††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# The effect of charm quark on chiral phase transition in $N_{f}=2+1+1$
holographic QCD

Hiwa A. Ahmed  School of Nuclear Science and Technology, University of Chinese
Academy of Sciences, Beijing 100049, China  Department of Physics, College of
Science, Charmo University, 46023, Chamchamal, Sulaymaniyah, Iraq  Mamiya
Kawaguchi  School of Nuclear Science and Technology, University of Chinese
Academy of Sciences, Beijing 100049, China  Mei Huang  School of Nuclear
Science and Technology, University of Chinese Academy of Sciences, Beijing
100049, China

###### Abstract

We investigate the effect of charm quark on the chiral phase transition of
light quarks at finite temperature based on the four-flavor soft-wall
holographic QCD model. In the massless limit, we find that the thermal chiral
phase transition is of the second order in the four-quark flavor system. In
the case with the massive charm quark and the massless light and strange
quarks, the order of the phase transition changes to the first order. This is
due to the quark flavor symmetry breaking which is associated with the
violation of the $U(1)$ axial symmetry. Once the light and strange quarks get
massive, the explicit chiral symmetry breaking becomes eminent, then the
crossover phase transition is realized at the physical quark masses. We also
map the order of the phase transition on a phase diagram in the quark mass
plane where the light- and strange-quark masses are degenerate but differ from
the value of the charm quark mass. This phase diagram is an extension of the
conventional Columbia plot to the four-quark flavor system.

##  I Introduction

Investigating the thermal QCD phase transition is an important subject for
understanding the hot environment relevant to the initial stage of heavy-ion
collisions and the early universe. To clarify its phase structure, the order
of the chiral phase transition, especially depending on the number of quark
flavors ($N_{f}$) and the current quark masses ($m_{f}$), has been extensively
investigated by the lattice QCD framework [1, 2, 3, 4, 5, 6, 7], the
functional renormalization group (FRG) method [8, 9, 10, 11, 12, 13, 14, 15],
Dyson-Schwinger equations [16, 17] and chiral effective model approaches [18,
19, 20].

In a pioneer work, the existence of universality classes has been investigated
by using the renormalization group analysis in the linear sigma model approach
at the one-loop calculation with the $\epsilon$-expansion. They have concluded
that the chiral phase transition is of the first order for $N_{f}\geq 3$ with
massless quarks [18]. Motivated by this first research, the order of the
chiral phase transition in the three quark flavor system has been mapped onto
the plane of the light- and strange-quark masses. This phase diagram is
referred to as the Columbia plot [1]. Interestingly, the first order phase
transition domain is anticipated to emerge around the small quark mass regions
and it is closely associated with the contribution of the $U(1)$ axial anomaly
in the effective model analyses [18, 19, 20]. However, the definite evidence
of the first order domain has not been observed in the lattice QCD simulation.
The specifics of the Columbia plot is still unclear.

Shifting our focus to the multi-flavor system ($N_{f}\geq 4$) could offer a
fresh perspective on the thermal QCD phase transition. Recently, the lattice
QCD study has endeavored to extend the Columbia plot to the multi-flavor
system incorporating more than four quark flavors [21]. According to this
lattice simulation, the chiral phase transition is estimated to be of the
second order in the massless limit with $N_{f}\leq 6$. This lattice estimation
differs from the first study based on searching the universality class in the
linear sigma model approach [18]. The discrepancy between the lattice QCD
observation and the effective model results becomes more apparent through the
investigation in the multi-flavor system.

Although the extension has been made by the lattice work [21], our
understanding of the phase structure in the multi-flavor system is still
limited even compared to the study of the conventional Columbia plot. In our
study, we address the thermal chiral phase transition in such a multi-flavor
system, specifically focusing on the four-quark flavor system, by using the
holographic QCD approach.

The anti-de Sitter/conformal field theory (AdS/CFT) correspondence and the
conjecture of the gravity/gauge duality is a powerful tool for investigating
the non-perturbative QCD property [22, 23]. There are two main approaches in
holographic QCD: the top-down approach and the bottom-up approach. Unlike the
top-down approach, the bottom-up holographic QCD aims to describe the QCD
phenomena observed in the experimental physics and also it is attempted to
build a realistic holographic QCD model. As a development, it has been pointed
out that the dilaton configuration plays an important role in providing the
realistic QCD phenomena. Taking into account the infrared (IR) behavior of the
dilaton, the soft-wall AdS/QCD model can successfully reproduce the linear
behavior of meson spectra [24]. Furthermore, the spontaneous chiral symmetry
breaking in the chiral limit is well described by the ultraviolet (UV)
behavior of the dilaton [25, 26, 27].

With the developed soft-wall model, the phase structure has been studied at
finite temperature and/or finite chemical potentials [28, 29, 30, 25, 26, 31,
32, 27, 33]. The Columbia plot has also been described by the previous study
in the holographic approach. In our study, we extend the previous works of the
soft-wall model to the four-quark flavor system and propose the extended
Columbia plot.

##  II $N_{f}$-quark flavor system based on holographic QCD approach

Our aim of this paper is to investigate the quark-flavor dependence on the
chiral phase diagram based on the holographic QCD approach. In this section,
we begin by giving a brief review of the soft-wall model which is applicable
to the chiral phase transition at finite temperature. This model is defined in
the AdS5 spacetime. The temperature is introduced through the geometries with
black holes and it is described by the background metric where there is no
dynamical contribution. In our work, we take the solution of the simple AdS-
Schwarzchild(AdS-SW) black hole:

$ds^{2}=e^{2A_{s}(z)}\left(-f(z)dt^{2}+\frac{1}{f(z)}dz^{2}+dx_{i}dx^{i}\right),$
(1)

where $z$ represents the fifth coordinate and $A_{s}(z)$ is
$A_{s}(z)=-\log(\frac{z}{L})$ scaled by the radius $L$ of the AdS curvature
($L$ is set to $1$ below for simplicity). The function $f(z)$ is given by

$f(z)=1-\frac{z^{4}}{z_{h}^{4}},$ (2)

where $z_{h}$ denotes the black hole horizon defined at $f(z_{h})=0$. Using
this function $f(z)$, we have the Hawking temperature formula at the black
hole horizon,

$T=\frac{1}{4\pi}\left|\frac{df(z)}{dz}\right|_{z\to z_{h}}=\frac{1}{\pi
z_{h}}.$ (3)

This $T$ is identified as the temperature of the hot QCD system within the
holographic model approach.

###  II.1 Soft-wall model with multi-quark flavors

Following the original work of the bottom-up holographic QCD model [24], we
employ a five-dimensional gauge model respecting the $U(N_{f})_{L}\times
U(N_{f})_{R}$ symmetry with $N_{f}$ being the number of the quark flavors. The
five-dimensional model is described by the bulk left- ($L_{M}$) and right-
($R_{M}$) gauge fields. To incorporate the quark condensation operator in the
bottom-up approach, we introduce a bulk scalar field $X$ which is a
$N_{f}\times N_{f}$ matrix and transforms as bifundamental representation
under the $U(N_{f})_{L}\times U(N_{f})_{R}$ symmetry. Furthermore, considering
the breaking of the conformal invariance, we also include the dilaton
background field $\Phi(z)$ which acts as a smooth cutoff. The five-dimensional
soft-wall action takes the form:

$\displaystyle S_{M}$ $\displaystyle=-\int
d^{5}x\sqrt{-g}e^{-\Phi}\left\\{\operatorname{Tr}\left[\left(D^{M}X\right)^{\dagger}\left(D_{M}X\right)+\frac{1}{4g_{5}^{2}}\left(L^{MN}L_{MN}+R^{MN}R_{MN}\right)\right]+V(X)\right\\},$
(4)

where $g$ denotes the determinant of metric $g_{MN}$:
$\sqrt{-g}=e^{5A_{s}(z)}=(1/z)^{5}$; the covariant derivative of $X$ is given
by $D^{M}X=\partial_{M}X-iL_{M}X+iXR_{M}$; $L_{MN}$ and $R_{MN}$ are the field
strength defined as
$L_{MN}=\partial_{M}L_{N}-\partial_{N}L_{M}-i[L_{N},L_{M}]$ and
$R_{MN}=\partial_{M}R_{N}-\partial_{N}R_{M}-i[R_{N},R_{M}]$ in terms of the
left and right hand gauge field respectively; $g_{5}$ denotes the gauge
coupling constant determined by matching with the UV asymptotic behavior of
the vector current correlator $g_{5}^{2}=12\pi^{2}/N_{c}$ [34]. $V(X)$
represents the scalar potential which takes the polynomial form of $X$:

$\displaystyle V(X)=V_{0}(X)+V_{\rm det}.$ (5)

$V_{0}$ is constructed by only the quark-flavor symmetric interactions, which
is actually invariant under the $U(N_{f})_{L}\times U(N_{f})_{R}$ symmetry,

$\displaystyle V_{0}(X)=M_{5}^{2}{\rm Tr}\left[X^{\dagger}X\right]+\lambda{\rm
Tr}\left[|X|^{4}\right]$ (6)

where the mass-parameter $M_{5}^{2}$ is given by
$M_{5}^{2}=(\Delta-p)(\Delta+p-4)=-3$ with $\Delta=3$ and $p=0$ through the
holographic dictionary, and $\lambda$ is a positive-dimensionless quartic
coupling. In the absence of the $X^{4}$ term, the quark condensate is linked
with the current quark mass. At the chiral limit, the condensate accidentally
vanishes. To avoid this issue, the $X^{4}$ term is included in the potential
$V_{0}(X)$ [35].

In addition, the potential part $V(X)$ also has the determinant term as in
Ref. [36] to provide the quark-flavor mixing term,

$\displaystyle V_{\rm det}=\gamma{\rm Re}\,[{\rm det}(X)],$ (7)

with $\gamma$ being a real parameter. Note that this determinant term is
anomalous under the $U(1)_{A}$ symmetry, but it is invariant under the
$SU(N_{f})_{L}\times SU(N_{f})_{R}$.

In this study, we mainly focus on the four flavor system. Hence, the following
analysis is based on the soft-wall model with $N_{f}=4$.

To determine the classical solution of $X$, we consider the static homogeneous
expectation value of the scalar field:

$X_{0}={\rm
diag}\left(\frac{\chi_{u}(z)}{\sqrt{2}},\frac{\chi_{d}(z)}{\sqrt{2}},\frac{\chi_{s}(z)}{\sqrt{2}},\frac{\chi_{c}(z)}{\sqrt{2}}\right)$
(8)

where the factor $\frac{1}{\sqrt{2}}$ is taken to get a canonical form of the
kinetic terms of $\chi_{f}$ ($f=u,d,s,c$). In this study, we impose the
isospin symmetry, $m_{l}=m_{u}=m_{d}$, so that the light sector of the scalar
field $X_{0}$ takes $\chi_{l}=\chi_{u}=\chi_{d}$ through the matching
condition with QCD. Since we are concerned with the chiral phase transition,
we set the bulk gauge fields $L_{M}$ and $R_{M}$ to zero. With this setup, the
action in Eq. (4) is evaluated as

$\displaystyle S\left[\chi_{l},\chi_{s},\chi_{c}\right]=$ $\displaystyle-\int
d^{5}x\sqrt{-g}e^{-\Phi}\left\\{g^{zz}\left(\chi_{l}^{\prime
2}+\frac{1}{2}\chi_{s}^{\prime 2}+\frac{1}{2}\chi_{c}^{\prime
2}\right)\right.$ (9)
$\displaystyle\left.+\left[-3\left(\chi_{l}^{2}+\frac{1}{2}\chi_{s}^{2}+\frac{1}{2}\chi_{c}^{2}\right)+v_{4}\left(2\chi_{l}^{4}+\chi_{s}^{4}+\chi_{c}^{4}\right)+3v_{\rm
det}\chi_{l}^{2}\chi_{s}\chi_{c}\right]\right\\}$

with $v_{\rm det}=\gamma/12$ and $v_{4}=\lambda/4$.

From this action, the equations of motion for $\chi_{f}$ ($f=l,s,c$) read

$\begin{gathered}\chi_{l}^{\prime\prime}+\left(3A_{s}^{\prime}-\Phi^{\prime}+\frac{f^{\prime}}{f}\right)\chi_{l}^{\prime}+\frac{e^{2A_{s}}}{f}\left(3\chi_{l}-3v_{\rm
det}\chi_{l}\chi_{s}\chi_{c}-4v_{4}\chi_{l}^{3}\right)=0,\\\
\chi_{s}^{\prime\prime}+\left(3A_{s}^{\prime}-\Phi^{\prime}+\frac{f^{\prime}}{f}\right)\chi_{s}^{\prime}+\frac{e^{2A_{s}}}{f}\left(3\chi_{s}-3v_{\rm
det}\chi_{l}^{2}\chi_{c}-4v_{4}\chi_{s}^{3}\right)=0,\\\
\chi_{c}^{\prime\prime}+\left(3A_{s}^{\prime}-\Phi^{\prime}+\frac{f^{\prime}}{f}\right)\chi_{c}^{\prime}+\frac{e^{2A_{s}}}{f}\left(3\chi_{c}-3v_{\rm
det}\chi_{l}^{2}\chi_{s}-4v_{4}\chi_{c}^{3}\right)=0,\\\ \end{gathered}$ (10)

where the prime denotes the derivative with respect to $z$ component. These
equations are a set of coupled equations, which is driven by the determinant
term associated with the quark-flavor violation ($m_{l}\neq m_{s}\neq m_{c}$).
Indeed, in the four quark flavor symmetric limit, $m_{l}=m_{s}=m_{c}$, the
three coupled equations reduce to a single equation.

In the soft wall holographic QCD model, a non-dynamical background field
called dilaton field $\Phi$, which is only a function of $z$ added to break
the conformality of the AdS space. This background field plays an important
role in the phenomenological observation and the chiral dynamics. Since the
dilaton modifies the equations of motion, it affects mesons spectra, form
factors, and any other property derived from the action. A positive dilaton
field at the infrared (IR) region $\phi(z\to\infty)\approx+z^{2}$ is necessary
to get the linear confinement [24]. In contrast to IR region, the dilaton
field has to be a negative quadratic $\phi(z\to 0)\approx-z^{2}$ in the
ultraviolet (UV) limit to describe the spontaneous symmetry breaking in the
chiral limit, as pointed out in Ref. [25, 26]. The specific profile of the
dilaton field, which respect both the IR and UV behavior, is proposed as [25]

$\Phi(z)=-\mu_{1}^{2}z^{2}+(\mu_{1}^{2}+\mu_{0}^{2})z^{2}\text{tanh}(\mu_{2}^{2}z^{2}),$
(11)

where the parameters $\mu_{0}=0.43$ GeV is fixed by the radial excitation of
the $\rho$ meson masses, and $\mu_{1}=0.83$ GeV and $\mu_{2}=0.176$ GeV are
chosen to get the critical temperature of the second order phase transition
around $150$ MeV for $N_{f}=2$ with massless quarks [25, 26].

###  II.2 Matching condition with QCD and numerical demonstration

Using the holographic correspondence, the quark condensate in QCD is extracted
from the UV asymptotic behavior of the scalar fields $\chi_{f}$ ($f=l,s,c$) in
the five dimensional model [34, 24], which serves as the order parameter of
the spontaneous chiral symmetry breaking. Indeed, $X/z$ is dual to the
$\bar{q}q$ operator within the holographic QCD approach. Through the matching
condition with QCD at the UV ($z\to 0$), $X/z$ is identified as the current
quark mass $m_{f}$ corresponding to the source of the quark condensate.

The UV solutions of $\chi_{f}$ can be obtained by using the Frobenius method
near the boundary $z=0$ [37]. At small $z$, the general expression of
$\chi_{f}$ are given by

$\begin{gathered}\chi_{l}=a_{l}z-\left(\mu_{1}^{2}-a_{l}^{2}v_{4}-\frac{3}{2}a_{s}a_{c}v_{det}\right)a_{l}z^{3}\log(z)+b_{l}z^{3}+.....,\\\
\chi_{s}=a_{s}z-\left(\mu_{1}^{2}a_{s}-a_{s}^{3}v_{4}-\frac{3}{2}a_{l}^{2}a_{c}v_{det}\right)z^{3}\log(z)+b_{s}z^{3}+.....,\\\
\chi_{c}=a_{c}z-\left(\mu_{1}^{2}a_{c}-a_{c}^{3}v_{4}-\frac{3}{2}a_{l}^{2}a_{s}v_{det}\right)z^{3}\log(z)+b_{c}z^{3}+.....,\\\
\end{gathered}$ (12)

where $a_{f}$ and $b_{f}$ are the integral constants of the three coupled
differential equations. Owing to the holographic dual of QCD at the UV
boundary, the coefficients of the $z$ terms in Eq. (12) correspond to the
current quark masses:

$\displaystyle a_{l}=m_{l}\zeta,\quad a_{s}=m_{s}\zeta,\quad
a_{c}=m_{c}\zeta.$ (13)

with $\zeta=\frac{\sqrt{N_{c}}}{2\pi}$ [38]. By considering the linear
response with respect to the current quark masses, the coefficients of the
$z^{3}$ terms in Eq. (12) are identified as the quark condensates,
$\sigma_{l}\equiv\langle\bar{u}u\rangle=\langle\bar{d}d\rangle$,
$\sigma_{s}\equiv\langle\bar{s}s\rangle$ and
$\sigma_{c}\equiv\langle\bar{c}c\rangle$:

$\begin{aligned} b_{l}=\frac{\sigma_{l}}{\zeta},\quad
b_{s}=\frac{\sigma_{s}}{\zeta},\quad
b_{c}=\frac{\sigma_{c}}{\zeta}\end{aligned}.$ (14)

In the numerical calculation, the integral constants $a_{f}$ are treated as
input parameters with the values of quark masses. On the other hand, $b_{f}$
linked with quark condensates are determined by solving the equations of
motion in Eq. (10) with appropriate boundary conditions.

To numerically solve the equations of motion, we first impose the boundary
condition near the UV,

$\lim_{\epsilon\to
0}\frac{\chi_{l}(\epsilon)}{\epsilon}=m_{l}\zeta,\quad\lim_{\epsilon\to
0}\frac{\chi_{s}(\epsilon)}{\epsilon}=m_{s}\zeta,\quad\lim_{\epsilon\to
0}\frac{\chi_{c}(\epsilon)}{\epsilon}=m_{c}\zeta,$ (15)

where $\epsilon$ is a small number. In addition, another boundary condition
arises from the black hole horizon defined at $f(z_{h})=0$. In fact, the
coupled equations of motion in Eq. (10) have a singularity at the horizon
$z=z_{h}$, resulting in the unexpected divergence of $\chi_{f}$. To avoid the
singularity at the horizon $z=z_{h}$, the following boundary condition should
be applied to numerically solve the equations of motion,

$\begin{gathered}f^{\prime}\chi_{l}^{\prime}+e^{2A_{s}}\left(3\chi_{l}-3v_{\rm
det}\chi_{l}\chi_{s}\chi_{c}-4v_{4}\chi_{l}^{3}\right)|_{z=z_{h}-\epsilon}=0,\\\
f^{\prime}\chi_{s}^{\prime}+e^{2A_{s}}\left(3\chi_{s}-3v_{\rm
det}\chi_{l}^{2}\chi_{c}-4v_{4}\chi_{s}^{3}\right)|_{z=z_{h}-\epsilon}=0,\\\
f^{\prime}\chi_{c}^{\prime}+e^{2A_{s}}\left(3\chi_{c}-3v_{\rm
det}\chi_{l}^{2}\chi_{s}-4v_{4}\chi_{c}^{3}\right)|_{z=z_{h}-\epsilon}=0.\\\
\end{gathered}$ (16)

With these boundary conditions, we perform numerical calculations in several
cases to provide practical examples, below.

The coupled equations of motion in Eq. (10) can be solved by utilizing the
shooting method with the boundary conditions specified in Eqs. (15) and (16).
By shooting from the near horizon limit $z=z_{h}-\epsilon$ to the near UV
boundary $z=\epsilon$, we obtain the numerical solutions of $\chi_{f}$ for the
case of the four-flavor symmetric massless limit, $m_{l}=m_{s}=m_{c}=0$, in
Fig. 1 where we take $T=0.100$ GeV and $T=0.150$ GeV corresponding to
$z_{h}\simeq 3.18\,{\rm GeV}^{-1}$ and $z_{h}\simeq 2.12\,{\rm GeV}^{-1}$
respectively. This figure shows that the numerical solutions of $\chi_{f}$
surely satisfy the boundary conditions and there is no singularity. Note that
the solutions of the scalar fields result in $\chi_{l}=\chi_{s}=\chi_{c}$,
owing to the four-flavor symmetry, which implies that the quark condensates
are equal for each other, $\sigma_{l}=\sigma_{s}=\sigma_{c}$. The value of the
parameter $v_{4}$ is fixed at $v_{4}=8$ by following the parameter set in the
previous analyses based on the soft-wall model with $N_{f}=2$ and $N_{f}=3$
[25, 26], which is also similar to the values of the quartic term couplings
presented in Ref. [39]. For $N_{f}=3$, the $v_{\rm det}$ parameters is defined
by the coupling of the determinant term as $v_{det}=\frac{\gamma}{6\sqrt{2}}$
and assigned to be $-3$ [25, 26]. As for $N_{f}=4$, the parameter $v_{\rm
det}$ is given by $v_{\rm det}=\frac{\gamma}{12}$ . Since one can go from the
four-quark flavor system to the three-quark flavor system by integrating out
the contribution of the charm quark with the heavy quark limit,
$m_{c}\to\infty$, the parameter $v_{\rm det}$ is redefined as
$v_{det}=\gamma/(12~{}\chi_{c}(z_{h})|_{m_{c}\to\infty})$. To match the value
in the three-flavor system, $v_{\rm det}$ in the four-flavor system is
assigned to be $-7.07$.

(a)

(b)

Figure 1: Solutions of $\chi_{l}$, $\chi_{s}$, and $\chi_{c}$ as functions of
holographic coordinate $z$ at massless limit $m_{l}=m_{s}=m_{c}=0$ for
$T=0.100$ GeV (a) and $T=0.150$ GeV (b).

The quark condensate $\sigma_{f}$ is determined by fitting the UV behavior of
$\chi_{f}$ in Eq. (12). In the case of the massless quarks, the solution of
$\sigma_{f}$ is identical for all flavors, resulting in equal condensates,
$\sigma_{l}=\sigma_{s}=\sigma_{c}$. The values of the condensates for
$T=0.100$ GeV and $T=0.150$ GeV are evaluated as
$\sigma_{l}=\sigma_{s}=\sigma_{c}=(0.3794\text{ GeV})^{3}$ and
$\sigma_{l}=\sigma_{s}=\sigma_{c}=(0.2276\text{ GeV})^{3}$, respectively.

Similarly, the equation of motions for $\chi_{f}$ in the case of massive
quarks can also be solved numerically. In Fig. 2, we display the solutions of
$\chi_{f}$ with the physical quark masses, $m_{l}=3.45\text{ MeV}$,
$m_{s}=93.4\text{ MeV}$, and $m_{c}=1.27\text{ GeV}$ [40], where the
temperature is taken as $T=0.150$ GeV. This figure clearly shows that the
solutions $\chi_{f}$ take different values owing to the violation of the
$SU(4)$ quark flavor symmetry. This violation is reflected in the quark
condensates, $\sigma_{l}=(0.3552\text{ GeV})^{3}$, $\sigma_{s}=(0.341\text{
GeV})^{3}$, and $\sigma_{c}=(0.3128\text{ GeV})^{3}$.

To verify the consistency of the model, the Gell-Mann-Oakes-Renner (GOR)
relation is used, $f_{\pi}^{2}m_{\pi}^{2}=2m_{q}\sigma$. The value of the
light quark condensate at zero temperature for physical quark masses is
obtained. With the values of the light quark mass and condensate, the pion
mass and the pion decay constant are evaluated (see Ref. [39] for details on
calculating the meson spectra and decay constants). For the physical quark
mass, we have the light quark condensate $\sigma_{l}=0.07095\text{ GeV}^{3}$,
the pion mass $m_{\pi}=0.17957$ GeV and the pion decay constant
$f_{\pi}=0.12314$ GeV. Actually, these values satisfy the GOR relation.

Figure 2: Solutions of $\chi_{l}$, $\chi_{s}$, and $\chi_{c}$ as functions of
$z$ at the physical quark masses $m_{l}=3.45\text{ MeV}$, $m_{s}=93.4\text{
MeV}$, and $m_{c}=1.27\text{ GeV}$ and temperature $T=0.150$ GeV.

##  III Chiral phase transition with four-quark flavors

With the above setup, we are now ready to explore the chiral phase transition
including the four-quark flavors. In this section, we eventually show the
extended Columbia plot for the four-quark flavor system through the analysis
of altering the current quark masses.

###  III.1  Chiral crossover with physical quark masses in three- and four-
quark flavor system

In this subsection, we discuss the chiral phase transition at physical quark
masses with three- and four-quark flavors. To exhibit the temperature
dependence on the light quark condensate (chiral condensate), we take the
following values of the physical quark masses in the case of the four-quark
flavors: $m_{l}=3.45\text{ MeV}$, $m_{s}=93.4\text{ MeV}$, and
$m_{c}=1.27\text{ GeV}$ [40]. On the other hand, to get from the soft-wall
model with $N_{f}=4$ to the three-quark flavor system, the value of the charm
quark mass is changed to heavy one which is taken as $m_{c}=3\text{ GeV}$ in
our study, while the light- and strange-quark masses are fixed as the physical
values.

In Fig. 3, we plot the light quark condensate $\sigma_{l}$ as a function of
temperature in the case of the three- and four-quark flavor system with the
physical quark masses. The panel (a) shows that the chiral condensate smoothly
decreases with the increase of the temperature, but does not take exact zero
even at high temperatures. This implies that the holographic QCD model in Eq.
(4) undergoes the chiral crossover and the chiral symmetry is approximately
restored at finite temperatures.

The chiral crossover is characterized by the pseudocritical temperature which
is evaluated by the reflection point of the quark condensate,
$d^{2}\sigma_{l}/dT^{2}\bigl{|}_{T=T_{\rm pc}}=0$. The panel (b) of Fig. 3
shows the variation of $\sigma_{l}$ with temperature, which has the peak
structure. The temperature at the peak point actually corresponds to the
pseudocritical temperature $T_{\rm pc}$. For the three-quark flavor system
with physical masses of light and strange quarks, the pseudocritical
temperature is evaluated as $T_{\rm pc}\bigl{|}_{\rm hQCD}=0.1729~{}{\rm
GeV}$. This value is smaller than the conventional effective model results
like the NJL evaluation: $T_{\rm pc}\bigl{|}_{\rm NJL}\simeq 0.189\,{\rm GeV}$
[41]. On the other hand, the estimate of the holographic QCD is larger than
the lattice observation of the three-flavor QCD with the physical quark
masses, $T_{\rm pc}\bigl{|}_{\rm lat.}\sim 0.1565\,{\rm GeV}$ [42]. In
comparison with the lattice QCD observation, there exits a $10\%$ deviation.
However, the holographic QCD model is defined at the large-$N_{c}$, so that
the deviation by about $30\%$ would be acceptable within the large-$N_{c}$
approximation. Thus, our result qualitatively aligns well with the lattice
observation.

We also evaluate the pseudocritical temperature for the four-quark flavor
system with the physical quark masses, which is $T_{\rm pc}\bigl{|}_{\rm
hQCD}=0.1703\,{\rm GeV}$. Although the physical charm quark mass reduces the
pseudocritical temperature, the decrement is relatively small.

(a)

(b)

Figure 3: Panel (a): The light quark condensate behavior for the four-quark
flavors (dashed blue line denoted with $N_{f}=2+1+1$) and the three-quark
flavors (dot-dashed red line with $N_{f}=2+1$) with the physical quark masses.
Panel (b): The variation of $\sigma_{l}$ with respect to temperature $T$ for
similar case as panel (a). The maximum points correspond to the pseudo-
critical temperatures ($T_{\rm pc}$). The pseudocritical temperatures are
$T_{c}=0.1703$ GeV and $0.1729$ GeV for the four- and three-quark flavors,
respectively.

###  III.2  Two-flavor chiral phase transition from the four-flavor framework

When the quark masses deviate from their physical values, the chiral crossover
would change to the first or second order phase transition, as was observed in
the previous holographic studies [28, 29, 30, 25, 26, 31, 32, 27, 33]. Using
the soft-wall model with $N_{f}=4$, we explore the influence of quark masses
on the chiral phase transition. To facilitate our discussion on the chiral
restoration, we focus on the two-quark flavor system in this subsection, which
is effectively realized by taking the heavy quark mass limit for the strange
and charm quarks.

In Fig. 4, we show the influence of the light quark mass on the chiral phase
transition for the two-quark flavor system, where the strange- and charm-quark
masses are taken as $m_{s}=1$ GeV and $m_{c}=3$ GeV to be effectively
integrated out from the four-flavor framework. Like in the case of the three-
and four-quark flavor system, we observe the chiral crossover with the
physical value of the light quark mass, as depicted in the panel (a) of Fig.
4. The pseudocritical temperature is determined to be $T_{\rm pc}=0.2124$ GeV
from the panel (b) of Fig. 4. When comparing the cases of the three- and four-
quark flavor system, the two-flavor pseudocritical temperature is higher. This
shows that the inclusion of the strange quark significantly reduces the
pseudocritical temperature in both the three- and four-quark flavor systems.

When taking the chiral limit ($m_{l}=0$), the chiral crossover is changed to
the chiral second-order phase transition. The critical temperature of the
second order phase transition reads $T_{\rm c}=0.2123$ GeV, determined from
the point of the divergence in the panel (b) of Fig. 4. After reaching the
critical temperature, the chiral condensate becomes zero, so that the chiral
symmetry is completely restored at high temperatures, $T>T_{\rm c}$. Moreover,
this second order is anticipated to belong to the $O(4)$ universality class
[18].

Indeed, the second order phase transition has also been observed in the
previous study for the massless two-quark flavor system based on the soft-wall
model with $N_{f}=2$ and $N_{f}=3$ (where strange quark is taken to be heavy)
[25, 26]. Here, our numerical analysis certainly shows that the soft-wall
model with $N_{f}=4$ can effectively replicate the two-flavor chiral phase
transition by taking the heavy mass limit of the strange and charm quarks.

(a)

(b)

Figure 4: Panel (a): Temperature dependence of the light quark condensate with
the large strange quark mass $m_{s}=1$ GeV and charm quark mass $m_{c}=3$ GeV.
This case corresponds to the two-quark flavor system for the massless (solid
blue line) and physical quark mass (dashed red line) of the light quark. Panel
(b): The variation of $\sigma_{l}$ with temperature $T$ with similar color
coding as panel (a).

###  III.3  Three-flavor first order phase transition and influence of charm
quark mass

The previous studies based on the soft-wall model with $N_{f}=3$ have also
provided the chiral first-order phase transition in the case of the massless
three-quark flavors and have shown that the first order phase transition is
driven by the quark-flavor symmetry breaking via the determinant term $V_{\rm
det}$ tagged with the parameter $v_{\rm det}$ [25, 26]. To address the first
order phase transition in the framework of the soft-wall model with $N_{f}=4$,
we move onto the three-quark flavor system in this subsection. In addition, we
also consider the influence of the finite charm quark mass on the three-flavor
first order phase transition.

The panel (a) of Fig. 5 shows the contribution of the parameter $v_{\rm det}$
on the chiral phase transition. Here, we set the massless three-quark flavor
system where the charm quark mass takes the heavy value ($m_{c}=3$ GeV) to be
integrated out. This figure clearly shows that the presence of the determinant
term surely induces the first order phase transition even when starting from
the framework of the soft-wall model with $N_{f}=4$.

Next, adjusting the charm quark mass to be small, we consider the influence of
the charm quark mass on the first order phase transition. The panel (b) shows
that the critical temperature is moved to the low temperature region with the
decrease of $m_{c}$, while the first order phase transition is intact for the
finite charm quark mass.

(a)

(b)

Figure 5: Panel (a): The light quark condensate as a function of temperature
without ($v_{det}=0$) and with ($v_{det}=-7.07$) the contribution of the
determinant term. Panel (b): Light quark condensate as a function of
temperature for a fixed value of $m_{l}=m_{s}=0$ and changing charm quark mass
as $m_{c}=0.50$ GeV (solid blue line), $m_{c}=1.20$ GeV (dashed red line),
$m_{c}=2$ GeV (dash-dotted green line), and $m_{c}=3$ GeV (dashed magenta
line).

Turning on the mass of the light quark, the chiral symmetry is explicitly
broken and is not completely restored even at high temperatures. Then, the
first order phase transition is contaminated by the current quark mass and
makes the phase transition crossover at the sufficient value of the quark
mass. As for the case of the $m_{c}=3$ GeV (see the panel (a) of Fig. 6), the
order of the phase transition is altered at $m_{l}=m_{s}\sim 10$ MeV. This
quark mass value represents the critical quark mass to separate the region
between the first-order and crossover domains. At this critical mass, the
phase transition is of the second order, which is characterized by the $Z(2)$
universality class [18]. As the charm quark mass decreases, the critical quark
mass is shifted to be small, as depicted in other panels of Fig. 6: the
critical masses are evaluated as $m_{l}=m_{s}\sim 10.5$ MeV, $m_{l}=m_{s}\sim
9$ MeV, $m_{l}=m_{s}\sim 7$ MeV and $m_{l}=m_{s}\sim 4$ MeV for $m_{c}=3$ GeV,
$m_{c}=2$ GeV, $m_{c}=1.2$ GeV and $m_{c}=0.5$ GeV, respectively.

(a)

(b)

(c)

(d)

Figure 6: Panel (a): The temperature dependence of the light quark condensate
for a fixed value of $m_{c}=3$ GeV and changing the degenerate light and
strange quark masses to see how the order of phase transition changes by
changing the mass, as $m_{l}=m_{s}=9.3$ MeV (dashed blue line),
$m_{l}=m_{s}=10.5$ MeV (solid red line), and $m_{l}=m_{s}=11.7$ MeV (dash-
dotted green line). The color online for panel (b), (c), and (d) is similar to
panel (a) with different quark masses.

To grasp the quark mass dependence on the chiral phase transition, we map the
order of the chiral phase transition on a phase diagram in the quark-mass
plane of $m_{l}$ and $m_{s}$, as shown in Fig. 7. This figure is a sort of the
Columbia plot, in which the charm quark mass contribution is included. Here,
we have considered the four different values of the charm quark mass. The four
solid lines corresponding to each charm quark mass represent the critical
quark masses of $m_{l}$ and $m_{s}$. The colored domain denotes the region
where the first order phase transition occurs, while the white regions signify
the occurrence of the chiral crossover.

In the three-quark flavor system, corresponding to the case of $m_{c}=3$ GeV,
the first order domain is distant from the physical point. Furthermore, when
compared to the lattice observation of the chiral crossover [5], our estimated
first-order domain does not touch the lattice data. Such a narrow first order
domain has been also reported in a mean-field level analysis of the NJL model
[20] and the FRG method with the constant anomaly coupling [13].

As the charm quark mass decreases, the charm quark contribution becomes
significant in the thermal system. The phase diagram in Fig. 7 shows that the
first order domain becomes shrink by adjusting $m_{c}\searrow 0$. The first
order domain is also contaminated by the charm quark contribution. This
contamination implies that the first order domain vanishes in the massless
limit of the charm quark. The fate of the first order domain in the four-
flavor system with the massless charm quark will be discussed in the
subsequent subsection.

Figure 7: The phase diagram for chiral phase transition in $m_{l}-m_{s}$ plane
with four different values of charm quark mass. The solid red line shows the
critical line (second order line) between the first order region (the bottom
left corner) and the crossover region(the upper right corner) at $m_{c}=0.50$
GeV. Similar critical line is shown for $m_{c}=1.20$ GeV (solid blue line),
$m_{c}=2.0$ GeV (solid green line), and $m_{c}=3.0$ GeV (solid magenta line).
Note that there exists the second order line in the $m_{s}$ axis above the
critical value of the strange quark mass corresponding to the tri-critical
point. The white region represents the crossover domain and the $m_{l}$ axis
above the critical value of the light quark mass also falls within the
crossover domain. The orange points are lattice QCD results taken from Ref.
[4, 5, 7] which are in the crossover region.

###  III.4  Emergence of second order phase transition and four-flavor phase
diagram

Moving onto the massless four-quark flavor system, we make a plot of the
chiral phase transition in a way similar to the case of the massless three-
flavor system in the panel (a) of Fig. 5. In this case, the second order phase
transition emerges, as shown in Fig. 8. The figure indicates that the presence
of the determinant term is irrelevant to the second order phase transition.
This is in contrast to the massless three-flavor system depicted in the panel
(a) of Fig. 5, and the critical temperature is not influenced by the
determinant term contribution, holding steady at $T_{c}=0.1515$ GeV.

As shown below, the mechanism of the second order phase transition with the
massless four-quark flavors is analogous to that with the massless two-quark
flavors. In the flavor-symmetric system for $N_{f}=4$, where
$\chi_{l}=\chi_{s}=\chi_{c}$, the potential $V(X)$ is expressed as

$\displaystyle V(\chi_{l})=a\chi_{l}^{2}+b\chi_{l}^{4},$ (17)

with $a=-6$ and $b=4v_{4}+3v_{\rm det}$. On the other hand, the potential for
$N_{f}=2$ goes like

$\displaystyle V(\chi_{l})=\bar{a}\chi_{l}^{2}+\bar{b}\chi_{l}^{4},$ (18)

where $\bar{a}$ and $\bar{b}$ are give by $\bar{a}=-6+\bar{v}_{\rm det}$ and
$\bar{b}=2v_{4}$ with $\bar{v}_{\rm det}=\gamma/2$ being the parameter of the
determinant term for $N_{f}=2$. The potentials in the soft-wall model exhibit
the same structure for both the two- and four-quark flavors: the potentials
are constructed by only the $\chi_{l}^{2}$ and $\chi_{l}^{4}$ terms. In these
cases, the contribution of the determinant term becomes invisible in the
potential structure. This is because the determinant term for $N_{f}=4$
($N_{f}=2$) is incorporated into the $\chi_{l}^{4}$ term (the $\chi_{l}^{2}$
term). Owing to the similarity to the potential form in the massless two-quark
flavor system, the second order phase transition also manifests in the
massless four-flavor system. Moreover, this similarity would indicate that the
universality class of the second order in the massless four-quark flavors may
align with that observed in the massless two-quark flavors.

Furthermore, the steady behavior of the critical temperature for $N_{f}=4$ can
also be understood as analogous to the observations in the two-flavor system.
The previous study has shown that the critical temperature with the massless
two-quark flavors remains unaffected by the choice of $v_{4}$ and is instead
determined by only the parameter set in the dilaton profile [26]. Given this
fact, the critical temperature with the massless four-quark flavors is not
shifted by the contribution of the determinant term because $v_{\rm det}$ is
absorbed into the parameter of the $\chi_{l}^{4}$ term.

Figure 8: The quark condensate as a function of temperature for the massless
quarks at $v_{det}=-7.07$ (solid blue line) and $v_{det}=0$ (dashed red line).

When the finite mass of the light quark is introduced in the four-quark flavor
system, the second order phase transition is contaminated, similar to the case
of the first order phase transition in the three-quark flavor system shown in
Fig. 6. Figure 9 clearly illustrates that the light quark mass makes the
second order phase transition crossover. In contrast to the three-flavor first
order phase transition, this system promptly undergoes the chiral crossover as
soon as the quark mass becomes massive, leading to the absence of a critical
mass of the light quark in the case of $m_{c}=0$.

Figure 9:  Light quark condensate as a function of temperature for massless
charm quark and different values of the light and strange quark masses as
$m_{l}=m_{s}=0$ (solid blue line), $m_{l}=m_{s}=0.5$ MeV (dashed red line),
and $m_{l}=m_{s}=1$ MeV (dash-dotted green line).

Next, we move onto the case of the massive charm quark from the massless four-
quark flavor system. Figure 10 illustrates that as the charm quark mass
increases, the second order phase transition shifts to the first order phase
transition around $m_{c}\sim 5$ MeV. This value represents the critical charm
quark mass that distinguishes between the crossover and the first-order phase
transition.

Figure 10: The light quark condensate as a function of temperature for a
massless light and strange quarks $m_{l}=m_{s}=0$ and changing charm quark
mass to see how the order of phase transition changes by changing the charm
quark mass as $m_{c}=2$ MeV (dash-dotted green line), $m_{c}=4$ MeV (dashed
red line), and $m_{c}=6$ MeV (Solid line). The critical charm quark mass that
changes the order of the chiral phase transition from second order to first
order is between $4-6$ MeV.

By combining the results of Figs. 8, 9 and 10, we depict the phase diagram at
the four-quark flavor system in Fig. 11 where $m_{l}$ and $m_{s}$ are
degenerate but differ from $m_{c}$. This phase diagram is on the
$m_{l=s}$-$m_{c}$ plane, which is mainly drawn by the first order domain and
the crossover domain. The coexistence of the two domains in the phase diagram
is formed by the competition between the contribution of the determinant term
associated with the flavor symmetry breaking and the current quark mass source
related to the explicit chiral symmetry breaking, similar to the phase diagram
on the $m_{l}$-$m_{s}$ plane shown in Fig. 7. Incidentally, to consistently
connect to the conventional Columbia plot in the three-quark flavors, the
second order boundary separating the first order and crossover domains would
be characterized by the $Z(2)$ universality class.

Furthermore, there exists the tri-critical point on the $m_{l=s}$-$m_{c}$
plane ($m_{l}$ degenerates with $m_{s}$), at which the three domains of
crossover, first- and second-order phase transitions merge. Of interest is
that the tri-critical point corresponds to the critical charm quark mass
estimated in Fig. 10 and stems from the massless four-quark flavor limit where
the second order phase transition observed in Fig. 8 emerges.

Figure 11: The phase diagram for chiral phase transition in $m_{l=s}$-$m_{c}$
plane. The solid red line shows the critical line (second order line) between
the first order region(light-green shaded area) and the crossover region(the
white region). The tri-critical point is located at $m_{c}\sim 5$ MeV for
which the order of chiral phase transition changes from second order to first
order phase transition. Below the tri-critical point, there is only second
order phase transition on the y-axis which is at massless $m_{l=s}=0$ and the
order changes to crossover for any deviation from the massless limit.

###  III.5 Critical temperatures with massless quarks

With the above results of the critical temperatures in the massless systems,
we finally make a comparison between the holographic estimates and other model
results based on the functional renormalization group (FRG) method in Fig. 12.
The FRG method [8, 9, 15] provides the second order phase transition for
$N_{f}=2,3,4$. However, our soft-wall model in the three-quark flavor system
provides the first order phase transition while the second order phase
transition arises in the two- and four-quark flavor system. Despite the
discrepancy in the phase transition order between the soft-wall model and
other model results, our estimated critical temperatures can be considered to
be comparable to those of other models when taking account of the
large-$N_{c}$ approximation.

Figure 12: The critical temperature as a function of number of flavors for
massless quarks. The blue diamond is the results from our calculations, and
the red open circles are the results of the FRG model [8, 9].

##  IV Summary and discussion

In this paper, we have explored the quark-flavor and -mass dependence on the
thermal chiral phase transition. In order to clarify the phase structure on
the quark-mass plane, we have employed the soft-wall holographic QCD model
with $N_{f}=4$, in which the quark-flavor mixing is incorporated through the
determinant term interaction. By using the holographic model framework, the
four-quark flavor system is described and is applicable to finite temperature
across the chiral phase transition. Furthermore, the holographic four-quark
flavor system can be reduced to the two- or three-quark flavor system by
taking the heavy mass limit of the charm and/or strange quark.

First, we have investigated the three-quark flavor system with the physical
quark masses. In this case, we have observed that the soft-wall model surely
undergoes the chiral crossover and its pseudocritical temperature aligns well
with the lattice QCD simulations within the large-$N_{c}$ approximation. In
addition, we have also confirmed that the four-flavor model framework is
capable of reproducing the observations of the phase transition order from
prior holographic studies with $N_{f}=2,3$: in the massless limit of the two-
quark (three-quark) flavor system, the holographic model with $N_{f}=4$ yields
the second (first) order phase transition, consistent with the prior studies
[25, 26]. This first order phase transition is driven by the contribution of
the determinant term, similar to the prior studies [25, 26].

Applying our framework to the three-quark flavor system for various values of
quark masses, we have illustrated the chiral phase diagram on the plane of the
light quark mass $m_{l}$ and the strange quark mass $m_{s}$, which is a part
of the Columbia plot. For the region where $m_{l}$ and $m_{s}$ are small
compared to their physical values, the phase diagram is dominated by the first
order transition owing to the presence of the determinant term. The first
order domain is also distant from the chiral crossover points measured in
lattice QCD simulations [4, 5, 7].

Moreover, we have also investigated the influence of the charm quark mass on
the chiral phase diagram on the $m_{l}$-$m_{s}$ plane, especially the first
order domain. It has been found that by adjusting the charm quark mass to be
small, the first order domain becomes shrink. It implies that the first order
domain disappears at the massless limit of the charm quark mass. To further
explore the disappearance of the first-order domain, we have investigated the
massless four-quark flavor system, resulting in the emergence of the second-
order phase transition. The second-order phase transition emerged with
massless four-quark flavors can be understood by the mechanism analogous to
that observed in the massless two-quark flavor system. Indeed, in the case of
the massless quark system for both $N_{f}=2$ and $N_{f}=4$, the determinant
term is absorbed into other interaction terms described by even functions of
the scalar field. Consequently, its contribution becomes irrelevant to
evaluating the order of the chiral phase transition.

Given the phase transition order observed in the massless four-quark flavor
system, we have proposed the new phase diagram with the four-quark flavors,
which is described on the $m_{l=s}$-$m_{c}$ plane. Even for the four-quark
flavor system, both the first-order domain and the crossover domain coexist in
the phase diagram. We have found that the tri-critical point exists in the
$m_{c}$ axis, specifically at $m_{c}\sim 5{\rm MeV}$ for $m_{l=s}=0$. This
tri-critical point stems from the massless four-quark flavor limit where the
second order phase transition occurs.

Incidentally, our estimated critical temperatures are comparable to these of
FRG method in the massless systems [8, 9, 15]. However, the phase transition
order of the holographic model, especially observed in the three-quark flavor
system, is in contrast to the other model observations.

Below, we provide some comments on our findings and their implications.

  * •

As the pioneer work for the phase diagram on the quark-mass plane, the linear
sigma model has predicted the first-order phase transition for the massless
four-quark flavor system [18]. In contrast, the soft-wall holographic QCD
model exhibits the second-order phase transition. This estimation is
consistent with the recent prospect of the lattice QCD simulations [21].

  * •

The conventional NJL model also provides the first order phase transition for
the massless three-quark flavors, which is triggered by the instanton induced
anomaly term. Actually, the instanton induced anomaly term is described by the
determinant term, which is called the KMT interaction [43, 44, 45, 46].
However, the determinant term in the soft-wall model would not be directly
linked with the KMT term and the instanton contribution is still unclear
within the holographic QCD framework at finite temperature. Hence, it is worth
to study the role of the determinant term and find out the holographic dual
associated with the instanton contribution that affect the order of the chiral
phase transition.

  * •

Figure 12 implies that by taking the extrapolation of the number of the quark
flavor, the critical temperature would be zero at the large number of the
quark flavor. Actually, other effective models have shown that the critical
temperature vanishes at the large number of the quark flavor and have pointed
out the appearance of the conformal window [47, 48, 49, 50, 51, 52].
Furthermore, the conformal window has been also studied in the lattice QCD
simulation at finite temperatures [21]. Therefore, it is interesting that the
soft-wall model approach is extended to the more multi-flavor system and is
addressed to the conformal window. At any rate, our new phase diagram with the
four-quark flavor in Fig. 11 is a milestone in investigating the chiral phase
structure in the multi-flavor system.

## Acknowledgments

We would like to thank Kazem Bitaghsir Fadafan, and Danning Li for the useful
discussions. This work is supported in part by the National Natural Science
Foundation of China (NSFC) Grant Nos: 12235016, 12221005, 12147150, 12305136
and the Strategic Priority Research Program of Chinese Academy of Sciences
under Grant No. XDB34030000, the start-up funding from University of Chinese
Academy of Sciences (UCAS), and the Fundamental Research Funds for the Central
Universities. H. A. A. acknowledges the ”Alliance of International Science
Organization (ANSO) Scholarship For Young Talents” for providing financial
support for the Ph.D. study.

## References

  * Brown  _et al._ [1990] F. R. Brown, F. P. Butler, H. Chen, N. H. Christ, Z.-h. Dong, W. Schaffer, L. I. Unger, and A. Vaccarino, On the existence of a phase transition for QCD with three light quarks, Phys. Rev. Lett. 65, 2491 (1990).
  * Aoki  _et al._ [2012] S. Aoki, H. Fukaya, and Y. Taniguchi, Chiral symmetry restoration, eigenvalue density of Dirac operator and axial U(1) anomaly at finite temperature, Phys. Rev. D 86, 114512 (2012), arXiv:1209.2061 [hep-lat] .
  * Bazavov  _et al._ [2017] A. Bazavov, H. T. Ding, P. Hegde, F. Karsch, E. Laermann, S. Mukherjee, P. Petreczky, and C. Schmidt, Chiral phase structure of three flavor QCD at vanishing baryon number density, Phys. Rev. D 95, 074505 (2017), arXiv:1701.03548 [hep-lat] .
  * Ding  _et al._ [2019a] H. T. Ding _et al._ (HotQCD), Chiral Phase Transition Temperature in ( 2+1 )-Flavor QCD, Phys. Rev. Lett. 123, 062002 (2019a), arXiv:1903.04801 [hep-lat] .
  * Ding  _et al._ [2019b] H.-T. Ding, P. Hegde, O. Kaczmarek, F. Karsch, A. Lahiri, S.-T. Li, S. Mukherjee, and P. Petreczky (HotQCD), Chiral phase transition in (2 + 1)-flavor QCD, PoS LATTICE2018, 171 (2019b), arXiv:1905.11610 [hep-lat] .
  * Kuramashi  _et al._ [2020] Y. Kuramashi, Y. Nakamura, H. Ohno, and S. Takeda, Nature of the phase transition for finite temperature $N_{\rm f}=3$ QCD with nonperturbatively O($a$) improved Wilson fermions at $N_{\rm t}=12$, Phys. Rev. D 101, 054509 (2020), arXiv:2001.04398 [hep-lat] .
  * Dini  _et al._ [2022] L. Dini, P. Hegde, F. Karsch, A. Lahiri, C. Schmidt, and S. Sharma, Chiral phase transition in three-flavor QCD from lattice QCD, Phys. Rev. D 105, 034510 (2022), arXiv:2111.12599 [hep-lat] .
  * Braun and Gies [2006] J. Braun and H. Gies, Chiral phase boundary of QCD at finite temperature, JHEP 06, 024, arXiv:hep-ph/0602226 .
  * Braun and Gies [2010] J. Braun and H. Gies, Scaling laws near the conformal window of many-flavor QCD, JHEP 05, 060, arXiv:0912.4168 [hep-ph] .
  * Fukushima  _et al._ [2011] K. Fukushima, K. Kamikado, and B. Klein, Second-order and Fluctuation-induced First-order Phase Transitions with Functional Renormalization Group Equations, Phys. Rev. D 83, 116005 (2011), arXiv:1010.6226 [hep-ph] .
  * Grahl and Rischke [2013] M. Grahl and D. H. Rischke, Functional renormalization group study of the two-flavor linear sigma model in the presence of the axial anomaly, Phys. Rev. D 88, 056014 (2013), arXiv:1307.2184 [hep-th] .
  * Fejős [2014] G. Fejős, Fluctuation induced first order phase transition in $U(n)\times U(n)$ models using chiral invariant expansion of functional renormalization group flows, Phys. Rev. D 90, 096011 (2014), arXiv:1409.3695 [hep-ph] .
  * Resch  _et al._ [2019] S. Resch, F. Rennecke, and B.-J. Schaefer, Mass sensitivity of the three-flavor chiral phase transition, Phys. Rev. D 99, 076005 (2019), arXiv:1712.07961 [hep-ph] .
  * Fejos [2022] G. Fejos, Second-order chiral phase transition in three-flavor quantum chromodynamics?, Phys. Rev. D 105, L071506 (2022), arXiv:2201.07909 [hep-ph] .
  * Braun  _et al._ [2023] J. Braun _et al._ , Soft modes in hot QCD matter, (2023), arXiv:2310.19853 [hep-ph] .
  * Gao and Pawlowski [2022] F. Gao and J. M. Pawlowski, Phase structure of (2+1)-flavor QCD and the magnetic equation of state, Phys. Rev. D 105, 094020 (2022), arXiv:2112.01395 [hep-ph] .
  * Bernhardt and Fischer [2023] J. Bernhardt and C. S. Fischer, QCD phase transitions in the light quark chiral limit, (2023), arXiv:2309.06737 [hep-ph] .
  * Pisarski and Wilczek [1984] R. D. Pisarski and F. Wilczek, Remarks on the Chiral Phase Transition in Chromodynamics, Phys. Rev. D 29, 338 (1984).
  * Lenaghan [2001] J. T. Lenaghan, Influence of the U(1)(A) anomaly on the QCD phase transition, Phys. Rev. D 63, 037901 (2001), arXiv:hep-ph/0005330 .
  * Fukushima [2008] K. Fukushima, Phase diagrams in the three-flavor Nambu-Jona-Lasinio model with the Polyakov loop, Phys. Rev. D 77, 114028 (2008), [Erratum: Phys.Rev.D 78, 039902 (2008)], arXiv:0803.3318 [hep-ph] .
  * Cuteri  _et al._ [2021] F. Cuteri, O. Philipsen, and A. Sciarra, On the order of the QCD chiral phase transition for different numbers of quark flavours, JHEP 11, 141, arXiv:2107.12739 [hep-lat] .
  * Maldacena [1998] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998), arXiv:hep-th/9711200 .
  * Witten [1998] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998), arXiv:hep-th/9802150 .
  * Karch  _et al._ [2006] A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, Linear confinement and AdS/QCD, Phys. Rev. D 74, 015005 (2006), arXiv:hep-ph/0602229 .
  * Chelabi  _et al._ [2016a] K. Chelabi, Z. Fang, M. Huang, D. Li, and Y.-L. Wu, Realization of chiral symmetry breaking and restoration in holographic QCD, Phys. Rev. D 93, 101901 (2016a), arXiv:1511.02721 [hep-ph] .
  * Chelabi  _et al._ [2016b] K. Chelabi, Z. Fang, M. Huang, D. Li, and Y.-L. Wu, Chiral Phase Transition in the Soft-Wall Model of AdS/QCD, JHEP 04, 036, arXiv:1512.06493 [hep-ph] .
  * Li and Huang [2017] D. Li and M. Huang, Chiral phase transition of QCD with $N_{f}=2+1$ flavors from holography, JHEP 02, 042, arXiv:1610.09814 [hep-ph] .
  * Colangelo  _et al._ [2012] P. Colangelo, F. Giannuzzi, S. Nicotri, and V. Tangorra, Temperature and quark density effects on the chiral condensate: An AdS/QCD study, Eur. Phys. J. C 72, 2096 (2012), arXiv:1112.4402 [hep-ph] .
  * Cui and Wu [2013] L.-X. Cui and Y.-L. Wu, Thermal Mass Spectra of Scalar and Pseudo-Scalar Mesons in IR-improved Soft-Wall AdS/QCD Model with Finite Chemical Potential, Mod. Phys. Lett. A 28, 1350132 (2013), arXiv:1302.4828 [hep-ph] .
  * Bartz and Jacobson [2016] S. P. Bartz and T. Jacobson, Chiral Phase Transition and Meson Melting from AdS/QCD, Phys. Rev. D 94, 075022 (2016), arXiv:1607.05751 [hep-ph] .
  * Fang  _et al._ [2016a] Z. Fang, S. He, and D. Li, Chiral and Deconfining Phase Transitions from Holographic QCD Study, Nucl. Phys. B 907, 187 (2016a), arXiv:1512.04062 [hep-ph] .
  * Fang  _et al._ [2016b] Z. Fang, Y.-L. Wu, and L. Zhang, Chiral phase transition and meson spectrum in improved soft-wall AdS/QCD, Phys. Lett. B 762, 86 (2016b), arXiv:1604.02571 [hep-ph] .
  * Bartz and Jacobson [2018] S. P. Bartz and T. Jacobson, Chiral phase transition at finite chemical potential in 2+1 -flavor soft-wall anti–de Sitter space QCD, Phys. Rev. C 97, 044908 (2018), arXiv:1801.00358 [hep-ph] .
  * Erlich  _et al._ [2005] J. Erlich, E. Katz, D. T. Son, and M. A. Stephanov, QCD and a holographic model of hadrons, Phys. Rev. Lett. 95, 261602 (2005), arXiv:hep-ph/0501128 .
  * Gherghetta  _et al._ [2009] T. Gherghetta, J. I. Kapusta, and T. M. Kelley, Chiral symmetry breaking in the soft-wall AdS/QCD model, Phys. Rev. D 79, 076003 (2009), arXiv:0902.1998 [hep-ph] .
  * Eshraim  _et al._ [2015] W. I. Eshraim, F. Giacosa, and D. H. Rischke, Phenomenology of charmed mesons in the extended Linear Sigma Model, Eur. Phys. J. A 51, 112 (2015), arXiv:1405.5861 [hep-ph] .
  * Li  _et al._ [2021] M.-W. Li, Y. Yang, and P.-H. Yuan, Analytic Study on Chiral Phase Transition in Holographic QCD, JHEP 02, 055, arXiv:2009.05694 [hep-th] .
  * Cherman  _et al._ [2009] A. Cherman, T. D. Cohen, and E. S. Werbos, The Chiral condensate in holographic models of QCD, Phys. Rev. C 79, 045203 (2009), arXiv:0804.1096 [hep-ph] .
  * Ahmed  _et al._ [2023] H. A. Ahmed, Y. Chen, and M. Huang, Electromagnetic form factors in the Nf=4 holographic QCD, Phys. Rev. D 108, 086034 (2023), arXiv:2308.14975 [hep-ph] .
  * Workman  _et al._ [2022] R. L. Workman _et al._ (Particle Data Group), Review of Particle Physics, PTEP 2022, 083C01 (2022).
  * Cui  _et al._ [2022] C.-X. Cui, J.-Y. Li, S. Matsuzaki, M. Kawaguchi, and A. Tomiya, New interpretation of the chiral phase transition: Violation of the trilemma in QCD, Phys. Rev. D 105, 114031 (2022), arXiv:2106.05674 [hep-ph] .
  * Bazavov  _et al._ [2019] A. Bazavov _et al._ (HotQCD), Chiral crossover in QCD at zero and non-zero chemical potentials, Phys. Lett. B 795, 15 (2019), arXiv:1812.08235 [hep-lat] .
  * Kobayashi and Maskawa [1970] M. Kobayashi and T. Maskawa, Chiral symmetry and eta-x mixing, Prog. Theor. Phys. 44, 1422 (1970).
  * Kobayashi  _et al._ [1971] M. Kobayashi, H. Kondo, and T. Maskawa, Symmetry breaking of the chiral u(3) x u(3) and the quark model, Prog. Theor. Phys. 45, 1955 (1971).
  * ’t Hooft [1976a] G. ’t Hooft, Symmetry Breaking Through Bell-Jackiw Anomalies, Phys. Rev. Lett. 37, 8 (1976a).
  * ’t Hooft [1976b] G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D 14, 3432 (1976b), [Erratum: Phys.Rev.D 18, 2199 (1978)].
  * Appelquist  _et al._ [1996] T. Appelquist, J. Terning, and L. C. R. Wijewardhana, The Zero temperature chiral phase transition in SU(N) gauge theories, Phys. Rev. Lett. 77, 1214 (1996), arXiv:hep-ph/9602385 .
  * Gies and Jaeckel [2006] H. Gies and J. Jaeckel, Chiral phase structure of QCD with many flavors, Eur. Phys. J. C 46, 433 (2006), arXiv:hep-ph/0507171 .
  * Ryttov and Shrock [2016] T. A. Ryttov and R. Shrock, Infrared Zero of $\beta$ and Value of $\gamma_{m}$ for an SU(3) Gauge Theory at the Five-Loop Level, Phys. Rev. D 94, 105015 (2016), arXiv:1607.06866 [hep-th] .
  * Antipin  _et al._ [2019] O. Antipin, A. Maiezza, and J. C. Vasquez, Resummation in QFT with Meijer G-functions, Nucl. Phys. B 941, 72 (2019), arXiv:1807.05060 [hep-th] .
  * Ryttov and Shrock [2018] T. A. Ryttov and R. Shrock, Physics of the non-Abelian Coulomb phase: Insights from Padé approximants, Phys. Rev. D 97, 025004 (2018), arXiv:1710.06944 [hep-th] .
  * Alvares  _et al._ [2012] R. Alvares, N. Evans, and K.-Y. Kim, Holography of the Conformal Window, Phys. Rev. D 86, 026008 (2012), arXiv:1204.2474 [hep-ph] .


* [22] D. R. M. Arvidsson-Shukur, J. Chevalier Drori, and N. Yunger Halpern.  Conditions tighter than noncommutation needed for nonclassicality.  Journal of Physics A: Mathematical and Theoretical, 54(28):284001, June 2021.
  * [23] S. De Bièvre.  Complete Incompatibility, Support Uncertainty, and Kirkwood-Dirac Nonclassicality.  Physical Review Letters, 127(19):190404, November 2021.
  * [24] S. De Bièvre.  Relating incompatibility, noncommutativity, uncertainty, and Kirkwood–Dirac nonclassicality.  Journal of Mathematical Physics, 64(2):022202, February 2023.
  * [25] J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber.  Direct measurement of the quantum wavefunction.  Nature, 474(7350):188, 2011.
  * [26] J. S. Lundeen and C. Bamber.  Procedure for Direct Measurement of General Quantum States Using Weak Measurement.  Physical Review Letters, 108(7):070402, February 2012.
  * [27] C. Bamber and J. S. Lundeen.  Observing Dirac’s Classical Phase Space Analog to the Quantum State.  Physical Review Letters, 112(7):070405, February 2014.
  * [28] G. S. Thekkadath, L. Giner, Y. Chalich, M. J. Horton, J. Banker, and J. S. Lundeen.  Direct Measurement of the Density Matrix of a Quantum System.  Physical Review Letters, 117(12):120401, September 2016.
  * [29] D. R. M. Arvidsson-Shukur, N. Yunger Halpern, H. V. Lepage, A. A. Lasek, C. H. W. Barnes, and S. Lloyd.  Quantum advantage in postselected metrology.  Nature Communications, 11(1):3775, December 2020.
  * [30] J. H. Jenne and D. R. M. Arvidsson-Shukur.  Unbounded and lossless compression of multiparameter quantum information.  Phys. Rev. A, 106:042404, October 2022.
  * [31] N. B. Lupu-Gladstein, B. Y. Yilmaz, D. R. M. Arvidsson-Shukur, A. Brodutch, A. O. T. Pang, A. M. Steinberg, and N. Halpern Yunger.  Negative quasiprobabilities enhance phase estimation in quantum-optics experiment.  arXiv:2111.01194, November 2021.
  * [32] N. Yunger Halpern.  Jarzynski-like equality for the out-of-time-ordered correlator.  Phys. Rev. A, 95:012120, January 2017.
  * [33] N. Yunger Halpern, A. Bartolotta, and J. Pollack.  Entropic uncertainty relations for quantum information scrambling.  Communications Physics, 2(1):1–12, 2019.
  * [34] J. R. González Alonso, N. Yunger Halpern, and J. Dressel.  Out-of-time-ordered-correlator quasiprobabilities robustly witness scrambling.  Physical Review Letters, 122(4):040404, February 2019.
  * [35] R. Mohseninia, J. R. González Alonso, and J. Dressel.  Optimizing measurement strengths for qubit quasiprobabilities behind out-of-time-ordered correlators.  Phys. Rev. A, 100:062336, December 2019.
  * [36] Y. Aharonov, D. Z. Albert, and L. Vaidman.  How the result of a measurement of a component of the spin of a spin-½ particle can turn out to be 100.  Phys. Rev. Lett., 60:1351–1354, April 1988.
  * [37] I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan.  The sense in which a “weak measurement” of a spin-½ particle’s spin component yields a value 100.  Phys. Rev. D, 40:2112–2117, September 1989.
  * [38] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd.  Colloquium: Understanding quantum weak values: Basics and applications.  Rev. Mod. Phys., 86:307–316, March 2014.
  * [39] M. F. Pusey.  Anomalous Weak Values Are Proofs of Contextuality.  Physical Review Letters, 113(20):200401, November 2014.
  * [40] J. Dressel and A. N. Jordan.  Significance of the imaginary part of the weak value.  Phys. Rev. A, 85:012107, January 2012.
  * [41] R. Kunjwal, M. Lostaglio, and M. F. Pusey.  Anomalous weak values and contextuality: Robustness, tightness, and imaginary parts.  Phys. Rev. A, 100:042116, October 2019.
  * [42] J. T. Monroe, N. Yunger Halpern, T. Lee, and K. W. Murch.  Weak measurement of a superconducting qubit reconciles incompatible operators.  Phys. Rev. Lett., 126:100403, March 2021.
  * [43] R. Wagner, Z. Schwartzman-Nowik, I. L. Paiva, A. Te’eni, A. Ruiz-Molero, R. Soares Barbosa, E. Cohen, and E. F. Galvão.  Quantum circuits measuring weak values and Kirkwood-Dirac quasiprobability distributions, with applications, 2023.
  * [44] M. Lostaglio, A. Belenchia, A. Levy, S. Hernández-Gómez, N. Fabbri, and S. Gherardini.  Kirkwood-Dirac quasiprobability approach to quantum fluctuations: Theoretical and experimental perspectives, June 2022.
  * [45] A. Levy and M. Lostaglio.  Quasiprobability distribution for heat fluctuations in the quantum regime.  PRX Quantum, 1:010309, September 2020.
  * [46] M. Lostaglio.  Certifying quantum signatures in thermodynamics and metrology via contextuality of quantum linear response.  Phys. Rev. Lett., 125:230603, December 2020.
  * [47] S. Hernández-Gómez, S. Gherardini, A. Belenchia, M. Lostaglio, A. Levy, and N. Fabbri.  Projective measurements can probe non-classical work extraction and time-correlations, 2023.
  * [48] T. Upadhyaya, Jr. W. F. Braasch, G. T. Landi, and N. Yunger Halpern.  What happens to entropy production when conserved quantities fail to commute with each other.  2023\.
  * [49] Y. Suzuki, M. Iinuma, and H. F Hofmann.  Violation of Leggett-Garg inequalities in quantum measurements with variable resolution and back-action.  New Journal of Physics, 14(10):103022, 2012.
  * [50] Y. Suzuki, M. Iinuma, and H. F Hofmann.  Observation of non-classical correlations in sequential measurements of photon polarization.  New Journal of Physics, 18(10):103045, 2016.
  * [51] J. J. Halliwell.  Leggett-6Garg inequalities and no-signaling in time: A quasiprobability approach.  Phys. Rev. A, 93:022123, February 2016.
  * [52] R. B Griffiths.  Consistent histories and the interpretation of quantum mechanics.  Journal of Statistical Physics, 36(1-2):219–272, 1984.
  * [53] H. F. Hofmann.  Uncertainty limits for quantum metrology obtained from the statistics of weak measurements.  Physical Review A, 83(2):022106, February 2011.
  * [54] J. Dressel.  Weak values as interference phenomena.  Physical Review A, 91(3):032116, March 2015.
  * [55] V. Fiorentino and S. Weigert.  Uncertainty relations for the support of quantum states.  Journal of Physics A: Mathematical and Theoretical, 55(49):495305, December 2022.
  * [56] N. Gao, D. Li, A. Mishra, J. Yan, K. Simonov, and G. Chiribella.  Measuring incompatibility and clustering quantum observables with a quantum switch.  Phys. Rev. Lett., 130:170201, April 2023.
  * [57] A. Budiyono and H. K. Dipojono.  Quantifying quantum coherence via kirkwood-dirac quasiprobability.  Phys. Rev. A, 107:022408, February 2023.
  * [58] J.-B. Hiriart-Urruty and C. Lemaréchal.  Fundamentals of Convex Analysis.  Springer, Berlin, Heidelberg, 2001.
  * [59] J. Xu.  Classification of incompatibility for two orthonormal bases.  Physical Review A: Atomic, Molecular, and Optical Physics, 106(2):022217, August 2022.
  * [60] M. G. Genoni, M. L. Palma, T. Tufarelli, S. Olivares, M. S. Kim, and M. G. A. Paris.  Detecting quantum non-Gaussianity via the Wigner function.  Physical Review A, 87(6):062104, June 2013.
  * [61] Z. Van Herstraeten and N. J. Cerf.  Quantum Wigner entropy.  Physical Review A, 104(4):042211, October 2021.
  * [62] A. Hertz and S. De Bièvre.  Decoherence and nonclassicality of photon-added/subtracted multi-mode Gaussian states.  arXiv:2204.06358, (arXiv:2204.06358), April 2023.
  * [63] D. Gross.  Hudson’s theorem for finite-dimensional quantum systems.  Journal of Mathematical Physics, 47(12):122107, December 2006.
  * [64] L. M. Johansen.  Quantum theory of successive projective measurements.  Physical Review A, 76(1):012119, July 2007.
  * [65] G. B. Folland.  A Course in Abstract Harmonic Analysis.  Number 29 in Textbooks in Mathematics Series. CRC Press/Taylor & Francis, Boca Raton, second edition edition, 2016.
  * [66] R. Takagi and Q. Zhuang.  Convex resource theory of non-Gaussianity.  Physical Review A, 97(6):062337, June 2018.
  * [67] J.J. Sakurai.  Modern Quantum Mechanics.  Addision-Wesley, 1994.
  * [68] P. O. Boykin, M. Sitharam, M. Tarifi, and P. Wocjan.  Real Mutually Unbiased Bases, September 2005.
  * [69] T. Banica.  Complex Hadamard matrices and applications.  arXiv:1910.06911, 2022.