# Chiral phase transition in soft-wall AdS/QCD with scalar-dilaton coupling

Sean P. Bartz Email<EMAIL_ADDRESS>Dept. of Chemistry and Physics,
Indiana State University, Terre Haute, IN 47809  Robert C. Meadows  Dept. of
Chemistry and Physics, Indiana State University, Terre Haute, IN 47809  Glenn
Brock  Dept. of Chemistry and Physics, Indiana State University, Terre Haute,
IN 47809

###### Abstract

The chiral phase boundary of nuclear matter is expected to have a critical
point where the rapid crossover of lattice methods at zero chemical potential
becomes a first-order phase transition. Phenomenological models based on the
AdS/CFT correspondence, known as AdS/ QCD, have succeeded in capturing many
features of nuclear matter, with recent progress in producing the expected
critical point. We study a model that produces a critical point in the chiral
phase diagram by introducing a coupling between the scalar chiral field and
the dilaton. We examine the effect of the scalar-dilaton coupling on the
critical point. We also study the zero-temperature chiral dynamics, which must
allow for spontaneous chiral symmetry breaking in the limit of zero quark
mass. We find that when the scalar-dilaton coupling is large enough to ensure
correct zero-temperature chiral dynamics, a critical point is present only if
the quark mass is greater than 12.8 MeV.

##  1 Introduction

The exploration of the phase structure of quantum chromodynamics (QCD) at
extreme temperature and density is an important project for both theory and
experiment [1, 2, 3]. Lattice QCD finds a crossover phase transition at zero
quark chemical potential [4, 5], while other models find a first-order phase
transition at high chemical potential [6]. In combination, these models
suggest the existence of a critical point, but its exact location in the phase
diagram remains an open question [7, 8].

Experimentally, the search for the critical point requires a reduction in
center of mass energy [9], motivating the recently-completed Beam Energy Scan
at the Relativistic Heavy Ion Collider [10], as well as current and future
fixed-target experiments [11, 12, 13]. From the theoretical perspective,
lattice methods suffer from a well-known sign problem at finite chemical
potential [14]. Extrapolation techniques allow lattice analysis up to a baryon
chemical potential $\approx 300$ MeV, with no evidence of a critical point
[15].

The AdS/CFT correspondence [16, 17, 18] has emerged as a powerful tool to
study various aspects of QCD, including the phase diagram [19, 20]. The soft-
wall AdS/QCD model, which uses a background dilaton field to encode
confinement, has been extensively used to analyze hadron spectra [21, 22, 23]
and the QCD phase diagram [24]. While there has been success in finding a
critical point in the deconfinement phase transition using holographic
techniques [25, 26, 27, 28, 29, 30, 31, 32], producing a critical point in the
chiral phase transition has been more elusive.

In this work, we consider a modified soft-wall AdS/QCD model with a coupling
between the scalar and dilaton fields. Prior work has shown that the
introduction of such a coupling can improve the resulting meson spectra and
introduce a critical point in the chiral phase diagram [33]. We focus on the
effect of scalar-dilaton coupling on the critical point in the QCD phase
diagram and the zero-temperature chiral dynamics.

##  2 Soft-wall model with scalar-dilaton coupling

We use an anti-de Sitter black hole metric

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

with the AdS curvature $L=1$ throughout the rest of this work. Following
established procedure [34, 35, 36], we model finite temperature and chemical
potential with a charged black hole described by the 5D AdS–Reissner-Nordström
blackness function

$f(z)=1-(1+Q^{2})\left(\frac{z}{z_{h}}\right)^{4}+Q^{2}\left(\frac{z}{z_{h}}\right)^{6},$
(2)

where $z_{h}$ is the black hole horizon, and $Q$ is related to the black hole
charge $q$ by $Q=qz_{h}^{3}$. The quark chemical potential and temperature are
determined by the charge and horizon position

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle\kappa\frac{Q}{z_{h}},$ (3)
$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{1}{\pi
z_{h}}\left(1-\frac{Q^{2}}{2}\right),$ (4)

where $0<Q^{2}<2$ and $\kappa=1$ [37]. Note that $\mu$ is the quark chemical
potential, with a value one third of the baryon chemical potential. These
relationships are invertible for $z_{h}$ and $q$.

In the soft-wall model of AdS/QCD, confinement is introduced via a dilaton
field, which is quadratic in the IR limit. In this work, we use a quadratic
dilaton

$\Phi(z)=\mu_{g}^{2}z^{2}$ (5)

where $\mu_{g}=440$ MeV sets the confinement scale. The relevant matter fields
are described by the action

$\mathcal{S}=\frac{1}{2k}\int
d^{5}x\sqrt{-g}e^{-\Phi(z)}\left\\{\mathrm{Tr}\left[|DX|^{2}+V_{m}(X,\Phi)\right]+\gamma\textrm{Re}\left[\textrm{det}(X)\right]\right\\},$
(6)

where $X$ contains the scalar and pseudoscalar meson fields. The t’Hooft
determinant term provides mixing between light and heavy flavors, and with
$\gamma<0$ it produces a first-order phase transition in the chiral limit
[38]. We omit the vector and axial-vector meson fields to focus on the chiral
dynamics.

A quartic term in the scalar potential is required for spontaneous chiral
symmetry breaking [39]. Including the coupling between the dilaton and scalar
field, the potential is

$V_{m}(X,\Phi)=m_{5}^{2}|X|^{2}+\lambda_{1}\Phi|X|^{2}+4\lambda_{4}|X|^{4},$
(7)

The AdS/CFT dictionary sets the mass of the chiral field $m_{5}^{2}=-3$ [18].
Following [40], we take $\lambda_{4}=4.2$ and $\gamma=-22.6$.

Figure 1: A sketch of the Columbia plot, which shows the expected order of the
chiral phase transition as a function of light and strange quark masses [41,
42].

The scalar-dilaton coupling term gives the chiral field an effective mass that
runs with energy scale

$m_{5}^{2}\rightarrow-3+\lambda_{1}\mu_{g}^{2}z^{2}.$ (8)

This running mass has been used to obtain the correct mass splitting between
excited states of meson chiral partners [43], to reproduce the Columbia plot
(Figure 1) at zero chemical potential [40], to obtain the correct chiral
transition behavior [44], and to produce a critical point in the chiral phase
diagram [33]. It is worth noting that models with a modified dilaton profile
but without the running mass (8) achieve the first three of these goals [38,
45] but do not produce the critical point [46].

The scalar field has a $z$-dependent vacuum expectation value (VEV) that
describes the chiral symmetry breaking of the model. In a three flavor model,
the VEV becomes

$\langle X\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}{\chi_{u}(z)}&0&0\\\
0&{\chi_{d}(z)}&0\\\ 0&0&{\chi_{s}(z)}\end{pmatrix}.$ (9)

In this work, we will focus on the flavor-symmetric case
$\chi_{u}=\chi_{d}=\chi_{s}$. Varying 6 with respect to $\chi$ yields the
following equation of motion,

$\chi^{\prime\prime}-\left(\frac{3}{z}+\Phi^{\prime}-\frac{f^{\prime}}{f}\right)\chi^{\prime}-\frac{1}{z^{2}f}\left[(-3-\lambda_{1}\Phi)\chi+4\lambda_{4}\chi^{3}+3\lambda_{3}\chi^{2}\right]=0,$
(10)

where $\gamma\rightarrow 6\sqrt{2}\lambda_{3}$ is defined for convenient
notation. As the chiral field is the source of the $\bar{q}q$ operator, the
AdS/CFT dictionary identifies its coefficients at the UV boundary with the
sources of chiral symmetry breaking,

$\chi(z\rightarrow 0)\sim m_{q}\zeta z+\frac{\sigma}{\zeta}z^{3},$ (11)

where $\zeta=\sqrt{N_{c}}/(2\pi)$ [47], the quark mass $m_{q}$ is the source
of explicit chiral symmetry breaking, and the chiral condensate $\sigma$ is
the source of spontaneous chiral symmetry breaking.

##  3 Numerical procedure

Finding the chiral condensate requires solving (10) numerically and using the
AdS/CFT dictionary to relate the solution for $\chi(z)$ to the parameters
$m_{q},\,\sigma$. The presence of a singular point at $z=z_{h}$ presents a
challenge to this procedure. A commonly-used numerical method begins with a UV
approximation for the chiral field and integrates toward the horizon [48, 45,
43]. While this method works well near the chiral transition temperature, it
is less reliable at low temperatures. Instead, we use a method that starts
with the asymptotic solution at the black hole horizon $z_{h}$ and integrates
toward the UV boundary [49]. A comparison between these two methods is
discussed in Appendix 3.

The near-horizon solution is approximated by the Taylor series

$\chi(u\rightarrow 1)=d_{0}+d_{1}(1-u)+d_{2}(1-u)^{2}+\ldots$ (12)

where $u=z/z_{h}$ and the higher-order coefficients are solved by substitution
into (10). The result is

$\displaystyle d_{1}$ $\displaystyle=$
$\displaystyle\frac{d_{0}}{2\left(Q^{2}-2\right)}\left(3+\lambda_{1}z_{h}^{2}\mu_{g}^{2}-3d_{0}\lambda_{3}-4d_{0}^{2}\lambda_{4}\right)$
(13) $\displaystyle d_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{{16\left(Q^{2}-2\right)^{2}}}\Big{\\{}6d_{1}(-6+Q^{2}+Q^{4})+4d_{0}^{3}(14-13Q^{2})\lambda_{4}$
(14) $\displaystyle+$ $\displaystyle
d_{0}^{2}\left[(42-39Q^{2})\lambda_{3}-24d_{1}\left(Q^{2}-2\right)\lambda_{4}\right]-2d_{1}\left(Q^{2}-2\right)(4Q^{2}-8-\lambda_{1})z_{h}^{2}\mu_{g}^{2}$
$\displaystyle+$ $\displaystyle
3d_{0}\left[-14+13Q^{2}+8d_{1}\lambda_{3}-4d_{1}Q^{2}\lambda_{3}+\lambda_{1}(3Q^{2}-2)z_{h}^{2}\mu_{g}^{2}\right]\Big{\\}}$

For each value of $T,\mu$, we vary $d_{0}$ and compare the numerical solution
to the UV expansion of the chiral field

$\displaystyle\chi(u\rightarrow 0)$ $\displaystyle\approx$ $\displaystyle
m_{q}\zeta
z_{h}u-3m_{q}^{2}\zeta^{2}\lambda_{3}z_{h}^{2}u^{2}+\frac{\sigma}{\zeta}z_{h}^{3}u^{3}$
(15) $\displaystyle+$
$\displaystyle\frac{1}{4}\left[m_{q}^{3}\zeta^{3}(\lambda_{4}-36\lambda_{3}^{2})+2m_{q}\zeta\mu_{g}^{2}(\lambda_{1}-2)\right]z_{h}^{3}u^{3}\log\left(z_{h}u\right)+\dots$

The terms of order $u$ and $u^{3}$ have their coefficients defined by the
AdS/CFT dictionary and the other coefficients are found by solving (10) order-
by-order.

We evaluate the numerical solution and its derivative at a small value
$u_{i}\approx 10^{-3}$ and calculate the coefficients by comparing to the UV
expansion (15). Keeping terms up to order $u^{3}$, the relationships are
analytically solvable,

$\displaystyle m_{q}$ $\displaystyle=$
$\displaystyle\frac{1-\sqrt{1-3\lambda_{3}(3\chi_{UV}-u_{i}\chi^{\prime}_{UV})}}{3\lambda_{3}\zeta
u_{i}z_{h}},$ (16) $\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\zeta\left(\frac{1-3\lambda_{3}(2\chi_{UV}+u_{i}\chi^{\prime}_{UV})-\sqrt{1-3\lambda_{3}(3\chi_{UV}-u_{i}\chi^{\prime}_{UV})}}{3u_{i}^{3}z_{h}^{3}\lambda_{3}}\right),$
(17)

where $\chi_{UV}=\chi(u_{i})$ and $\chi^{\prime}_{UV}=\chi^{\prime}(u_{i})$.
The quadratic relationship has another set of solutions, which produce
unphysical values of $m_{q},\sigma<0$. In the two-flavor case, $\lambda_{3}=0$
and the above relationships cannot be used. Instead we find

$\displaystyle m_{q}$ $\displaystyle=$
$\displaystyle\frac{3\chi_{UV}-u_{i}\chi^{\prime}_{UV}}{2\zeta z_{h}u_{i}},$
(18) $\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\zeta\frac{u_{i}\chi^{\prime}_{UV}-\chi_{UV}}{2z_{h}^{3}u_{i}^{3}}.$
(19)

##  4 Results

Figure 2: The relationship between $\sigma$ and $m_{q}$ for a variety of
values of the scalar-dilaton coupling $\lambda_{1}$ with two flavors at zero
temperature and chemical potential. For $\lambda_{1}\geq 6.0$, the chiral
condensate is present even in the chiral limit $m_{q}=0$.

In this section, we find the dependence of the chiral condensate $\sigma$ on
quark mass $m_{q}$ for various values of the scalar-dilaton coupling
$\lambda_{1}$. We show that there is a minimum value of $\lambda_{1}$ that
allows for spontaneous chiral symmetry breaking in the chiral limit
$m_{q}\rightarrow 0$. We show how the (pseudo-) critical temperature is found
and how crossover and first-order transitions are distinguished. Finally, we
show how the location of the critical point is affected by the value of
$\lambda_{1}$.

###  4.1 Spontaneous chiral symmetry breaking

Separate sources of spontaneous and explicit chiral symmetry breaking are
required in the theory. The original soft-wall model did not achieve this,
finding $\sigma\sim m_{q}$ instead [21]. Including a quartic term in the
scalar potential allows these quantities to be independent, crucially
maintaining spontaneous chiral symmetry breaking in the chiral limit $m_{q}=0$
[39].

The relationship between $m_{q}$ and $\sigma$ in this model depends on the
strength of the scalar-dilaton coupling $\lambda_{1}$. We check this
relationship in both the 2-flavor ($\lambda_{3}=0$) and 3-flavor
($\lambda_{3}\neq 0$) cases. In the two-flavor case, the relationship between
$\sigma$ and $m_{q}$ is one-to-one for quark mass $m_{q}\geq 0$. For small
values of the scalar-dilaton coupling, the spontaneous chiral symmetry
breaking vanishes as $m_{q}\rightarrow 0$, but at higher values of
$\lambda_{1}\geq 6.0$, $\sigma$ is non-zero in the chiral limit, as seen in
Figure 2.

Figure 3: The relationship between $\sigma$ and $m_{q}$ for a variety of
values of the scalar-dilaton coupling $\lambda_{1}$ with three flavors. When
$\sigma$ is multi-valued, the lowest value is thermodynamically favored. For
$\lambda_{1}\geq 6.0$, the chiral condensate is present even in the chiral
limit $m_{q}=0$.

In the three-flavor case, we find the same requirement $\lambda_{1}\geq 6.0$
for $\sigma$ to remain finite as the quark mass goes to zero. In Figure 3, it
is evident that $\sigma$ becomes multi-valued for intermediate values of
$\lambda_{1}$. In these cases, the smaller value of $\sigma$ is
thermodynamically favored. This means that in the case of e.g.
$\lambda_{1}=5.7$ it appears that there are finite solutions of $\sigma$ at
zero quark mass, but these solutions are unphysical, and the lower branch of
the graph shows $\sigma\sim m_{q}$ at small quark mass for these values of
$\lambda_{1}$.

###  4.2 Chiral phase transition

Figure 4: These plots illustrate the difference between a crossover phase
transition and a first-order phase transition. At zero chemical potential
(left), the transition is smooth, but at higher $\mu$ (right), the value of
$\sigma$ becomes multi-valued, indicating a first-order phase transition. Both
plots use $m_{q}=15$ MeV and $\lambda_{1}=6$.

For a given value of the chemical potential $\mu$ and quark mass $m_{q}$, we
find the values of $\sigma$ for a range of temperatures. The order of the
chiral phase transition is determined by the way in which $\sigma$ transitions
to a smaller value. Smooth transitions are considered crossover, while a
first-order phase transition is characterized by the chiral condensate
becoming multi-valued, as illustrated in Figure 4.

Figure 5: The two-flavor results in the chiral limit for chiral condensate
$\sigma$ as a function of temperature at zero chemical potential. For scalar-
dilaton coupling $\lambda_{1}\leq 6$, $\sigma$ vanishes at low temperatures.

In crossover transitions, the pseudo-critical temperature is the temperature
where the chiral susceptibility $|d\sigma/dT|$ is maximized. For first-order
transitions, the critical temperature occurs at the lowest temperature where
$\sigma$ is multi-valued.

The values of $\lambda_{1}$ that produce unphysical chiral dynamics at zero
temperature also show unphysical behavior in the chiral phase transition. The
chiral condensate is plotted as a function of temperature at $\mu=0$ in the
chiral limit for 2 flavors (Figure 5) and 3 flavors (Figure 6). Note that the
two-flavor case allows negative values of $\sigma\leftrightarrow-\sigma$ as
solutions. This symmetry is broken in the three-flavor case, and also when
$m_{q}>0$, although negative solutions are still present [38]. These non-
physical solutions are ignored in the rest of the analysis.

The chiral phase transition has the expected low $T$ behavior in the chiral
limit. As this parameter is decreased, the “bump” below the critical
temperature becomes more pronounced. When $\lambda_{1}$ is below a certain
value, the chiral condensate disappears at low temperature. This unphysical
result is further evidence for a minimum value for the scalar-dilaton coupling
in this model.

Figure 6: The three-flavor results in the chiral limit for chiral condensate
$\sigma$ as a function of temperature at zero chemical potential. For scalar-
dilaton coupling $\lambda_{1}\leq 5.5$, $\sigma$ vanishes at low temperatures.

###  4.3 Phase diagram and critical point

The chiral phase diagram is produced by plotting the (pseudo-) critical
temperature as a function of the chemical potential. A critical end point is
found for combinations of $m_{q}$ and $\lambda_{1}$ that produce a crossover
phase transition at zero chemical potential. At sufficiently large $\mu$, the
phase transition becomes first order.

Figure 7: The phase diagram is shown for $m_{q}=15$ MeV for various values of
the scalar-dilaton coupling parameter $\lambda_{1}$. Dashed lines indicate a
crossover and solid lines indicate a first-order phase transition. The
critical points are indicated by a large dot. Increasing $\lambda_{1}$ moves
the critical point to smaller chemical potential values while also increasing
the crossover temperature at $\mu=0$.

To examine the effect of scalar-dilaton coupling on the location of the
critical point, we show in Figure 7 phase boundaries for a sample quark mass
$m_{q}=15$ for varying $\lambda_{1}$ with three symmetric quark flavors. The
critical point occurs at smaller chemical potential as $\lambda_{1}$ is
increased. At the same time, the (pseudo-) critical temperature is increased
at all values of $\mu.$

When the scalar-dilaton coupling is sufficiently large, the phase transition
is first order at $\mu=0$, and the critical point vanishes. This is seen in
Figure 8, where the location of the critical point is plotted for several
values of the quark mass. When the quark mass is large, a critical point can
still be found when $\lambda_{1}\geq 6.0$, as required for the proper chiral
dynamics detailed in Section 4.1. At low values of the quark mass, obtaining a
critical point requires $\lambda_{1}<6.0$. We find that the minimum quark mass
with a critical point in the phase diagram when $\lambda_{1}=6.0$ is
$m_{q}=12.8$ MeV. In the chiral limit, we find that the chiral phase
transition is always first-order, regardless of the value of $\lambda_{1}$,
and no critical point is present.

Figure 8: The locations of the critical point for different values of the
quark mass and scalar-dilaton coupling $\lambda_{1}$ are shown. The curves are
labeled by the value of the quark mass.

##  5 Discussion

In this work, we used a soft-wall holographic QCD model with a scalar-dilaton
coupling to study the chiral phase transition at finite temperature and
density. Our analysis shows that a coupling $\lambda_{1}\geq 6.0$ is necessary
for achieving the correct chiral dynamics with either two or three symmetric
quark flavors, in line with the requirement that holographic models mirror the
chiral symmetry breaking mechanism of QCD [22, 23]. Furthermore, the presence
of a critical point in this model’s chiral phase diagram contributes to the
ongoing effort to comprehend the phase structure of QCD, a topic of
considerable theoretical and experimental interest [50, 51].

Previous soft-wall AdS/QCD models that achieved correct chiral dynamics at
zero chemical potential by using a UV-modified dilaton [38, 46] could be
extended to include a scalar-dilaton coupling term. It may be interesting to
examine whether this will circumvent the problems shown in the current work at
small values of the scalar-dilaton coupling.

Looking ahead, we will allow the strange quark mass to differ from the light
quark masses and explore the 2+1 flavor results at a range of scalar-dilaton
coupling. These results will be compared to the Columbia plot shown in Figure
1. Another goal is to combine the analysis of the chiral transition with the
deconfinement phase transition. Previous work combining a scalar chiral field
with the dynamical Einstein-Maxwell-Dilaton model have shown some promise in
the case of two quark flavors [52]. Including the scalar-dilaton mixing term
produces the crossover chiral transition that is expected for two quark
flavors with $m_{q}>0$ [53]. Considering these extensions will allow a more
thorough exploration of all aspects of the holographic QCD phase diagram.

## Acknowledgments

Work on this project was partially funded by a grant from the University
Research Committee at Indiana State University. Further support came from
Indiana State’s Summer Undergraduate Research Experience. SB thanks Alfonso
Ballon-Bayona and Diego Rodrigues for useful discussion about numerical
methods.

## References

  * [1] H. Elfner and B. Müller, “The exploration of hot and dense nuclear matter: introduction to relativistic heavy-ion physics,” Journal of Physics G: Nuclear and Particle Physics 50 no. 10, (Sep, 2023) 103001. https://dx.doi.org/10.1088/1361-6471/ace824.
  * [2] X. An, M. Bluhm, L. Du, G. V. Dunne, H. Elfner, C. Gale, J. Grefa, U. Heinz, A. Huang, J. M. Karthein, D. E. Kharzeev, V. Koch, J. Liao, S. Li, M. Martinez, M. McNelis, D. Mroczek, S. Mukherjee, M. Nahrgang, A. R. Nava Acuna, J. Noronha-Hostler, D. Oliinychenko, P. Parotto, I. Portillo, M. S. Pradeep, S. Pratt, K. Rajagopal, C. Ratti, G. Ridgway, T. Schäfer, B. Schenke, C. Shen, S. Shi, M. Singh, V. Skokov, D. T. Son, A. Sorensen, M. Stephanov, R. Venugopalan, V. Vovchenko, R. Weller, H.-U. Yee, and Y. Yin, “The BEST framework for the search for the QCD critical point and the chiral magnetic effect,” Nuclear Physics A 1017 (2022) 122343. https://www.sciencedirect.com/science/article/pii/S0375947421002086.
  * [3] A. Pandav, D. Mallick, and B. Mohanty, “Search for the QCD critical point in high energy nuclear collisions,” Progress in Particle and Nuclear Physics 125 (July, 2022) 103960. http://dx.doi.org/10.1016/j.ppnp.2022.103960.
  * [4] Y. Aoki, G. Endrődi, Z. Fodor, S. D. Katz, and K. K. Szabó, “The order of the quantum chromodynamics transition predicted by the standard model of particle physics,” Nature 443 no. 7112, (Oct., 2006) 675–678. http://dx.doi.org/10.1038/nature05120.
  * [5] G. Aarts, J. Aichelin, C. Allton, A. Athenodorou, D. Bachtis, C. Bonanno, N. Brambilla, E. Bratkovskaya, M. Bruno, M. Caselle, C. Conti, R. Contino, L. Cosmai, F. Cuteri, L. Del Debbio, M. D’Elia, P. Dimopoulos, F. Di Renzo, T. Galatyuk, J. N. Guenther, R. Houtz, F. Karsch, A. Y. Kotov, M. P. Lombardo, B. Lucini, L. Maio, M. Panero, J. M. Pawlowski, A. Pelissetto, O. Philipsen, A. Rago, C. Ratti, S. M. Ryan, F. Sannino, C. Sasaki, P. Schicho, C. Schmidt, S. Sharma, O. Soloveva, M. Sorba, and U.-J. Wiese, “Phase transitions in particle physics: Results and perspectives from lattice quantum chromo-dynamics,” Progress in Particle and Nuclear Physics 133 (2023) 104070. https://www.sciencedirect.com/science/article/pii/S0146641023000510.
  * [6] K. Fukushima and C. Sasaki, “The phase diagram of nuclear and quark matter at high baryon density,” Progress in Particle and Nuclear Physics 72 (2013) 99–154. https://www.sciencedirect.com/science/article/pii/S0146641013000550.
  * [7] R. Bellwied, S. Borsányi, Z. Fodor, J. Günther, S. Katz, A. Pásztor, C. Ratti, and K. Szabó, “Towards the QCD phase diagram from analytical continuation,” Nuclear Physics A 956 (Dec, 2016) 797–800, arXiv:1601.00466. http://www.sciencedirect.com/science/article/pii/S037594741600097X?via{%}3Dihubhttp://arxiv.org/abs/1601.00466http://dx.doi.org/10.1016/j.nuclphysa.2016.02.010http://linkinghub.elsevier.com/retrieve/pii/S037594741600097X.
  * [8] P. J. Gunkel and C. S. Fischer, “Locating the critical endpoint of QCD: Mesonic backcoupling effects,” Phys. Rev. D 104 (Sep, 2021) 054022. https://link.aps.org/doi/10.1103/PhysRevD.104.054022.
  * [9] L. Adamczyk et al., “Energy Dependence of Moments of Net-Proton Multiplicity Distributions at RHIC,” Physical Review Letters 112 no. 3, (Jan., 2014) 032302. http://dx.doi.org/10.1103/PhysRevLett.112.032302.
  * [10] STAR Collaboration, M. M. Aggarwal, et al., “An Experimental Exploration of the QCD Phase Diagram: The Search for the Critical Point and the Onset of De-confinement,”. https://arxiv.org/abs/1007.2613.
  * [11] K. Meehan, “STAR Results from Au+Au Fixed-Target Collisions at $\sqrt{s_{NN}}=4.5$ GeV,” Nuclear Physics A 967 (Nov., 2017) 808–811. http://dx.doi.org/10.1016/j.nuclphysa.2017.06.007.
  * [12] T. Ablyazimov et al., “Challenges in QCD matter physics –The scientific programme of the Compressed Baryonic Matter experiment at FAIR,” The European Physical Journal A 53 no. 3, (Mar., 2017) 60. http://dx.doi.org/10.1140/epja/i2017-12248-y.
  * [13] D. Almaalol et al., “QCD Phase Structure and Interactions at High Baryon Density: Continuation of BES Physics Program with CBM at FAIR,”. https://arxiv.org/abs/2209.05009.
  * [14] O. Philipsen, “The QCD equation of state from the lattice,” Progress in Particle and Nuclear Physics 70 (May, 2013) 55–107. http://dx.doi.org/10.1016/j.ppnp.2012.09.003.
  * [15] S. Borsanyi, Z. Fodor, J. N. Guenther, R. Kara, S. D. Katz, P. Parotto, A. Pasztor, C. Ratti, and K. K. Szabó, “QCD Crossover at Finite Chemical Potential from Lattice Simulations,” Phys. Rev. Lett. 125 (Jul, 2020) 052001. https://link.aps.org/doi/10.1103/PhysRevLett.125.052001.
  * [16] J. Maldacena, “The large $N$ limit of superconformal field theories and supergravity,” Advances in Theoretical and Mathematical Physics 2 no. 2, (1998) 231–252, arXiv:hep-th/9711200.
  * [17] E. Witten, “Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,” Advances in Theoretical and Mathematical Physics 2 no. 3, (1998) 505–532, arXiv:hep-th/9803131.
  * [18] E. Witten, “Anti de Sitter space and holography,” Advances in Theoretical and Mathematical Physics 2 no. 2, (1998) 253–291, arXiv:hep-th/9802150.
  * [19] S. S. Gubser and A. Nellore, “Mimicking the qcd equation of state with a dual black hole,” Physical Review D 78 no. 8, (Oct., 2008) 086007. http://dx.doi.org/10.1103/PhysRevD.78.086007.
  * [20] S. S. Gubser, A. Nellore, S. S. Pufu, and F. D. Rocha, “Thermodynamics and bulk viscosity of approximate black hole duals to finite temperature quantum chromodynamics,” Physical Review Letters 101 no. 13, (Sept., 2008) 131601. http://dx.doi.org/10.1103/PhysRevLett.101.131601.
  * [21] A. Karch, E. Katz, D. Son, and M. Stephanov, “Linear confinement and AdS/QCD,” Physical Review D 74 no. 1, (7, 2006) 015005, hep-ph/0602229.
  * [22] A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, “Linear confinement and AdS/QCD,” Physical Review D 74 no. 1, (7, 2006) 015005, arXiv:hep-ph/0602229.
  * [23] J. Erlich, E. Katz, D. T. Son, and M. A. Stephanov, “QCD and a Holographic Model of Hadrons,” Physical Review Letters 95 no. 26, (12, 2005) 261602, arXiv:hep-ph/0501128.
  * [24] R. Rougemont, J. Grefa, M. Hippert, J. Noronha, J. Noronha-Hostler, I. Portillo, and C. Ratti, “Hot QCD phase diagram from holographic Einstein–Maxwell–Dilaton models,” Progress in Particle and Nuclear Physics 135 (Feb., 2024) 104093. http://dx.doi.org/10.1016/j.ppnp.2023.104093.
  * [25] O. DeWolfe, S. S. Gubser, and C. Rosen, “A holographic critical point,” Physical Review D 83 no. 8, (Apr., 2011) 086005. http://dx.doi.org/10.1103/PhysRevD.83.086005.
  * [26] O. DeWolfe, S. S. Gubser, and C. Rosen, “Dynamic critical phenomena at a holographic critical point,” Physical Review D 84 no. 12, (Dec., 2011) 126014. http://dx.doi.org/10.1103/PhysRevD.84.126014.
  * [27] Z. Li, Y. Chen, D. Li, and M. Huang, “Locating the QCD critical end point through peaked baryon number susceptibilities along the freeze-out line,” Chinese Physics C 42 no. 1, (Jan., 2018) 013103. http://dx.doi.org/10.1088/1674-1137/42/1/013103.
  * [28] R. Critelli, J. Noronha, J. Noronha-Hostler, I. Portillo, C. Ratti, and R. Rougemont, “Critical point in the phase diagram of primordial quark-gluon matter from black hole physics,” Physical Review D 96 no. 9, (Nov., 2017) 096026, arXiv:1706.00455. http://dx.doi.org/10.1103/PhysRevD.96.096026.
  * [29] R. Rougemont, R. Critelli, and J. Noronha, “Nonhydrodynamic quasinormal modes and equilibration of a baryon dense holographic QGP with a critical point,” Physical Review D 98 no. 3, (Aug., 2018) 034028. http://dx.doi.org/10.1103/PhysRevD.98.034028.
  * [30] J. Grefa, J. Noronha, J. Noronha-Hostler, I. Portillo, C. Ratti, and R. Rougemont, “Hot and dense quark-gluon plasma thermodynamics from holographic black holes,” Physical Review D 104 no. 3, (Aug., 2021) 034002. http://dx.doi.org/10.1103/PhysRevD.104.034002.
  * [31] R.-G. Cai, S. He, L. Li, and Y.-X. Wang, “Probing QCD critical point and induced gravitational wave by black hole physics,” Physical Review D 106 no. 12, (Dec., 2022) L121902. http://dx.doi.org/10.1103/PhysRevD.106.L121902.
  * [32] M. Hippert, J. Grefa, T. A. Manning, J. Noronha, J. Noronha-Hostler, I. P. Vazquez, C. Ratti, R. Rougemont, and M. Trujillo, “Bayesian location of the QCD critical point from a holographic perspective,” arXiv:2309.00579 [nucl-th].
  * [33] Z. Fang, Y.-L. Wu, and L. Zhang, “Chiral phase transition and QCD phase diagram from AdS/QCD,” Phys. Rev. D 99 (Feb, 2019) 034028. https://link.aps.org/doi/10.1103/PhysRevD.99.034028.
  * [34] A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, “Charged AdS black holes and catastrophic holography,” Physical Review D 60 no. 6, (8, 1999) 064018, arXiv:hep-th/9902170.
  * [35] C. Park, “Dissociation of a heavy meson in the quark medium,” Physical Review D 81 no. 4, (2, 2010) 045009, arXiv:0907.0064 [hep-ph].
  * [36] P. Colangelo, F. Giannuzzi, and S. Nicotri, “Holography, heavy-quark free energy, and the QCD phase diagram,” Physical Review D 83 no. 3, (2, 2011) 035015, arXiv:1008.3116 [hep-ph].
  * [37] P. Colangelo, F. Giannuzzi, S. Nicotri, and V. Tangorra, “Temperature and quark density effects on the chiral condensate: an AdS/QCD study,” The European Physical Journal C, Volume 72, Issue 8, 2096 (2012) 72 no. 8, (2012) 2096, arXiv:1112.4402 [hep-ph]. https://link.springer.com/article/10.1140/epjc/s10052-012-2096-9.
  * [38] K. Chelabi, Z. Fang, M. Huang, D. Li, and Y.-L. Wu, “Chiral phase transition in the soft-wall model of AdS/QCD,” Journal of High Energy Physics 2016 no. 4, (4, 2016) 36, arXiv:1512.06493 [hep-ph].
  * [39] T. Gherghetta, J. Kapusta, and T. Kelley, “Chiral symmetry breaking in the soft-wall AdS/QCD model,” Physical Review D 79 no. 7, (4, 2009) 076003, arXiv:hep-ph/0902.1998.
  * [40] Z. Fang, Y.-L. Wu, and L. Zhang, “Chiral phase transition with $2+1$ quark flavors in an improved soft-wall AdS/QCD model,” Phys. Rev. D 98 (Dec, 2018) 114003. https://link.aps.org/doi/10.1103/PhysRevD.98.114003.
  * [41] F. R. Brown, F. P. Butler, H. Chen, N. H. Christ, Z. 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 (Nov, 1990) 2491–2494. https://link.aps.org/doi/10.1103/PhysRevLett.65.2491.
  * [42] P. de Forcrand and O. Philipsen, “The Chiral critical line of N(f) = 2+1 QCD at zero and non-zero baryon density,” JHEP 01 (2007) 077, arXiv:hep-lat/0607017 [hep-lat].
  * [43] Z. Fang, Y.-L. Wu, and L. Zhang, “Chiral phase transition and meson spectrum in improved soft-wall AdS/QCD,” Physics Letters B 762 (Nov., 2016) 86–95, arXiv:1604.02571 [hep-ph]. http://dx.doi.org/10.1016/j.physletb.2016.09.009.
  * [44] Z. Fang and Y.-L. Wu, “Equation of state and chiral transition in soft-wall AdS/QCD with a more realistic gravitational background,” Chinese Physics C 44 no. 10, (Oct, 2020) 103101. https://dx.doi.org/10.1088/1674-1137/abab90.
  * [45] S. P. Bartz and T. Jacobson, “Chiral Phase Transition and Meson Melting from AdS/QCD,” Phys. Rev. D94 (2016) 075022, arXiv:1607.05751 [hep-ph].
  * [46] 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 (Apr, 2018) 044908. https://link.aps.org/doi/10.1103/PhysRevC.97.044908.
  * [47] A. Cherman, T. D. Cohen, and E. S. Werbos, “The Chiral condensate in holographic models of QCD,” Phys. Rev. C79 (2009) 045203, arXiv:0804.1096 [hep-ph].
  * [48] D. Li, M. Huang, and Q.-S. Yan, “A dynamical soft-wall holographic QCD model for chiral symmetry breaking and linear confinement,” The European Physical Journal C 73 no. 10, (10, 2013) 2615, arXiv:1206.2824 [hep-th].
  * [49] A. Ballon-Bayona, L. A. H. Mamani, and D. M. Rodrigues, “Spontaneous chiral symmetry breaking in holographic soft wall models,” Phys. Rev. D 104 (Dec, 2021) 126029. https://link.aps.org/doi/10.1103/PhysRevD.104.126029.
  * [50] K. Rajagopal, “Mapping the qcd phase diagram,” Nuclear Physics A 661 no. 1, (1999) 150 – 161. http://www.sciencedirect.com/science/article/pii/S0375947499850179.
  * [51] E. Laermann and O. Philipsen, “The Status of lattice QCD at finite temperature,” Ann. Rev. Nucl. Part. Sci. 53 (2003) 163–198, arXiv:hep-ph/0303042 [hep-ph].
  * [52] Y. Yang and P.-H. Yuan, “QCD phase diagram by holography,” Physics Letters B 832 (2022) 137212. https://www.sciencedirect.com/science/article/pii/S037026932200346X.
  * [53] X.-Y. Liu, X.-C. Peng, Y.-L. Wu, and Z. Fang, “A holographic study on QCD phase transition and phase diagram with two flavors,”. https://arxiv.org/abs/2312.01346.
  * [54] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing.  Cambridge University Press, 3 ed., 2007.  http://www.amazon.com/Numerical-Recipes-3rd-Scientific-Computing/dp/0521880688/ref=sr_1_1?ie=UTF8&s=books&qid=1280322496&sr=8-1.

##  Appendix A Comparison of numerical methods

Figure 9: This comparison of the numerical methods shows agreement between
them, but more spurious points when integrating from the UV. The data shown is
for two representative values of the scalar-dilaton coupling $\lambda_{1}$
with $m_{q}=1$ MeV and $\mu=0$.

Another commonly-used numerical method for solving the chiral equation of
motion (10) uses the UV expansion of (15) as one boundary condition and the
regularity of the chiral field at the horizon $z=z_{h}$ is used as the other.
Regularity is difficult to check numerically, so a “test” function is defined
that includes all the potentially singular parts of the equation of motion,

$-z^{2}\frac{f^{\prime}(z)}{f(z)}\chi^{\prime}(z)+\frac{1}{f(z)}\left[(-3-\lambda_{1}\Phi)\chi+4\lambda_{4}\chi^{3}+3\lambda_{3}\chi^{2}\right].$
(20)

This collection of terms must be zero at the horizon, otherwise there will be
a divergence as $f\rightarrow 0$. Ensuring the test function is zero becomes
the second boundary condition.

The shooting method [54] is implemented with the required quark mass given as
an input parameter and $\sigma$ varied until the boundary condition is met
near the black hole horizon. This method is used to find the allowed values of
$\sigma$ for a given $T,\mu$ and, $m_{q}$.

The limitations of this method are revealed at low temperature $T\ll T_{c}$.
At low temperature, $z_{h}\sim T^{-1}$ becomes large, and numerical
instabilities in the numerical solution to (10) make it difficult to determine
the correct value of $\sigma$. This has not been a problem in previous
publications that focus on the chiral phase transition. Typically, the value
of $\sigma$ approaches a constant value as temperature is decreased below the
transition temperature. However, this is not always the case in this model,
where $\sigma=0$ at low temperatures for some values of $\lambda_{1}$, as seen
in Figure 9.

The method of integrating from the UV boundary is trustworthy near the
transition temperature. However, in this work we are also interested in the
low temperature chiral dynamics, particularly in ensuring separate sources of
explicit and spontaneous chiral symmetry breaking, as discussed in Section
4.1. By starting near the singular point and integrating away from it, the
method of Section 3 is more numerically stable at lower temperatures.