text:
\section{Introduction} \label{sec:Introduction}

Recently and in the near future, there is and will be big experimental effort to explore the phase diagram of strongly-interacting matter be it by the energy scan program at RHIC \cite{Ye:2018oqi}, the NA61/SHINE experiment at CERN \cite{Andronov:2018ccl} and or at the future facilities NICA at JINR \cite{Kekelidze:2016wkp} and FAIR at the GSI site \cite{Senger:2017oqn}. These laboratory experiments at low and intermediate baryon densities are complemented by the detection of gravitational waves of inspiraling neutron stars that allow to learn about the nature of QCD matter at the largest densities \cite{TheLIGOScientific:2017qsa}.
All these experimental measurements require a theoretical counterpart to interpret and analyse their results. First principle calculations are not yet able to be this match in the medium and large baryon density region. Calculations on a discretised space-time lattice face the infamous sign problem \cite{Soltz:2015ula} and methods which circumvent it as complex Langevin dynamics are still limited to fundamental theoretical investigations but are not yet connected to phenomenology \cite{Aarts:2016qrv,Kogut:2019qmi}. First-principle continuum calculations using the Functional Renormalization Group, perturbation theory or variational approaches are still in progress towards the true number of quark flavours and quark masses \cite{Mitter:2014wpa,Maelger:2018vow,Maelger:2019cbk,Quandt:2018bbu}.

Therefore, widely used alternatives are frameworks which are based on chiral symmetry and center symmetry and eventually scale symmetry. These symmetries of the QCD Lagrangian are related to two fundamental properties of strongly interacting matter, namely to constituent quark masses and confinement in the hadronic phase and the liberation of light quarks in the transition to the quark-gluon plasma.
The interaction between constituent quarks that gives them their mass due to spontaneous breaking of chiral symmetry can be described in two ways, either as a point interaction or by exchange of a meson. The former leads to what is called the Nambu--Jona-Lasinio (NJL) model \cite{Nambu:1961tp} and the latter to the Quark-Meson (QM) model \cite{GellMann:1960np,Metzger:1993cu,Jungnickel:1995fp}. Extended by the order parameter of center symmetry breaking, the Polyakov loop, these frameworks allow to explore the phase diagram of strongly-interacting matter \cite{Mocsy:2003qw,Fukushima:2003fw}.

Even though the Lagrangian of such models itself is invariant under chiral transformations for massless quarks, the appearance of a non-vanishing quark condensate $\langle\bar{q} q\rangle$ breaks chiral symmetry spontaneously. Therefore, the quark condensate $\langle\bar{q} q\rangle$ is a order parameter for chiral symmetry breaking.
The relation between quark masses, chiral symmetry and quark condensate can be exploited to explain the generation of constituent quark masses by spontaneous chiral symmetry breaking. The constituent quark mass of up and down quarks that are confined in protons or neutrons is of the order of one third of the mass of these nucleons, $\mc{O}\sim300\,\mrm{MeV}$ which is significantly larger than their current quark masses, $\mc{O}\sim\mrm{MeV}$.
While the non-zero current quark masses are responsible for explicit chiral symmetry breaking the constituent quark masses are dynamically generated by spontaneous chiral symmetry breaking, $m\sim\langle\bar{q} q\rangle$. With increasing temperature and/or density of the quarks the dynamic chiral-symmetry breaking becomes restored. In the chiral limit of vanishing current quark masses this is a transition of second order which turns at a tricritical point into a first order one. With the explicit chiral-symmetry breaking when the non-zero current quark masses are taken into account the precise second order phase transition becomes washed out into a crossover that can turn in a critical point to a first order transition. The chiral phase structure of these models is for example discussed in Refs.~\cite{Scavenius:2000qd,Buballa:2003qv,Costa:2008yh}.

The Polyakov-loop field $\Phi(\vec x)$ is the appropriate order  parameter to study the SU$(N_c)$ phase structure and is associated with $\mathbb{Z}_{N_c}$, center of
SU$(N_c)$ \cite{Polyakov:1978vu}. In the pure gauge sector, the phase transition that occurs is related to the deconfinement of colour at high temperature such that the Polyakov loop is an order parameter for the deconfinement transition.
Coupling dynamical quarks to the Polyakov-loop field  and adding a potential that represents the effective glue potential at finite temperature to drive spontaneous center symmetry breaking gives also information on the deconfinement of quarks. But in this way quarks are not confined in the strict sense since dynamical quarks add an explicit breaking of center symmetry such that the contributions coming from one and two (anti-)quarks are suppressed below the transition temperature but are not vanishing \cite{Hansen:2006ee}.

In this work we will discuss several aspects of the correlation between quarks and the Polyakov loop and investigate how to disentangle effects from restoration of chiral symmetry and from breaking of center symmetry to have a deeper knowledge on how both are correlated in the results of the model. This allows to better understand the current status of comparing results of these kind of models with those of lattice QCD calculations and to shed further light to certain less known results of the model.

In the next section we will give a summarised introduction to the Polyakov-loop--extended Nambu--Jona-Lasinio (PNJL) model, putting special emphasis on the correlation between quarks and the Polyakov loop. In \Sec{sec:transition_scales} we discuss the correlation of the transition scales of restoration of chiral symmetry and of breaking of center symmetry and how it could be possible to reproduce the splitting that is seen in lattice QCD calculations at vanishing densities. Section \ref{sec:thermaleffect} contains the analysis of the effect of the kinetic contribution of quarks to the correlation between quarks and Polyakov loop by comparing results for medium dependent and constant quark mass. The analysis of the effects that can be found in such an investigation are extended in Secs.~\ref{sec:MFPhaseDiag} and \ref{sec:CondensPloopCorr}. In \Sec{sec:FermiMomEffect} we analyse and discuss the correlation between the quark contribution and Polyakov loop by the chemical potential. Section \ref{sec:CombindedEffects} discusses the combination of both effects which is completed by an analysis of a confined chirally restored phase in \Sec{sec:CCS}.

We  will present  results of  the model for  two flavors. While including the strange quark is important for quantitative predictions, focusing on two light flavours allows us to disentangle non-trivial correlations of the ingredients of the framework giving a more comprehensive view of the physical phenomenon.

\section{The PNJL model in SU(2)} \label{sec:PNJLmodel}

In this section we introduce the Polyakov-loop--extended Nambu--Jona-Lasinio (PNJL) model to study the thermodynamics of QCD at
finite density and temperature, in the grand canonical ensemble.

The lack of confinement in the NJL models does not allow the study of
the very important transition of deconfinement at non-zero temperature.  As explained in
Sec. \ref{sec:Introduction} the Polyakov loop can be used
as an order parameter for this transition. In this section we will introduce it in
the NJL model and describe its features.

Throughout  this  section we  will  in  particular  discuss that  if  the
Polyakov  loop   brings  some   sort  of   confinement  (``statistical
confinement''), it  is not  a real  confinement. The Fock space
structure of this model still  contains  quark  degrees  of  freedom and only the quark occupation numbers will be modified.
As a result when  we discuss  confinement, we  will often not  be very
rigorous  on  the vocabulary.

\subsection{Pure gauge sector at finite temperature}

In the pure gauge sector, the phase transition that occurs is related to the
deconfinement     of     color     at    high     temperature.
Following the arguments given in
\cite{Meisinger:1995ih,Pisarski:2000eq,Meisinger:2001cq,Mocsy:2003qw} to study the SU$(N_c)$ phase structure, the
appropriate order  parameter (associated with $\mathbb{Z}_{N_c}$,  center of
SU$(N_c)$) is the Polyakov loop field $\Phi(\vec x)$:
\begin{eqnarray}
L(\vec      x)      &\equiv&     {\mathrm{Tr}}_c
\mathcal{P}\exp\left[i\int_{0}^{\beta}d\tau\,
A_4\left(\vec{x},\tau\right)\right], \\
\Phi(\vec  x) &\equiv&  \frac 1  {N_c} \,  \langle\!\langle L(\vec  x)
\rangle\!\rangle_\beta .
\end{eqnarray}
In the above, $A_4 = i A^0$ is the temporal component of the Euclidean
gauge field  $(\vec{A}, A_4)$, in  which the strong  coupling constant
$g_S$ has been  absorbed, $\mathcal{P}$ denotes path  ordering and the
usual notation  for the thermal  expectation value has  been introduced
with $\beta = 1/T$ with the Boltzmann constant set to one ($k_B \equiv 1$).

An effective potential that respects the  $\mathbb{Z}_3$ symmetry of
the original  Lagrangian may be built and it is conveniently chosen  to reproduce results
obtained  in lattice  calculations. In  this approximation,
the Polyakov loop field  $\Phi(\vec{x})$ is simply set to be equal to its expectation
value $\Phi=$const., which minimizes the potential

\begin{eqnarray}
\frac{\mathcal{U}_{Poly}\left(\Phi,\bar\Phi;T\right)}{T^4} &=&-\frac{b_2\left(T\right)}{2}\bar\Phi \Phi
-\frac{b_3}{6}\left(\Phi^3+
{\bar\Phi}^3\right)\nonumber\\
&+&\frac{b_4}{4}\left(\bar\Phi \Phi\right)^2\;,
\label{Ueff}
\end{eqnarray}
where
\begin{eqnarray}
b_2\left(T\right)=a_0+a_1\left(\frac{T_0}{T}\right)+a_2\left(\frac{T_0}{T}
\right)^2+a_3\left(\frac{T_0}{T}\right)^3.
\label{eq:b2}
\end{eqnarray}
$T_0 = 270$ MeV is the critical temperature for the deconfinement phase
transition   according  to   pure   gauge   lattice  findings \cite{Kaczmarek:2002mc}.

A fit of the coefficients $a_i,~b_i$ has been performed in
Ref. \cite{Ratti:2005jh} to reproduce the lattice data \cite{Boyd:1996bx} for thermodynamics in the pure gauge sector: $a_0 = 6.75$, $a_1 =
-1.95$, $a_2 = 2.625$, $a_3 = -7.44$, $b_3 = 0.75$, $b_4 = 7.5$.  By
minimizing this potential, it was possible to compute the Polyakov-loop expectation value in good agreement with lattice gauge findings \cite{Kaczmarek:2002mc}.

A popular alternative to this potential is the logarithmic one \cite{Roessner:2006xn}
that reads:
\begin{multline}
    \frac{\mathcal{U}_{Log}\left(\Phi,\bar\Phi;T\right)}{T^4}
    =-\frac{a\left(T\right)}{2}\bar\Phi \Phi\\
		+b(T)\mbox{ln}[1-6\bar\Phi \Phi  +4(\bar\Phi^3+ \Phi^3)-3(\bar\Phi
    \Phi)^2],
    \label{UeffLog}
\end{multline}
where
\begin{eqnarray}
    a\left(T\right)&=&a_0+a_1\left(\frac{T_0}{T}\right)+a_2\left(\frac{T_0}{T}
  \right)^2\\\mbox{ and }\,\,b(T)&=&b_3\left(\frac{T_0}{T}\right)^3.
\end{eqnarray}

The parameters of the effective potential $\mathcal{U}_{Log}$ are given by $a_0=3.51$, $a_1=
-2.47$, $a_2=15.2$ and $b_3=-1.75$.

This  effective  potential with these parameters  exhibits   the  feature  of  a  phase
transition from color confinement to color deconfinement through a stronger first-order transition than $\mathcal{U}_{Poly}$ with the parameters of \Ref{Ratti:2005jh}, see e.g.~Fig.~8 of \Ref{Haas:2013qwp}.

\subsection{Coupling between quarks and the gauge sector}
The  PNJL  model  aims at describing in a  simple way two of the characteristic phenomena of QCD, namely deconfinement and chiral symmetry breaking.
We start from the two flavor  NJL  description of  quarks
(global   SU$_c(3)$  symmetric   and  chirally   invariant  point-like
interaction), coupled  in a minimal way  to the Polyakov loop  via the
following Lagrangian \cite{Ratti:2005jh} (the range of applicability of
this model is limited to temperatures $T\leq 2.5~T_c$
):
\begin{eqnarray}
\mathcal{L}_{PNJL}=\bar{q}\left(i\gamma_{\mu}D^{\mu}-\hat{m}_0
\right)q&+&
G_S \left[\left(\bar{q}q\right)^2+\left(\bar{q}i\gamma_5
\vec{\tau}q
\right)^2\right]\nonumber\\
&-&\mathcal{U}\left(\Phi[A],\bar\Phi[A];T\right),
\label{lagr:PNJL}
\end{eqnarray}
where $q=\left(q_u,q_d\right)$ are the quark fields and where the
covariant derivative reads $ D^{\mu} = \partial^\mu - i A ^ \mu$ with
$A^\mu= g_{S} {\cal A}^\mu_a(x) \frac {\lambda_a}2
\delta^{\mu}_{0}A^0$ and $A^\mu=\delta^{\mu}_{0}A^0$ (Polyakov gauge),
with $A^0 = -iA_4$.
The two-flavor  current quark  mass matrix  is $\hat{m}_0=\mbox{diag}(m_u,m_d)$
which breaks chiral symmetry explicitly (we work in the isospin symmetric limit
and consequently $m_u=m_d=m_0$).
$G_S$ is the coupling strength of the chirally symmetric
four-fermion interaction.
Finally, also a tridimensional ultraviolet cut-off $\Lambda$ will be introduced in \Eq{omega} in order
to regularize divergent integrals.

This minimal coupling between quarks and the Polyakov loop is a first way to
take into account back-reactions of the quark sector to the gluonic sector.

The parameters of the pure NJL sector in Eq.~(\ref{lagr:PNJL}) are fixed at zero
temperature and density and have the following values:
$\Lambda =  651 $ MeV,  $G_S = 5.04$ GeV$^{-2}$  and $m_0 =  5.5$ MeV.
They are chosen to  reproduce the mass of the pion  $m_\pi = 139.3$ MeV
and its  decay constant $f_\pi  = 92.3$ MeV  (obtained in a  Hartree +
Ring  (RPA) calculation)  as well  as  the chiral  condensate
(order  parameter  of the  chiral symmetry) $|\langle{\bar q}q\rangle|^{1/3}  = 251$ MeV.
The constituent (Hartree) quark mass  is $m = 325$ MeV. With $M_N$ the
nucleon mass this means $m \simeq  M_N/3$.  The PNJL model  gives a
simple  explanation of  the nucleon  as being  a state  composed of  three
quarks but their  mass is the corresponding dynamical  mass. Even with $m_0  = 0$, $m$
has the same  magnitude: essentially the baryonic  mass is due
to the ``glue'',  the interaction energy carried out by  the gauge fields.
The mechanism of  mass generation via the spontaneous symmetry breaking  was one of
the first interests of the NJL model.

\subsection{Statistical confinement at  finite temperature and density
in the Hartree approximation}

\subsubsection{Grand canonical potential, mean  field  equations  and the  modified  Fermi--Dirac
distribution functions}
\label{sec:GrandPotential}
The usual  techniques can be used  to obtain the PNJL  grand canonical potential
from  the Hartree  propagator (see for instance Refs.~\cite{Hansen:2006ee,Klevansky:1992qe,Ratti:2005jh}),
\begin{multline}
\Omega(\Phi,\bar\Phi,m;T,\mu)={\cal U}\left(
\Phi,\bar{\Phi},T\right)+\frac{(m - m_0)^2}{4G_S}
-\\- 2 N_c N_f\int_\Lambda\frac{\mathrm{d}^3p}{\left(2\pi\right)^3}\,{E_p}  \\
- 2N_f\,T\int_{+\infty}\frac{\mathrm{d}^3p}{\left(2\pi\right)^3}
\left\{\mathrm{Tr}_c\ln\left[1+ L^\dagger \mathrm{e}^{-\left(E_p-\mu
\right)/T}\right] \right.\\
+\left.\mathrm{Tr}_c\ln\left[1+ L\mathrm{e}^{-\left(E_p+\mu\right)/T}\right]
\right\}. \\
\label{omega}
\end{multline}
In the above  equation $E_p=\sqrt{\vec{p}\,^2+m^2}$  is  the  Hartree
single  quasi-particle  energy  (which  includes  the  constituent  or
dynamical quark mass $m$ and not  the current mass $m_0$).
Besides for $N_c = 3$ we have,
\begin{multline}
z^+_\Phi(E_p)=\mathrm{Tr}_c\ln\left[1+ L^\dagger
         \mathrm{e}^{-\left(E_p-\mu\right)/T}\right]
         =\\=
         \ln\left\{ 1 + 3\bar\Phie^{-\left(E_p-\mu\right)/T} + 3 \Phi e^{-2\left(E_p-\mu\right)/T}
         \right.\\+\left. e^{-3\left(E_p-\mu\right)/T} \right\},
\label{zplus}
\end{multline}
\begin{multline}
z^-_\Phi(E_p)=\mathrm{Tr}_c\ln\left[1+ L
         \mathrm{e}^{-\left(E_p+\mu\right)/T}\right]
         =\\= \ln\left\{ 1 + 3\Phie^{-\left(E_p+\mu\right)/T}  + 3\bar\Phie^{-2\left(E_p+\mu\right)/T} \right.\\+\left.
       e^{-3\left(E_p+\mu\right)/T} \right\}.
\label{zmoins}
\end{multline}

The solutions of  the mean field equations are  obtained by minimizing
the grand potential with respect to $m$, $\Phi$ and $\bar\Phi$, namely
$\frac{\partial\Omega}{\partial\Phi}          =          0          $,
$\frac{\partial\Omega}{\partial\bar\Phi}      =      0      $      and
$\frac{\partial\Omega}{\partial m}  = 0  $. The latter equation coincides with
the  gap   equation,
\begin{multline}
  m-m_0=2G_S N_f
  N_c\int_{+\infty}\frac{\mathrm{d}^3p}{\left(2\pi\right)^3}\frac{2m}{E_p}\left[\theta(\Lambda^2
  - p^2)\right. \\\left.- f^+_\Phi(E_p)-f^-_\Phi(E_p)\right]\,,
\end{multline}
where the  modified Fermi--Dirac distribution  functions $f^+_\Phi$
and $f^-_\Phi$ have  been introduced:
\begin{multline}
  f^+_\Phi(E_p)
          =\\\frac{ \left( \Phi + 2\bar\Phi e^{-\beta\left(E_p-\mu\right)} \right) e^{-\beta\left(E_p-\mu\right)} + e^{-3\beta\left(E_p-\mu\right)} }
  {1 + 3\left( \Phi + \bar\Phi e^{-\beta\left(E_p-\mu\right)} \right) e^{-\beta\left(E_p-\mu\right)} + e^{-3\beta\left(E_p-\mu\right)}} \label{fpPhi},
\end{multline}
\begin{multline}
  f^-_\Phi(E_p)
         =\\\frac{ \left( \bar\Phi + 2\Phi e^{-\beta\left(E_p+\mu\right)} \right) e^{-\beta\left(E_p+\mu\right)} + e^{-3\beta\left(E_p+\mu\right)} }
  {1 + 3\left( \Phi + \bar\Phi e^{-\beta\left(E_p+\mu\right)} \right) e^{-\beta\left(E_p+\mu\right)} + e^{-3\beta\left(E_p+\mu\right)}}. \label{fmPhi}
\end{multline}

The equations presented above, which were introduced for the first time in
\cite{Hansen:2006ee}, allow to straightforwardly generalize the results for thermodynamics of the NJL model to those of the PNJL model by replacing the usual Fermi--Dirac occupation numbers
by the modified ones given by Eqs.~(\ref{fpPhi}) and (\ref{fmPhi}).

The explicit form of the mean field equations for  $\Phi$ and $\bar\Phi$,
which can be found for  example in \cite{Hansen:2006ee},
will be given in \secref{sec:backreaction} where  we will discuss  in detail
the correlation between quarks and gauge fields.

Here, we want to point out that we choose not to use the Fock term because, if added, we
would obtain the same equation with the replacement $G_S \rightarrow G_S(1 + 4/N_c)$.
Since $G_S$ is a parameter to be fixed, it is not very important to include the Fock term.
We can also  notice that the Hartree term is  of order $O(N_c^1)$, the
Fock term is  $O(N_c^0)$ and the ring approximation can  be shown to
be also $O(N_c^0)$. As discussed in Ref. \cite{Blin:1987hw} the Hartree term
may  be  seen as  the  first  term in  a  $1/N_c$  expansion. Then, for
coherence the  Fock term should  be added  when going beyond  the mean
field.

\subsubsection{The influence of quarks on the gauge fields} \label{sec:backreaction}

To perform the study of the influence of quarks on the gauge fields we will push
the model in later sections to a very high density scale, probably well beyond the range of
validity of the model. However, we find these calculations interesting as they
lead us to understand what ingredients are probably missing in the model
to get a more faithful representation of QCD in the high density region.

Obviously, in QCD quark fields act upon gauge field via quark loops in
the gluon  self energy. Such an effect is lacking in the NJL model since
there are no gluon fields (although, in some sense gluons  are present in
the model  via the  NJL contact interaction  as we  mentioned earlier).

In the PNJL model there is still no  dynamical gluons but there is an interaction
between  the  static gauge  field  and  the  quarks via  the  covariant
derivative $ D^{\mu} =  \partial^\mu - i A ^ \mu$. As  a result, the mean
field equation for $\Phi$  does have a dependence on quarks
(see previous section). Explicitly for two flavors, we have for $\Phi$
(the discussion for $\bar\Phi$ goes along the same lines):
\begin{multline}
\frac{\partial\Omega}{\partial\Phi} = 0 \quad\Leftrightarrow\quad
0  =   {T^4}  \frac{\partial{\cal  U}}{\partial  \Phi}
\\-            6           T            \sum_{\left\{i=u,d\right\}}
\int\frac{\mathrm{d}^3p}{\left(2\pi\right)^3}  \left( \frac  {e^{-2\beta\left(E_p+\mu\right)}}{
e^{z^+_\Phi(E_p)} } +  \frac  {e^{-\beta\left(E_p-\mu\right)}}{e^{z^-_\Phi(E_p)}  }
\right)
\label{eq:domegadfi}
\end{multline}
with $z^+_\Phi(E_p)$ and $z^-_\Phi(E_p)$ given by Eqs. (\ref{zplus}) and
(\ref{zmoins}) respectively.

The second  term in Eq. (\ref{eq:domegadfi}) adds some  quark  corrections  and  interesting
information can be extracted from this equation as we will discuss later.
Without quarks  the solution  would be the  pure gauge  one, i.e., (${\partial
{\cal U}}/{\partial\Phi} =  0$).
The model would then have a first-order phase transition.

\subsubsection{Statistical confinement} \label{sec:StatisticalConf}

In  the  modified Fermi--Dirac  functions, Eqs.\ (\ref{fpPhi}) and (\ref{fmPhi})
entering to the  grand  potential, Eq.\ (\ref{omega}),
the confinement mechanism  that exists in Polyakov models can be seen.  It is not  a true  confinement (the
quarks are still asymptotically free and we will  see that they can be produced e.g.\ by the
sigma meson decay in vacuum) but has been dubbed ``statistical confinement''.

Indeed, it can be seen that in  the  grand  potential, Eq. (\ref{omega}), the contributions coming from one
and two (anti-)quarks are suppressed below $T_c$ (when $\Phi,\bar\Phi\to 0$, confined phase of  the model) due to their coupling with $\Phi$ and $\bar\Phi$ but the 3-(anti-)quarks Boltzmann factor is not.
The interpretation is that  there are still unconfined (anti-)quarks in the  vacuum part (the Fermi
sea) of the grand potential but in the thermal bath only 3-(anti-)quarks contributions (a reminder
of the  fact that in  QCD only colorless  combinations can exist  in the confined  phase) are
present.  This  reduces significantly  the number  of (anti-)quarks  in the  thermal bath  since it
requires three times more energy for the (anti-)quarks to be thermodynamically active.

It is known that,  at a given value  of $T$ and $\mu$,  the pure NJL model
always overestimates the density (see Ref. \cite{Ratti:2005jh}),
even if it merges for large  temperatures with the PNJL model (when
$\Phi\to1$).
At fixed  values of $T$ and $\mu$, the
PNJL value for $n_q$  is much lower than in the NJL  case. In fact, all the possible contributions to the latter turn out
to be somehow suppressed: the one- and two-quark contributions because
of $\Phi,\bar\Phi\to 0$,  while the thermal excitation  of three quarks
has a negligible  Boltzmann factor. We would be  tempted to identify
these  clusters  of three  dressed  (anti-)quarks  with precursors  of
(anti-)baryons but no binding for  these structures is provided by the
model. In any case it is  encouraging that coupling the NJL Lagrangian
(which parameters are chosen to reproduce zero temperature properties)
with  the Polyakov-loop field  (described by  a pure  gauge effective
potential) leads to results pointing into the right direction.

Another important impact of statistical confinement is  that the transition will occur in a
shorter range of temperature than in the NJL model. It is  seen from  Eqs.\ (\ref{zplus})  and
(\ref{zmoins}) that before the transition, the pressure (for example)
is kept low by the effect just discussed. Then, quark  degrees of freedom are liberated in  a short
temperature range  when $\Phi  \rightarrow 1$.  As  seen in Ref. \cite{Costa:2010zw} the
transition in the PNJL model is indeed much ``faster'' that the one in the pure NJL model.

The  mesonic contribution to the pressure is also important (about 10\%) and  this effect has
to be included in the mean-field description (e.g. with the  Beth-Uhlenbeck formalism \cite{Hufner:1994ma} which is another way of doing RPA) or to a model with
mesonic  degrees of freedom such as  the Polyakov-loop--extended Quark-Meson (PQM) model \cite{Stiele:2014}.
This allows to fit nowadays well the most recent
lattice results with physical quark masses.
A  detailed  discussion  of the  results  in  mean-field  approximation can  be
found  in \cite{Costa:2010zw}.

\section{Coincidence between chiral and deconfinement transition}

One of  the first successes of the PNJL model was the observation that without
any  additional  tuning, chiral  and deconfinement transitions almost coincide
in a small range of temperatures at zero density (see Fig.~\ref{fig:PNJLPhases}).
It  was  an interesting  test as LQCD calculations confirmed this coincidence
(even if recent lattice results show there is no perfect coincidence \cite{Borsanyi:2010bp}).

What is not  obvious is that by mixing the  deconfinement scale $T_0 =
270$ MeV  and the  chiral restoration  scale (of about  $220$ MeV  in the NJL
model)  the  two  transitions   automatically  coincide  at  a  lower
temperature.  The PNJL minimal coupling is enough to have some sort of
back-reaction effect that produces this matching.

This is also a quite stable feature \cite{Ratti:2005jh}: we can take
$T_0 = 190$ MeV and still get a quite good coincidence. The motivation to
change the value of $T_0$ is to get a better agreement with the value of the
LQCD chiral transition temperature. As pointed out, it can be justified since in
an effective model we are mixing physical sectors with different scales in
a symmetry based  (Landau) framework.  It is then understandable that
the absolute scale of the two sectors can be slightly adjusted
to  get a  better agreement  with the phenomenology.\\
In  other words, we authorize ourselves to consider $T_0$ as a free parameter of the model
with a loose constraint on it coming  from LQCD, exactly as it can be done in the chiral sector with
the constituent quark masses despite that there is some estimate of these from measurements.

\section{Adjusting chiral and deconfinement phase scales \label{sec:transition_scales}}

The most problematic aspect for PNJL type models is probably the fact that the difference between the chiral transition and the raise of $\Phi$ in the QGP in recent lattice
data is larger than previously seen, see Fig. \ref{fig:PQMvsLQCD}.  To show this, we use  a recent PQM model \cite{Stiele:2014} because we need a good agreement between the model and LQCD data.
\begin{figure*}
  \centering
  \includegraphics[width=0.47\textwidth]{pressure_muQ0_best}
  \hfill
  \includegraphics[width=0.47\textwidth]{e3p_muQ0_best}\\
  \includegraphics[width=0.47\textwidth]{DeltaLS_muQ0_best}
  \hfill
  \includegraphics[width=0.47\textwidth]{Ploop_muQ0_best}
  \caption{Pressure, trace anomaly, chiral condensate and Polyakov loop as a function of
  $T$ in a PQM model \cite{Stiele:2014} compared with LQCD data of Refs.
  \cite{Borsanyi:2010bp,Bazavov:2012bp,Bazavov:2013yv}.
  }
  \label{fig:PQMvsLQCD}
\end{figure*}
In  this  figure, $t$  is  the  reduced  temperature adjusted  to  the
chiral crossover temperature  $T_c$: $t = T/T_c  - 1$.  The
pressure plot, or the quark number density one, shows that at $T_c$  ($t=0$) quark
degrees  of  freedom  are  liberated.  But  there  are  two  mechanisms
responsible  for  this  liberation:  the chiral restoration  and  the
deconfinement transition. This can be easily studied in effective models since
we have a control on how these phenomena are coupled.

In Fig.~\ref{fig:PQMvsLQCD}, bottom left, it is possible to see that at $t=0$
chiral symmetry goes toward its restoration. There is a liberation of
thermodynamical degrees of freedom simply because  $m \rightarrow m_0$
hence the  quark Boltzmann factor has a bigger contribution  to the
pressure. This occurs at the chiral transition scale.
On the bottom right panel, it is seen that the results for $\Phi$ from LQCD and
PQM model calculations do not agree.
The (statistical)  deconfinement occurs  at a
higher temperature
than in the PQM model (at $t=0$, the PQM model  gives $\Phi \simeq 0.5$ whereas it is
only $0.1$ for LQCD results). I.e.~the (statistical) deconfinement scales differ in LQCD
data and in the PQM model.

One subtlety concerning the Polyakov loop is that there are different order parameters. The standard order parameter is  the expectation value $\langle\Phi\slr{A_0}\rangle$. The functional dependence indicates that the Polyakov loop derives from the temporal component of the gauge field $A_\mu$. The expectation value of the later $\langle A_0\rangle$ relates to another Polyakov-loop order parameter, $\Phi\slr{\langle A_0\rangle}$.

The LQCD results are for $\langle\Phi\rangle$ since this is the easily accessible quantity in these results.
But in Polyakov-loop extended effective models for QCD the Polyakov-loop potential is adjusted to lattice results on $\langle\Phi\rangle$ in pure gauge theory while it is actually the gauge field that appears in the fermionic determinant initially, see \Eq{omega} which is then rewritten into a dependence on the Polyakov loop $\Phi$ in \Eqs{zplus} and \eq{zmoins}.
Both Polyakov-loop order parameters are related by Jensen's inequality in the following way: $\langle\Phi\slr{A_0}\rangle\leq\Phi\slr{\langle A_0\rangle}$ \cite{Braun:2007bx,Marhauser:2008fz}. This condition shows that the transition scales of both Polyakov-loop order parameters might differ with that of $\langle\Phi\rangle$ being larger than that of $\Phi\slr{\langle A_0\rangle}$. The results shown in \Fig{fig:PQMvsLQCD} are consistent considering this condition.

The origin of these both Polyakov-loop order parameters, their derivations and their relation is discussed in detail in \Ref{Herbst:2015ona}.
The temperature dependence of $\Phi\slr{\langle A_0\rangle}$ is calculated in continuum approaches such as the Functional Renormalization Group, with Dyson-Schwinger equations and the 2PI formalism as well as in the Hamiltonian approach and covariant variational approach and in perturbative approaches.
For pure gauge theory, available results on $\Phi\slr{\langle A_0\rangle}$ in different continuum approaches are shown together with those for $\langle\Phi\slr{A_0}\rangle$ of different lattice calculations in \Fig{fig:PloopYM_latticeVScont}.
\begin{figure
	\centering
	\includegraphics[width=0.47\textwidth]{PloopYM_lattice_and_continuum}
	\caption{Comparison of the temperature dependence of the Polyakov-loop order parameter $\langle\Phi\slr{A_0}\rangle$ on the lattice \cite{Kaczmarek:2002mc,Gupta:2007ax,Lo:2013hla} and $\Phi\slr{\langle A_0\rangle}$ in different continuum approaches \cite{Herbst:2015ona,Quandt:2016ykm,Reinosa:2014ooa,Reinosa:2015gxn}.}
	\label{fig:PloopYM_latticeVScont}
\end{figure}
Interestingly enough, for the perturbation theory calculations, the leading order result agrees with the results of the other continuum approaches while evaluating the next-to-leading-order result with the leading-order relation between expectation value of the gauge field and that of the Polyakov loop (see also \Ref{Reinosa:2015gxn}), it is very close to the lattice results.

The information given by LQCD calculations is that the scales of chiral symmetry restoration and center symmetry breaking do not match
perfectly. In  general, in  the PNJL/PQM  model families,  this
scale  separation
is not achieved  or  worse,  with
common
parametrization as the one we use in PNJL ($T_0 = 190$ MeV), the scale
hierarchy is inverted, see \Fig{fig:PNJLPhases}.

In effective  models, one  can have
control  over
this  hierarchy as we show in the following.
The  shortcoming however  is that the
output of the model for the absolute temperature of the phase transition will
not be  correct. This  is a  problem but  not to  the point that the model
becomes  meaningless.  Indeed, in  the spirit  of  Landau theory,  two
systems with the  same universal behavior cannot  be directly compared
but they have to be compared relatively to a given  scale. This is of
course not a complete justification for what we will do in the following but it will illustrate
which information is possible to obtain about some mechanism that is probably
missing in the model. We recall  that a scale adjustment was already
done at the beginning of the PNJL model: for example, in Ref. \cite{Ratti:2005jh}
the $T_0$ scale is modified to get a better absolute agreement with the
chiral transition temperature given by LQCD results.

So the point  is that we want to have in the model the correct scale
hierarchy  for  the transitions. In the PNJL model we have two  sectors  to
determine  the respective scales; the  NJL one (fitted  to chiral symmetry
breaking phenomenology in vacuum, a scale related to the strength of the condensate)
and the gluonic one (fixed by pure gauge results at finite temperature
with  the  scale $T_0$).   The  coupling  is  done via  the  covariant
derivative but if we allow to vary $T_0$ then we can control the
relative scales of the  transitions.
This  is done in Fig. \ref{fig:T0variation}:
we see  that the correct  hierarchy can  indeed be achieved  with $T_0
\simeq  400$  MeV  but the  price  to  pay  is  that both  chiral  and
deconfinement transitions occur at too high temperatures.

To get a  better phenomenology one could compute the PNJL properties not relatively
to an absolute scale but relatively to $t$ where
$T_c$  is  the chiral  transition  temperature. Then, one adjusts  the  NJL
parameters  to  get  a  better mesonic  phenomenology  when  expressed
relatively  to this  scale. The  latter adjustment  is needed  because
without it,  the pion mass  would be too low.  Indeed, if we have to rescale
everything to get the correct  absolute scale, it would be needed a correction
of about a half to send  $T_c \simeq 400$ MeV
to  the correct  $T_c \simeq  200$ MeV  hence the  pion mass  would be
around 65 MeV.

\begin{figure}
  \centering
\includegraphics[width=0.47\textwidth]{T0variation}
  \caption{Chiral (black line) and  deconfinement (green line and pink
dots) transition temperatures (together with the size of the transition region
and  for deconfinement, the  full  (pink) and  fixed mass  (green)
calculations  are  presented)   as  a  function  of   the  pure  gauge
deconfinement scale  $T_0$. We use  the polynomial Polyakov-loop potential.  We have
the same feature with the Log potential but at a much higher scale.
}
  \label{fig:T0variation}
\end{figure}

We are  not sure if such  a full reparametrization is  possible but
anyway the message is that some mechanism is missing to get the
correct scale  hierarchy and we should try to understand which mechanism
would allow to have the same effect than our rather artificial
increase of  $T_0$. Since up to now no model can reproduce this fact, we probably
need to look outside of the existing mechanisms of the PNJL/PQM models.
Our guess is that introducing a minimal dynamics to the gauge
sector \cite{Alba:2014lda} may be the correct way.

We  also  notice  that in the PQM model calculation shown in Fig.~\ref{fig:PQMvsLQCD} the pressure and quark density
are probably a little bit overestimated for  $t>0$ when compared to LQCD.
If the  correct  scale hierarchy could be implemented, maybe the later
liberation of quarks via statistical deconfinement could reduce this excess.
This is an a bit far fetched conclusion since the excess is low (statistically it seems
around a  small but constant 1  sigma deviation) and in  PQM model the
mesonic degrees of freedom tend to survive in the QGP phase \cite{Herbst:2013ufa}

Finally,  for what concerns this aspect of  quark   and  Polyakov  loop
correlations, we also notice in Fig. \ref{fig:T0variation} that the full calculation
for  $\Phi$ from \Eq{eq:domegadfi} and the one keeping the mass  $m$ at a fixed value in \Eq{eq:domegadfi} coincide,  except  when  the
difference  between   the  chiral   temperature  and  the   fixed  $m$
deconfinement temperature  is smaller  than about  25 MeV.  This means
that the  order parameters $\Phi$ and the chiral condensate  are weakly
coupled by the  mean field equations because except for a small 25 MeV
$\ll \Lambda_{QCD}$ scale difference, the decorrelated calculation (fixed $m$) gives the
same result as the correlated one. We will discuss this kind of analysis in more detail in
the next section.

For the quarks, at low $T_0$ the chiral transition is  below its NJL
value (around 220 MeV) and it happens at a large (but below 1) value
of the Polyakov loop while at high $T_0$ the transition occurs for
small value (around zero) of the Polyakov loop. Thus, in the PNJL model
the chiral transition always  behave differently from what happens
in the  NJL model due to  the action of the  Polyakov loop, explaining
the success of  the model. The shortcoming of the
model is essentially in the behavior of the Polyakov with respect to quarks but
the converse seems to be fine.

The general  conclusion is that quite certainly this class  of models
underestimate the correlation for the  lack of dynamical  quark loop
effect on  the Polyakov  loop. The only effect present is due to the
presence of thermodynamical quarks (via  the quark Boltzmann factor in
the mean field  equation) that are due to the  minimal coupling $\bar q
A_\mu  q$. But  since  $A_\mu$ is  not dynamical  this  term will  not
generate quark  loops in a  diagrammatic approach ($A_\mu$ acts  as an
external constant gauge field only).

\section{Dynamical thermal quarks effect} \label{sec:thermaleffect}
\begin{figure*}[t]
  \centering
  \includegraphics[width=0.47\textwidth]{phi_mfix_fence_mu_0_2}
  \includegraphics[width=0.47\textwidth]{phi_mfix_fence_mu_0_5_2}
  \caption{ $\Phi$ and $\chi_\Phi$ as functions  of $T$  for different  values of the constituent quark mass $m$. At zero
density  (left) and at  finite  density (right) where the Fermi momentum contributes to the kinetic energy.
Varying the mass changes the strength of explicit breaking of  the $\mathbb{Z}_3$
symmetry: the lower the mass
the more
crossover like is  the transition. The higher  the chemical potential,
the higher  mass is needed  to restore the  symmetry and to have  a first-order transition.}
 \label{fig:Z3ExplicitBreaking_m}
\end{figure*}
In this Section we will consider the effect of the kinetic term of quarks in
Eq.~(\ref{eq:domegadfi}).
At zero  density ($\mu=0$), and with  infinite quark mass, the  second term
disappears as  it should.  Indeed,  quarks are then no more  dynamical (they
are  quenched) and  the system  is in  its pure  gauge or  Yang--Mills
limit.  The $\mathbb{Z}_3$ symmetry is  then exact (for the grand potential)
but spontaneously broken at $T > T_0$.

\begin{figure*}
  \centering
  \includegraphics[width=0.47\textwidth]{phi_mfix_0_mu_fence_2}
  \includegraphics[width=0.47\textwidth]{phi_mfix_2_mu_fence_2}
  \caption{$\Phi$ and $\chi_\Phi$
as   functions  of   $T$  for  different   values  of   the  chemical
potential. At zero quark mass (left) and large quark mass $m=2\Lambda$ (right).
A higher chemical potential implies a larger kinetic term and explicit
$\mathbb{Z}_3$-symmetry  breaking. A  larger  mass is  required to  get  back a  weak
explicit breaking and a first-order transition.}
\label{fig:Z3ExplicitBreaking_mu}
\end{figure*}
When quarks are considered dynamical again, by lowering  their mass approaching the respective
physical values, the $\mathbb{Z}_3$   symmetry    gets   also    an   explicit    breaking   term
Sec.~2.2.2 and 2.3.2 in \Ref{Stiele:2014}) because of the kinetic term in the
Lagrangian.
This mechanism can be understood in detail by solving the  mean field equations,
not self-consistently  as usual, but simply by fixing  in
Eq.~(\ref{eq:domegadfi}) the mass $m$ to a given value. In Figs. \ref{fig:Z3ExplicitBreaking_m} and
\ref{fig:Z3ExplicitBreaking_mu} we plot the behavior of  the Polyakov loop for
different  masses, and hence changing the ratio  between the mass energy $E_m$ and
the kinetic energy $E_k$ of a  quark: increasing the mass reduces  the ratio
$E_k/E_m$;  when this ratio is  zero (quenched quarks) the symmetry is restored.
To better illustrate if there is a stronger or weaker explicit
breaking of the $\mathbb{Z}_3$ symmetry, we use as Polyakov-loop potential a parametrization that has a strong first-order transition in
the  pure  gauge sector,  (the $U_{Log}$ potential, Eq.~(\ref{UeffLog})). If the  transition becomes  smoother (crossover  like
transition) it indicates that we have a stronger explicit breaking of $\mathbb{Z}_3$
symmetry.

These figures  also show the  impact of  varying the chemical  potential.
Indeed, there is a contribution to  the kinetic energy due to the Fermi
momentum  of quarks:  increasing  $\mu$ increases  the  ratio $E_k/E_m$ and  the
symmetry is  more broken. We will  discuss further this effect  in
Sec.~\ref{sec:CondensPloopCorr}
but for now, we focus on the zero density situation
where there is only one contribution to the kinetic energy.

We observe  that a lowering of the mass term introduces  a stronger
explicit breaking of $\mathbb{Z}_3$ symmetry as  seen by the large crossover obtained
and the respective smooth Polyakov loop susceptibility.
We also  notice that for $m$  below the scale $\Lambda_{QCD}$  the qualitative behavior of the Polyakov
loop  becomes  relatively  insensitive  to   the  value  of  $m$.  In
Fig. \ref{fig:PhiMFix} it is seen better that $\Phi$ can be computed with
$m = m_0$ or $m = m_{vac}\simeq 325$ MeV without significant difference.

We notice that, except very close  to the chiral transition (where $m$
changes quickly) $\chi_{\Phi}$ is not much altered by keeping the
mass  constant.  So,  even  the transition  properties  are almost not affected
by  the precise value of  the mass. It is  probable that this lack of sensitivity
to the mass of the quark is  a sign of a missing mechanism  to take into account
the quark back-reaction on $\Phi$ because of  the lack of dynamical quark loops in
its calculation.   For this  reason, in  the  literature it  is
proposed to  add phenomenologically  more sources of back-reaction by  using for
example    a    scalar    coupling     $G$    that    depends on  $\Phi$, the entangled PNJL
(EPNJL) model \cite{Sakai:2010rp}. Fitting this dependence is quite difficult (due
to the  lack of  enough data)  and some authors  proposed to  use the
          Roberge--Weiss
transition at imaginary          chemical          potential \cite{Roberge:1986mm} to fit  this dependence.  Other sources
of  back-reaction may  be  added  by  using  a $\mu$-dependent  gauge
transition temperature  $T_0(\mu)$ \cite{Schaefer:2007pw} or even a  $\mu$-dependent
Polyakov potential ${\cal U}(\Phi ;  T, \mu)$ \cite{Dexheimer:2009hi}.

We will not discuss further the EPNJL model
but here we want  to stress that,
before going to this
kind  of model  with reinforced  back-reaction, it  is interesting  to
understand  the multiple aspects of back reaction already in the PNJL model.  Furthermore, models with
many  sources   of  back-reaction  ($G(\Phi)$,   $T_0(\mu)$) introduce many
new parameters and it is not easy  to  consistently   parametrize  them
without overestimating the back-reaction. In this regard, LQCD data at finite
density would  be very helpful  or high density constraints  (e.g. the
measurement of the position of the CEP  in experiments).  It is interesting to
note  that LQCD  data  at finite  imaginary chemical  potential
exists \cite{deForcrand:2002hgr} and  it is believed that the  continuation $i \mu
\rightarrow \mu$  is analytical  (it can  be seen  easily in  the PNJL
model grand potential) so constraints  at imaginary $\mu$ are relevant
for models to be used at real $\mu$.\\

It is also worth mentioning a last remark concerning Fig.~\ref{fig:PhiMFix}:
as already discussed in Sec.~\ref{sec:StatisticalConf}, the main effect of the Polyakov loop onto the quark condensate is to obtain a
steeper crossover  transition and in  Fig.~\ref{fig:PhiMFix} (lower left panel)  we see
indeed  that when  $\chi_{\Phi}$  is  at  its maximum  it also introduces a dominant peak in
$\chi_{\sigma}$.

\begin{figure*}
  \centering
  \includegraphics[width=0.47\textwidth]{composite_m_fix_mu0_1}
  \includegraphics[width=0.47\textwidth]{composite_m_fix_mu0_3_1}
  \caption{Quark   condensate   normalized   to   its   vacuum   value
($\sigma/\sigma_0$) and the Polyakov loop as a function of $T$ at zero
and non-vanishing chemical potential (left  and right).
The Polyakov is computed either with  a fixed mass or by fully solving
the mean field equations.
}
\label{fig:PhiMFix}
\end{figure*}

\section{Mean field phase diagram} \label{sec:MFPhaseDiag}

Using  the techniques  described  in \cite{Costa:2010zw}  we
investigate the phase structure of a  common two flavours PNJL model,
computed in this case with the logarithmic Polyakov-loop potential\footnote{The results in
this section do not depend on the precise details of the model.
They are features shared by all Polyakov type models (see Refs.
\cite{Costa:2008gr,Costa:2009ae} for results with the polynomial
Polyakov potential).}. To investigate the deconfinement transition we study the Polyakov
loop susceptibility,
\begin{eqnarray}
\chi_\Phi = \frac {d\Phi}{dT},
\end{eqnarray}
and extract from it the characteristic transition temperature of center symmetry breaking as a sign of deconfinement.

\begin{figure
  \centering
  \includegraphics[width=0.47\textwidth]{PhaseDiag_mu_fix_mu_2}
  \caption{Phase diagram  obtained in  the ($T,  \mu$) plane.
  The red dashed and solid lines represent the chiral crossover and critical line. The blue dashed line marks the deconfinement crossover and is within the transition region which is outlined by the blue solid lines (and defined within the main text).
  The yellow  line represents  the minimum  of the Polyakov loop susceptibility,
  $\chi_\Phi$.
  The transition region of the chiral crossover is within the green solid lines.
  }
\label{fig:PNJLPhases}
\end{figure}

Some of the details of the numerical calculations for a similar model, that are not of importance for our quantitative discussions here can be found in \cite{Biguet:2014sga}.
To obtain the susceptibilities, the ``easy way'' is to compute them as numerical derivatives.
This is both inefficient (it is quite long since for each step in temperature the mean field
equations have to be  solved several  times) and  inaccurate.
The inaccuracy will show up as a numerical  fluctuation in the transition line plot which shows the maxima of the susceptibilities.
A more accurate and faster way is to compute analytically (or at least semi-analytically in
the sense  that the  non analytic thermal  integrals are kept  as numerical  integrals) all
first  and second order partial derivatives  of the  grand potential  with respect  to the  order
parameters and $T$  and $\mu$ and then combine them to get the desired susceptibilities (see
\cite{Wagner:2009pm,Biguet:2014sga}).

\begin{figure
  \centering
  \includegraphics[width=0.47\textwidth]{phi_MFE_with_der_mu_fix_3}
    \caption{Polyakov loop $\Phi$ and its susceptibility  for chemical potentials below (red),
    at  (green),   and  above  (blue)  the one of the  chiral  CEP.
    }
\label{fig:PhiSusc}
\end{figure}

Results   for   the  order   parameters   $m$   and  $\Phi$   are presented in
Fig. \ref{fig:PNJLPhases}. Concerning the chiral properties, they behave as discussed in the
introduction and we observe a first-order transition line that
starts at zero temperature and ends with increasing temperature at the critical end point (CEP), where a
second-order phase transition occurs,  continued by a crossover for further increasing temperatures.

For what concerns the deconfinement transition we see that, except for chemical potentials $\gtrsim250\,\mrm{MeV}$ where
$\chi_\Phi$ has several
maxima, the order  parameter seems  to always  follow a  crossover
transition.  These multiple maxima are simply due to the presence of $m$ in the mean field
equation for $\Phi$: when $dm/dT$ has multiple maxima
also has $\Phi$.
If we look at the Polyakov loop susceptibility in Fig.~\ref{fig:PhiSusc}, we see that  there is a peak  at the  chiral transition temperature but physically this is
not the signal of the deconfinement transition.
We see that in the region between the different maxima,
$\Phi$ increases only about 5\%  and stays below 0.2: this  is not a (first-order)
deconfinement transition because quarks are still statistically confined when $\Phi = 0.2$.
There is also a bump in $\chi_\Phi$, and there $\Phi$  changes between 0.4 and 0.7 and the system reaches the statistically deconfined phase. Hence, we will consider this temperature as the one of the deconfinement crossover.

For sake of completeness, we have plotted in the phase diagram in \Fig{fig:PNJLPhases} besides the maximum of
$\chi_{\Phi}$ and its inflection points (that estimates the ``size'' of the crossover region), also the
minimum of $\chi_{\Phi}$, when it exists, between the deconfinement crossover bump and
the first-order chiral symmetry phase transition peak.

It is worth noticing that both transitions seem  to be strongly correlated below $\mu\lesssim250\,\mrm{MeV}$ and then they become decorrelated at higher chemical potentials, a  sign of the mutual influence
between chiral and deconfinement sectors.
The topic of the  quark - Polyakov loop  correlation deserves therefore special
attention as discussed in \secref{sec:backreaction}.

\section{Quark condensate and Polyakov loop correlation \label{sec:CondensPloopCorr}}

We continue our study focusing on finite density phases.

At zero density, the pseudocritical temperatures for partial chiral symmetry restoration
and deconfinement are very close, differing by approximately 20 MeV \cite{Borsanyi:2010bp}.
At finite  density the  status  is not  clear (different  model
parametrizations  give very  different results).
Experimentally, there is  a lack of high  temperature and density  experiments.
Furthermore, it is also difficult  to extract information
about the chiral sector  only or
the deconfinement sector only.

In  our  model  (this  feature  is  model  dependent  and will eventually
disappear by considering a $T_0(\mu)$ parametrization, see Ref. \cite{Kahara:2010wh})
we will study the high density phases and the possible opening of a new
phase near the CEP. Close to the CEP the  chiral and deconfinement transitions decouple
as seen in Fig. \ref{fig:PNJLPhases}.

\begin{figure}
  \centering
\includegraphics[width=0.49\textwidth]{suscettivita}
  \caption{
Polyakov-loop  susceptibility as  a  function  of $T$  for  different
chemical potentials (given in GeV).  The dashed lines are the calculation with fixed $m =
m_{mean}$ (see definition in the main text); the full line is the fully consistent calculation.
Notice in the region of low temperature that for  high chemical potentials,  $\chi_{\Phi}$
has no more a maximum.}
\label{fig:dPhiMFix}
\end{figure}

In Fig. \ref{fig:dPhiMFix} the  Polyakov loop susceptibility is computed
with the full mean field equations, and with a mass fixed  to the mean value
of its value in the vacuum and the current quark mass $m_0$, denoted $m_{mean}$ (dashed lines).
We can see that for a chemical potential that gives a first-order phase transition
for  the  chiral transition (red full line),  there is a peak  in the Polyakov
loop susceptibility at the chiral transition.
Having in mind the discussion in the previous section,
at this peak, the Polyakov only  varies from 0.2  to 0.3
There is also a bump in $d\Phi/dT$ at
higher  temperature  and  around  this  bump,  $\Phi$  goes  from  0.4
(essentially  confined phase)  to 0.7  (essentially deconfined).
Our interpretation is the following: it is not
possible to say that a first-order phase transition occurs
for $\Phi$ even if there is a peak in $\chi_{\Phi}$ because the value
of the Polyakov loop remains small.  However,  there  is a  smooth crossover
towards the deconfined phase at  higher temperatures.  These two transitions must be
considered  and this interpretation opens the possibility to
describe an intermediate phase, sometimes called quarkyonic phase, that
we prefer to call confined chirally restored phase (CCS).

Indeed, it can be explicitly shown that in this phase quark matter
is  still  (statistically)  confined. However, we postpone  the detailed
discussion to \secref{sec:CCS}.

In Fig. \ref{fig:dPhiMFix} it can be seen that  when using a constant mass, the
chiral  peak   in  the Polyakov loop susceptibility  disappears (dashed lines).
This is understandable since we argued that there is not a significant variation in
$\Phi$ at these temperatures but only a brutal mass change that creates
the large derivative. $\Phi$ itself does not vary much since we saw that
it is  not very  sensitive to mass variations below $\Lambda_{QCD}$.

When the mass is fixed at a chemical potential lower that
the one of the CEP, the maximum is approximately the same. This shows that
taking $m$ fixed really allows to decorrelate the chiral condensate
effects because the shape of the Polyakov loop susceptibility does not change significantly
when the mass varies strongly  (in this  interval the mass goes from  320 MeV to
150  MeV). All  the  remaining  effects are  only  due  to the  chemical
potential, an aspect that we will discuss in
the next section about the Fermi momentum effect.

Another important aspect is that  the use of a fixed  mass is also
very useful  numerically. If we want  to draw a phase  diagram but the
exact behavior of the Polyakov loop near the CEP is not so relevant, we can
use this trick and  get a faster algorithm since only two
mean field equations have to be solved\footnote{We also expect that in
some reasonable limits an analytical  solution can be found.}.
This  method can  be  helpful  even  to compute  the  full
solution: it can  greatly increase the calculation speed if we  start to compute
the  approximated fixed  mass  value and  then incorporate  it  in the solving
algorithm  for  the full  equations.   This  kind of  algorithm  (root
polishing  when the approximate solution is known)  is particularly
helpful when  the dimension  of the  problem increases (here  three dimensions since there are
three mean field  equations; including the strange  sector and without isospin
symmetry it would be five dimensions).

It is possible to qualitatively compare both methods, fixed mass and full calculation, by
looking  at  the phase  diagram  in  Fig. \ref{fig:mfixPhaseD} (here the Polyakov potential is  the Polynomial one but qualitatively
our results are independent on the choice of potential
).
The conclusion is that except  in the region of the CEP  it does  not differ  from the
previous one found in Fig. \ref{fig:PNJLPhases}, \secref{sec:MFPhaseDiag}. The
extra features will be discussed in  the next section since they concern
large density effects.

\begin{figure}
  \centering
  \includegraphics[width=0.49\textwidth]{fase_fi_mu}
  \caption{Phase   diagram   using   a  constant   mass   to   compute $\Phi$.
  Other  features are added: the pink dashed line represents $\Phi =0.5$;
  the blue line is given by the minimum of the Polyakov loop susceptibility $-$
  when this minimum and the maximum (the crossover curve, represented by the
  green line) meet, it is no  more possible to strictly define even  a crossover
  transition at higher chemical potentials. $\mu_0$ is the  chemical potential where $\chi_\Phi(T=0)$ start
  to be non zero, see \Fig{fig:dPhiMFix}. The red dashed lines represent the transition region and metastable region of the chiral transition and the dash-dotted green lines the transition region of the deconfinement crossover.
  }
\label{fig:mfixPhaseD}
\end{figure}

\section{Fermi momentum effect \label{sec:FermiMomEffect}}

In the previous section, we have  seen  in  detail one important aspect  of  the  chiral
transition/deconfinement  transition  correlation  by carefully  considering  the
effect of  the quark masses on the  field equations.  As mentioned,  there is another quark effect, in the mean field equation via the quark chemical
potential,  that originates  from a non zero Fermi  momentum of the quarks.

Because of the covariant derivative and  the inclusion of $\mu$ in the
Lagrangian, there  is a  dependence on $\mu$  in Eq.~(\ref{eq:domegadfi})
that  can have a strong  effect on the system.  In fact, the Fermi  energy will  act
exactly as any kinetic energy (in the Lagrangian $\mu\gamma^0$ acts as
$\partial_0\gamma^0$) and increases the explicit breaking of $\mathbb{Z}_3$ symmetry.  In
Fig.~\ref{fig:dPhiMFix} we computed $\chi_{\Phi}$ for  several
chemical potentials.  It is verified that, at some  point, the breaking of
the symmetry is so  strong that  the transition  cannot be  defined even as a crossover
anymore (at least according to our rigorous definition
).
From the moment on where the Polyakov loop susceptibility does not
have a maximum anymore, our interpretation is that the transition goes
so  fast from  $\Phi  = 0$  to  $1$
(and  without  the characteristic of a crossover  transition where  correlation  lengths  quickly
increase  then  decrease)
that  we   can  almost  say  that the  system  is deconfined even at zero temperature
(in Fig. \ref{fig:mfixPhaseD}, $\Phi$ is already 0.5 at $T=50$ MeV for large values of $\mu$).
$\Phi$ is forced to be zero  at zero temperature
but
in the large $\mu$ limit, $\Phi=1$ at any non-vanishing temperature.
Hence, the  model basically tell us that, apart a small missing ingredient to allow $\Phi$ to
be non zero at zero temperature, a statistical deconfinement induced only by the density is possible.
This  missing ingredient is again probably related  to quark back-reaction on the gauge fields.

Here we  want to point out that it is quite  remarkable that, in its
simple  version,  the  PNJL model  already  allows  a  deconfinement
transition caused only by the density (with a small temperature),
without any sort of {\it ad hoc} back-reaction added.

In the phase diagram presented in Fig.~\ref{fig:mfixPhaseD} more features
can be read.
It is interesting to have an idea of the value of the order parameter, $\Phi$, with
respect to  the transition  lines, so we add the $\Phi  = 0.5$
characteristic line (pink dashed curve).  More importantly, we show in
this diagram that there is a  kind of ``crossover  end point'' in this
model calculation. At very high $\mu$ (see also Fig. \ref{fig:dPhiMFix})
the explicit breaking of $\mathbb{Z}_3$ symmetry is so strong that $\chi_\Phi$
has  no longer  a maximum  (except at $T=0$).
Somehow this explicit symmetry breaking is  dominant in the Lagrangian and after
this point it is no longer
possible to talk about a transition, even
a crossover one. The symmetry  is completely destroyed by this breaking
term.

As a conclusion, we emphasize that only the careful assessment of
the sources of  correlations in the mean field equations allowed us
to see  those effects. Again, we  advocate that  even in  this simple
model some  mechanism as the Fermi momentum action can be  seen.  The
use of more realistic models (e.g.  SU(3)  PNJL model) would also be
interesting to get more faithful results. However, we think
that those  models, with many  interplays  between  physical  sectors
whose  contributions are controlled by parameters
adjusted to phenomenology, are complementary to the simpler models and both have their merit.

For example, if we want to answer a qualitative question about a physical
phenomenon,  like if the existence of a CCS phase  will favor  or
disfavor the  existence of compact  star with quark matter cores, we believe
that the results given by a SU(2) PNJL model are much more useful as a
guide, while a quantitative answer must be given by a more realistic model where
all physical  mechanism contribute.

\section{Combined effects of chiral condensate and Fermi momentum\label{sec:CombindedEffects}}

To complement our discussion on the influence of changing the chiral condensate
(by varying  $m$) and the Fermi  momentum (by varying $\mu$) on $\Phi$,
Fig.~\ref{fig:Tmumfix}  presents the results for  the deconfinement
temperature obtained when varying  both $\mu$  and the  fixed  mass $m_{fix}$.

\begin{figure}%
\begin{center}
\includegraphics[width=0.49\textwidth]{T_deconf_mfix}
\end{center}
\caption{\label{fig:Tmumfix} Deconfinement  characteristic temperature
as  a  function   of  the  fixed  mass  and   for  different  chemical
potentials. The full  line is the crossover temperature  and the dotted
lines are the metastable limit of a first-order phase transition.
}
\end{figure}

Taking $\mu  = 0$,  the deconfinement  transition for $m_{fix} = 0$  is a
crossover with a treansition temperature about the  same as the chiral one (for this
model parametrization  it is about  220 MeV). This corresponds  to the
usual situation  of the PNJL  model when both transitions almost coincide  at zero
density.       As      already     explained      when      discussing
Fig.~\ref{fig:Z3ExplicitBreaking_m},  by  increasing the  mass  (thus
decreasing the  explicit $\mathbb{Z}_3$ symmetry breaking) the  temperature increases
until  a  value  where  the   crossover  becomes a first-order phase transition,  at  a
temperature  of about  270  MeV (which  is  the value  of  $T_0$ in  this
parametrization).

Now, looking at $m_{fix} = 0$ and  increasing $\mu$, we see a lowering of
the  deconfinement  crossover  temperature  until a  value  where  the
crossover cannot be defined anymore.  The symmetry breaking induced by
the Fermi momentum is so strong  that after this point, $\mathbb{Z}_3$ is no
more a valid symmetry of the model.

Finally, by varying both $\mu$ and $m_{fix}$ it is always possible to go to a
first-order phase transition if the  mass is sufficiently high compared to
$\mu$ in order to counteract the explicit breaking of $\mathbb{Z}_3$ symmetry.

\section{Confined chirally restored (CCS) phase}\label{sec:CCS}

In this section we will take a closer look to the CCS phase, a phase where
chiral symmetry and deconfinement are decorrelated.

In the PNJL model it is always necessary to remember that quarks are not
confined in the strict sense
and when we talk about confinement
we mean statistical confinement in the sense given by the definition in
\secref{sec:StatisticalConf}. These type of models cannot tell us the true nature
of quark confinement.

To explore the CCS phase  we want to count the number of thermal  quark degrees of
freedom  (the only one affected by statistical  confinement).
For this reason we need to subtract the quarks in the vacuum, present due to the lack
of confinement (we have seen that their contribution is quite reasonable  by looking
to the quark  number density in  \secref{sec:StatisticalConf}).
Hence, we calculate a relative pressure difference by subtracting
the pressure at zero temperature and normalizing it to the Stefan-Boltzmann pressure:
\begin{equation}
  \label{eq:DeltaP}
  \Delta P \equiv \frac{P(T, \mu) - P(T=0, \mu)} {P_{SB}(T, \mu) - P_{SB}(T=0, \mu)}.
\end{equation}

This quantity  is adequately related  to the thermal quark  degrees of
freedom and  indeed it shows  signs of statistical confinement  in the
CCS phase (see Fig.~\ref{fig:DeltaP}).
\begin{figure}
  \centering
\includegraphics[width=0.49\textwidth]{DeltaP_all}
  \caption{ \label{fig:DeltaP}
Relative  pressure difference (color map), in  the ($T,  \mu$) plane, super  imposed  on the  phase diagram of \Fig{fig:PNJLPhases} .
}
\end{figure}
In the hadronic phase, $\Delta P$  is zero (the black area) meaning  that the PNJL
model will  not thermodynamically create  any quarks in  the hadronic phase  (besides those
already present in vacuum). In the QGP phase (the yellow region), thermal quarks are very
active thermodynamically. Finally, in the CCS phase  a quite stable ``plateau''
at around 0.2 is found.
At a given (small) temperature, there is a discontinuous jump from 0 to 0.2
when increasing $\mu$ across  the first-order transition.  So, in the  CCS phase some quarks
have been liberated  by the system due to  the chiral transition but their contribution to thermodynamics remains low.
It seems that in this phase, where the chiral symmetry is restored, the model tells us
that quarks still are far from being asymptotically  free. Phenomenologically, this could  have
rather  important  effects on  experimental  observables  in HICs  (viscosity,  hadronic
abundances, etc).
If this hypothetical phase exists in QCD, the PNJL model cannot say if quarks  are
really confined (no asymptotic  states) or if they are deconfined but the interaction
is still strong.
Performing the same calculation in the pure NJL model, i.e.~without statistical  confinement, one observes
at small  temperature  that  $\Delta P$  has the
same  jump at  the chiral transition  and  then  increases slowly  without a  ``plateau'' behavior.

It is also relevant to notice that, at those densities, an important missing effect is the presence of diquark condensates (the strange sector is  also important).
This point deserves further investigations. \\

To confirm this finding and that it may lead to significant
differences in HICs, we study a more direct observable, namely we check
if mesonic probes act as it is expected in a confined but chirally
symmetric phase.
\begin{figure*}
  \centering
  \includegraphics[width=0.49\textwidth]{mu0t015_1}
  \hfill
  \includegraphics[width=0.49\textwidth]{mu04t015_1}\\
	\vspace{0.25cm}
  \hspace{9.1cm}\includegraphics[width=0.49\textwidth]{mu04t005_1}
  \caption{Upper panels: Pion and  sigma spectral function in the hadronic phase
  (left) and the QGP phase (right). Confining regime: $\sigma$ is a narrow resonance;
  the chiral symmetry is broken and $\langle\bar qq\rangle  \neq 0$; $\Phi \simeq 0 $.
  Deconfinement: $\sigma$  and $\pi$  spectral functions are  almost the same  as an
  uncorrelated $\bar  q q$  pair;  the chiral  symmetry is restored and
  $ \langle\bar qq\rangle \simeq 0$; $\Phi \neq 0 $.\\
  Lower panel: Pion and  sigma spectral function in the CCS phase.
  It looks like a non asymptotically free regime (statistically confined at least)
  since $\sigma$ and  $\pi$ are still narrow  resonances.
  But chiral symmetry  is restored: $ \langle\bar qq\rangle \simeq 0$;  $\Phi \simeq 0$.
  }
\label{fig:MesonicCCS1}
\end{figure*}

We compute the spectral function (see \cite{Hansen:2006ee}) in the
meson to quarks channel at zero momentum and report them on
Fig.~\ref{fig:MesonicCCS1}. The finite width of the sigma in vacuum
indicates that it is not confined (this channel is open) but there is a
strong interaction and thus it is a sharp resonance.

The expected behavior for mesons in a CCS is observed in
Fig.\ref{fig:MesonicCCS1}: chiral symmetry is effectively restored when the
sigma and pion spectral functions essentially coincide \cite{Hansen:2006ee},
the statistical confinement is still strong as it can be seen by the fact
that their width is small (right upper panel) compared to the full deconfined phase
(lower panel).

Finally, we remark that other parametrizations, for example with a chemical potential dependence of $T_0$, will essentially close this CCS phase \cite{Kahara:2010wh}.
Yet, we think it is still interesting to try to understand better the CCS phase.
We also think that $T_0(\mu$) is a too strong {\it a priori} generalization
to introduce in the model.
This dependence is  justified theoretically in Ref. \cite{Schaefer:2007pw}, where
it is argued that this minimal necessary generalization is needed because, without
a $\mu$-dependent $b_2$ in Eq.~(\ref{eq:b2}), the confinement-deconfinement phase
transition has a higher critical temperature than the chiral phase transition,
being an unphysical scenario since QCD with dynamical massless
quarks in the chirally restored phase cannot be confining because the string
breaking scale would be zero.
However, the problem is that it is badly constrained.
By using this parametrization  the hypothesis we have is basically that the
deconfinement and chiral  restoration  will coincide
Of course the CCS phase will then not exist anymore but in our opinion this is not
an argument against CCS existence.
It can eventfully be an argument against the CCS phase only if this {\it a priori}
generalization is correct but which has to be  put on more firm basis
(in particular better constrained).

In the  meantime we think it is still interesting to consider models with a
CCS phase even if in the future it will be contradicted by experimental data, the ultimate judge.
Anyway, it is still a nice exercise to see what physical information can be
gathered by using effective models.

\section{Conclusions and outlook}\label{sec:CaO}

We started this working by showing the current status
on how far effective models based on the restoration of
chiral symmetry and the breaking of center symmetry
(to describe the deconfinement transition of QCD) are
able to reproduce the result of lattice QCD (LQCD)
calculations.
Based on the observation that there is a substantial quantitative difference in the temperature dependence of the Polyakov loop,  regarding the chiral
condensate one, we have analysed several aspects of the correlation between the chiral sector and the Polyakov
loop.\\
Examining the pseudo-critical temperatures of chiral and Polyakov-loop transition in dependence of the critical scale of the Polyakov-loop potential $T_0$ we see that the order of chiral and Polyakov-loop transition and their difference in temperature as seen in LQCD calculations would be realised with a $T_0$ larger than the one in pure gauge theory. The absolute temperature scale of the QCD transition would then be larger than in LQCD but in the spirit of Landau theory all scales could be rescaled to arrive at the correct value for $T_\mrm{c}$.\\

For studying the effect of the kinetic contribution of quarks to the correlation between quarks and Polyakov loop we compared the behaviour of the later between calculations using a self-consistent medium-dependent quark mass and for several values of constant quark masses. We observe that for values of the dressed quark mass $m$ below the scale $\Lambda_\mrm{QCD}$ the qualitative behaviour of the Polyakov loop becomes relatively insensitive to the exact value of $m$. Except very close to the chiral transition (where $m$ changes quickly) the Polyakov-loop susceptibility is not much altered by keeping the mass constant so that even the transition properties are almost not affected by the precise value of the mass. We interpret this lack of sensitivity to the mass of the quark as sign of a missing mechanism to take into account the quark back-reaction on the Polyakov loop because of the lack of dynamical quark loops in its calculation.
Comparing the Polyakov-loop susceptibility for medium-dependent and constant mass at finite density allows to disentangle a smooth crossover proper to breaking of center symmetry from a peak in the susceptibility introduced purely by a chiral first order transition but at which the value of the Polyakov loop remains small, so in the confined regime. This picture opens the possibility to discuss an intermediate phase, general known as quarkyonic phase, that is in our model a confined chirally restored phase.
Probing the number of thermal quark degrees of freedom within this phase, indeed one sees signs of statistical confinement with a quite stable ``plateau'' where some quarks have been liberated due to the chiral transition but where their contribution to thermodynamics remains low. Furthermore, meson spectral functions behave as it is expected in a confined but chirally symmetric phase: the sigma and pion spectral functions essentially coincide but their width is small.\\
Another contribution to the correlation between quarks and Polyakov loop at non-vanishing density comes from the fact that the Fermi energy acts exactly as any kinetic energy and increases the explicit breaking of center symmetry. From some large chemical potential on this explicit breaking of the symmetry is so strong that the Polyakov-loop susceptibility does not have a maximum anymore except at $T = 0$. Hence, apart a missing ingredient to allow the Polyakov loop to be non-zero at zero temperature, a statistical deconfinement induced only by the density seems to be realisable within the model. This missing ingredient is again probably related to quark back-reaction on the gauge fields.

Overall, the goal of this work is to shed light to some less known aspects of Polyakov-loop extended chiral models to motivate certain paths for future improvements and detailed quantitative analysis.

\begin{acknowledgments}
This work was supported by FCT (Fundação para a Ciência e Tecnologia), Portugal, under the Grant No. SFRH/BPD/102273/2014 (PC) and by an annual INFN Post Doctoral Fellowship (competition INFN notice n.~18372/2016, RS).
RS acknowledges hospitality at the Laboratoire de Math\'ematiques et Physique Th\'eorique (LMPT), today's Institut Denis Poisson at Universit\'e de Tours, France and thanks Szabolcs Borsanyi, Gergo Endrodi, Tina K.~Herbst, Peter Petreczky, Markus Quandt and Urko Reinosa for providing their respective data and correspondence.
This work was supported by STSM Grants from the COST Action CA15213 ``Theory of hot matter and relativistic heavy-ion collisions'' (THOR) and HH and RS thank for the support and hospitality of CFisUC (programme UID/FIS/04564/2016).
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


EXCEED AND EXCEL (AUS) (Bay 2000-Stud 2004). 7 wins-2 at 2, VRC Newmarket H., Gr.1. Champion Australian Sire 2012-13. Sire of 1748 rnrs, 1219 wnrs, 142 SW, inc. Overreach (ATC Golden Slipper S., Gr.1), Guelph, Excelebration, Mr Stunning, Earthquake, Margot Did, Amber Sky, Flamberge, Helmet, Reward for Effort, Outstrip, Sidestep, Fulbright, Best Terms, Heavy Metal, Priceless, Championship, Kuroshio, Masamah, Wilander, Buratino, Calliope, etc.
KHAN (g. by Exceed and Excel). Winner. See below.
Mansera. Winner at 2400m, 3d Chantilly Prix de la Grange des Pres.
QUE BUENA (Nedawi). 3 wins from 1400m to 1600m in Brazil, Cidade Jardim Clássico Emerald Hill, L, 2d Cidade Jardim Prova Especial Joaquim da Cunha Bueno - C.
Race Record: 4 Starts 1-0-0 A$118,725.
Winner at 2, A$118,725, in 2016-17, ATC Breeders' P., L.


Truth be told, and I know I'm gonna hear about this, I've never been a big fan of the Three Stooges. I like them. I watched them in TV as a kid but they were never my favorites. I much preferred The Marx Brothers or Abbott and Costello. But when the opportunity arose for me to paint these guys for a comic book cover, I jumped at the chance!! I rarely find myself "clever" but when this idea of breaking the fourth wall popped into my head, I just HAD to do it!! I'm also thrilled that this is seeing print and I was jazzed that a picture of it as well as my name was in the recent issue of Previews! The comic comes out in October by the way.


This article originally appeared on Map Happy.
Fees are the reality of flying today and it's left us with a whole bunch of to-pay-or-not-to-pay questioning. So, which extras are actually worth paying for?
It all comes down to a industry-wide move to the fee-based model, which flyers have experienced the slow and costly shift to. First came the charges for checked bags. Then meals disappeared on flights and now someone tells me that seats have actually been shrinking.
The airlines, and some economists, argue that the rise of the fee model is good for travellers (sic). You only pay for what you want, and you can therefore save money if you, for instance, don't mind sitting in middle seats in the back, waiting in line to board, or bringing your own food.
Unfortunately, whether you like it or not, not all fees are worth it. Here's our take.
Ideally not but sometimes it can't be avoided. Here's our list of U.S. airlines' baggage fees for a better idea of what we're all up against. Keep things to one bag because the big fees come after bag number one. For the wise packers: When gate-checking, there's no fee. Make sure the look and size is right (carry-on) to get it past check-in.
Yes—at least for long flights. Like for those really, really long international flights. I weigh the cost-value benefit based on flight time and my projected mental and emotional state for the duration of the flight. If it's an overnighter or I'm going to be spending quite a bit of time in that seat, those extra inches at my feet really do make a difference. This is one area where it's an advantage to be short.
We're not just getting bigger, either. Seats these days are smaller than they've ever been. It doesn't mean there aren't cost-free methods to get more room though.Get a wingman and snag a whole row with this method or this trick if you're flying solo.
No. Now it really comes down to personal taste and preference but overpaying for mediocre (at best) food is something that really grates on me. There are lots of really good snacks for traveling. And I've reported before, bringing home-cooked food through security is kosher with the TSA, too, as long as it's packaged properly.
If anything, buy at the airport before or after, where there are at least non-boxed options. (I also consider ending up at a place like Chicago O'Hare where Rick Bayless has his Tortas Frontera outpost, a real food win. Erica loves this place too.) In fact, T Magazine says we are living in the golden age of airport restaurants, so eat up.
As for alcohol, mini liquor bottles actually can be carried on, so mixed drinks can come free. As an extra, here's the super duper crazy list of airlines that serve booze. It might not be the best tasting but you ain't paying.
Sometimes this can't be avoided but there are ways to minimize the damage. In fact, in some cases just buying a new flight can be cheaper than changing it. For those d'oh moments when a ticket is booked, there's a grace period of 24 hours during which the ticket can be changed—that also can be hacked to 48 hours. And when it's too late for that, we've found that waiting until the very last moment to change it—we're talking the day of—is cheapest.
Not if you just want to use basic Google apps. But if you need a little bit more, there's also a way to mod your browser to get cheaper in flight Wi-Fi. Dude, there's even a whole guide on this stuff, courtesy of Erica. But just to know before you jump through all this hoopla, this is exactly how much in-flight Wi-Fi will cost you on your next flight.
Depending on the day, airport and layover you might hear something different from me. Sometimes I've just got to a get shower and want to splay out on a comfortable chair with complimentary beverages. And if I don't have access already granted with status and I'm facing a doozy of a layover, I'll potentially shell out. This post goes a little bit more in-depth on the topic.
There's just no right answer here but a lot of it is going to depend on the lounge you're thinking about. Here's a hint: If you can just pay for a separate standalone shower, that's usually the best amenity to give in to.
Why British Airways Is the Best Frequent Flyer Program for Free Flights in the U.S.


Awarded as best fashion designer 2002/2005 from France's ESMOD School of design, Jeffry Tan graduated top of his class. During his studies he also participated in the Young ASEAN Fashion Design Contest in Singapore, inline with Singapore Fashion Week 2003.
Jeffry's passion lies in the art of pattern drafting and the construction process. He is constantly experimenting to reach his next innovative creation.
Modern in concept, he passionately remains true to his roots by working with weavers, craftsmen and manufacturers throughout Indonesia.
Every collection is worked to perfection, mixing, blending and marrying elements and matter, exploring every one of their possibility and function. to produce cutting edge and sophisticated clothes. He shares his emotions and communicates through his designs, in a poetic,yet functional.


When Christ Church was consecrated in 1876, shortage of funds meant that it lacked the planned bell tower over the main entrance. After a fundraising drive begun in 1885, work to complete the tower was commissioned in mid-1887, being completed in October that year.
but the top two-thirds of the tower just 18 years old.
The resulting eight bells, weighing about 2¾ tons, carried the following dedications.
By convention, bells' weights are shown in the imperial measures of hundredweights (112 pounds or one twentieth of a ton), quarters (28 pounds) and pounds.
"On Thursday, the last day of July, a large congregation assembled at Christ Church, Epsom, for the purpose of taking part in the service of consecration of the new ring of eight bells. The Vicar (the Rev. Archer G. Hunter) having commenced the service, the act of consecration was performed by the Suffragan of the Bishop of Winchester (Bishop Sumner, of Guildford). This took place at the altar, after which the Bishop gave a short and very appropriate address on the use of Church Bells. After his address, a short touch (a course of Grandsire Triples) was rung; and very impressive it was, the bells being acknowledged by all to be particularly sweet toned and musical."
This dedication - and the first major (immediately post-service) peal - is commemorated on the oldest board in the ringing chamber - see below.
Board commemorating the 1887 completion of the Tower and the 1980 dedication of the Bells.
The ringing chamber contains a number of other chime boards recording particularly significant peals. The next four oldest are pictured below.
Boards commemorating some early major peals on Christ Church's Bells.
Top left: 13 January 1891 - Union Triples in 2 hours 55 minutes.
Top right: 23 November 1893 - Grandsire Triples in 2 hours 59 minutes.
the Revd Archer Hunter) - Grandsire Triples in 2 hours 53 minutes.
Grandsire Triples in 2 hours 54 minutes.
As noted above, the fundraising had been intended to cover the cost of not only the new bells but also a clock. However, even when the "Bells & Clock Fund" Committee published its final accounts in May 1891, there was a deficit of £16 1s 9d just in relation to the bells and their installation. In its covering report, the Committee invited further donations to meet that shortfall and said, "As regards the Clock, we have great pleasure in informing you that a Parishioner has most kindly and generously offered to present this to the church." That Parishioner was Basil Braithwaite, and the fine clock (by Gillett & Johnson of Croydon) was given in memory of his father Isaac (1810-90), from whom he had just inherited Hookfield Grove.
In addition to driving the external clock faces, the mechanism was linked (as is its electrical successor) to small hammers by the bells to sound the hours - and the quarters - with the familiar Westminster chimes. However, for normal - and traditional - "change" ringing, the team of ringers use the bell ropes, as in the following animation.
Animation of traditional bell ringing.
Notwithstanding some significant maintenance of the bell mechanism in 1930, by the late 1980s the near century's use was beginning to tell on not only the bells' fittings but, more importantly, the tower's structural integrity - the latter meaning that a lighter peal of bells was needed.
Initially, all options seemed prohibitively expensive: Christ Church was already committed to very substantial expenditure to "re-order" its effectively unchanged Victorian interior (to meet modern liturgical and other needs) as well as a major rebuild of its original William Hill pipe organ.
Serendipitously, a considerably less expensive option became available through the complementary needs of Holy Trinity Church, Hawley (near Camberley) which was also wanting a lighter peal of eight bells. Through the agency of Guildford Diocesan Guild of Church Bell Ringers, it was arranged that Christ Church's bells should, after re-tuning, be refitted in Holy Trinity's tower, and that Holy Trinity's bells should be melted down and recast to make the lighter set needed at Christ Church, with the unused metal helping to offset the costs.
Left: Lifting one of the old bells from its pit.
After the old bells had gone, a few volunteers hauled an industrial cleaner up into the belfry and took a few days to clear out the mass of dead pigeons and pigeons droppings that still remained behind and under the frame. (Live pigeons has been cleared from the belfry by a contractor three years previously, and the louvres - through which the bells sound over the surrounding area - made bird-proof.) The frame and floors were then washed with disinfectant ready for the new bells.
These arrived, in all their pristine glory, on 24 February and were left in the church for a day to enable the sponsors and others to see the new bells. Greatly reinforced by the under-16 rugby squad of Glyn School, the eight new bells were then hauled back up the tower ready for careful work by the experts in installing the bells in their pits with their attendant fittings.
Right: The new bells ready to be rung, 25 March 1992.
The specification of the new bells and the dedications (which were cast into the bells) are set out below.
2. The dedication on the 7th reflected the Guildford Diocesan Guild Of Church Bell Ringers' significant practical and financial help with the whole operation.
On Sunday 29 March 1992, the bells were dedicated by the Archdeacon of Dorking, the Ven Christopher Herbert, who climbed to the belfry to perform the ceremony which was broadcast to the congregation by radio. The new bells were, of course, rung during the service and their first full peal (of Plain Bob Triples, in 2 hours 44 minutes) was rung on Sunday 2 August 1992 - all as recorded on a chime board in the ringing chamber, see below.
Chime board recording the new bells' dedication and first full peal.
As for the last century and more, Christ Church's bells continue to be rung before Sunday morning services and for other special occasions, including weddings. Click here for more details.


text:
Reviewing the Redshirts: LB Michael Deeb
Analyzing the Irish freshmen who retained a year of eligibility in 2013 and what role they may play in 2014.
By Eric Murtaugh@OneFootDown Jan 21, 2014, 8:01am EST
Share All sharing options for: Reviewing the Redshirts: LB Michael Deeb
Jeff Zelevansky
Reviewing the Redshirts is an off-season series meant to resurrect discussion around the current Irish freshmen who did not play in 2013.
There were 12 this year who retained a year of eligibility and today we take a look at a picture perfect middle linebacker who is looking to play a lot of minutes this fall.
LB, Michael Deeb
Hometown: Plantation, Florida
247Sports COMPOSITE RANKING:
3-star, No. 28 ILB, No. 64 FL, No. 437 USA, 0.8737 score
Need at Position: High
Expected Spot on the 2014 Depth Chart: Possible Starter
I'm real excited to see what Deeb brings to the table this fall. Here's what I wrote about him shortly after National Signing Day last February:
Deeb's foot speed improved and that was my biggest knock on him last year. Overall he's much faster and will be a great fit at middle linebacker for the Irish. Best of all, this kid seems like a tireless worker who puts everything in to the game. I love having that from a recruit that we know is going to play middle linebacker. Height might be a little concerning but Deeb is a major steal.
We've been looking at prospects and kids who sign with Notre Dame in depth for a few years now and I don't think I've ever seen a recruit improve from junior to senior seasons like Deeb did. This may be controversial, but I think Deeb's film as a senior is even better than Nyles Morgan's, and I'm still shocked how small of a bump Deeb received in the recruiting rankings during his final year of high school.
If I had to guess if Deeb is a starter next year, I would put very favorable odds on it. Jarrett Grace is going to be nursing a broken leg well into the summer, so the spring will see some combination of Deeb/Moore/Schmidt and I wouldn't be surprised if Deeb comes out of that group as the favored player.
If Deeb comes out of the spring in such a position he's likely to be a game one starter.
Everyone knows my worry by now. I hope he doesn't show up at 252 pounds and instead of all that explosiveness we saw at American Heritage, his change of direction is labored and mechanical. If the numbers are to be believed he was about 10 pounds heavier last fall than he was as a high school senior. His listed weight of 242 should be good enough.
Both Dan Fox and Carlo Calabrese played considerable minutes as redshirt freshmen and I don't see how Deeb plays any less than those guys. I see a little bit of Fox's game in Deeb but I think Deeb has much more potential. I know it can be difficult for some people to accept such glowing reports on someone who is "just a 3-star recruit" but look at this kid's film. For what Notre Dame is looking at for middle linebacker, he's pretty much the whole package.
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OFD Podcast: As we wait for Notre Dame's bowl game, other things are happening

redpajama_set_name: RedPajamaCommonCrawl


The objective of refining is to remove objectionable co-constituents in oil with the least possible damage to glycerides and minimal loss of desirable constituents. Ways of refining depend on nature and quantity of impurities contained in crude oil, application of RBD oil.Classical chemical refining is used through successive steps: it must be degumming, neutralizing, bleaching, deodorizing and sometimes winterizing (Mainly removing free fatty acid and other minor constituents).
The set of equipment has obtained national patent (patent No. isLD99 25288.7), it is our developed third generation product with more advanced and mature technology, and itwon GoldMedal on 8th China Fair of Inventions and New Technologies and Silver Medal on 6th International Invention Exhibition. It adopts intermittent refining craft, and has advantages of convenient transportation, small covering area, less operation staff and saving energy.


text:
Coldirodi is een plaats (frazione) in de Italiaanse gemeente Sanremo.

Plaats in Imperia
Voormalige gemeente in Ligurië

redpajama_set_name: RedPajamaWikipedia


text:
Melbourne Origin tickets on sale Friday
Thu 18 Aug 2011, 01:45 PM
He may have been a reluctant tourist but NSW VB Blues coach Ricky Stuart is promising Victorians will be treated to a 2012 Harvey Norman State of Origin Series spectacle unlike any they have ever seen before.
With tickets for the Etihad Stadium match next May going on sale to the general public from 10am tomorrow, Stuart today joined with one of the Queensland stars who shattered his Origin dreams this season – Storm captain Cameron Smith – to outline his plans to finally wrest the prized Origin Shield back from the Maroons.
Stuart did not back down from his claim that he wanted to play the match in Sydney and take every possible advantage the Blues could to stop the Maroons from winning a seventh straight series.
But he is now more determined than ever to turn Etihad Stadium into a new Blues fortress – and with Blatchy's Blues founder Dan Blatch joining him to unveil plans for an interstate invasion of blue-wigged fans, Stuart said Victorian supporters had a unique opportunity to be part of the greatest rivalry in sport.
"It's no secret I would have loved to have played this game in Sydney. That's nothing against Melbourne, it's just such a prize to host the opening match," Stuart said today.
"We face an enormous challenge to beat this Queensland side. We took some big steps towards achieving that this year and a big part of that was the atmosphere the fans created at ANZ Stadium.
"Origin games have been played in Melbourne before but not like this, not with the colour and atmosphere that the Blatchy's Blues will bring to Etihad Stadium next year.
"We are going to have thousands of fans travelling down from Sydney for this game here and I would love to see all the fans across Victoria join them in their blue wigs and be part of a great sporting experience."
In typical Origin spirit, Smith was quick to establish the Maroons' claims to the hearts of Victorian fans.
The Wally Lewis Medal winner as the best player in this year's Harvey Norman State of Origin Series, Smith said that he and fellow 2011 Storm Origin stars Cooper Cronk, Billy Slater and Dane Neilsen would love nothing more than to have the chance to play in next year's series opener in front of their loyal fans.
"Victorian sports fans are among the most passionate in the world and I know they will come out in force to support Rugby League and in particular Origin," Smith said.
"The Queensland side has always received a great reception in Victoria and I have no doubt there will be plenty of Maroon support down here again.
"From a Storm perspective, it's great to have the pinnacle of Rugby League in Melbourne. The match will have a positive impact on our ongoing commitment to grow the game across Victoria.
"It's a great reward for the fans here who support the Storm week in, week out, to see an Origin game live in Melbourne and I'd be proud to run out in front of them."
ARL Chief Executive Geoff Carr, Victorian Minister for Tourism and Major Events, the Hon. Louise Asher MP, Victorian Minister for Sport and Recreation, the Hon. Hugh Delahunty MP and Storm CEO, Mr Ron Gauci, were among the guests joining Stuart and Smith at the Sofitel on Collins in Melbourne today to begin the countdown to the 2012 series.
The 2012 clash will be the second series opener and the sixth Harvey Norman State of Origin match to be staged in Melbourne, continuing a partnership between the Australian Rugby League, the Victorian Major Events Company and the Victorian Government that has also featured Rugby League World Cup matches and the VB Test that officially opened AAMI Park last year.
"Since 2005 our partnership with the VMEC and Victorian Government has brought some truly memorable events in Australian sport to Melbourne and they have attracted sell-out crowds," ARL Chief Executive, Mr Geoff Carr, said today.
"To once again host a series opener here shows just how successful that partnership has been and once again fans from not just Victoria but across Australia will converge on Melbourne to be part of something that they are really going to enjoy."
Victorian Minister for Tourism and Major Events, the Hon Louise Asher MP, said today: "Victoria has long been recognised as the sporting capital of Australia and this State of Origin game will be a great addition to our world-class major events calendar."
Victorian Minister for Sport and Recreation, the Hon Hugh Delahunty MP, added: "We are proud of our partnership with the ARL and this game will attract thousands of Rugby League fans from two of our key interstate markets and deliver significant economic benefits for the State."
Game 2 of the 2012 Harvey Norman State of Origin Series will be held at Sydney's ANZ Stadium with Game 3 to be played at Brisbane's Suncorp Stadium.
2012 Harvey Norman State of Origin Series
Game 1, Wednesday, May 23, Etihad Stadium, Melbourne
Tickets are available through Ticketmaster on 136100, at www.ticketmaster.com.au/origin or at Ticketmaster outlets.
Level 1 Supporter Bay Ticket Prices*
*Tickets include a blue t-shirt and wig for NSW fans and a maroon t-shirt and headwear for Queensland fans.
Presented By EISS Super Now


text:
Q: How to select random 2 percent rows in mysql? How to get a random 2% sample from a table in MySQL?
I tried the following queries
SELECT orderID
    FROM orders
    ORDER BY rand()
    LIMIT 1 OFFSET (SELECT convert(0.02 * count(*), unsigned) FROM orders)
;

SELECT orderID
    FROM orders
    ORDER BY rand()
    LIMIT (SELECT convert(0.02 * count(*), unsigned) FROM orders)
;

Both of them give errors. Could someone explain error and alternate methods.
Database: Northwind
MySQL version: 8.0

A: I would use ROW_NUMBER here:
WITH cte AS (
    SELECT *, ROW_NUMBER() OVER (ORDER BY RAND()) rn, COUNT(*) OVER () cnt
    FROM orders
)

SELECT *
FROM cte
WHERE rn < 0.02 * cnt;  -- select first 2% of a random sample

A: If approximately 2% is sufficient, then you can just use:
SELECT orderID
FROM orders
WHERE rand() <= 0.02;

Because no sorting is required, this is probably the fastest method.
Do note that the result is not exactly 2% but on a large table it will be close enough.

A: You can use MySQL Stored Procedure to create a function inside our MySQL.
https://www.mysqltutorial.org/getting-started-with-mysql-stored-procedures.aspx/
CREATE PROCEDURE GetRand2PercentSample()
BEGIN
    DECLARE nLimit INT DEFAULT 0;

    SELECT COUNT(*) * 0.2
    INTO nLimit
    FROM orders;

    SELECT * FROM orders ORDER BY RAND() LIMIT nLimit;
END

// to call the function
call GetRand2PercentSample()

redpajama_set_name: RedPajamaStackExchange


David Stephen Bennett is currently a PhD candidate at Michigan State University studying Modern American History with Dr. Michael Stamm and Dr. Kirsten Fermaglich. He is also studying African American History with Dr. Pero Dagbovie, and History of Science, Medicine, and Technology with Dr. Helen Veit. He earned his MA from the University of Louisiana at Lafayette where he served as editor-in-chief of the department journal and received a number of honors and awards. His thesis, Birth of a Virtual Battleground: Television and the Desegregation Crises of 1957 and 1960, was nominated to represent the university for the Conference of Southern Graduate Schools Master's Thesis Award in Social Sciences.
His teaching experience is varied. He has taught a number of courses for Michigan State University, Lake Superior State University, University of Louisiana at Lafayette, and South Louisiana Community College. He has been awarded with assistantships from both Michigan State University and the University of Louisiana at Lafayette. He has worked directly with fellow scholars Dr. Michael Stamm, Dr. Peter Beattie, Dr. Emily Conroy Krutz, and Dr. Charles Keith. He assisted in the digitization of physical records for the Vietnam Project Archives, which was funded by the National Endowment of the Humanities.
His current research focuses on investigating the nature of news media's representation urban identity during the civil rights era. This research has garnered a number of awards including the Madison A. Kuhn Award, Rose Library Fellowship, and the Hugh F. Rankin Prize. His first article, "The Televised Revolution: 'Progressive' Television Coverage of the 1960 New Orleans School Desegregation Crisis," is forthcoming for publication in the Louisiana History Journal. This research has also spawned a number of digital projects, including Visualizing Southern Television Visualizing Southern Television, which projects changes in television station distribution throughout the south through the years between 1946 and 1966. It is accompanied with a table that I hope over the years will facilitate future scholarship in this field. Another piece of the project is focused specifically on Louisiana Television Stations, and their own timeline and geography.
You can follow me on my website Technological Borderlands, or my project site located at Visualizing Southern Television.