Anirban Lahiri

# Aspects of finite temperature QCD towards the chiral limit

###### Abstract

QCD under extreme conditions has been studied for a long time, and the chiral
limit has been a grey area mostly. In this write-up of my talk, I review some
of the recent developments made by the community to unveil various features of
QCD towards the chiral limit, which includes calculation of the chiral
critical temperature and determination of the order of chiral phase transition
for various numbers of flavors. Acknowledging the importance of the studies
regarding the effective restoration of $U_{A}(1)$, I try to give a
comprehensive overview about the various studies done in the last few years in
a comparative manner to realize the current status of the community in this
regard. I also discuss very recent efforts about the relevance of various
energy-like observables w.r.t. the chiral phase transition.

##  1 Why this talk?

Let me start with some prelude about why it is interesting to study the
Quantum Chromo-Dynamics (QCD) in/towards the chiral limit, defined by the
vanishing quark masses, which can’t be realized in nature.

Two fundamental features of QCD are spontaneous breaking of chiral symmetry
and confinement. More than four decades ago, there were studies to understand
their interplay. One set of studies claimed that the spontaneous breaking of
the chiral symmetry happens in a confining environment [1, 2]. Almost at the
same time other studies tried to answer the question by calculating and
comparing the corresponding transition temperatures [3, 4], but this remains
somewhat inconclusive w.r.t. the behavior of theories with different number of
colors. Although not very sophisticated w.r.t. present day calculations, the
above-mentioned works shows the importance of the chiral transition
temperature, which in itself defines a fundamental scale.

Figure 1: Schematic phase diagram of QCD in the space of temperature ($T$) $-$
baryon chemical potential ($\mu_{B}$) $-$ degenerate mass of $u$ and $d$
quarks. Figure is taken from ref. [5].

On the phenomenological side also, the chiral transition temperature and the
nature of the chiral transition is very important to establish the global
phase structure of strongly interacting matter. To illustrate this point, I
show the conjectured phase diagram of QCD in the space of temperature ($T$)
$-$ baryon chemical potential ($\mu_{B}$) $-$ degenerate mass of $u$ and $d$
quarks in fig. 1, which I borrowed from ref. [5]. Since chiral symmetry is
exact in the chiral limit, spontaneous breaking of chiral symmetry must happen
through a phase transition [6]. The zero temperature transition is expected to
be of first order and it extends in the $T-\mu_{B}$ plane and bends toward the
$T$-axis. This is shown by the thick black line in fig. 1. For vanishing
chemical potential, on the other side, it was argued in a seminal work [7]
that the chiral phase transition for massless $u$ and $d$ quarks will be of
second order belonging to the $O(4)$ universality class (red dot in fig. 1).
This also extends for finite $\mu_{B}$ and bends towards the $\mu_{B}$-axis
(represented by the thick red line in fig. 1) and meets the before-mentioned
first order line in a tri-critical point (magenta dot in fig. 1). This
completes the picture in the chiral limit and indicates
$T_{c}>T_{\text{tri}}$. When one leaves the chiral plane, the first order
transition at low $T$ and high $\mu_{B}$ remains first order but the
transition at vanishing and small but non-vanishing values of $\mu_{B}$
becomes a crossover which is represented by the black dashed line in fig. 1.
The first order line in the $T-\mu_{B}$ plane for finite quark masses ends in
a (bi-)critical point which shifts to smaller $T$ (so
$T_{\text{tri}}>T_{\text{cep}}$) and larger $\mu_{B}$ [8], which defines the
so-called wing line(s), represented by thick blue line in fig. 1. The critical
end-point at finite masses are connected to the temperature axis through a
crossover, and this crossover temperature at $\mu_{B}=0$ for physical pion
mass has been recently determined very precisely [9, 10]. It is not hard to
realize that the QCD phase diagram at physical value of pion mass is nothing
but a slice of the more general phase diagram represented in fig. 1. These
interlinked pieces of information gives rise to a very important inequality:
$T_{\text{cep}}<T_{c}$, i.e. the chiral critical temperature puts an upper
bound on the position of the conjectured critical end point in the physical
world QCD.

The picture presented in the last paragraph is not undisputed. In the before-
mentioned work by Pisarski and Wilczek [7], it is also mentioned that if the
strength of the anomaly decreases with increasing temperature and $U_{A}(1)$
gets effectively restored at the critical temperature then the chiral
transition for two massless flavors can be of first order. Then the scenario
will be quite different compared to what is shown in fig. 1. Although later it
was proposed [11] that the chiral transition for two massless flavors can
still be continuous even if $U_{A}(1)$ is restored, but then the universality
class will be different.

Figure 2: Two most celebrated versions of the Columbia plot. Figures are taken
from ref. [12].

In this context, two widely celebrated versions of a flag diagram, known as
the Columbia plot [13] which presents the order of the chiral transition in
the plane of light (degenerate $u$ and $d$) and strange quark masses, is shown
in fig. 2. In the left panel the second order scenario for $2$\- and
($2+1$)-flavor is shown, whereas the right panel shows the scenario where the
chiral transition is of first order. I will come to more details of these
diagrams later. Various other aspects about the Columbia plot can be found in
[14].

I hope by now I convinced the reader that why/how the calculation of the
chiral critical temperature and determination of the order of the chiral phase
transition is of fundamental necessity. In this talk I shall try to map the
progress made by the community to shed light on various aspects of QCD towards
the chiral limit. My apologies if I miss any reference; it goes without saying
that would be completely unintentional.

##  2 Results: a few mentions

In this part I shall try to mention various works that have tried and are
trying to answer various fundamental questions about the chiral phase
transition. First I mention the calculations regarding the chiral transition
temperature or to determine the order of the chiral transition for two
degenerate light flavors ($N_{f}=2$), and then I mention the same for three
($N_{f}=3$) and larger than three ($N_{f}>3$) numbers of degenerate flavors.
Finally I come back to $N_{f}=2(+1)$ and mention very recent developments
regarding the energy-like observables.

###  2.1 $N_{f}=2(+1)$: the chiral transition temperature

Let me start with a well known scaling formula for the pseudo-critical
temperatures, defined by the peak position of order parameter susceptibility
for various quark masses:

$T_{\text{p}}(H)=T_{c}\left(1+\frac{z_{p}}{z_{0}}H^{1/\beta\delta}\right)+\text{sub-
leading}\;,$ (1)

where $T_{p}(H)$ is the peak position for the symmetry breaking field $H$,
which in QCD is proportional to the light quark masses. The chiral transition
temperature $T_{c}$ and the scale of the scaling variable $z_{0}$ are non-
universal parameters, and $z_{p}$ specifies the peak position of the scaling
function of the order parameter susceptibility. For the relevant universality
classes regarding the chiral transition for two massless flavors, the value of
$z_{p}$ is order unity [15] which gives rise to a significant drop in a non-
linear fashion of the pseudo-critical temperature towards the chiral limit,
w.r.t. e.g. physical values of the light quark masses. Authors of [15] came up
with two novel estimators of pseudo-critical temperatures by the following
conditions;

$\frac{H\chi_{M}(T_{\delta}),H}{M(T_{\delta},H)}=\frac{1}{\delta}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}\chi_{M}(T_{60},H)=0.6~{}\chi_{M}(T_{p},H)\;,$
(2)

where $M$ and $\chi_{M}$ are the order parameter (proportional to the chiral
condensate) and its susceptibility (proportional to the chiral susceptibility)
for the chiral phase transition. The values of the scaling variable
corresponding to the conditions in eq. 2, named as $z_{\delta}$ and $z_{60}$,
are two orders of magnitude smaller compared to $z_{p}$ [15], which results in
way less variation of the pseudo-critical temperatures towards the chiral
limit.

Figure 3: Chiral extrapolation of various pseudo-critical temperatures. Left:
calculation with HISQ and the plot is taken from [16]. Right: calculation with
twisted mass Wilson fermion and the plot is taken from [17].

In fig. 3 the chiral extrapolation of various pseudo-critical temperatures
from two different groups is shown. The left panel shows the continuum
extrapolated results of $T_{\delta}$ and $T_{60}$, and they are compared with
the conventional estimator of $T_{p}$ (also continuum extrapolated),
calculated with highly improved staggered quarks (HISQ). The stability of the
new estimators is vivid in the plot. In ref. [15] it is found that the
behavior of various pseudo-critical estimators is consistent with $O(4)$
scaling ansätze. Finally the chiral critical temperature, in the continuum, is
quoted as $T_{c}=132^{+3}_{-6}$ MeV [15]. In the right panel the chiral
extrapolation of various pseudo-critical estimators, calculated with twisted
mass Wilson fermions [17], are shown. In contrary to ref. [15], this
calculation is performed within a fixed scale approach. Still the quoted
chiral critical temperature, $T_{c}=134^{+6}_{-4}$ MeV, agrees well with the
other calculation. Consistency with $O(4)$ scaling is found. Moreover, the
chiral extrapolations of pseudo-critical estimators are found not quite
sensitive to the difference on the universality classes.

###  2.2 $N_{f}=2(+1)$: order of the chiral transition

Coming to the nature of the chiral transition for $N_{f}=2$, there exist two
major ways to look into it: either from the scaling perspective or through
direct determination of the fate of $U_{A}(1)$. In the following I will take
the tour in both ways.

####  2.2.1 $N_{f}=2(+1)$: order of the chiral transition from scaling

Close to the critical point the ratio of the order parameter and its
susceptibility is given by [16]

$\left.\frac{M}{\chi_{M}}\right|_{T_{X},H}=\left.\frac{f_{G}(z)}{f_{\chi}(z)}\right|_{z_{X}}H~{}+~{}\text{sub-
leading}.$ (3)

where $X=\text{p},~{}\delta$ or 60. It can be clearly seen that in absence of
the sub-leading corrections the RHS is determined solely through the universal
contribution. This fact has been exploited in the left panel of fig. 4.
Parameter free comparison of the data to the scaling expectation clearly
depicts that the data prefers an $O(N)$ scenario ($O(4)$ in continuum and
$O(2)$ for staggered calculation at a finite temporal extent, $N_{\tau}$) over
a first order transition in the chiral limit. Little deviations of the data
w.r.t. the scaling expectation towards larger quark masses can be accommodated
using a regular term, which is proportional to quark mass in the order
parameter. Similar results are found for $z_{60}$. More details about this
analysis can be found in [16].

Figure 4: Left: Continuum extrapolated ratio of the chiral order parameter and
its susceptibility evaluated at the peak of the chiral susceptibility $T_{p}$
plotted as function of the scaled light quark masses. Plot is taken from [16].
Right: Bare critical quark mass at which a second order phase transition
belonging to $Z_{2}$ universality class is found, shown as a function of
number of flavors $N_{f}$ for various $N_{\tau}$. Plot is taken from [18].

There is another attempt to determine the order of the chiral transition using
tri-critical scaling. In ref. [18] the number of degenerate quark flavors was
treated as a real continuous parameter and found the phase boundary in an
extended parameter space. In the right panel of fig. 4 only the projection in
the plane of quark mass and number of flavors is shown. Each data point
implicitly specifies a critical coupling. The chiral transition is first order
below this set of points for a given $N_{\tau}$ and above it is crossover.
Each of these critical points corresponds to second order phase transitions
belonging to the $Z_{2}$ universality class. These are usually calculated on
the basis of the finite size scaling arguments of the kurtosis of the chiral
condensate. These critical points, creating a line, are eventually expected to
go to a tri-critical point at some critical value of $N_{f}$, as can be
realized from the left panel of fig. 2 comparing the situation for $N_{f}=3$
and 2. In the right panel of fig. 4, one can clearly see the sizeable shift of
the critical value of $N_{f}$ w.r.t. $N_{\tau}$, and it is quite clear that in
the continuum the critical $N_{f}$ will be larger than 2, implying that the
chiral transition for $N_{f}=2$ will be of second order.

####  2.2.2 $N_{f}=2(+1)$: effective restoration of $U_{A}(1)$

The $U_{A}(1)$ symmetry, which is broken in nature upon quantization, is
expected to get effectively restored at high temperature because of the
deceasing number of non-perturbative topological configurations [7]. So
investigation regarding the effective restoration of $U_{A}(1)$ is extremely
important to establish the nature of the chiral transition for 2-flavor and
(2+1)-flavor QCD. Before I go to mentioning different studies in this
direction, let me start with a brief introduction how effective restoration of
$U_{A}(1)$ is usually studied in the literature. In this context I borrowed
fig. 5 from ref. [19]. In this schematic diagram various mesonic channels are
connected through either the flavor non-singlet chiral transformation or by
the flavor singlet axial transformation, $U_{A}(1)$. The well known expected
degeneracy between iso-triplet pseudo-scalar meson ($\pi$) and iso-scalar
scalar meson ($\sigma$ or $f_{0}$) can be seen through the horizontal
connection between the top two entries of fig. 5. Similarly the iso-triplet
pseudo-scalar meson ($\pi$) is related to the iso-triplet scalar meson
($\delta$ or $a_{0}$) by the flavor singlet axial transformation, $U_{A}(1)$,
which is represented by the vertical connection between the two left side
entries of fig. 5. To investigate the effective restoration of $U_{A}(1)$,
usually the degeneracy between the masses or susceptibilities is checked, both
of which is actually based upon the degeneracy between the correlation
functions in those channels. Specifically most of the literature calculates
differences like $m_{a_{0}/\delta}-m_{\pi}$ or
$\chi_{\pi}-\chi_{a_{0}/\delta}$ as a measure of the $U_{A}(1)$ breaking.

Figure 5: Various mesonic channels of QCD and their relation through either
the flavor non-singlet chiral transformation or by the flavor singlet axial
transformation. Figure is taken from ref. [19].

It is interesting to note that in the chirally symmetric background one can
use the degeneracy between the diagonal entries of fig. 5 to check the
effective restoration of $U_{A}(1)$. When one chooses to connect the diagonal
containing $\delta$ and $\sigma$, it can be immediately realized that the
corresponding susceptibilities are given by the connected part and the total
chiral susceptibility. As a result of this identification one can write [20]

$\displaystyle\chi_{\pi}-\chi_{a_{0}/\delta}$ $\displaystyle=$
$\displaystyle\left(\chi_{\pi}-\chi_{\sigma}\right)+\chi_{\text{disc}}$ (4)
$\displaystyle=$
$\displaystyle\chi_{\text{disc}},~{}~{}\text{in~{}the~{}chirally~{}symmetric~{}phase,}$

which implies the condition for the effective restoration of $U_{A}(1)$ boils
down to the condition whether in the chirally symmetric background
$\chi_{\text{disc}}$ vanishes or not.

A plethora of literature can be found about the effective restoration of
$U_{A}(1)$ and is not possible to cover all of those here. In the following I
rather list some of the recent efforts which are almost equally divided into
the groups which either favors or disfavors the effective restoration of
$U_{A}(1)$. I show one key plot from each study and mention few features of
the calculations belonging to either world.

Figure 6: Temperature variation of symmetry breaking measures from calculation
with (2+1)-flavor domain-wall fermions (DWF) for $N_{\tau}=8$. Plot taken from
- Left: ref. [19], Middle: ref. [21], Right: ref. [22].

In the left panel of Fig. 6 the temperature variation of symmetry breaking
measures is shown from a calculation with (2+1)-flavor domain-wall fermions
(DWF) for $N_{\tau}=8$ [19] with a $\sim$ 45% heavier pion compared to the
physical one. $\chi_{\pi}-\chi_{\delta}$ remains non-zero for the entire
temperature interval of the study suggesting that $U_{A}(1)$ is effectively
broken even after restoration of chiral symmetry. Moreover degeneracy of
$\chi_{\pi}-\chi_{\delta}$ with the disconnected scalar/chiral susceptibility
$\chi_{\text{disc}}$ and disconnected pseudo-scalar susceptibility
$\chi_{5,\text{disc}}$ at high temperatures ensures that the non-vanishing
$\chi_{\pi}-\chi_{\delta}$ originates indeed from $U_{A}(1)$ breaking and not
from explicit breaking due to quark masses.

Figure 7: Eigenvalue spectrum of the massless 2-flavor overlap Dirac operator
for $N_{\tau}=8$ [23]. Lighter shed represents smaller masses.

A follow-up study [21] with the same set up as above but with larger volumes
confirms that the finite volume effects are under control. As can be seen from
the middle panel of fig. 6 the $U_{A}(1)$ breaking measures have a clearly
non-vanishing value, and the near equality between the different measures
ensures that the explicit chiral symmetry breaking is tiny around the highest
temperatures of this study. Further analyses of the volume dependence and the
chirality of the near-zero modes agrees with the expectations from the dilute
instanton gas approximation picture, again confirming breaking of $U_{A}(1)$
above the pseudo-critical temperature.

In the right panel of fig. 6 the temperature variation of
$\chi_{\pi}-\chi_{\delta}$ from calculation with (2+1)-flavor domain-wall
fermions (DWF) for $N_{\tau}=8$ [22] with a physical pion is shown. Again the
$U_{A}(1)$ breaking measure remains non-zero over the entire temperature range
of the study. At low temperature the expected increase of
$\chi_{\pi}-\chi_{\delta}$ with decreasing quark mass can be seen. The
apparent mass independence of $\chi_{\pi}-\chi_{\delta}$ at high temperatures
confirms that the non-vanishing value comes due to the breaking of $U_{A}(1)$
and not due to small explicit breaking through non-vanishing quark masses.

In fig. 7 the eigenvalue spectrum of the massless 2-flavor overlap Dirac
operator for $N_{\tau}=8$ [23] is shown. Lighter sheds represent smaller
masses. The upper panel corresponds to $T<T_{\text{pc}}$, where reduction of
the mass reduces the density of near zero modes $\rho(0)$, which is compatible
with finite volume effects. In the infinite volume limit $\rho(0)$ is expected
to be finite. The middle panel corresponds to $T\sim T_{pc}$, and here
decreasing the mass resulted in suppression of eigenvalues up to $\sim 40$
MeV, which is consistent with a vanishing $\rho(0)$ in the chiral limit. This
suppression is even more prominent for $T>T_{\text{pc}}$ and indicates towards
a gap in the spectrum in the chiral limit. Such a gapped spectrum eventually
leads to the degeneracy between correlators in $\pi$ and $\delta$ channels,
implying effective restoration of $U_{A}(1)$ in the chiral limit.

Figure 8: Left: Renormalized eigenvalue spectra calculated with overlap
fermions on (2+1)-flavor HISQ ensembles [24] for $N_{\tau}=6,8$. Right: Chiral
extrapolation of the difference between the pseudo-scalar and scalar masses,
calculated using non-perturbatively ${\cal O}(a)$-improved 2-flavor Wilson
fermions [25] with $N_{\tau}=16$. Figure 9: Chiral extrapolation of
$\chi_{\text{disc}}$ calculated using (2+1)-flavor HISQ action with
$N_{\tau}=8,~{}12$ and 16, for $T=1.6~{}T_{c}$ [26].

In the left panel of fig. 8 renormalized eigenvalue spectra calculated with
overlap fermions on (2+1)-flavor HISQ ensembles [24] for $N_{\tau}=6,8$ are
shown. Accumulation of near-zero modes being independent of the cutoff and
being stable under smearing of the gauge fields ensures that the accumulation
is not due to lattice artifacts. Since these near-zero modes will result into
a non-vanishing $\chi_{\pi}-\chi_{\delta}$, apparently $U_{A}(1)$ seems to be
broken even above the pseudo-critical temperature. Detailed study of the
spatial structure and localization properties of these near-zero modes reveals
agreement with the expectation from weakly interacting instantons and anti-
instantons.

In the right panel of fig. 8 a chiral extrapolation of the difference between
the pseudo-scalar and scalar masses at the pseudo-critical temperature of the
respective masses is shown, calculated using non-perturbatively ${\cal
O}(a)$-improved 2-flavor Wilson fermions [25] with $N_{\tau}=16$.
Extrapolations with linear and square-root ansätze w.r.t. quark masses lead to
a vanishing value (within uncertainties) of the mass difference shown in the
plot, implying an effective restoration of $U_{A}(1)$ in the chiral limit.

In fig. 9 chiral extrapolation of $\chi_{\text{disc}}$ calculated using the
(2+1)-flavor HISQ action with $N_{\tau}=8,~{}12$ and 16 is shown for
$T=1.6~{}T_{c}$, where $T_{c}$ is the chiral critical temperature in the
continuum [26]. The blue band represents a joint chiral and continuum
extrapolation using an ansätz that is quadratic in quark mass, and the
coefficients have a quartic correction in lattice spacing. The magenta band,
on the other hand, shows the result from the ‘proper’ order of limits for
staggered calculations where the continuum limit is taken first with a
quadratic correction in lattice spacing using the two largest $N_{\tau}$ and
then mass extrapolation using a quadratic ansätz. Both extrapolations lead to
a non-vanishing value of $\chi_{\text{disc}}$ with more than 95% confidence,
implying that $U_{A}(1)$ is broken in continuum even at $1.6~{}T_{c}$. Similar
results are obtained from the analysis of $\chi_{\pi}-\chi_{\delta}$. Analyses
of the correlation function of the eigenvalue density reveals that the
microscopic origin of the axial anomaly at high temperature can be described
within the weakly interacting (quasi)instanton gas picture. A similar analysis
at somewhat lower temperature, closer to $T_{\text{pc}}$, on the other hand
shows that the eigenvalue spectrum is quite different from that at higher
temperatures and is not consistent with the dilute instanton gas picture [27].

Figure 10: Eigenvalue histograms of domain-wall (top), partially quenched
overlap (middle) and reweighted overlap (bottom) Dirac operators on ensembles
generated with 2-flavor domain-wall sea quarks [28].

In fig. 10 eigenvalue histograms of domain-wall (top panel), partially
quenched overlap (middle panel) and reweighted overlap (bottom panel) Dirac
operators on ensembles generated with 2-flavor domain-wall sea quarks [28] are
shown. The apparent suppression of near-zero modes for domain-wall and
reweighted overlap spectra towards the chiral limit seems to be stable under
changing lattice spacing and spatial volume implying that the observed effect
is not a lattice artifact. Eigenvalues in the lowest bin monotonically
decrease w.r.t. decreasing quark mass and become consistent with zero in the
chiral limit. A highly contrasting behavior can be observed for the partially
quenched overlap spectrum - a sharp peak is present in the lowest bin and
persists against the decrease of quark mass. Since this peak does not appear
in the domain-wall and reweighted overlap spectra, this was considered as an
artifact of partial quenching. After removing this artifact
$\chi_{\pi}-\chi_{\delta}$ is found to be consistent with zero in the chiral
limit, implying effective restoration of $U_{A}(1)$ in the chiral limit.

Figure 11: Left: Renormalized eigenvalue density of the appropriately mass-
tuned valence overlap operator on (2+1)-flavor HISQ ensembles for three
different light quark masses [29]. Right: Disconnected part of the chiral
susceptibility from a calculation from reweighted overlap Dirac spectrum on
Möbius domain-wall sea quarks, within a fixed scale approach [30].

In the left panel of fig. 11 renormalized eigenvalue densities of
appropriately mass-tuned valence overlap operator on (2+1)-flavor HISQ
ensembles for three different light quark masses [29] are shown. The
occurrence of a near-zero peak seems to be stable under a change of quark
mass. The calculation of the renormalized $U_{A}(1)$ breaking measure using
these eigenvalues confirms that in the chiral limit $U_{A}(1)$ remains broken
just above the transition temperature.

In the right panel of fig. 11 the disconnected part of the chiral
susceptibility, obtained from a calculation of the reweighted overlap Dirac
spectrum on Möbius domain-wall sea quarks within a fixed scale approach [30],
is shown. $\chi_{\text{disc}}$ is consistent with zero for the lowest quark
mass and no significant volume dependence is seen, leading to the conclusion
that $U_{A}(1)$ is restored in the chiral limit.

One important thing to note from all the studies mentioned above is that most
of the those probe the temperature range above the pseudo-critical temperature
$T_{\text{pc}}$ and consequently well above the chiral critical temperature
$T_{c}$. One has to keep in mind that at high temperatures the $U_{A}(1)$
breaking anyway becomes small being consistent with the dilute instanton gas
approximation. The effective restoration or breaking of $U_{A}(1)$ thus needs
to be checked close to the chiral critical temperature while going towards the
chiral limit. This is difficult!

In the left panel of fig. 12 the temperature variation of
$\chi_{\pi}-\chi_{a_{0}}$ is shown calculated using the meson correlation
function with HISQ action with $N_{\tau}=8$ [31]. The apparent increase with
smaller quark masses at low temperature is expected from the Ward identity. At
high temperatures this difference becomes small according to the expectation,
and upon looking carefully one can realize that the mass dependence is
inverted w.r.t. the dependence at low temperature. The exact same behavior can
also be noticed in the temperature variation of the disconnected part of the
chiral susceptibility, $\chi_{\text{disc}}$ [20]. It is not hard to realize
that a given fixed (intermediate) temperature simultaneously belongs to the
chirally restored phase and chirally broken phase for comparatively large and
small masses, respectively. This makes the mass dependence of any $U_{A}(1)$
breaking measure highly non-monotonic for temperatures above but close to the
chiral critical temperature, $T_{c}$. This can be seen in a much clearer
manner through the right panel of fig. 12 where the mass variation of the
renormalized $\chi_{\text{disc}}$ is plotted for various temperature above but
close to $T_{c}$ which is $\sim$ 144 MeV for $N_{\tau}=8$ [15]. It is clearly
seen for comparatively higher temperatures, $T\gtrsim 165$ MeV, in the right
panel of fig. 12. $\chi_{\text{disc}}$ monotonically decreases towards the
chiral limit which is easier to handle w.r.t. chiral extrapolation. On the
other hand, for lower temperatures one can clearly see the onset of the non-
monotonic behavior w.r.t. the quark mass - with decreasing quark masses
$\chi_{\text{disc}}$ first increases to a certain point in mass then turns and
start decreasing for even lower masses. Moreover it can be seen that this
turning around towards the low values shifts to smaller masses and the slope
in quark mass becomes larger when the temperature approaches towards $T_{c}$
from above which makes it really hard to comment about the chiral limit value
of $\chi_{\text{disc}}$ without having calculations at really small masses.
Exactly same behavior has also been observed in the study of $U_{A}(1)$
breaking/restoration through mesonic susceptibilities [31]. This is one of the
main difficulties in the study of effective restoration of $U_{A}(1)$ for
temperatures above but close to $T_{c}$.

Figure 12: Left: Temperature variation of $\chi_{\pi}-\chi_{a_{0}}$,
calculated using the meson correlation function using the HISQ action with
$N_{\tau}=8$. Plot is taken from [31]. Right: Variation of
$\chi_{\text{disc}}$ w.r.t. light quark mass for various (fixed) temperatures
above but close to the chiral critical temperature. Calculations have been
performed with the (2+1)-flavor HISQ action with $N_{\tau}=8$.

Interestingly the above mentioned behaviors of the $U_{A}(1)$ breaking
measures have a striking similarity with scaling expectation of the order
parameter susceptibility and the difference between so-called the transverse
and the longitudinal susceptibilities [32]. The latter one has a leading
quadratic dependence on the symmetry breaking field and the former has also a
quadratic dependence but in the next-to-leading order. It can also be shown,
that this quadratic behavior can only be realized for arbitrarily small values
of the symmetry breaking field when one approaches the critical temperature
from above. At this point it is tempting to connect the discussed quadratic
behavior with the same found in ref. [26] and could be seen in fig. 12 very
close to the chiral limit, although one has to admit, it will be a bit
ambiguous to associate the scaling behavior with the part(s) of the order
parameter susceptibilities which appears in the $U_{A}(1)$ breaking measures.
Another difficulty comes through the fact that the pseudo-critical
temperatures $T_{\text{pc}}$ depend on $N_{\tau}$, meaning for a fixed mass a
given temperature can simultaneously be below and above the pseudo-critical
temperature for a smaller and larger $N_{\tau}$, respectively, which makes the
continuum extrapolations very difficult. I hope I could convince the reader
why it is way more difficult to study the breaking/restoration of $U_{A}(1)$
closer to $T_{c}$ compared to away from it.

###  2.3 $N_{f}=3$

Pisarski and Wilczek originally argued that for three massless flavors the
chiral transition will be of first order, which seems natural given that a
cubic term in an effective chiral framework can be realized for $N_{f}=3$ or
through the critical fluctuations, even when the strength of the anomaly is
quite small near the chiral transition temperature [7]. This finding was
seconded by a study based on renormalization-group flow where the chiral
transition for $N_{f}=3$ is found to be of first order irrespective of the
effective restoration of $U_{A}(1)$ [11]. In this regard, it appeared that the
assumption of the irrelevance of the gauge degrees of freedom in the Landau-
Ginzburg-Wilson type approaches also should be given a serious second thought
[33, 34]. Given these apparent concerns it is important to ask the questions
what is the order of the chiral phase transition in QCD for three massless
flavors? Moreover, if it is first order, then what is the critical pion mass
at which the chiral transition is of second order belonging to $Z_{2}$
universality class?

The community is trying to answer this question for quite a while and since it
is impossible to mention all of them in the following I try to mention some of
the recent efforts in this direction. Let me start with a calculation with
unimproved staggered actions which found a finite critical pion mass [35],
although it was realized that the critical pion mass is very sensitive to
cutoff effects. In a later calculation with the HISQ action [36] there was no
evidence of a first order transition down to pion mass of 80 MeV, and scaling
arguments pushed the bound on critical pion mass down to 50 MeV. Still, a
possibility of a small first order region could not be ruled out. Almost at
the same time, a calculation based on stout-smeared rooted staggered quarks
[37] observed a significant decrease in the critical pion mass with increasing
amount of smearing, raising the possibility of having a second order phase
transition in the three flavor corner of the Columbia plot.

Figure 13: Left: Ratio of the non-subtracted chiral order parameter and its
susceptibility calculated at the peak of the susceptibility shown as a
function of mass for three degenerate flavors and compared with the scaling
expectations. The comparison with the singular part is parameter free as
expected from eq. 3. Plot is taken from ref. [38]. Right: Columbia-like plot
in terms of hadron masses to show the critical point belonging to the $Z_{2}$
universality class from calculations with ${\cal O}(a)$ improved Wilson
fermions. The strong cutoff dependence of the critical pion mass is vivid.
Plot is taken from ref. [39].

Recently in a calculation using the standard staggered formulation treating
$N_{f}$ as a real continuous parameter [18], it was found that the critical
pion mass for having a second order transition decreases with increasing
$N_{\tau}$, and the scaling analysis suggests that this critical mass
eventually vanishes for finite values of $N_{\tau}$, implying that the chiral
transition for three massless flavors will be of second order in the
continuum.

In a very recent study with the HISQ action no direct evidence of a first
order phase transition was found [38]. Moreover it was found that the behavior
of various chiral observables made out of the chiral condensate and its
susceptibility can be very satisfactorily described assuming a second order
phase transition at the three flavor chiral point. Further support arises from
an analysis described through Eq. 3 also for $N_{f}=3$. It can be clearly seen
from the left panel of fig. 13 that even a calculation with a fixed value of
$N_{\tau}$ seems to prefer a second order chiral phase transition with the
correct universality class, which is $O(2)$ in this case. The critical
temperature was found to be $T_{c}=98^{+3}_{-6}$ MeV, within the scale setting
of (2+1)-flavor QCD.

Coming to the Wilson fermion side, the chronology has a similar character. In
all of the calculations with ${\cal O}(a)$-improved Wilson fermions, a finite
value of critical mass is found either assuming or establishing the $Z_{2}$
universality class through the calculation of finite size scaling of the
kurtosis of the chiral condensate [40, 41, 39]. It is found that although the
critical temperature does not suffer much from cutoff effects, the critical
pion mass suffers from huge cutoff effects. In the right panel of fig. 13, I
borrowed a Columbia-like plot from ref. [39], which clearly shows the strong
cutoff dependence of the critical pion mass leaving only a tiny room for a
first order phase transition in the chiral limit. An interesting observation
is that although of different nature, the transition temperature found in
staggered calculations [38] is quite comparable with what is calculated from
the Wilson studies [41, 39]. Recently it has been shown [18] that the existing
result for critical pion masses from Wilson fermion calculations also show
tri-critical scaling and leads to vanishing pion masses at a finite value of
$N_{\tau}$ before reaching the continuum, implying a second order chiral phase
transition.

I would like to end this part with the remark that at this point discussions
regarding the chiral transition for $N_{f}=3$ are standing at a very
interesting and crucial crossing. In the following I try to list a few
comments or questions regarding the chiral transition for three massless
flavors which I collected during the discussion:

  * •

An increasing amount of evidence is coming up in favor of a second order
chiral phase transition.

  * •

What is the reason that the first order transition, expected for a long time,
doesn’t show up in QCD calculations?

  * •

Related to the earlier question, what is the possible reason that the tri-
linear coupling in the chiral effective framework originated from the anomaly
contribution becomes unimportant?

  * •

The origin of the observed tri-critical scaling may lie in a hexa-linear
coupling in Landau-Ginzburg-Wilson type theories, as suggested in ref. [18].

  * •

What is the role of gauge symmetries in the Landau-Ginzburg-Wilson type
approaches?

Hopefully these will motivate many interesting studies in future.

###  2.4 $N_{f}>3$

Theories with large number of quark flavors have also been a point of interest
of the community for various reasons. According to Pisarski and Wilczek [7]
the chiral transition is also expected to be of first order for $N_{f}>3$. In
the following I try to mention some recent studies and my excuses if I miss
some.

Figure 14: Left: Critical pion mass (in units of the critical temperature
$T_{c}$) for various $N_{\tau}$ from a $N_{f}=4$ calculation with unrooted
standard staggered fermions. For comparison the critical pion mass for
$N_{f}=3$ calculations are also shown. Plot is taken from ref. [42]. Right:
Critical pion mass for various $N_{\tau}$ from a $N_{f}=4$ calculation with
the on-perturbatively improved Wilson-Clover fermion action. Plot is taken
from ref. [43].

It is usually expected that with increasing number of flavors the first order
region gets larger, which indicates to a larger value of the critical pion
mass. This expectation seems to be fulfilled in calculations with $N_{f}=4$,
performed with unrooted standard staggered fermions [42]. The strong reduction
of the critical pion mass has been realized also for $N_{f}=4$, and the cutoff
dependence seems to be even larger compared to $N_{f}=3$, which can be seen
from the left panel of fig. 14. This study also clarifies one important
criticism about calculations with staggered fermions that the strong reduction
of the extent of the first order region in the Columbia plot is not due to
rooting.

Coming to the calculations with Wilson fermions the situation is quite similar
to staggered one, meaning the critical pion mass for $N_{f}=4$ is found to be
larger compared to the same for $N_{f}=3$ [43], which can be seen from the
right panel of fig. 14, where apparently a non-zero critical pion mass can be
seen in the continuum limit. Since the cutoff effects are not fully under
control in ref. [43] an update has been planned [44] to this analysis.

Going over to even higher numbers of flavors it can be seen from the right
panel of fig. 4 that the cutoff effects become larger with increasing number
of flavors. It is found that the critical pion masses calculated for $N_{f}=5$
also prefers tri-critical scaling, and a clear tension can be realized while
trying to describe the critical masses with an ansatz compatible to a first
order scenario. This can only be avoided given a change in the curvature
arbitrarily close to the continuum, which would be rather surprising. Given
all these observations and realizations it is argued that for $N_{f}\lesssim
6$ the first order transition can not persist to the continuum implying that
in the continuum the chiral transition for $N_{f}\lesssim 6$ is going to be of
second order [18]. This finding may have a bigger perspective w.r.t. to the
discussions about a conformal window of QCD when $N_{f}$ exceeds some critical
value [45, 46].

###  2.5 Energy-like observables for $N_{f}=2+1$.

For the universality classes of our interest, namely $O(N)$ or $Z_{2}$, in the
infinite volume limit, there exits two relevant scaling fields in the RG
sense. One, corresponding to the ‘magnetic’-direction, breaks the symmetry
explicitly and is proportional to the symmetry breaking fields/parameters in
the leading order of Taylor expansion around the critical point. Derivatives
of the partition function w.r.t. this symmetry breaking field give rise to the
“magnetization-like” observables e.g. the order parameter and its
susceptibility. The other relevant scaling field, called “temperature-like”,
corresponds to the ‘thermal’-direction in RG space and does not break the
symmetry. This “temperature-like” scaling field is proportional to the so-
called reduced temperature to the leading order. Important consequences arise
from the realization that any parameter which does not break the corresponding
symmetry of the theory under consideration must appear in the temperature-like
scaling field. A derivative of the the partition function w.r.t. any such
parameter, eventually being proportional to a derivative w.r.t. the
temperature-like scaling field, defines an “energy-like” observable. The
behavior of these observables proportional to the first and second derivatives
of the partition function w.r.t. the temperature-like scaling field, can be
related to that of the energy and specific heat in the spin model [32],
respectively.

Let’s start with the Polyakov loop (PL), which was first proposed as an order
parameter of the confinement-deconfinement phase transition in the quenched or
infinite quark mass limit of QCD [47, 48, 49]. Later on there were attempts to
use the inflection point of PL or the peak of the PL susceptibility to comment
about the deconfinement crossover. Some of these studies found that the
crossover temperature defined through observables derived from the chiral
condensate are quite close to the same defined through the various PL
observables [50, 51, 52, 53], whereas others found that the latter estimators
give higher temperatures compared to the former ones [54, 55, 56, 57].

Figure 15: Left: Scaling fit to the quark mass derivative of the HQFE. Right:
Description of the PL using the fit parameters from the scaling fits of the
mixed-susceptibility and the HQFE. Calculations are performed with HISQ action
with $N_{\tau}=8$. Both plots taken from ref. [58].

Interestingly at some point it was thought that the PL is not directly
sensitive to the singular structure and none of its features are related to
any critical exponent of chiral criticality [52]. However, evolving thoughts
with a gap of more than a decade gave a completely fresh perspective of the PL
w.r.t. the chiral phase transition [58]. Being a purely gluonic operator the
PL remains invariant under global chiral rotations. Since the PL does not
appear in the QCD action directly, it will be helpful to think of an effective
theory of QCD near the critical point. In such an effective theory the PL will
appear as an energy-like operator because it does not explicitly break the
chiral symmetry. Given this expectation the behavior of the PL and the related
heavy quark free energy (HQFE) are expected to behave as energy-like
observables w.r.t. the chiral criticality of QCD. Being energy-like, the HQFE
itself is not expected to diverge at the chiral limit. Rather a ‘mixed’
susceptibility, defined through the quark mass derivative of the HQFE which is
proportional to the correlation between the PL and the chiral condensate,
diverges moderately (slower compared to the magnetic susceptibility) in the
chiral limit [58, 59]. In the left panel of Fig. 15 the behavior of this
mixed-susceptibility is shown towards the chiral limit and it can be clearly
seen that the mass dependence can very well be described by the scaling
expectations. Non-universal parameters extracted from the scaling fits of the
mixed-susceptibility and the regular part parameters extracted from the HQFE
scaling fit describe the mass and temperature dependence of the PL very
satisfactorily, as is evident from the right panel of Fig. 15 ensuring the
fact that the PL is indeed an energy-like observable w.r.t. the chiral phase
transition of QCD which calls into question treating the PL as an ‘indicator’
of the deconfinement at physical and lower than physical pion masses. I shall
come back later to this point while discussing “specific heat-like”
observables.

Following the discussion in the beginning of this section, one can realize
that the baryon chemical potential ($\mu_{B}$) does not break the chiral
symmetry and it should appear in the temperature like scaling field [60, 61]:

$t\propto\left(\frac{T}{T_{c}}-1\right)+\kappa^{0}_{B}\left(\frac{\mu_{B}}{T_{c}}\right)^{2}\Rightarrow\kappa^{0}_{B}\frac{\partial}{\partial
T}=\frac{T_{c}}{2}\frac{\partial^{2}}{\partial\mu_{B}^{2}},\text{ within the
scaling window}\;,$ (5)

where the first term in the RHS of the proportionality relation is the usual
reduced temperature and the quadratic dependence on $\mu_{B}$ is to regard the
${\cal CP}$ conjugation. The non-universal parameter $\kappa^{0}_{B}$ is the
curvature of the critical line in the $T-\mu_{B}$ plane. This proportionality
relation implies that within the scaling window, one temperature derivative is
equivalent (with correct dimensions, of course) to a second order derivative
w.r.t. $\mu_{B}$, which immediately tells us that the second order baryon
number susceptibility will be an energy-like observable w.r.t. the chiral
phase transition. Singular contributions to the energy-like observables, being
scaled as $H^{(1-\alpha)/\beta\delta}$ with $H$ being proportional to the
light quark masses, vanish in the chiral limit [60, 61].

Figure 16: Left: Temperature variation of the baryon number susceptibility for
various light quark masses, calculated with the HISQ action for $N_{\tau}=8$.
The yellow band represents the chiral critical temperature, $T_{c}=144(2)$ MeV
for $N_{\tau}=8$ [15, 16, 20]. Right: Scaling fit to the quark mass variation
of the baryon number susceptibility from the left panel, evaluated at $T_{c}$,
for various quark masses.

In the left panel of Fig. 16, the baryon number susceptibility,
$\chi_{2}^{B}$, for various light quark masses, calculated with HISQ action
for $N_{\tau}=8$, is shown. In the right panel of Fig. 16, it can be seen that
the scaling expectation of $\chi_{2}^{B}$ being proportional to
$H^{(1-\alpha)/\beta\delta}$ is satisfied very well. The same expectation also
holds for strangeness susceptibility since the strangeness chemical potential
does not break the chiral symmetry either. Other aspects of this analysis can
be found in ref. [61, 60].

Similar to chemical potentials, in (2+1)-flavor QCD, the strange quark mass
$m_{s}$ also does not break the 2-flavor chiral limit. Hence $m_{s}$ should
also appear in the temperature like scaling variable and one derivative w.r.t.
$m_{s}$ should also be equivalent (again, with proper dimensions) to one
temperature derivative in the scaling regime. As a result of this conjecture,
the strange quark condensate should behave as energy-like observable w.r.t.
the chiral transition for two massless flavors which can be ensured by
calculating another mixed-susceptibility that is essentially the correlation
between condensates of light and strange quarks [61]. Note that the fact that
strange quark condensate is energy-like, may have an impact on the scaling
analysis of the subtracted chiral condensate.

Unill now I discussed some aspects of various energy-like observables. Going
to the next level, observables related to fourth order derivatives w.r.t.
chemical potentials or one temperature and two chemical potential derivatives
or second order derivative w.r.t. strange quark mass of logarithm of the
partition function, should behave like a specific heat, which is obtained by
two temperature derivatives of the logarithm of the partition function. All
these observables are expected to show the characteristic cusp of the $O(N)$
systems at $T_{c}$ whereas for small but finite quark masses they may show a
rounded peak near the corresponding $T_{\text{pc}}$. There is of course a
subtlety in the form that the appearance of peaks at small masses are highly
subjected to the regular background which differs among various observables
significantly. Such a background from energy density rising with temperature
was pointed out [62] and confirmed [63] in a calculation of the specific heat
itself. An example can also be found from the conserved charge fluctuations
where the fourth order baryon number susceptibility shows a peak around
$T_{\text{pc}}$ but the fourth order strangeness susceptibility monotonically
rises across the $T_{\text{pc}}$. It is also difficult to locate a peak in the
PL susceptibility in calculations done with improved actions, close to the
continuum limit whereas the $\mu_{B}$ response of the HQFE [64] clearly
resolves a peak already at the physical mass [65, 66]. These apparently
contradicting behaviors of various specific heat-like observables regarding
the peak structure needs more detailed analyses, probably focused to
understanding namely whether it is some kind of background contribution which
is present in one class of observables and not in the other.

I end the discussion on the specific heat-like observables by discussing a
special one - the temperature derivative of HQFE. When calculated through
temperature interpolation performed mass-by-mass, this shows a peak close to
the corresponding $T_{\text{pc}}$, and this peak height decreases with quark
mass. This behavior was proposed as a possible indicator of deconfinement and
interpreted as a hint for the possible coincidence of the chiral and
deconfinement crossover [67, 68]. On the other hand the consistent description
of mass and temperature variation of the HQFE and the mixed-susceptibility by
the corresponding scaling functions, renders a scenario where peaks at finite
quark masses eventually grow (expected at quark masses which are at least two
orders of magnitude smaller than what is being used in the current day
simulations) towards the chiral limit and show the characteristic $O(N)$ cusp
at $T_{c}$ [58]. This asks for a serious reconsideration of the temptation to
carry the interpretation of deconfinement through the free energy or entropy
of a test charge from the quenched corner to the regime of physical and lower
than physical pion masses.

I drop the scene by discussing recent efforts of calculating the curvature of
the (pseudo-)critical lines. There are mainly three avenues to calculate the
curvature - using Taylor expansion, analytic continuation of calculations done
at imaginary chemical potentials or using scaling arguments. In the former
method one can choose a physical condition at $\mu_{B}=0$ and follow that
condition over the $T-\mu_{B}$ plane through the Taylor expansion using the
parametrization of the of $T_{\text{pc}}$ on $\mu_{B}$;

$T_{\text{pc}}\left(\mu_{B}\right)=T_{\text{pc}}\left(0\right)\left[1-\kappa^{H}_{B}\left(\frac{\mu_{B}}{T_{\text{pc}}\left(0\right)}\right)^{2}\right].$
(6)

Then collecting terms in various orders in $\mu_{B}$ will define the curvature
within the same order of the Taylor expansion in terms of the Taylor
coefficients; e.g. the quadratic term will give the expression of
$\kappa^{H}_{B}$ for this specific value of quark mass (proportional to $H$).
The choice of the physical condition may have a wide variety, either by fixing
the value of pressure, energy density or entropy to its value at $\mu_{B}=0$
and $T_{\text{pc}}\left(0\right)$ [69] or choosing a pseudo-critical condition
e.g. the inflection point of the chiral condensate or peak of the chiral
susceptibility [70, 9]. This calculation, when performed mass-by-mass, can
directly show the quark mass dependence of the curvature of the pseudo-
critical lines (represented by the black dashed line in the $T-\mu_{B}$ plane
in fig. 1). A direct calculation of the curvature for imaginary chemical
potentials using a parametrization of eq. 6 gives compatible results [71, 10]
which rely on the validity of the analytic continuation.

Figure 17: Estimators of the curvature of the critical line in the $T-\mu_{B}$
plane. Calculations are done with HISQ action for $N_{\tau}=8$. The vertical
range enclosed by the pair of black lines shows the continuum extrapolated
estimate of the curvature of the pseudo-critical line for physical pion mass
[9].

The third method, which can give directly the curvature of the critical line
(represented by the red solid line in fig. 1) by exploiting the scaling
relation of the mixed-susceptibility, involves the mixture of magnetic and
thermal derivatives [72]. There also exists a simpler way to calculate
estimators of $\kappa_{B}^{0}$ using the implied equality in eq. 5; taking the
ratio between two $\mu_{B}$ derivative of any chiral observables and one $T$
derivative of of the same gives an estimator of $\kappa_{B}^{0}$ for a given
$H$. In fig. 17 such a calculation with non-subtracted chiral condensate is
shown for various quark masses. The calculations have been done with HISQ
action for $N_{\tau}=8$. The value of the ratio around $T_{c}$, shown by a
yellow band, is what should be looked for. In the same plot the continuum
extrapolated estimate of the curvature of the pseudo-critical line for
physical pion mass, obtained through the Taylor expansion method [9], is
indicated by the enclosed vertical interval between two black lines. This
preliminary comparison tends to suggests that the curvature may not change
dramatically towards the chiral limit [73].

An interesting point to note is that the ratio shown in fig. 17 to calculate
the estimators of $\kappa_{B}^{0}$ for a given $H$ can also be used used to
estimate $\kappa_{B}^{H}$ for the same $H$ from the Taylor expansion method,
corresponding to the condition that the chiral condensate is fixed to its
$\mu_{B}=0$ value at $T_{\text{pc}}\left(0\right)$, over the $T-\mu_{B}$ plane
[69]. Although then one has to focus around the pseudo-critical temperature,
$T_{\text{pc}}$, for that particular value of $H$, not around $T_{c}$ as is
done fig. 17. This suggests that the change of $\kappa_{B}^{H}$ in the scaling
regime will be $\sim$ 10% while going to its chiral limit value
$\kappa_{B}^{0}$.

##  3 Take home messages

I hope that through this write-up of my talk I could convince the reader that
study of QCD towards the chiral limit is one of the most interesting and
important fields of research. Various groups in the lattice community have
been actively working over the years on a variety of problems and hopefully
this will continue in future. In this write-up I tried to review some of the
aspects of this very dynamic field. With this said I would like to conclude by
pointing out some take home messages in the following:

  * •

There have been some progress ….

    * –

The chiral transition temperature for $N_{f}=2$ has been found to be around
130 MeV.

    * –

The CEP for physical world has to be searched for $T<130$ MeV and
correspondingly for $\mu_{B}>400$ MeV.

    * –

The curvature of (pseudo-)critical lines seems to have very weak quark mass
dependence towards the chiral plane.

    * –

The Polyakov loop behaves as an energy-like observable w.r.t. the chiral phase
transition, calling into question its relation with deconfinement, even at
physical mass.

    * –

Various conserved charge fluctuations and the strange condensate also behave
as energy-like observables.

  * •

Feel the heat ….

    * –

Effective restoration of $U_{A}(1)$ is yet to be settled among various fermion
discretizations.

    * –

Significantly more attention needed for temperatures higher than but close to
$T_{c}$.

    * –

The possibility of having a $1^{\text{st}}$-order region in the $N_{f}=3$
corner getting is feeble.

    * –

Studies of many flavor QCD are going to be interesting in future, especially
w.r.t. the existence of a conformal window.

## Acknowledgement

I acknowledge support from the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through Project No. 315477589-TRR 211 and from the German
Bundesministerium für Bildung und Forschung through Grant No. 05P18PBCA1.

I would like to thank Frithjof Karsch for many enlightening discussions and
for a critical reading of this manuscript. I would also like to thank David A.
Clarke for a careful reading of the write-up.

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