa^{4})]$
$\displaystyle h_{2}$ $\displaystyle=$
$\displaystyle\frac{N_{\tau}\kappa^{2}}{N_{c}}\quantity(1+2\frac{u-u^{N_{\tau}}}{1-u}+8u^{5}+16\kappa^{3}u^{4}).$

The expressions appearing in the third line in Eq. (4) are defined as

$\displaystyle
W^{f}_{nm}\quantity(\mathbf{x})=\Tr\frac{\quantity(h_{1f}W_{\mathbf{x}})^{m}}{\quantity(\mathbb{I}+h_{1f}W_{\mathbf{x}})^{n}},\quad\overline{W}^{f}_{nm}\quantity(\mathbf{x})=\Tr\frac{\quantity(\overline{h}_{1f}W^{\dagger}_{\mathbf{x}})^{m}}{\quantity(\mathbb{I}+\overline{h}_{1f}W^{\dagger}_{\mathbf{x}})^{n}}.$
(5)

The effective theory in Eq. (4) is structured as follows: the first line
represents the pure gauge contribution, the second line contains the static
determinant, and the last line is the leading order of the kinetic quark
determinant. All contributions can be expressed fully in terms of Polyakov
loops, if desired. An important feature of this effective theory is that it
has a much weaker sign problem than full QCD and can therefore be simulated
using reweighting techniques or complex Langevin methods [5, 8]. Moreover, it
can also be treated by analytic linked-cluster expansion methods in the
effective couplings [9, 10, 11].

##  3 Mean field analysis of the effective theory

Since the effective theory resembles a spin model, it is natural to also
consider mean field methods as a short cut to a first evaluation and to get an
impression of the overall phase structure. We proceed in analogy to early
investigations of similar models [12, 13]. The basic idea is to consider
fluctuations of the Polyakov loop around its mean field value,
$L_{\mathbf{x}}=\overline{L}+\delta L_{\mathbf{x}}$ and
$L^{*}_{\mathbf{x}}=\overline{L}+\delta L^{*}_{\mathbf{x}}$, and to expand the
effective action up to linear order in the fluctuations,

$\displaystyle S_{\text{eff}}\quantity[L]\approx
S_{\text{eff}}\quantity[\overline{L}]+\sum_{\mathbf{x}}\quantity(\partialderivative{S_{\text{eff}}}{L}\bigg{|}_{\overline{L}}\delta
L_{\mathbf{x}}+\partialderivative{S_{\text{eff}}}{L^{*}}\bigg{|}_{\overline{L}^{*}}\delta
L^{*}_{\mathbf{x}})+\dots\;.$

Inserting this mean field approximation, the path integral for the effective
theory simplifies to

$\displaystyle
Z_{\mathrm{mf}}=f\quantity(\overline{L})\quantity[\int\differential
W\exp\quantity{-\partialderivative{S_{\text{eff}}}{\overline{L}}L-\partialderivative{S_{\text{eff}}}{\overline{L}^{*}}L^{*}}]^{V}.$
(6)

The expression $f\quantity(\overline{L})$ represents the saddle point
contribution to the path integral and reads

$\displaystyle
f\quantity(\overline{L})=\exp\quantity[-S_{\text{eff}}[\overline{L}]+\partialderivative{S_{\text{eff}}}{\overline{L}}\overline{L}+\partialderivative{S_{\text{eff}}}{\overline{L}^{*}}\overline{L}^{*}]\;.$
(7)

The remaining integration in Eq. (6) is reduced to one-site integrals, which
can be done after expanding down the exponential, as shown in the example
below. After that we have an explicit formula to calculate the free energy in
the mean field approximation,

$F_{\mathrm{mf}}=-\ln Z_{\mathrm{mf}}\;.$ (8)

##  4 A test case: deconfinement transition in pure gauge theory

To illustrate our concrete calculations, we list the resulting expressions for
the effective theory representing finite temperature Yang-Mills theory, i.e.,
the partition function Eq. (4) reduced to its first line with one effective
coupling only. This will also provide a test of our mean field procedure. For
the linearised corresponding effective action we have

$\displaystyle-S_{\text{eff}}[L]\approx
d\sum_{\mathbf{x}}\quantity[\ln\quantity(1+2\lambda_{1}\absolutevalue{\overline{L}}^{2})+\frac{2d\lambda_{1}}{\quantity(1+2\lambda_{1}\absolutevalue{\overline{L}}^{2})}\quantity(\bar{L}^{*}\delta
L_{\mathbf{x}}+\bar{L}\delta L^{*}_{\mathbf{x}})]\;,$ (9)

where $d=3$ denotes the number of space dimensions. The saddle point
contribution then is

$\displaystyle
f(\overline{L})=\exp\quantity[Vd\ln\quantity(1+2\lambda_{1}\absolutevalue{\overline{L}}^{2})+\frac{2Vd}{\quantity(1+2\lambda_{1}\absolutevalue{\overline{L}}^{2})}]$
(10)

and the partition function in the mean field approximation

$\displaystyle Z_{\mathrm{mf}}$
$\displaystyle=f(\overline{L})\quantity[\int\differential
W\exp\left\\{\frac{2d\lambda_{1}\overline{L}^{*}}{1+2\lambda_{1}\absolutevalue{\overline{L}}^{2}}~{}L+\frac{2d\lambda_{1}\overline{L}}{1+2\lambda_{1}\absolutevalue{\overline{L}}^{2}}~{}L^{*}\right\\}]^{V}$
$\displaystyle=f(\overline{L})\quantity[\sum_{n,m}\frac{\quantity(z_{1})^{n}\quantity(z_{2})^{m}}{n!m!}\int\differential
W~{}L^{n}\quantity(L^{*})^{m}]^{V}$ (11)

with

$z_{1}\equiv\frac{2d\lambda_{1}\overline{L}^{*}}{1+2\lambda_{1}\absolutevalue{\overline{L}}^{2}}\;,\quad
z_{2}\equiv\frac{2d\lambda_{1}\overline{L}}{1+2\lambda_{1}\absolutevalue{\overline{L}}^{2}}\;.$
(12)

To proceed we need to evaluate the Polyakov loop integrals, for which we we
use the formula [14]

$\displaystyle\int\differential
W~{}L^{n}\quantity(L^{*})^{m}=\sum^{\lfloor\frac{n}{3}\rfloor}_{j=\max\quantity(0,\frac{n-m}{3})}\frac{T(n-m)~{}2n!~{}m!~{}\binom{3\quantity(n-j-\frac{n-m}{3}+1)}{n-3j}\binom{2j-\frac{n-m}{3}}{j}}{\quantity(n-j-\frac{n-m}{3}+1)!\quantity(n-j-\frac{n-m}{3}+2)!\quantity(2j-\frac{n-m}{3})!}\;,$
(13)

with binomial coefficients and the triality function $T(n)=1$ if
$n\;\mbox{mod}\;3=0$, and $T(n)=0$ otherwise.

It is well known from lattice simulations that 4d $SU(3)$ pure gauge theory at
finite temperature features a first-order deconfinement transition due to
spontaneous center symmetry breaking, which is faithfully reproduced by the 3d
effective theory [6]. In Fig. 1 we display the mean field free energy density
as a function of the real part of the Polyakov loop. For decreasing effective
coupling we indeed observe a non-trivial second minimum to form at non-
vanishing expectation values, which triggers the center symmetry breaking
transition in the effective theory. Since there is a hill between the minima,
this is a first-order transition. The critical coupling for the transition is
the value where the two minima are degenerate, which happens at
$\lambda_{1c}=0.152$. This is to be compared with a Monte Carlo simulation of
the 3d effective theory, which gives $\lambda_{1c}=0.188$ [6]. We conclude
that the mean field treatment reproduces the correct order of the
deconfinement phase transition, and the predicted critical coupling is within
a reasonable 20% of the true answer. For an extension of the deconfinement
transition to the situation with dynamical quarks, which is also reproduced,
as well as further refinements of the mean field approach, see [15].

Figure 1: Pure Gauge Case: Mean-Field free energy $\mathcal{F_{\text{mf}}}$ as
a function of the expectation value of the Polyalov-Loop
$\expectationvalue{L}$, with different values around the critical values of
$\lambda_{1}$.

##  5 Phase structure for zero temperature and finite chemical potentials

After this successful test, we proceed to our case of interest, namely QCD
with dynamical quarks at low temperatures with non-vanishing chemical
potentials. Now Eq. (4) will be considered with all contribution up to
$\kappa^{2}$ and $N_{f}=2$. For finite baryon chemical potential,
$\mu_{u}=\mu_{d}=\mu_{B}/3$, the effective theory has a sign problem, which
however is much milder than the original one. It has been demonstrated that
the effective theory can be simulated with a choice of different algorithms.
As expected, the theory displays an onset transition to a medium with net
baryon density at $\mu_{B}\approx m_{B}$ [16, 8]. Like the deconfinement
transition, our mean field treatment reproduces this baryon onset to be first-
order for sufficiently large $N_{\tau}$, i.e. low temperatures, by our mean
field treatment.

Here, we investigate for the first time how the effective theory behaves when
isospin chemical potenital is introduced,
$\mu_{I}=\quantity(\mu_{u}-\mu_{d})/2$. In Fig. 2 we display the mean field
free energy density for the QCD lattice parameters $\beta=5.7,\kappa=0.0004$
and $N_{\tau}=10000$. Upon increasing isospin chemical potential, we observe a
flattening of the potential well and the formation of a minimum for a non-
vanishing value of the Polyakov loop expectation value. In contrast to the
previous transitions, there is no hill in the free energy density separating
minima, but instead a critical value of the chemical potential where the
curvature vanishes. This feature predicts the phase transition to be second
order. Once again, this result is fully compatible with what is found in
simulations of full QCD at the physical point with an isospin chemical
potential [1].

Figure 2: Variation of the mean field free energy density with the isospin
chemical potential, lattice parameters: $\beta=5.7$ and $N_{\tau}=10000$

In contrast to any other method, our effective theory now permits us to
consider non-vanishing baryon and isospin chemical potential at the same time,
and to follow what happens to the observed first-order baryon onset and
second-order isospin condensation transitions, respectively. That is, we keep
$N_{\tau}=10000$ fixed and identify the critical combinations
$(\mu_{B}^{c},\mu_{I}^{c})$ by either coexistence of two degenerate minima
(first order) or a vanishing curvature (second order) in the mean field free
energy density. The result is shown in Fig. 3, where the chemical potentials
are given in terms of the baryon and pion masses, which are here crudely
approximated by their leading order values in the hopping expansion,
$am_{B}=-3\ln(2\kappa),\;am_{\pi}=-2\ln(2\kappa)$.

In Fig. 3 we observe both transitions to curve towards each other, and to join
in a special point, where the order of the transition changes. That the baryon
and pion onset transitions must connect somewhere is also to be expected
before an explicit calculation. The region below the transition line
represents vacuum, whereas above it the ground state of the system is either
baryon matter or a Bose-Einstein condensate of pions. In both of the latter
cases there is a medium whose rest frame breaks Lorentz symmetry. Thus a true
phase transition must exist, independent of the direction in the plane of
chemical potentials. Similarly, a Bose-Einstein condensate $\langle
m_{\pi}^{\pm}\rangle\neq 0$ and a non-vanishing baryon expectation value
$\langle n_{B}\rangle\neq 0$ are distinguished by quantum numbers pertaining
to different symmetries. Consequently, baryon matter must be separated from
pion matter by a true phase transition. Indeed, we observe a second-order
transition line separating those phases to emanate from the meeting point of
the two transition lines from vacuum to matter. Unfortunately, we cannot yet
follow this line to larger chemical potentials, which is precluded by lattice
saturation.

Figure 3: $\quantity(\mu_{I},\mu_{B})$-Phase diagram for $T\rightarrow 0$.

##  6 Conclusions

In this work, we have studied three-dimensional effective lattice theories for
heavy quark QCD via a mean field approximation. This approach is promising
because the effective theories represent $SU(3)$ spin models, and one expects
at least qualitatively correct results for the phase structure of the
effective theories. Indeed, for the simplest case representing the finite
temperature pure gauge theory, a first-order deconfinement transition is
observed, and the critical effective coupling is reproduced within $\sim$20%
of the true answer.

We have then studied the situation close to the zero temperature limit, when
both a baryon chemical potential and an isospin chemical potential are
switched on. We obtain a clear picture with a first-order baryon onset
transition for $\mu_{B}\approx m_{B},\mu_{I}=0$ and a second-order transition
to a phase with isospin or Bose-Einstein condensate at $\mu_{I}\approx
m_{\pi}/2,\mu_{B}=0$. These transitions are continuously connected and
separate vacuum from matter. They meet in a branch point, where an additional
second-order line emerges to separate the baryon region from the pion
condensate region.

## Acknowledgments

The authors acknowledge support by the Deutsche Forschungsgemeinschaft (DFG)
through the grant CRC-TR 211 “Strong-interaction matter under extreme
conditions” and by the State of Hesse within the Research Cluster ELEMENTS
(Project ID 500/10.006).

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